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abstract: 'We report on the use of elliptical pump spatial modes to increase the observed brightness of spontaneous parametric downconversion in critically phase matched crystals. Simulations qualitatively predict this improvement which depends on the eccentricity and orientation of the pump ellipse. We experimentally confirm a factor of two improvement in brightness when compared to the traditional circular-symmetric pump spatial modes. These results support previous theoretical work that proposes the use of elliptical pump modes to enhance the performance of parametric processes in anisotropic materials.'
author:
- Aitor Villar
- Arian Stolk
- Alexander Lohrmann
- Alexander Ling
bibliography:
- 'bibliography.bib'
title: Enhancing SPDC brightness using elliptical pump shapes
---
Entangled photon pair sources play a crucial role in fundamental tests of nature [@bouwmeester1997experimental; @yin2017satellite] and emerging quantum applications [@poppe2004practical; @bell2013multicolor]. Spontaneous parametric downconversion[@burnham1970observation] (SPDC) is the most common method to generate these photon pairs, with a variety of designs[@kwiat1999ultrabright; @fedrizzi2007wavelength; @villar2018experimental] and materials used. In SPDC, a photon (pump, $p$) in a nonlinear crystal can downconvert into two photons (signal, $s$; idler, $i$) obeying energy and momentum conservation.
Early studies identified the influence of pump parameters on parametric brightness[@boyd1966theory]. The most comprehensive review on pump optimization was made by Boyd and Kleinman[@boyd1968parametric], who suggested the use of circular symmetric spatial modes for pumping nonlinear processes. Though used most often, assumptions about circular (isotropic) symmetry might not optimize frequency conversion given that the vast majority of parametric processes take place in anisotropic materials. A specific case of interest, where the anisotropy is important, is when using critical phase matching in birefringent materials. When the pump is extraordinarily polarized it experiences spatial walk-off, as depicted in Fig. \[fig:experiment\](a). The interplay of pump spatial modes with walk-off was first studied by Volosov[@volosov1970effect] and Kuizenga[@kuizenga1972optimum]. They showed that elliptical pump shapes could improve frequency conversion performance in second-harmonic generation (SHG) and other parametric processes. Their main observation was that the spatial walk-off limits the overlap of any parametric emission (see Fig. \[fig:experiment\](b)). This restricts the benefits of tight pump focusing in the direction of walk-off. In contrast, this restriction is absent in the non-walk-off direction, which allows for tighter focusing. Hence, it is natural to consider different focusing conditions for different directions, leading to an elliptical (astigmatic) spatial mode.
![(a) Side view of the pump spatial walk-off (blue line) within the crystal. The collinear SPDC emission is depicted by red arrows. The dotted line is the crystal’s optical axis. (b) Sketch adapted from[@kuizenga1972optimum] representing the overlap between the pump beam (solid lines) and the downconversion beams (dotted lines), depicting the importance of controlling the pump focusing in order to maximize the overlap. (c) Collinear SPDC emission profile at the exit face of the crystal when the major axis of the pump ellipse (inset) is parallel to the walk-off direction. (d) Same as (c) but the ellipse major axis is rotated by 90$\degree$. (e) Experimental setup, 1: single-mode fiber pump output, 2: collimation lens, 3-4: cylindrical lenses, 5: focusing lens, 6: BBO crystal, 7-8: excess pump removal optics, 9: single-mode fiber, 10: photon separation via a dichroic mirror followed by coincidence detection setup.[]{data-label="fig:experiment"}](pics/experiment.pdf)
Experimental studies on SHG using elliptical pump beams verified the predicted gain, but the general observation was that the additional experimental complexity involving cylindrical focusing appeared to offset any enhancement[@steinbach1996cw; @freegarde1997general]. This observation stems from the challenges of operating elliptical beams in a resonant cavity. However, single-pass parametric processes such as SPDC are exempt from these difficulties.
In this work, we study the effect of combining elliptical pump and circular-symmetric collection modes in type-I SPDC with negative birefringent materials. Our approach is outlined in Fig. \[fig:experiment\]. We distinguish between two pump orientations; parallel (Fig. \[fig:experiment\](c)) and orthogonal (Fig. \[fig:experiment\](d)) to the walk-off plane (defined by the propagation and walk-off direction).
To systematically investigate the effect of eccentricity the pump is launched from a single-mode fiber before undergoing cylindrical focusing. The eccentricity of the pump is defined via the aspect ratio $r=\frac{\omega_y}{\omega_x}$, where $\omega_y$ ($\omega_x$) is the vertical (horizontal) waist of the ellipse. In our discussion, the vertical direction is associated with the walk-off. The change in pump aspect ratio was achieved by tuning only the waist of the horizontal axis. For the ideal pump shape, Kuizenga suggests that the size of the pump in the two directions are determined separately by considering whether walk-off is present.
To optimize the pump shape in each direction, we should utilize the birefringence parameter $B$[@boyd1968parametric]: $$B=\frac{\rho}{2}(lk_{0})^{1/2},
\label{eq:b_param}$$ where $\rho$ is the walk-off angle, $l$ is the length of the crystal and $k_0$ is the pump wave vector. In the direction where there is no walk-off, $B=0$.
The experimental layout for investigating elliptical pump performance is shown in Fig. \[fig:experiment\](e). A $\beta$-Barium Borate (BBO) crystal (cut-angle $\theta = 28.76\degree$) is pumped with light to generate type-I, collinear, non-degenerate SPDC wavelengths (signal and idler wavelengths at and , respectively). Excess pump power was removed by a dichroic mirror and a long-pass filter. Finally, the SPDC photons were coupled into a single-mode fiber and sent to a detection setup.
A range of pump aspect ratios were realized with different cylindrical lenses. Once a target aspect ratio was achieved, a plano-convex lens focused down the pump ellipse.
According to the literature [@boyd1968parametric], the optimal pump waist in the walk-off direction (given our crystal properties) is , whereas the optimal pump waist in the direction with no walk-off is . This gives the optimal aspect ratio of $r=2.9$.
These optimal values, however, do not take into account experimental limitations, such as aberrations introduced by thick lenses. Empirically, we found that in our setup a pump waist of optimized the brightness and collection efficiency when the aspect ratio was 1.0, with a collection waist of .
Therefore, to investigate the effect of the pump aspect ratio we fixed the pump size in the walk-off direction at . Correspondingly, the pump size in the non-walk off direction ranged from to . Thus, the range of aspect ratios that were implemented ranged from 4.44 to 0.66. The experiment was then repeated for the case where the major axis of the pump beam was orthogonal to the walk-off direction. In this case, the experimental aspect ratio investigated ranged from 2.94 to 0.65.
To provide better insight on the interplay of elliptical pump and circular collection modes a model was developed. Within the model, the SPDC intensity is governed by the conventional phase matching conditions of the respective ($p$, $s$ and $i$) fields. The classical $p$ field is treated as a probability distribution of $\vec{k}_p$, that follows the angular and spatial intensity distributions of a Gaussian beam profile.
For a given wavelength $\lambda_p$, starting position within the crystal $\{x,y,z\}$ and propagation direction $\vec{k}_{p}$, the weighting function $A$ can be obtained: $$A= e^{-\frac{( \omega_{x} \Delta k_{x} )^2+( \omega_{y} \Delta k_{y})^2}{2}} \cdot\operatorname{sinc}\bigg(\frac{l\Delta k_{z}}{2}\bigg)^2 .
\label{eq:sinc}$$ Here $\Delta \vec{k} = \Delta k_x\hat{x} + \Delta k_y \hat{y} + \Delta k_z \hat{z}$ is the phase mismatch in Cartesian coordinates, $\omega_{x,y}$ the beam waist in horizontal (*x*) and vertical (*y*) directions, and $l$ the crystal length. This weighting function determines the probability of generating a signal/idler pair with wavelengths $\lambda_{s}$ and $\lambda_{i}$ in the propagation directions $\hat{k}_{s}$ and $\hat{k}_{i}$. The individual treatment for each $\vec{k}_p$ accounts for effects of the wavefront curvature.
Using conventional ray-tracing techniques the $\vec{k}_{s},\vec{k}_{i}$ pair was propagated through the downstream optics towards a single-mode fiber (numerical aperture of $0.1$ and core diameter ). The overlap of the SPDC rays with the collection mode of the fiber determines the in-fiber photon pair brightness. The model is available online at: <https://github.com/arianstolk/SPDC_test/>.
The simulated and experimental results are shown in Fig. \[fig:results\_1\] and Fig. \[fig:results\_2\]. When the major axis of the pump ellipse is parallel to the walk-off direction, the experimental data agrees qualitatively with the model, which predicts increased brightness for larger pump aspect ratio. The best observed improvement over the circular symmetric mode is a factor of 2. When the major axis of the pump ellipse is orthogonal to the walk-off direction, there is a dramatic difference as the brightness shows little response when the aspect ratio changes.
In the case where the major axis is aligned with the walk-off direction, increasing the aspect ratio monotonically increased the photon pair brightness that can be observed. Due to experimental difficulties, the optimal aspect ratio of 2.9 was not achieved. However, it is instructive to consider the results of the two aspect ratios that bracket this value. These results showed a strong dependence of brightness on the orientation of the pump major axis, validating Kuizenga’s proposal.
In Fig. \[fig:results\_1\](c), there is a strong linear relationship between the brightness and collection waist, which is not present in the model.This is attributed partly to the lack of precision within the model which uses the thin lens approximation for modeling the single-mode fiber photon collection and ignores the lens thickness. Furthermore, the observed $M^2$ value of the pump beam in the walk-off direction deviates from the ideal case for large aspect ratio (see Tab. \[tab:params\] in Appendix) due to lens aberrations when using thick cylindrical lenses.
Improvements in the model (e.g. improved approximations for ray tracing of the curved optics) will help to match the experimental results and provide better predictions about the overall source performance. Additionally, an extension of this work would be the use of elliptical collection modes that are tailored to the pump profile and crystal anisotropy.
When considering asymmetric effects of the phase matching conditions on the SPDC profile, the use of a correctly oriented elliptical pump mode showed improved in-fiber brightness over the traditional circular symmetric modes. This deviates from experimental results in SHG in optical parametric oscillators, where the marginal improvement in performance did not warrant the expense of increased experimental complexity. Elliptical pump modes have already improved the brightness of entangled photon pairs sources where the crystal geometry allows an elliptical pump mode to be implemented[@villar2018experimental; @lohrmann2018high].
In many modern SPDC sources the pump is typically obtained from a laser diode which inherently exhibits an elliptical profile. Taking into account crystal anisotropy and beam walk-off it might be unnecessary to correct the elliptical pump profile, simplifying the construction of photon pair sources while improving performance.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP12-2013-02). This program was also supported by the Ministry of Education, Singapore.
The Boyd-Kleinman optimal circular symmetric pump beam calculation
==================================================================
In order to calculate the optimal circular symmetric pump beam suggested by Boyd-Kleinman, first the $B$ parameter of the nonlinear crystal needs to be calculated. In our case, the properties of the BBO crystal: $\rho=\SI{3.83}{\degree}$, $l=\SI{5}{\milli\metre}$ and $\lambda_0 = \SI{405}{\nano\metre}$. From Eq. \[eq:b\_param\] in the main text, $B=9.3$.
Knowing the $B$ parameter, the optimal $\xi$ value can be inferred via the optimization curves shown in Fig. \[fig:bk-curves\]:
![Optimization curves for circular symmetric beams reported in [@boyd1968parametric]. For different walk-off parameters (the $B$ number is shown above each curve) $\xi$ (x-axis) and the figure of merit $h_m$ (y-axis) is shown. Resorting to this curves and knowing the walk-off angle and the thickness of the crystal, the optimal circular symmetric beam for a particular wavelength $\lambda_0$ can be determined.[]{data-label="fig:bk-curves"}](pics/BK-single.pdf)
For $B=9.3$, $\xi$ is calculated to be 0.6579. From here, the ideal beam waist can be calculated, since $\xi=\frac{l}{b}=\frac{l\cdot\lambda}{\omega^2 \cdot 2\pi}$. In our case, $\omega=\SI{22.13}{\micro\metre}$.
Pump beam parameters
====================
Pump beam parameters used in both simulation (*Sim*) and experiment (*Exp*) are depicted in Tab. \[tab:params\]. Due to experimental factors (e.g., lens aberrations) when shaping the pump mode, the experimental Rayleigh length values are lower than the ideal values used in the simulation ($M^{2}_{\perp},M^{2}_{{\mathbin{\!/\mkern-5mu/\!}}} = 1$). This explains the difference in decaying rates between simulation and experiment.
[|c| c | c | c | c | c | c | c | c |]{} & & &$M^{2}_{{\mathbin{\!/\mkern-5mu/\!}}}$ & &$M^{2}_{\perp}$\
&Sim&Exp&Sim&Exp&Exp&Sim&Exp&Exp\
& 0.67 & 0.66 & 174.5 & 145.6 &1.1& 77.6 & 69.1&1.1\
& 1.0 & 1.03 & 77.6& 65.2 &1.1 & 77.6 &70.9 &1.1\
& 2.0 &1.85 & 19.4& 20.9 &1.0 & 77.6 &49.1&1.5\
& 3.0 & 4.44 & 8.4& 5.0 &1.0& 77.6& 30.8 &1.7\
& 0.67 & 0.65 & 77.6 & 76.0 &1.0& 174.5 & 155.5&1.1\
& 1.00& 0.97 & 77.6 & 65.2&1.1 & 77.6& 70.9&1.1\
& 2.0 &2.02 & 77.6& 66.2&1.1 & 19.4 &17.3&1.2\
& 3.0 &2.94 & 77.6 &70.9&1.0 & 8.4 &8.2&1.0\
\[tab:params\]
|
---
abstract: 'In this paper, we investigate characteristic polynomials of matrices in min-plus algebra. Eigenvalues of min-plus matrices are known to be the minimum roots of the characteristic polynomials based on tropical determinants which are designed from emulating standard determinants. Moreover, minimum roots of characteristic polynomials have a close relationship to graphs associated with min-plus matrices consisting of vertices and directed edges with weights. The literature has yet to focus on the other roots of min-plus characteristic polynomials. Thus, here we consider how to relate the $2$nd, $3$rd, $\dots$ minimum roots of min-plus characteristic polynomials to graphical features. We then define new characteristic polynomials of min-plus matrices by considering an analogue of the Faddeev-LeVerrier algorithm that generates the characteristic polynomials of linear matrices. We conclusively show that minimum roots of the proposed characteristic polynomials coincide with min-plus eigenvalues, and observe the other roots as in the study of the already known characteristic polynomials. We also give an example to illustrate the difference between the already known and proposed characteristic polynomials.'
---
Two characteristic polynomials corresponding to
graphical networks over min-plus algebra
Sennosuke WATANABE$^a$, Yuto TOZUKA$^b$, Yoshihide WATANABE$^c$,
Aito YASUDA$^d$, Masashi IWASAKI$^e$
[ ${}^a$ *Department of General Education, National Institute of Technology,* ]{}
[*Oyama College, 771 Nakakuki, Oyama City, Tochigi, 323-0806 Japan*]{}
[$^b$*Graduate School of Science and Engineering, Science of Environment and*]{}
[*Mathematical Modeling, Doshisha University, 1-3 Tatara Miyakodani,*]{}
[*Kyotanabe, 610-0394 Japan*]{}
[$^c$*Faculty of Science and Engineering, Department of Mathematical Sciences,*]{}
[*Doshisha University, 1-3 Tatara Miyakodani, Kyotanabe, 610-0394 Japan*]{}
[${}^d{}^e$ *Faculity of Life and Environmental Sciences, Kyoto Prefectural University,*]{}
[ *1-5 Nakaragi-cho, Shimogamo, Sakyo-ku, Kyoto, 606-8522 Japan*]{}
[*E-mail addresses:* $^a$sewatana@oyama-ct.ac.jp, $^c$yowatana@mail.doshisha.ac.jp,]{}
[ $^d$imasa@kpu.ac.jp]{}
[**Keywords** ]{} Circuit, Directed and weighted graph, Eigenvalue problem, Faddeev-LeVerrier algorithm, Min-plus algebra.
Introduction
============
Various fields of mathematics consider min-plus algebra, which is an abstract algebras with idempotent semirings. The arithmetic operations of min-plus algebra are $\min(a,b)$ and $a+b$ for $a, b\in\mathbb{R}_{\min}:=\mathbb{R}\cup \{\infty\}$ where $\mathbb{R}$ is the set of all real numbers. Although it has different operators to the well-known linear algebra, the eigenvalue problem is fundamental both types of algebra. The min-plus eigenvalue problem was shown in Gondran-Minoux [@BCOQ] and Zimmermann [@GM] to have a close relationship with the shortest path problem on graphs consisting of vertexes and edges, where every edge links to two distinct vertices. Directions are added to edges in directed graphs, and a value is associated with each edge in weighted graphs. Matrices whose entries are min-plus algebra figures are sometimes considered with respect to directed and weighted graphs. Such matrices are called min-plus matrices, and are practically defined by assigning the weights of edges from the vertices $i$ to $j$ to the $(i,j)$ entries. According to Gondran-Minoux [@BCOQ] and Zimmermann [@GM], if a min-plus matrix has an eigenvalue, this eigenvalue reflects a significant feature in the network on a directed and weighted graph that is associated with the min-plus matrix. There exist circuits whose average weights are the eigenvalue, where a circuit signifies a closed path without crossing; its average weight is given by the ratio of the sum of all weights to the vertex number. Conversely, in the network involving circuits, the minimum of the average weights of circuits coincides with the eigenvalue of the corresponding min-plus matrix. Moreover, the eigenvalues of min-plus matrices are the minimum roots of the characteristic polynomials, defined using tropical determinants, which correspond to the determinants over linear algebra, over min-plus algebra [@MS]. However, the $2$nd, $3$rd, $\dots$ minimum roots have not yet been related to graphical features. Thus, the first goal of this paper is to identify graphical significance of the $2$nd, $3$rd, $\dots$ minimum roots.
Over linear algebra, the $QR$, qd and Jacobi algorithms are representative numerical solvers for eigenvalue problem [@D; @GL; @R]. The divided-and-conquer and bisection algorithms are also the famous linear eigenvalue solvers [@D; @GL]. In contrast, few procedures for min-plus eigenvalues have been studied, with the exception of work by Maclagan-Sturmfels [@MS]. Moreover, to the best of our knowledge, min-plus eigenvalue procedures based on linear equivalents have not been yet addressed in the literature. Thus, the second goal of this paper is to propose new eigenvalue algorithms for min-plus algebra by emulating the Faddeev-LeVerrier algorithm [@Fad] in linear algebra. The Faddeev-LeVerrier algorithm employs only linear scalar and matrix arithmetic, which can be intuitively replaced with min-plus one. Strictly speaking, the Faddeev-LeVerrier algorithm generates not eigenvalues but characteristic polynomials of square matrices. In other words, our second goal is to essentially show how to derive new characteristic polynomials of min-plus matrices.
The remainder of this paper is organized as follows. Section 2 describes elementary scalar and matrix arithmetic over the min-plus algebra. Section 3 and 4 explain the relationships between min-plus matrices and the corresponding networks, and linear factorizations of min-plus polynomials, including an effective preconditioning, respectively. In Section 5, we elucidate not only minimum roots, but also the other roots of given min-plus characteristic polynomials given using tropical determinants from the perspective of graphical networks. In Section 6, by considering an analogue of the Faddeev-LeVerrier algorithm, we derive new characteristic polynomials of min-plus matrices, then clarify their features in the comparison with known characteristic polynomials. Finally, in Section 7, we provide concluding remarks.
Min-plus arithmetic
===================
In this section, we present elementary definitions and properties concerning min-plus algebra. We first focus on scalar arithmetic over the min-plus algebra, and then present the matrix arithmetic.
For $a,b\in\mathbb{R}_{\min}$, min-plus algebra has only two binary arithmetic operators, $\oplus$ and $\otimes$, which have the following definitions, $$\begin{aligned}
& a\oplus b=\min\{a,b\},\\
& a\otimes b= a+b.\end{aligned}$$ We can easily check that both $\oplus$ and $\otimes$ are associative and commutative, and $\otimes$ is distributive with respect to $\oplus$, namely, for $a,b,c\in\mathbb{R}_{\min}$, $$\begin{aligned}
a\otimes (b\oplus c)=(a\otimes b)\oplus (a\otimes c).\end{aligned}$$ Moreover, we may regard $\varepsilon =+\infty$ and $e=0$ as identities with respect to $\oplus$ and $\otimes$, respectively because for any $a\in\mathbb{R}_{\min}$, $$\begin{aligned}
& a\oplus\varepsilon =\min\{a,+\infty\}=a,\\
& a\otimes e=a+0=a.\end{aligned}$$ Using the identity $e$, we can uniquely define the inverse of $a\in\mathbb{R}_{\min}\!\setminus\!\{\varepsilon\}$ with respect to $\otimes$, denoted by $b$, as $$\begin{aligned}
a\otimes b=e.\end{aligned}$$ Since it holds that $$\begin{aligned}
a\otimes\varepsilon =a+\infty=\varepsilon,\end{aligned}$$ the identity $\varepsilon =+\infty$ with respect to $\oplus$ is absorbing for $\otimes$. Here, we consider matrices whose entries are all $\mathbb{R}_{\min}$ numbers to be min-plus matrices. For positive integers $m$ and $n$, we designate the set of all $m$-by-$n$ min-plus matrices as $\mathbb{R}_{\min}^{m\times n}$. Since the min-plus matrices appearing in the later sections are all square matrices, we hereinafter limit discussion to $n$-by-$n$ min-plus matrices. For $A=(a_{ij}),B=(b_{ij})\in\mathbb{R}_{\min}^{n\times n}$, the sum $A\oplus B =([A\oplus B]_{ij})\in\mathbb{R}_{\min}^{n\times n}$ and product $A\otimes B=([A\otimes B]_{ij})\in\mathbb{R}_{\min}^{n\times n}$ are respectively given as: $$\begin{aligned}
[A\oplus B]_{ij}=a_{ij}\oplus b_{ij}=\min\{a_{ij},b_{ij}\},\end{aligned}$$ and $$\begin{aligned}
[A\otimes B]_{ij}=\bigoplus^k_{\ell=1}(a_{i\ell}\otimes b_{\ell j})=\underset{\ell=1,2,\dots,k}{\min}\{a_{i\ell}+b_{\ell j}\}.\end{aligned}$$ Moreover, for $\alpha\in\mathbb{R}_{\min}$ and $A=(a_{ij})\in\mathbb{R}_{\min}^{n\times n}$, the scalar multiplication $\alpha\otimes A =([\alpha\otimes A]_{ij})\in\mathbb{R}_{\min}^{n\times n}$ is defined as $$\begin{aligned}
[\alpha\otimes A]_{ij}=\alpha\otimes a_{ij}.\end{aligned}$$
Graphs and min-plus eigenvalues
===============================
In this section, we first give a short explanation for min-plus matrices corresponding to graphs which are not functional graphs. Then, we review the relationships between the eigenvalues of min-plus matrices and the corresponding graphs.
Let $v_1,v_2,\dots, v_m$ denote vertices on the graph $G$, and let $e_{i,j}=(v_i,v_j)$ be edges which link the vertices $v_i$ and $v_j$. The edge $e_{i,i}=(v_i,v_i)$ is called a loop. Moreover, let $V:=\{v_1,v_2,\dots,v_m\}$ and $E:=\{e_{i,j} | (i,j)\in\sigma\}$ where $\sigma$ is the set of all pairs of $i$ and $j$ such that the edge $e_{i,j}$ exists. Then, two sets $V$ and $E$ uniquely determine the graph $G$. Thus, such $G$ is often expressed as $G=(V,E)$. If $G$ is a directed graph, then $e_{i,j}$ are directed edges whose tail and head vertices are $v_i$ and $v_j$, respectively. Further, if $G$ is a directed and weighted graph, then the real number $w(e_{i,j})$ is assigned to each edge $e_{i,j}$, and is called the weight. The pair $\mathcal{N}=(G,w)$ is often called the network on the graph $G$. The following definition gives the so-called weighted adjacency matrices associated with networks.
For the network $\mathcal{N}$ involving $m$ vertices, an $m$-by-$m$ weighted adjacency matrix $A(\mathcal{N})=(a_{ij})$ is given using $\mathbb{R}_{\min}$ numbers as $$\begin{aligned}
a_{ij}=\left\{\begin{array}{ll}
w((v_i,v_j)) & \text{if}\ (v_i,v_j)\in E,\\
+\varepsilon & \text{otherwise} .
\end{array}\right.\end{aligned}$$
It is emphasized here that the weighted adjacency matrix $A(\mathcal{N})$ is a min-plus matrix. Conversely, for any matrix $A\in\mathbb{R}_{\min}^{n\times n}$, there exists a network whose weighted adjacency matrix coincides with $A$. We hereinafter denote such a network by $\mathcal{N}(A)$.
If the vertex indices $i(0),i(1),\dots,i(s)$ are different from each other, and edges $e_{i(0),i(1)}, e_{i(1),i(2)},\dots,e_{i(s-1),i(s)}$ exist, then $P=(v_{i(0)},v_{i(1)},\dots,v_{i(s)})$ is a path on the network $\mathcal{N}$. For the path $P$, the length $\ell (P)$ denotes the edge number $s$, and the weight sum $\omega (P)$ designates the sum of the edge weights: $$\begin{aligned}
\omega(P)=\sum_{k=0}^{s-1}w((v_{i(k)},v_{i(k+1)}))=\sum_{k=0}^{s-1}a_{i(k)i(k+1)}
=\bigotimes_{k=0}^{s-1}a_{i(k)i(k+1)}. \end{aligned}$$ Moreover, the path $P$ with $i(0)=i(s)$ is just a circuit, and its length and weight are calculated in the same manner as those in path $P$. The following definition describes the average weight of the circuit $C$.
For the circuit $C$, the average weight $\text{ave}(C)$ is given by $$\begin{aligned}
\text{ave}(C)=\dfrac{\omega(C)}{\ell(C)}.\end{aligned}$$
The eigenvalues and eigenvectors of matrices play important roles in both linear algebra and min-plus algebra. The following definition determines the eigenvalues and eigenvectors of the min-plus matrix.
For the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, if there exist $\lambda\in\mathbb{R}_{\min}$ and $\bm{x}\in\mathbb{R}_{\min}^n\!\setminus\!\{(\varepsilon,\varepsilon,\dots,\varepsilon)^{\top}\}$ satisfying $$\begin{aligned}
A\otimes\bm{x}=\lambda\otimes\bm{x},\end{aligned}$$ then $\lambda$ and $\bm{x}$ are an eigenvalue and its corresponding eigenvector.
The eigenvalues of the min-plus matrix were shown in Baccelli et al. [@BCOQ] and Gondran-Minoux [@GM] to have interesting relationships with circuits in the network.
If the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$ has an eigenvalue $\lambda\not=\varepsilon$, there exists a circuit in the network $\mathcal{N}(A)$ whose average weight is equal to $\lambda$.
\[prp;mineigen\] The minimum of the average weights of circuits in the network $\mathcal{N}(A)$ coincides with the minimum eigenvalue of the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$.
In particular, Theorem \[prp;mineigen\] suggests that we can algebraically compute the minimum of average weights of circuits in $\mathcal{N}(A)$ without grasping pictorial situations.
Factorization of min-plus polynomials
=====================================
In this section, we briefly review Maclagan-Sturmfels [@MS] with regards to linear factorization over the min-plus algebra, and then describe a preconditioning algorithm in linear factorizations which will be helpful in later sections.
We now consider the so-called min-plus polynomial of degree $n$ with respect to $x$, $$\begin{aligned}
p(x)=x^{n}\oplus c_1\otimes x^{n-1}\oplus\cdots\oplus c_{n-1}\otimes x\oplus c_n. \end{aligned}$$ where $x^k:=\underbrace{x\otimes x\otimes\cdots\otimes x}_{k\text{ times}}=kx$ and $c_1,c_2,\dots,c_n\in\mathbb{R}_{\min} $ are the coefficients. The following proposition gives the necessary and sufficient condition for factorizing the min-plus polynomial $p(x)$ into linear factors as $p(x)=(x\oplus c_1)\otimes [x\oplus (c_2-c_1)]\otimes\cdots\otimes [x\oplus (c_n-c_{n-1})]$.
\[prp1\] The min-plus polynomial $p(x)$ can be completely factorized into linear factors if, and only if, the coefficients $a_0,a_1,\dots,a_{n-1}\in\mathbb{R}_{\min}$ satisfy the following inequality, $$\begin{aligned}
c_1\le c_2-c_1\le\cdots\le c_n-c_{n-1}.
\label{eq-lin-fact}\end{aligned}$$
Regarding $p(x)$ as the min-plus function with respect to $x$, we see that $p(x)$ is piecewise linear. This is because $$\begin{aligned}
p(x)=\min\{nx,c_1+(n-1),\dots,c_{n-1}+x,c_n\}.\end{aligned}$$ Thus, the functional graph consists of line segments and rays. Figure \[graph1\] shows an example of the functional graph of $x^2\oplus 2\otimes x\oplus 6$.
![The functional graph of $x^2\oplus 2\otimes x\oplus 6$.[]{data-label="graph1"}](fig01.eps){width="80mm"}
The piecewise linearity implies that $p(x)$ has a finite number of break points. Roots of the min-plus function $p(x)$ coincide with the values of the $x$-coordinates of break points. From Figure \[graph1\], we can thus factorize $p(x)$ as $p(x)=(x\oplus 2)\otimes (x\oplus 4)$. It is emphasized here that two distinct min-plus polynomials, $p(x)$ and $p'(x)$, are sometimes factorized using common linear factors, which differs from over linear algebra. If the linear factorizations of $p(x)$ and $p'(x)$ are the same, then we recognize that $p(x)$ is equivalent to $p'(x)$. To distinguish $p(x)=q(x)$, namely, $p(x)$ is completely equal to $p'(x)$, we express $p(x)\equiv p'(x)$ if $p(x)$ is equivalent to $p'(x)$.
We later need to find equivalent min-plus polynomials to observe the characteristic polynomials of min-plus matrices. Although it is not so difficult to derive equivalent min-plus polynomials, we show how to reduce them to equivalent ones that can be directly factorized into linear factors. Such algorithms, to the best of our knowledge, have not previously been presented. Therefore, we describe an algorithm for constructing an equivalent polynomial that can be factorized into linear factors.
\[alg1\] Constructing $p'(x)= x^n\oplus c'_1\otimes x^{n-1}\oplus\cdots\oplus c'_{n-1}\otimes x\oplus c'_n$ which is equivalent to $p(x)=x^n\oplus c_1\otimes x^{n-1}\oplus\cdots\oplus c_{n-1}\otimes x\oplus c_n$, namely, $p(x)\equiv p'(x)$.\
[**Input**]{}: The coefficients $c_1,c_2,\dots,c_n$ in the min-plus polynomial $p(x)$.\
[**Output**]{}: The coefficients $c'_1,c'_2,\dots,c'_n$ in the equivalent min-plus polynomial $p'(x)$.\
01: Set $c_1=0$ and $i:=0$.\
02: Set $c_j=\varepsilon$ if $p(x)$ does not involve $x^{n-j}$.\
03: Compute $T_k:=(c_k -c_i)/(k-i)$ for $k=i+1,i+2,\dots,n$.\
04: Find integer $m$ such that $T_m=\displaystyle\min_{k=i+1,i+2,\dots,n} T_k$.\
05: Compute $c'_{i+1},c'_{i+2},\dots,c'_{m}$ as $$\begin{aligned}
c'_\ell=\left\{\begin{array}{l}
c_i+(\ell-i)\dfrac{c_m-c_i}{m-i},\quad\ell=i+1,i+2,\dots,m-1,\\
c_m,\quad\ell=m.
\end{array}\right.\end{aligned}$$ 06: Overwrite $i:=m$.\
07: Set $c'_n:=c_n$ if $i=n$. Otherwise, go back to line 03.
All roots of characteristic polynomials of min-plus matrices
============================================================
Characteristic polynomials of matrices over linear algebra have roots which are just the eigenvalues. However, to the best of our knowledge, the characteristic polynomials of min-plus matrices have not yet been strictly defined. Min-plus characteristic polynomials can be, for example, given using the tropical determinant. Such characteristic polynomials have minimum roots which coincide with minimum eigenvalues and the minimums of average weights in the corresponding networks. The literature has not discussed whether the other roots are eigenvalues or not, nor whether they are meaningful features or not in the network. In this section, we thus clarify the relationship between the 2nd, 3rd,…, minimum roots and the average weights of circuits in a special network.
We first review characteristic polynomials of min-plus matrices using the tropical determinant [@MS]. For the min-plus matrix $A=(a_{ij})\in\mathbb{R}_{\min}^{n\times n}$, the tropical determinant, denoted $\text{tropdet}(A)$, is defined by: $$\begin{aligned}
\text{tropdet}(A)=\bigoplus_{\sigma\in S_n}a_{1\sigma(1)}\otimes a_{2\sigma(2)}\otimes\cdots\otimes a_{n\sigma(n)},\end{aligned}$$ where $S_n$ is the symmetric group of permutations of $\{1,2,\dots,n\}$. The following definition then determines the characteristic polynomial of $A$.
\[trop\] For the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, the characteristic polynomial $g_A(x)$ is given by $$\begin{aligned}
g_A(x)=\text{tropdet}(A\oplus x\otimes I), \end{aligned}$$ where $I$ is the $n$-by-$n$ identity matrix whose $(i,j)$ entries are $0$ if $i=j$, or $\varepsilon$ otherwise.
To distinguish the distinct circuits in the network $\mathcal{N}(A)$, that are associated with the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, we hereinafter use the notation $C(\ell_i,p_i)$ as the circuit of length $\ell_i$ and with the average weight $p_i$. Moreover, we prepare a set of circuits with a length sum of $\tilde{\ell}_i$ in the network $\mathcal{N}(A)$. Here, we regard the extended circuit of length $\tilde{\ell}_i$, and designate it as $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)$ where $\tilde{p}_i$ is the average weight. Of course, simple circuits are members of extended circuits, and the weight sum of $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)$ is $\tilde{\ell}_i\tilde{p}_i$ for each $i$. According to Maclagan-Sturmfels [@MS], we can easily derive a proposition concerning the relationships between coefficients of the characteristic polynomial and the weight sums of extended circuits in the network.
\[prp-eigen\] For the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, let us assume that the characteristic polynomial $g_A(x)$ is expanded as $$\begin{aligned}
g_A(x)\equiv x^n\oplus c_1\otimes x^{n-1}\oplus\cdots\oplus c_{n-1}\otimes x\oplus c_n.\end{aligned}$$ Then, each coefficient $c_j$ coincides with the minimum of the weight sums of the extended circuits in the set of the separated and extended circuits $\mathcal{C}_j:=\{\tilde{C}(\tilde{\ell}_i,\cdot)\mid\tilde{\ell}_i=j\}$ in the network $\mathcal{N}(A)$ that are associated with $A$.
Now, we consider the case where $k$ separate circuits $C(\ell_1,p_1),C(\ell_2,p_2),\dots,$ $C(\ell_k,p_k)$ existin the network $\mathcal{N}$. Strictly speaking, $C(\ell_1,p_1),C(\ell_2,p_2),\dots,$ $C(\ell_k,p_k)$ are distinct to each other and every vertex belongs to at most one circuit in the network $\mathcal{N}$. Without loss of generality, we may assume that $p_1\le p_2\le\cdots\le p_k$. Moreover, we recognize that the extended circuit $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)$ is homogeneous if all simple circuits in $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)$ have the same average weight $\tilde{p}_i$. We then see that, in the case where $C(\ell_1,p_1),$ $C(\ell_2,p_2),\dots,C(\ell_k,p_k)$ are separate circuits, $j$ homogeneous extended circuits $\tilde{C}(\tilde{\ell}_1,\tilde{p}_1),\tilde{C}(\tilde{\ell}_2,\tilde{p}_2),\dots,$ $\tilde{C}(\tilde{\ell}_j,\tilde{p}_j)$ exist where $\tilde{p}_1< \tilde{p}_2<\cdots<\tilde{p}_j$ and $j\le k$ in the network $\mathcal{N}$. This is key role to deriving the following two main theorems in this section.
\[thm1\] Let us assume that all circuits are separated in the network $\mathcal{N}(A)$ associated with the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$. Then the characteristic polynomial $g_A(x)$ can be factorized into linear factors of the form $g_A(x)\equiv (x\oplus\tilde{p}_1)^{\tilde{\ell}_1}\otimes (x\oplus\tilde{p}_2)^{\tilde{\ell}_2}\otimes\cdots
\otimes (x\oplus\tilde{p}_k)^{\tilde{\ell}_k}\otimes x^r$, where $r:=n-(\tilde{\ell}_1+\tilde{\ell}_2+\cdots+\tilde{\ell}_k)$.
Without loss of generality, we may assume $\tilde{p}_1<\tilde{p}_2<\cdots<\tilde{p}_k$. We first prove that $\tilde{p}_1,\tilde{p}_2,\dots,\tilde{p}_k$ are roots of $g_A(x)=x^n\oplus c_1\otimes x^{n-1}\oplus\cdots\oplus c_{n-1}\otimes x\oplus c_n$. It is obvious that the leading term $x^n$ becomes $n\tilde{p}_1$ at $x=\tilde{p}_1$. From Proposition \[prp-eigen\], the coefficient $c_{\tilde{\ell}_1}$ is equal to the minimum of weight sums of the extended circuits in the set $\mathcal{C}_{\tilde{\ell}_1}$. Since $\tilde{p}_1$ is the minimum average weight, $c_{\tilde{\ell}_1}=\tilde{\ell}_1\tilde{p}_1$. Thus, we can simplify the term $c_{\tilde{\ell}_1}\otimes x^{n-\tilde{\ell}_1}$ as $\tilde{\ell}_1\tilde{p}_1+(n-\tilde{\ell}_1)\tilde{p}_1=n\tilde{p}_1$ at $x=\tilde{p}_1$. Similarly, for all $i\not=\tilde{\ell}_1$, $c_{i}\otimes x^{n-i}=c_i+(n-i)\tilde{p}_1$ at $x=\tilde{p}_1$. If $c_i+(n-i)\tilde{p}_1<n\tilde{p}_1$, namely, $c_i/i<p_1$, then the homogeneous extended circuit $\tilde{C}(i,\tilde{p}_0)$ exists where $\tilde{p}_0<\tilde{p}_1$. This contradicts the assumption that $\tilde{p}_1$ is the minimum average weight. Thus, we conclude that $x=\tilde{p}_1$ is a root of $g_A(x)$. Moreover, we can easily derive $c_{\tilde{\ell}_1+\tilde{\ell}_2}\otimes x^{n-\tilde{\ell}_1-\tilde{\ell}_2}
=\tilde{\ell}_1\tilde{p}_1+(n-\tilde{\ell}_1)\tilde{p}_2$ at $x=\tilde{p}_2$. This is because Proposition \[prp-eigen\] immediately leads to $c_{\tilde{\ell}_1+\tilde{\ell}_2}=\tilde{\ell}_1\tilde{p}_1+\tilde{\ell}_2\tilde{p}_2$. Simultaneously, we can observe that $c_{\tilde{\ell}_1}+x^{n-\tilde{\ell}_1}
=\tilde{\ell}_1\tilde{p}_1+(n-\tilde{\ell}_1)\tilde{p}_2$. Thus, to prove that $\tilde{p}_2$ is a root of $g_A(x)$, it is necessary to show that, for all $i\not=\tilde{\ell}_1,\tilde{\ell}_1+\tilde{\ell}_2$, $c_i\otimes x^{n-i}\ge\tilde{p}_1\tilde{\ell}_1
+ (n-\tilde{\ell}_1) \tilde{p}_2$ at $x=\tilde{p}_2$, namely, $c_i-i\tilde{p}_2 \ge\tilde{\ell}_1(\tilde{p}_1-\tilde{p}_2)$. Recalling here that $\tilde{p}_1<\tilde{p}_2$, we see that $c_i-i\tilde{p}_2<0$, namely, $c_i/i<\tilde{p}_2$ if $c_i-i\tilde{p}_2<\tilde{\ell}_1(\tilde{p}_1-\tilde{p}_2)$. This implies that $c_i/i=\tilde{p}_1$ for $i\not=\tilde{\ell}_1$, but $c_i/i\not=\tilde{p}_1$ for $i\not=\tilde{\ell}_1$. Therefore, we recognize that $\tilde{p}_2$ is also a root of $g_A(x)$. Along the same lines, we observe that, for $m=2,3,\dots,k$, only two terms: $\tilde{\ell}_1\tilde{p}_1\otimes\tilde{\ell}_2\tilde{p}_2\otimes\cdots\otimes\tilde{\ell}_{m-1}\tilde{p}_{m-1}
\otimes x^{n-\tilde{\ell}_1-\tilde{\ell}_2-\cdots-\tilde{\ell}_{m-1}}$ and $\tilde{\ell}_1\tilde{p}_1\otimes\tilde{\ell}_2\tilde{p}_2\otimes\cdots\otimes\tilde{\ell}_m\tilde{p}_m
\otimes x^{n-\tilde{\ell}_1-\tilde{\ell}_2-\cdots-\tilde{\ell}_{m}}$ become both $\tilde{\ell}_1\tilde{p}_1+\cdots+\tilde{\ell}_{m-1}\tilde{p}_{m-1}
+(n-\tilde{\ell}_1-\tilde{\ell}_2-\cdots-\tilde{\ell}_{m-1})$ at $x=\tilde{p}_m$, and are the minimum among all terms in $g_A(x)$. This suggests that $x=\tilde{p}_m$ is a root of $g_A(x)$.
Next, we examine the linear factorization of $g_A(x)$. We can update $c_1,c_2,\dots,c_{\tilde{\ell}_1}$ as $\tilde{p}_1,2\tilde{p}_1,\dots,\tilde{\ell}_1\tilde{p}_1$, respectively, using Algorithm \[alg1\]. We then see that $x^n,c_1\otimes x^{n-1},\dots,c_{\tilde{\ell}_1}\otimes x^{n-\tilde{\ell}_1}$ are equal to each other at $x=\tilde{p}_1$. Similarly, Algorithm \[alg1\] updates $c_{\tilde{\ell}_1+1},c_{\tilde{\ell}_1+2},\dots,
c_{\tilde{\ell}_1+\tilde{\ell}_2}$ as $\tilde{\ell}_1\tilde{p}_1+\tilde{p}_2,\tilde{\ell}_1\tilde{p}_1
+2\tilde{p}_2,\dots,\tilde{\ell}_1\tilde{p}_1+\tilde{\ell}_2\tilde{p}_2$, then, it holds that $c_{\tilde{\ell}_1}\otimes x^{n-\tilde{\ell}_1}=c_{\tilde{\ell}_1+1}\otimes x^{n-\tilde{\ell}_1-1}=\cdots
=c_{\tilde{\ell}_1+\tilde{\ell}_2}\otimes x^{n-\tilde{\ell}_1-\tilde{\ell}_2}$ at $x=\tilde{p}_2$. Applying Algorithm \[alg1\] repeatedly, we see that $$\begin{aligned}
c_{i+1}-c_i=\left\{\begin{array}{ll}
\tilde{p}_1, & i=0,1,\dots,\tilde{\ell}_1-1, \\
\tilde{p}_2, & i=\tilde{\ell}_1,\tilde{\ell}_1+1,\dots,\tilde{\ell}_1+\tilde{\ell}_2-1,\\
& \qquad \vdots \\
\tilde{p}_k, & i=\tilde{\ell}_1+\tilde{\ell}_2+\cdots+\tilde{\ell}_{k-1},
\dots,\tilde{\ell}_1+\tilde{\ell}_2+\cdots+\tilde{\ell}_{k-1}+\tilde{\ell}_k-1,
\end{array}\right.\end{aligned}$$ where $c_0=0$. If $r=n-(\tilde{\ell}_1+\tilde{\ell}_2+\cdots+\tilde{\ell}_k)=0$, then it immediately follows from Proposition \[prp1\] that $g_A(x)=(x\oplus\tilde{p}_1)^{\tilde{\ell}_1}\otimes (x\oplus\tilde{p}_2)^{\tilde{\ell}_2}
\otimes\cdots\otimes (x\oplus\tilde{p}_k)^{\tilde{\ell}_k}$. If $r>0$, then there is no extended circuits greater in length than $n-r$ in the network $\mathcal{N}(A)$. This is because there exist $r$ vertices that do not belong to any circuits. Thus, we can overwrite the coefficients $c_{n-r+1},c_{n-r+2},\dots,c_{n-1}$ and the constant term $c_n$ with 0. Therefore, we have $g_A(x)=(x\oplus\tilde{p}_1)^{\tilde{\ell}_1}\otimes
(x\oplus\tilde{p}_2)^{\tilde{\ell}_2}\otimes\cdots\otimes (x\oplus\tilde{p}_k)^{\tilde{\ell}_k}\otimes x^r $.
\[thm2\] For the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, assume that all circuits are separated in the network $\mathcal{N}(A)$ associated with $A$. If the characteristic polynomial $g_A(x)$ can be factorized into linear factors of the form $g_A(x)\equiv (x\oplus\tilde{p}_1)^{\tilde{\ell}_1}\otimes (x\oplus\tilde{p}_2)^{\tilde{\ell}_2}\otimes\cdots
\otimes (x\oplus\tilde{p}_k)^{\tilde{\ell}_k}\otimes x^r$, then there exist homogeneous extended circuits $\tilde{C}(\tilde{\ell}_1,\tilde{p}_1),\tilde{C}(\tilde{\ell}_2,\tilde{p}_2),
\dots,\tilde{C}(\tilde{\ell}_k,\tilde{p}_k)$.
Similarly to prove Theorem \[thm1\], assume that $\tilde{p}_1<\tilde{p}_2<\cdots<\tilde{p}_k$. Here, we focus on the case $r=0$. Going over the proof of Theorem \[thm1\], we see that $g_A(x)$ is equivalent to $$\begin{aligned}
\hat{g}_A(x)&=x^n\oplus\tilde{\ell}_1\tilde{p}_1\otimes x^{n-\tilde{\ell}_1}\oplus
(\tilde{\ell}_1\tilde{p}_1\otimes\tilde{\ell}_2\tilde{p}_2)\otimes
x^{n-\tilde{\ell}_1-\tilde{\ell}_2}\oplus\cdots\\
&\quad\oplus (\tilde{\ell}_1\tilde{p}_1\otimes\cdots\otimes\tilde{\ell}_{k-1}\tilde{p}_{k-1})
\otimes x^{\tilde{\ell}_k}\oplus (\tilde{\ell}_1\tilde{p}_1\otimes\cdots\otimes\tilde{\ell}_k\tilde{p}_k)\end{aligned}$$ The coefficients $\tilde{\ell}_1\tilde{p}_1,\tilde{\ell}_1\tilde{p}_1\otimes\tilde{\ell}_2\tilde{p}_2,
\dots,\tilde{\ell}_1\tilde{p}_1\otimes\tilde{\ell}_2\tilde{p}_2\otimes\cdots\otimes\tilde{\ell}_{k-1}\tilde{p}_{k-1}$ and the constant term $\tilde{\ell}_1\tilde{p}_1\otimes\tilde{\ell}_2\tilde{p}_2
\otimes\cdots\otimes\tilde{\ell}_k\tilde{p}_k$ imply that the network $\mathcal{N}(A)$ includes the homogeneous extended circuits $\tilde{C}(\tilde{\ell}_1,\tilde{p}_1),
\tilde{C}(\tilde{\ell}_2,\tilde{p}_2),\dots,\tilde{C}(\tilde{\ell}_k,\tilde{p}_k)$.
From Theorems \[thm1\] and \[thm2\], we can conclude that the $2$nd, $3$rd, …$k$th minimum roots of the characteristic polynomial $g_{A}(x)$ are equal to the average weights $\tilde{p}_2,\tilde{p}_3,\dots,\tilde{p}_k$, respectively, if, and only if, the circuits $C(\ell_1,\tilde{p}_1),$ $C(\ell_2,\tilde{p}_2),\dots,C(\ell_k,\tilde{p}_k)$ are all separated.
New characteristic polynomials
==============================
In this section, we propose new characteristic polynomials of min-plus matrices by imagining the analogue of the Faddeev-LeVerrier algorithm [@Fad], which is an algorithm for generating characteristic polynomials of matrices in linear algebra.
In the Faddeev-LeVerrier algorithm, only the sums and products of scalars and matrices construct the characteristic polynomials of linear matrices. In fact, for a linear matrix $A\in\mathbb{R}^{n\times n}$, the coefficients $c_1,c_2,\dots,c_n$ appearing in the characteristic polynomial $x^n+c_1x^{n-1}+\cdots+c_{n-1}x+c_n$ is recursively given as $$\begin{aligned}
& c_1=-\text{Tr}(A),\\
& c_2=-\frac{1}{2}\text{Tr}(A^2+c_1A),\\
& \quad\vdots\\
& c_n=-\frac{1}{n}\text{Tr}(A^n+c_1A^{n-1}+\cdots+c_{n-1}A).\end{aligned}$$ Thus, we can derive new characteristic polynomials of min-plus matrices based on this method.
\[frame\] For the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, the characteristic polynomial $\hat{g}_A(x)$ $$\begin{aligned}
\hat{g}_A(x)=x^n\oplus c_1\otimes x^{n-1}\oplus\cdots\oplus c_{n-1}\otimes x\oplus c_n, \end{aligned}$$ is recursively given as $$\begin{aligned}
& c_1=\text{Tr}(A),\\
& c_2=\text{Tr}(A^2\oplus c_1\otimes A),\\
& \quad\vdots\\
& c_n=\text{Tr}(A^n\oplus c_1\otimes A^{n-1}\oplus\cdots\oplus c_{n-1}\otimes A),\end{aligned}$$ where $A^k=A^{k-1}\otimes A$ for $k=2,3,\dots,n$.
It is remarkable that the new characteristic polynomial $\hat{g}_A(x)$ usually differs from the already known characteristic polynomial $g_A(x)$. The following theorem gives the relationship between the minimum root of the characteristic polynomial $\hat{g}_A(x)$ and the eigenvalue of the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$.
\[thm3\] For the min-plus matrix $A\in\mathbb{R}_{\min}^{n\times n}$, the minimum root of the characteristic polynomial $\hat{g}_A(x)$ is equal to the eigenvalue of $A$.
With the help of Proposition \[prp;mineigen\], we may prove that the minimum root, denoted $p_{\min}$, is just the minimum of average weights of circuits in the network $\mathcal{N}(A)$. Regarding $\hat{g}_A(x)$ as the function with respect to $x$, we recall that $p_{\min}$ coincides with the minimum of the $x$-coordinates of breakpoints on the corresponding $xy$ functional graph. It is worth noting here that the breakpoints with the minimum $x$-coordinate are the intersection of two lines $y=nx$ and $y=c_i(n-i)x$ for some $i$. Thus, we derive $p_{\min}=c_i/i$.
It remains to proven that $c_i/i$ becomes the minimum of the average weights of circuits in the network $\mathcal{N}(A)$. Obviously, the coefficient $c_1= \text{Tr}(A)$ is equal to the minimum of the weight sums of circuits in the set $\mathcal{C}_1$. Taking into account that the diagonals of $A^2$ and $c_1\otimes A$ are the weight sums of all extended circuits in $\mathcal{C}_2$, we see that $c_2=\text{Tr}(A^2\oplus c_1\otimes A)$ expresses the minimum of the weight sums of all extended circuits in $\mathcal{C}_2$. Similarly, $c_i$ signifies the minimum of the weight sums of all extended circuits in $\mathcal{C}_i$. Thus, $p_{\min}=\min_ic_i/i$ is equal to the minimum of the average weights of all extended circuits in $\mathcal{C}_i$. Simultaneously, we see that the average weights of all extended circuits in the network $\mathcal{N}(A)$ are equal to or larger than $p_{\min}$. Moreover, if the minimum of the average weights of extended circuits in $\mathcal{C}_{\tilde{\ell}_i}$ is $p_{\min}$, then $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)$ is a simple circuit. This is because, if $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)$ is not a simple circuit, namely, $\tilde{C}(\tilde{\ell}_i,\tilde{p}_i)= \{C(\ell_1,p_1),C(\ell_2,p_2),\dots,C(\ell_k,p_j)\}$, then the average weight $\min_{i=1,2,\dots,j}\{\text{ave}(C(\ell_j,p_j))\}$ is smaller than $p_{\min}$. Therefore, we conclude that $p_{\min}$ becomes the minimum of the average weights of the circuits in the network $\mathcal{N}(A)$.
Although two characteristic polynomials, $g_A(x)$ and $\hat{g}_A(x)$ are essentially distinct, they are equivalent to each other in a special case.
\[crl1\] If all circuits are simple and separated in the network $\mathcal{N}(A)$, then two characteristic polynomials $g_A(x)$ and $\hat{g}_A(x)$ satisfy $g_A(x)\equiv\hat{g}_A(x)$.
Corollary \[crl1\] can be provided via the proofs of Theorems \[thm1\], \[thm2\] and \[thm3\]. We here give an example to illustrate the difference between two characteristic polynomials $g_A(x)$ and $\hat{g}_A(x)$. For the min-plus matrix $$\begin{aligned}
A=\begin{pmatrix}
\varepsilon & \varepsilon & 2 & \varepsilon & \varepsilon & \varepsilon & \varepsilon \\
3 & \varepsilon & \varepsilon & 2 & \varepsilon & \varepsilon & \varepsilon \\
\varepsilon & 1 & 3 & 9 & 1 & \varepsilon & \varepsilon \\
\varepsilon & 6 & \varepsilon & \varepsilon & \varepsilon & 2 & \varepsilon \\
\varepsilon & \varepsilon & \varepsilon & \varepsilon & \varepsilon & 2 & 1 \\
\varepsilon & \varepsilon & \varepsilon & \varepsilon & \varepsilon & \varepsilon & 1 \\
\varepsilon & \varepsilon & \varepsilon & \varepsilon & \varepsilon & \varepsilon & \varepsilon
\end{pmatrix},\end{aligned}$$ we obtain two characteristic polynomials $$\begin{aligned}
& g_A(x)=x^7\oplus 3\otimes x^6\oplus 8\otimes x^5\oplus 6\otimes x^4\oplus 20\otimes x^3,\\
& \hat{g}_A(x)=x^7\oplus 3\otimes x^6\oplus 6\otimes x^5\oplus 6\otimes x^4\oplus 9\otimes x^3
\oplus 12\otimes x^2\oplus 12\otimes x\oplus 15.\end{aligned}$$ Using Algorithm \[alg1\], we can factorize $g_A(x)$ and $\hat{g}_A(x)$ as $$\begin{aligned}
& g_A(x)\equiv (x\oplus 2)^3\otimes (x\oplus 14)\otimes x^3,\\
& \hat{g}_A(x)\equiv (x\oplus 2)^6\otimes (x\oplus 3).\end{aligned}$$ As shown in Theorems \[thm1\], \[thm2\] and \[thm3\], the minimum roots of $g_A(x)$ and $\hat{g}_A(x)$ are certainly both the eigenvalue of $A$. However, since the 2nd minimum roots of $g_A(x)$ and $\hat{g}_A(x)$ are 14 and 3, respectively, they are not equal to each other. In actuality, limited to computing the eigenvalue of $A$, we can equivalently simplify $\hat{g}_A(x)$ as $$\begin{aligned}
\breve{g}_A(x)=x^4\oplus 3\otimes x^3\oplus 6\otimes x^2\oplus 6\otimes x\oplus 9
\equiv (x\oplus 2)^3\otimes (x\oplus 3).\end{aligned}$$ In other words, it is not necessary to determine the coefficients $c_5=c_2\oplus 6=12,c_6=c_3\oplus 6=12,c_7=c_4\oplus 6=15$ to compute the eigenvalue. Obviously, the linear factorization of $\breve{g}_A(x)$ is easier than that of $g_A(x)$. New characteristic polynomials are thus expected to gain more advantage, as the matrix-size increases.
Concluding remarks
==================
In this paper, we focused on all the roots of the already known characteristic polynomials of matrices, which are given from the links of vertices in networks on graphs, over min-plus algebra, and presented distinct new characteristic polynomials. First, we briefly explained scalar and matrix arithmetic over min-plus algebra, the eigenvalues of min-plus matrices and the minimum average weights of circuits in networks, and the linear factorizations of min-plus polynomials. We then described a preconditioning algorithm for performing effective linear factorizations. Of course, the eigenvalues of min-plus matrices are the minimum roots of the already known characteristic polynomials. In other words, the minimum roots coincide with the minimum average weights of circuits in the corresponding networks. Restricting the case to one where all circuits are completely separated in networks, we next showed that the $2$nd, $3$rd, $\dots$ minimum roots of the already known characteristic polynomials are just the $2$nd, $3$rd, $\dots$ minimum average weights, respectively. Finally, we propose new characteristic polynomials whose minimum roots are also the eigenvalues of min-plus matrices, and showed that they are equivalent to the already known characteristic polynomials if all circuits are completely separated in networks. We provided an example to verify the difference between the already known and proposed characteristic polynomials. The example simultaneously suggests that the proposed characteristic polynomials can be substantially reduced if the edge number is not large in the corresponding networks. Thus, the proposed characteristic polynomials may be, so to speak, minimal polynomials. Future work will focus on examining this aspect and designing reduction algorithms.\
[**Acknowledgements**]{}\
This was partially supported by Grants-in-Aid for Scientific Research (C) No. 26400208 from the Japan Society for the Promotion of Science.
[99]{} K. Ahuja, L. Magnanti and B. Orlin, Network Flows, Prentice-Hall, 1993. F. Baccelli, G. Cohen, G.J. Olsder and J.P. Quadrat, Synchronization and Linearity, Wiley, 1992. J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997. D.K. Faddeev and V.N. Faddeeva Computational Methods of Linear Algebra, W.H.Freeman & Co Ltd, 1963. G.H. Golub, C.F. Van Loan, Matrix Computations, 4th edn., Johns Hopkins University Press, Baltimore, 2013. M. Gondran and M. Minoux, Graph, Dioids and Semiring, Springer Verlag, 2008. D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, AMS, 2015. H. Rutishauser, Lectures on Numerical Mathematics, Birkhäuser, Boston, 1990. U. Zimmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures, North-Holland Publishing Company, 1947.
|
---
abstract: 'Motivated by recent works of Sen [@Sen:2002nu; @Sen:2002in] and Gibbons [@Gibbons:2002md], we study the evolution of a flat and homogeneous universe dominated by tachyon matter. In particular, we analyse the necessary conditions for inflation in the early roll of a single tachyon field.'
author:
- |
Malcolm Fairbairn[^1]\
Michel H.G. Tytgat[^2]\
\
Service de Physique Théorique, CP225\
Université Libre de Bruxelles\
Bld du Triomphe, 1050 Brussels, Belgium
date: 'April 9, 2002'
title: 'Inflation from a Tachyon Fluid ?'
---
Introduction
============
Despite numerous efforts, it seems difficult to reconcile string theory with the highly successfull paradigm of inflation. In this brief note, following Gibbons [@Gibbons:2002md], we investigate whether some form of Sen’s tachyonic matter [@Sen:2002nu; @Sen:2002in] might provide the necessary ingredients for a phase of inflationary expansion in the early universe.
According to Sen [@Sen:2002in], (see also [@Gibbons:2001hf]), a rolling tachyon condensate in either bosonic or supersymmetric string theory can be described by a fluid which in the homogeneous limit has energy density $$\rho = {V(T)\over\sqrt{1 - \dot T^2}}$$ and pressure \[eos\] p = - V(T) - (T) (1 - T\^2) with $T$ the tachyon field and $V(T)$ the tachyon potential. These expressions are obtained from the tachyon matter effective lagrangian \[lag\] [L]{} = - V(T) . Given the generic properties of $V(T)$ for $T\geq 0$, a most remarkable feature of Sen’s equation of state (\[eos\]) is that tachyon matter interpolates smoothly between p = - w = -1 for $\dot T=0$ and p = 0 w = 0 as $\dot T$ reaches its limiting value $\dot T=1$. As already emphasized by Gibbons [@Gibbons:2002md], if the tachyon condensate starts to roll down the potential with small intial $\dot T$, a universe dominated by this new form of matter will smoothly evolve from a phase of accelerated expansion to a phase dominated by a non-relativistic fluid. It is tempting to speculate that the latter could contribute to some new form of dark matter. However, the topic of this paper is whether or not the tachyon condensate could be relevant for inflation. (For related speculations, see for instance [@speculations][@anupam].)
The shape of the tachyon condensate effective potential depends on the system under consideration. In bosonic string theory for instance, this potential has a maximum $V=V_0$ at $T=0$, where $V_0$ is the tension of some unstable bosonic D-brane, a local minimum with $V=0$, generically at $T \rightarrow + \infty$, corresponding to a metastable closed bosonic string vacuum, and a runaway behaviour for negative $T$. An exact classical potential ([*[i.e.]{}*]{} exact to all orders in $\alpha^\prime$, but only tree level in $g_s$) encompassing these properties has been computed [@Kutasov:2000qp], \[potential\] V(T) = V\_0(1+ [T/T\_0]{}). Note that the curvature at the top of the potential (\[potential\]) is $
{d^2V/ dT^2} = - {V_0/T_0^2}
$ . As the tachyon field has dimension $[T] = E^{-1}$, if $\dot T \ll 1$, from Eq.(\[lag\]) we see that it is natural to rescale $T$ by $\sqrt{V_0} T \equiv \phi$. At $V=V_0$, the mass of the canonically normalized $\phi$ is then $M_\phi^2 =
-1/T_0^2$. In [@Kutasov:2000qp], $T_0 \sim l_s$ and $V_0$ is the tension of a bosonic Dp-brane, $V_0 \sim 1/g_s
l_s^{p+1}$. We shall see that these values of $V_0$ and $T_0$ seem marginally incompatible with inflation: the potential is simply too steep. However, the values for $V_0$ and $T_0$ required to obtain slow-roll inflation are within an order of magnitude of these values.
Tachyon matter cosmology
========================
As shown by Gibbons, for a Roberston-Walker tachyon matter dominated universe, the Friedman equation takes the standard form H\^2 = [\^2 3]{} = [\^2 3]{}[V]{} with $\kappa^2= 8 \pi G= 8 \pi /M_{pl}^2$ and where we have assumed spatial flatness[^3] and have chosen to put the cosmological constant to zero. Entropy conservation gives as usual = - 3 H (+ p) the latter being equivalent to the equation of motion for the tachyon field $T$, \[eom\] [V T1 - T\^2]{} + 3 H VT + V\^= 0 where $V^\prime = dV/dT$. We would like to use these equations to determine the slow-roll conditions for inflation. The first condition to be satisfied is that the expansion is accelerating (see for instance [@liddle] or [@Lyth:1998xn]) $$\begin{aligned}
{\ddot a\over a} \equiv H^2 + \dot H &=& - {\kappa^2 \over 6}(\rho + 3 p) \;>\; 0\nonumber\\
&=& \frac{\kappa^2 }{3}{V\over \sqrt{1 - \dot T^2}}\left (1 - {3\over 2} \dot T^2\right)\;>\; 0\end{aligned}$$ which requires that[^4] \[inflation\] T\^2 < [23]{}. In order to have a sufficiently long period of inflation, the tachyon field should start rolling with small initial $\dot T$. To relate the condition (\[inflation\]) to the shape of the potential, we have to calculate the slow-roll conditons. Following a standard procedure [@liddle], we can express the evolution of the universe as a function of $T$ rather than time. Using H = - [\^2 2]{} [V T\^2]{} and because of the monoticity of $T$ with respect to time, we can rewrite this equation as H\^(T) = - [\^2 2]{} [V T]{} where the prime denotes derivation with respect to $T$. Taking the square of the Friedmann equation we get \[HJ\] [H\^]{}\^2 - [94]{} H\^4(T) + [\^4 4]{} V(T)\^2 = 0. This first order differential equation is the analog for tachyon matter of the Hamilton-Jacobi form of the Friedmann equation with a single inflaton field [@Salopek:1990jq]. It expresses the fact that $H^\prime$ is negligible as long as H\^2 V. Solving (\[HJ\]) for $H(T)$, we can then get $\dot T$ as a function of $T$ = 1 - ([\^2 V3 H\^2]{})\^2. Using the potential (\[potential\]) one can solve (\[HJ\]) numerically. We have chosen the field $T=T_i$ to be slightly displaced from the maximum of the potential with initial $\dot T_i=0$. We also assume $T_i>0$ for obvious reasons. The amount of inflation obtained depends upon the variable $T/T_0$ and on a dimensionless parameter $X_0$ which characterizes the flatness of the potential close to its peak X\_0\^2 = \^2 T\_0\^2 V\_0. In Figure \[dotT2\] we plot $\dot T^2$ for different values of $X_0$.
The pertinent feature of this figure is that for increasing $X_0$, inflation ends at increasing values of $T$. This is equivalent to requiring that as the rolling of the field commences, the $\ddot T$ term is negligible in the equation of motion for the tachyon field (\[eom\]), so that 3 H V T - V\^. \[slow\] Another quantity of interest is the number of e-folds during the inflationary phase, N(T) = \_t\^[t\_[end]{}]{} H dt = - \_T\^[T\_[end]{}]{} [H\^2 VV\^]{} dT This is shown in Figure \[efolds\]. To successfully use inflation to solve the horizon problem and bring us a suitably flat spectrum of perturbations we require $N~\geq~50-60$ e-folds of inflation whilst the field is rolling. As always, there is some uncertainty in the number of e-folds related to the choice of initial conditions. However, if we assume that $\dot T^2$ is small initially and that $T/T_0$ starts at less than about $0.1$ then the total number of e-folds becomes insensitive to the exact initial conditions. We choose the value of $X_0$ so as to accommodate enough e-folds and the normalization to the COBE spectrum, which we shall describe in the next section. For the time being, we simply note that inflation lasts longer if $X_0$ is larger, as shown in Figure \[dotT2\].
Once inflation is over the expansion of the tachyon-matter dominated universe slows rapidly which precludes the possibility of the modes generated during the slow roll regime being pushed outside our present day horizon.
Slow-rolling tachyon matter
===========================
The numerical analysis shows that inflation requires that $X_0 \gsim 3$. Once this is satisfied the field $T$ evolves in a framework analogous to the slow-roll approximation for scalar fields. In order for the conditions H\^2 V and H V T - V\^to hold, the following inequalities must be satisfied: \[epsilon\] 1 and \[eta\] - [V”\^2 V\^2]{} 1. These are just the usual definitions of the slow-roll parameters $\epsilon$ and $\eta$. Their form is different from the usual expressions because we are taking derivatives with respect to $T$. Near $V_0$, for small $\dot T$, one can use the canonically normalized field $\phi = \sqrt{V_0} T$ which brings (\[epsilon\]) and (\[eta\]) to their usual form [@liddle]. As we assume that the tachyon is rolling from the top of the potential, generically $\epsilon \ll \eta$ for small $T/T_0$, so that the second condition is the most stringent one. Using the potential (\[potential\]) to be specific, (\[eta\]) becomes \^2 V\_0 T\_0\^2 X\_0\^2 1 \[condition1\] a condition which is consistent with the numerical analysis of the preceding section.
Finally we must estimate the size of the fluctuations and compare this estimate with the magnitude of the density pertubations observed by COBE. Assuming that the slow-roll approximation can be made to hold over a significant range of cosmic scales, the spectral index $n(k)$ is taken to be scale independent with n 1 - 2 1. \[spectrum\] We also follow the standard procedure to estimate the density perturbations ||T where . We then use the slow roll condition ($\ref{slow}$) to eliminate $V^{\prime}$ leading to the expression || and we obtain the expression for the density perturbation during inflation when $T/T_0$ is of order one || which correponds to roughly 60 efolds before the end of inflation for $5\lsim X_0 \lsim 7$. Since $\delta\rho/\rho \approx 2\times 10^{-5}$ we see that \^[3]{} V\_0 T\_0 610\^[-4]{}. \[condition2\] In order to calculate the magnitude of our perturbations, we assumed that $\dot{T}$ is negligible and that we are close to the peak of the potential so that we can substitute $T$ for the canonical scalar field $\sqrt{V_0}\phi$. However, without a full derivation of the perturbations starting with the Lagrangian ($\ref{lag}$) we cannot say at what point our approximations break down, and what kind of behaviour replaces the normal lore at that point.
Discussion
==========
We have not mentioned the precise origin of the tachyon field in question, but given the string theory origin of Sen’s equation of state, it is tempting to write the parameters $V_0$ and $T_0$ in terms of the string length $l_s$ and the open string coupling constant $g_s$, V\_0=,T\_0=\_0 l\_s. where $v_0$ and $\tau_0$ are dimensionless parameters such that $V_0/v_0$ is the tension of a D3-brane and $\tau_0 l_s$ is the inverse tachyon mass[@strings]. The gravitational coupling in 4 dimensions is given in terms of the stringy parameters by $$\kappa^2\equiv 8 \pi G_N =\pi g_s^2 l_s^2\left(\frac{l_s}{R}\right)^{6}.$$ Here $R$ is the compactification radius of the compact 6 dimensional manifold, taken here to be a 6-torus.[^5] For the $D=4$ effective theory to be applicable one usually requires that $ R \gg l_s$ since $R=l_s$ denotes the self T-dual point where the mass spectrum of KK and winding modes become degenerate [@strings]. The volume of the compact space therefore must satisfy the inequality $V > (2\pi R)^6$.
In order for the gravitational waves at the end of inflation to be compatible with CMB observtions, there is a further condition $\cite{linde}$ $$\frac{H_{end}}{M_{pl}}\le 3.6\times 10^{-5}$$ which together with the requirement for enough e-folds ($\ref{condition1}$) and the magnitude of the perturbations as given by equation ($\ref{condition2}$) leads to three conditions which can be written as $$\begin{aligned}
\left(\frac{l_s}{R}\right)^{12}v_0 g_s^3 &\lsim& 1.4\times 10^{-6} \;\;\; \mbox{\rm (no grav. waves)}\label{ngw}\\
\left(\frac{l_s}{R}\right)^{9}v_0 \tau_0 g_s^2 &\sim& 2.7\times 10^{-2}\;\;\; \mbox{\rm(COBE normalization)}\\
\left(\frac{l_s}{R}\right)^{6}v_0 \tau_0^2 g_s &\gsim& 7.1\times 10^{2} \;\;\;\mbox{\rm (inflationary cond.)}\label{ic}\end{aligned}$$ These conditions cannot all be satisfied simultaneously with the values one would choose as a first guess, $v_0 \sim \tau_0 \sim 1$, for any $R \gsim l_s$.[^6]
For $R \sim l_s$ and insisting on $v_0 \sim 1$, (\[ngw\]) gives $g_s \lsim 10^{-2}$ while (\[ic\]) imposes $\tau_0 \gsim 10^{2}$, corresponding to a rather light tachyon mass $m_T \lsim 10^{-2}/l_s$. This is phenomenologically the simplest solution, but is at odds with expectations from string theory. (See however [@Kim:2002rv].)
If one insists on a tachyon mass $\sim 1/l_s$, the only alternative without entering the strong coupling regime is to increase the energy density, for instance by increasing the number of branes, in the spirit of [@anupam] where the potential arises from the combined tachyonic potential of a number of brane/anti-brane systems. The problem here is that in order for each individual brane to act in the way described by Sen’s effective action the typical seperation between branes within the compact directions would have to be larger than $l_s$.[^7] A miminal, although [*ad hoc*]{}, setting is to take $v_0$ to be equal to the number of string length size volumes within the compact space, i.e. $$v_0=\left(\frac{2\pi R}{l_s}\right)^6$$ Then it is possible to fulfill the above requirements with $g_s\sim 2\times 10^{-2}$, $R/l_s\sim 10$, corresponding to a very large number of branes ($\sim 10^{11}$). This initial condition is quite baroque and at first sight rather unattractive. However we take note that such a large number is not foreign to speculations on the dS/CFT correspondence [@Strominger:2001pn], which suggests that the number of light degrees of freedom of the putative Euclidean CFT dual to the inflationary phase of the universe should be very large $c \sim 10^8$ [@Larsen:2002et].
One last issue we would like to comment on is that of reheating at the end of inflation. In this paper our approach has been rather phenomenological. We took Sen’s equation of state together with a string-theory motivated effective potential. Both ingredients rest on quite severe approximations, but are supposed to capture at least some of the physics of tachyon condensation. Remarkable features are that 1) the minimum of the tachyon potential is at $T \rightarrow \infty$ and that 2) the energy of the brane stays confined to its hyperplane in the form of a pressureless fluid. As far as we understand, the first feature is supposed to be generic. As has been recently emphasized by Kovman and Linde [@linde], this is quite worrisome for the purpose of reheating the universe at the end of inflation. They also note that this is potentially a problem for [*all*]{} string-inspired models of inflation based on the annihilation or decay of branes.
From a phenomenological point of view the simplest resolution would be to have a potential that vanishes at finite $T$ so that reheating can proceed through oscillations of the tachyon fluid around its minimum. Either way, this raises the issue of the decay of the tachyon fluid. The standard lore in the string-community is that unstable D-branes should ultimately decay into closed string modes. To make this process manifest one should take into account $g_s$ corrections to the tachyon effective action, a program which has not been completed yet. We believe that a related issue will be to understand the precise nature of the tachyon fluid at the minimum of the potential. It is expected to be stable only in the limit $g_s \rightarrow 0$[@Sen] and should in principle be a good zeroth order approximation to the problem of tachyon matter decay.
An intruiging characteristic of Sen’s effective action is that at the minimum of the potential, the tachyon fluid equations of motions reduced to, in Hamiltonian form and in flat space-time, [@Sen:2002an] $$\begin{aligned}
1 &=& (\dot T)^2 - (\partial_i T)^2\\
\dot \Pi &=& \partial_i \left(\Pi {\partial_i T\over \dot T}\right) \end{aligned}$$ where $\Pi$ is the momentum conjugate to $\dot T$. These are precisely the equation of motion of a relativistic pressureless fluid of partons of mass unity, with velocity field $\vec v = \nabla T/\dot T \equiv \vec k/\omega(k)$ and energy density $\Pi(x,t)$. Solutions of these equations of motions are quite trivial to work out. A time-independent but otherwise arbitrary configuration of the energy-density $\Pi(x)$ corresponds in the parton picture to a bunch of massive partons [*at rest*]{}. However, generic solutions have caustic singularities and the free parton picture has probably a limited range. It makes nevertheless clear why there are no propagating tachyon waves at the minimum of the potential, simply because there are no collective excitations in a pressureless fluid.
In the limit of $g_s\rightarrow 0$, the tachyon fluid is stable and behaves as some new form of dark matter. The problem with this is that the universe would never become radiation dominated and for tachyon matter to be relevant in cosmology requires some fine-tuning [@Shiu:2002qe; @linde]. For finite $g_s$ however tachyon matter should be unstable and decay. If we are willing to take the parton picture seriously, it is plausible that their decay could solve the issue of reheating of the universe at the end of tachyon rolling, however, not being string theorists, we have no idea how this question could be addressed. As an illustration we simply parametrize the decay rate as $$\Gamma \sim \frac{g_s^n}{l_s} \sim \left(\frac{l_s}{R}\right)^3 g_s^{n+1}M_{pl}$$ The reheat temperature is then (presumably over-) estimated to be T\_[RH]{}\~ \~()\^g\_s\^M\_[pl]{} which for generic powers of $n$ is high enough for most cosmological puposes. Obviously it would be interesting, both for string theory and for phenomenological applications to develop further insights on this issue.
Conclusions
===========
In this paper we have investigated the possibility of using Sen’s equation of state to see if we can obtain viable cosmological inflation. We have described the conditions which the potential and the tachyon mass must fulfil in order to provide enough e-folds of inflation and density perturbations of the correct magnitude. Our approach has been essentially phenomenological. In particular, the conditions for slow-roll are essentially independent of the precise shape of the potential and whether it vanishes at finite or infinite $T$. One attractive feature of Sen’s equation of state is that it give an explicit realization of so-called k-inflation [@Armendariz-Picon:1999rj]. A related interesting property is that the exit from inflation is automatic for generic string-inspired effective potentials due to the limiting value $\dot T= 1$ for homogeneous configurations. As already emphasized numerous times a straightforward matching with string theory is problematic but perhaps not altogether impossible. An important issue, both for cosmological applications and, we presume, for string theory is to understand the ultimate fate of the tachyon matter at the bottom of the potential.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Laurent Houart, Samuel Leach and Ashoke Sen for useful conversations or comments.
[99]{}
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[^1]: mfairbai@ulb.ac.be
[^2]: mtytgat@ulb.ac.be
[^3]: As tachyon matter turns into a non-relativistic fluid for large $T$, a closed Universe will eventually recollapse. We assume flatness for simplicity. Alternatively, the recollapse can be supposed to be in the far future of our model universe.
[^4]: This condition for inflation is in contrast to that obtained for a normal scalar field, $\dot \phi^2 < V(\phi)$.
[^5]: Like in most higher dimensional scenarios, we assume that some unknown mechanism freezes the moduli associated to the extra dimensions. Also, for simplicity we assume that we can neglect the evolution of other fields, like the dilaton.
[^6]: In the first version of this paper we overlooked a factor of $g_s$ in the expression of the Planck mass which led us to underestimate strong gravity effects.
[^7]: If the branes are coincident, there are extra tachyonic degrees of freedom which all together transform either in the adjoint of the gauge group that lives on a stack of non-BPS D-brane or in the bi-fundamental for $Dp-\bar Dp$ pairs [@Horava:1998jy].
|
---
abstract: 'We present a simple (stationary) mechanism capable of generating the auroral downward field-aligned electric field [that is]{} needed for [accelerating the ionospheric electron component up into the magnetosphere and confining the ionospheric ions at low latitudes (as is required by observation of an ionospheric cavity in the downward auroral current region). The lifted ionospheric electrons carry the downward auroral current. Our model is based on the assumption of collisionless reconnection in the tail current sheet. It makes use of the dynamical difference between electrons and ions in the ion inertial region surrounding the reconnection [X]{}-line which causes Hall currents to flow. We show that the spatial confinement of the Hall magnetic field and flux to the ion inertial region centred on the [X]{}-point generates a spatially variable electromotive force which is positive near the outer inflow boundaries of the ion inertial region and negative in the central inflow region. Looked [at]{} from the ionosphere it functions like a localised meso-scale electric potential.]{} The positive electromotive force gives rise to upward electron flow from the ionosphere [during substorms (causing ‘black aurorae’)]{}. A similar positive potential is identified on the earthward side of the fast reconnection outflow region which has the same effect, explaining the observation that auroral upward currents are flanked from both sides by narrow downward currents.'
author:
- 'R. A. Treumann[^1]'
- 'R. Nakamura'
- 'W. Baumjohann'
title: 'Downward auroral currents from the reconnection Hall-region'
---
Introduction {#introduction .unnumbered}
------------
[There is no consensus yet on the origin of the substorm auroral field-aligned current system. Observations *in situ* the Earth’s auroral region have shown that it consists of a spatially separated combination of upward and downward field aligned currents which close in the ionosphere via Pedersen currents flowing perpendicular to the magnetic field at altitudes where the perpendicular ionospheric conductivity is large [@elphic1998; @carlson1998]. These currents are carried by electrons of different energies which have been accelerated along the magnetic field by field-aligned electric potential drops. ]{}
[The origin of the downward current electrons is clearly ionospheric, implying that the (cold) electrons have been accelerated upward [@carlson1998; @carlson1998a] along the magnetic field by the presence of an electric field that points down from the magnetosphere into the ionosphere[see also the review in @paschmann2003], being responsible for the absence of any optical aurora, i.e. causing ‘black aurora’ [for a review cf., e.g., @marklund2009 who also provides a timely account of electron fluxes and field-aligned currents based on Freja and Fast observations in the downward current region]. At the same time this strong magnetically parallel electric field keeps the ionospheric ion component in this region (of the auroral downward field-aligned current) down, confining it to low ionospheric altitudes thus causing a so-called ionospheric trough or plasma cavity. Conversely the upward (or return) currents are carried by high energy (hot) downward flowing electrons which have been accelerated somewhere in the magnetosphere and are responsible for the known auroral phenomena.]{}
[These two regions of field-aligned currents and electron fluxes during aurorae are spatially separated. Usually one observes that the downward current-upward electrons form the latitudinally narrow northern (high-latitude) edge of the auroral disturbances while the upward current-downward electrons are found in a latitudinally much broader region adjacent to the former on its low-latitude side but separated from it by a narrow latitudinal gap where both field-aligned currents and electron fluxes are lacking. In addition, in most cases this latitudinally extended upward current-downward electron region is flanked on it southern side by another latitudinally narrow region of downward current-upward electron fluxes. This sequence seems to be observationally established though frequently obscured by the high dynamics of a substorm where many such regions can occur in a chain. Examples (cf. also Figure \[aurorec-fig1a\]) can be found in the above cited literature.]{}
Recently we argued that the downward electron fluxes originate in the tailward reconnection region and that the complex structure of fluxes observed in the auroral region is caused by multiple reconnection sites in the magnetotail current sheet [@treumann2009]. [In this Letter we investigate just one such reconnection site. We argue that reconnection in the geomagnetic tail may indeed generate both the fluxes of accelerated downward electrons which cause the active aurora and also may generate, which so far has neither been realised nor suspected, the field aligned electric fields that are needed if one wished to extract the ionospheric electrons from the ionosphere upward into the magnetosphere that carry the downward field-aligned auroral currents. For this to work, we find that reconnection must indeed be collisionless, occur in a sufficiently thin current sheet of width at most a few ion inertial lengths $\lambda_i$ (for the definition see below), and Hall currents must flow in the reconnection environment. These statements are not completely independent since Hall currents can flow in reconnection[^2] only under complete non-collisionality [@sonnerup1979; @fujimoto1997; @oieroset2001; @nagai2001; @treumann2006].]{}
[ ]{}
[ ]{}
Electromotive force in the reconnection-Hall region {#electromotive-force-in-the-reconnection-hall-region .unnumbered}
---------------------------------------------------
[Reconnection under collisionless conditions in thin current sheets of width the order of the ion-inertial scale $\lambda_i=c/\omega_{pi}$ has observationally been confirmed [see, e.g., @fujimoto1997; @nagai2001; @oieroset2001; @nakamura2006] to occur inside a region of Hall current flow. These currents are carried by the magnetised plasma sheet electrons that flow across the non-magnetised plasma sheet ions [for a pedagogically instructive sketch the reader may consult the paper by @treumann2006]. ]{}
[The physics behind the generation mechanism of the Hall current under collisionless conditions is extraordinarily simple and may be summarised in a few sentences as follows: Hall currents require the presence of a magnetically perpendicular electric field ${\bf E}_\perp$ which forces the particles into an ${\bf E}_\perp\times{\bf B}$-drift perpendicular to both the magnetic and electric fields. This drift under non-collisionality is normally performed by both particle components and is current-free. (In the presence of collisions the ions are braked, only the electrons can flow, and a Hall current is caused, for instance in two-dimensional metals or in space in the ionosphere and probably as well in the solar chromosphere). In the collisionless state the mechanism of separation between electron and ion motions is due to the presence of a reconnection [X]{}-point (or line) where the two plasma counter-streaming plasmas ‘collide’. Under these conditions classical Hall currents are secondary to reconnection, being just a side effect. Within a distance of one ion gyro-radius (or ion inertial length $\lambda_{ci}=c/\omega_{pi}$ with $\omega_{pi}^2=e^2N/\epsilon_0m_i$ the square of the ion plasma frequency) away to both sides of the [X]{} line produced in reconnection, the ions behave non-magnetic. Transport of the magnetic field is the solely due to the ${\bf E}\times{\bf B}$ drift of the electrons against the ions. Hence, the electrons, in this region, carry a Hall current of strength $|{\bf J}_H|=e NE_\perp/B$. Due to their inertia, the ions are still slowly flowing in but do not perform an electric field drift anymore on this scale, rather they become accelerated along ${\bf E}_\perp$ thereby amplifying the sheet current (which may support the onset of reconnection).]{}
[In the magnetic [X]{}-point geometry at the reconnection site these Hall currents generate a secondary quadrupolar Hall-magnetic field the geometry of which had been inferred thirty years ago by @sonnerup1979 who also argued that the Hall current system necessarily included field-aligned currents in order to ‘close’ it, connecting the Hall currents to the ionosphere. In fact, these field aligned closure currents *do not belong* to the Hall current system; they only exist in the magnetospheric geometry where the field lines are tied to the conducting ionosphere. Hall currents are always exactly transverse to both the magnetic and the perpendicular electric fields. In an infinitely extended plane homogeneous collisionless current sheet field-aligned currents would be absent leaving reconnection independent even if including Hall currents. The divergence of the Hall currents in this case would be taken care of by their vanishing at the boundaries of the reconnection region. Reconnection in such plane current sheets has been studied extensively with the help of numerical PIC simulations [e.g., by @zeiler2000; @zeiler2002; @scholer2003 and others]. In recent papers [@baumjohann2010; @treumann2010; @treumann2011] we discussed the generation of seed-[X]{} points and some aspects of its micro-scale physics. ]{}
In the Earth’s Magnetosphere-Ionosphere system connection between the Hall region and the ionosphere is, however, unavoidable [because of the presence of the conducting ionosphere with its high perpendicular conductivity. Figure \[schematic-fig\] shows a very crude schematic of the magnetic connection between the tail reconnection site and the auroral region, i.e. the ion inertial domain at the reconnection site and the observed upward/downward electron fluxes in the auroral zone, both indicated in the figure. ]{}
The downward auroral electron fluxes respectively upward (so called return) currents can in the reconnection model be understood as having their source in the acceleration mechanism acting at the reconnection site. No convincing reason could so far been given for the generation of the strong observed upward electron fluxes. Any near Earth models simply assume that a ‘battery effect’ exists at the upper boundary of the ionosphere causing the required large field-aligned potential drops. This battery is assumed to be wave-driven, for instance by kinetic or shear Alfvén waves, or shear-flow driven, lacking any convincing reason for the appearance of shear flows at the top of the ionosphere.
[The conventional non-resistive model of how a field-aligned potential drop can be created under auroral conditions is based on the assumption that at the upper altitude boundary of the ionosphere some mechanism causes $\mathbf{E}_\perp\times\mathbf{B}$-shear flows [@carlson1998; @carlson1998a; @elphic1998]. Shear flows under collisionless conditions correspond to a spatial dependence of the perpendicular electric fields $\mathbf{E}_\perp(\mathbf{x})$. If these electric fields diverge in some place such that $\nabla\cdot\mathrm{E}<0$, their spatial dependence corresponds to a net ‘positive space charge’ (potential) which attracts electrons and repulses positive ions along the magnetic field. Otherwise, if the electric field converges, the correspondence is to a net ‘negative space charge’ which reflects electrons and attracts positive ions. In fact, shear flows of this case have barely been observed in the topside ionosphere. The interesting question that arises is, whether they may exist at the reconnection site in die near-Earth magnetosphere.We suggest here that the mere existence of the Hall (ion inertial) region at the near-Earth tail reconnection site is sufficient for producing the required ‘shear flow’ electromotive forces, i.e. the field-aligned potential drop for extracting and upward accelerating the ionospheric electron component.]{}
[In order to attract electrons upward from the ionosphere as suggested by the model that has been decribed above (Figure \[schematic-fig\]), one needs to generate the equivalent of a positive space charge at the lobe boundary of the Hall region.]{} In this section we demonstrate that the Hall region naturally produces such an induced equivalent space charge. Proof of this conjecture is quite easy to perform and proceeds along the following lines.
Assume that we are dealing with a *stationary* reconnection pattern in the tail current sheet as shown in Figure \[figrecon-1\] which is part of the tail region in Figure \[schematic-fig\]. [The reconnection site is centred inside a three-dimensional ion-inertial region (in the figure shown in white) with extension $\lambda_i\lesssim r_{ci}$ in the two directions perpendicular to the magnetic field (where $r_{ci}=v_i/\omega_{ci}$ is the thermal ion gyro-radius, $v_i=\sqrt{T_i/2m_i}$ ion thermal velocity, $T_i$ ion temperature in energy units, $\omega_{ci}=eB/m_i$ ion cyclotron frequency, $m_i$ ion mass) and a distance $\lambda_\|\lesssim v_{i\|}/\omega_{ci}\equiv\beta_{i\|}\lambda_i$ along the magnetic field, where $\beta_{i\|}=2\mu_0NT_{i\|}/B^2$ is the (parallel) ion-plasma-$\beta$ being the ratio of parallel ion thermal $NT_{i\|}$ to magnetic $B^2/2\mu_0$ energy densities. The latter condition takes into account that the ions remain to be unmagnetised along the magnetic field *only* over a distance they can travel with their average parallel thermal speed $v_{i\|}$ within one ion gyro-period.]{}
[In Figure \[figrecon-1\] let the secondary quadrupolar Hall magnetic field (for the global geometry see Figure \[schematic-fig\]) be ${\bf B}_{\rm H}(x,y)$ which is a function of space and, as noted earlier, is of quadrupolar structure, indicated in the figure by the symbols $\bigodot,\bigotimes$. ]{}Then the Hall-magnetic flux is given by the surface integral of the Hall magnetic field $$\Phi_{\rm H}(x,y)=\int\, {\bf B}_{\rm H}\cdot{\rm d}{\bf f}$$ which itself is clearly a function of space as well, and ${\rm d}{\bf f}$ is the surface element perpendicular to ${\bf B}_{\rm H}$ (i.e. perpendicular to the plane in Figure \[figrecon-1\]). The induced electromotive force ${\cal E}_{\rm H}(x,y)$ the Hall magnetic flux $\Phi_{\rm H}(x,y)$ may exert on the plasma is $${\cal E}_{\rm H}(x,y)=\int\,{\bf E}_{\rm H}(x,y)\cdot{\rm d}{\bf s} =-\frac{{\rm d}}{{\rm d}t}\Phi_{\rm H}(x,y)$$ the line integral of the Hall electric field ${\bf E}_{\rm H}$, which is expressed as the total time derivative of the Hall magnetic flux. Under stationary conditions (non-stationary conditions lead to more complicated pictures and are less transparent; here we assume that the process of reconnection is substantially faster than any typical variation period during a substorm) the plasma convects at velocity ${\bf V}$ in the frame of the tail current sheet across the Hall region, and the total time derivative reduces to ${\bf V}\cdot\nabla$. [This velocity is to be distinguished from the $\mathbf{E}\times\mathbf{B}$ drift of the electrons which is responsible for the generation of the Hall current. In fact, $\mathbf{V}$ is the velocity difference between the latter and the residual cross-magnetic field inertial velocity of the unmagnetised ions inside the ion inertial region.]{} One thus has $$\label{eq:emf}
{\cal E}_{\rm H}(x,y)=-{\bf V}\cdot\nabla\Phi_{\rm H}(x,y)$$ This electromotive force plays the r$\hat{\rm o}$le of an induced electric potential that is caused by the mere presence of the Hall magnetic field inside the ion inertial region, [as seen from outside the reconnection site.]{}
In the following we show by using a substantially simplified analytical model of both the Hall magnetic field and Hall magnetic flux that the quadrupolar structure of the Hall field just produces the wanted electric potential structure inside the Hall region that maps down to the ionosphere in a way to generate the auroral field aligned electron fluxes.
[ ]{}
Simple analytical model {#simple-analytical-model .unnumbered}
-----------------------
It requires little sophistication only to see that the Hall magnetic flux itself generates an electromotive force (induced electric potential) of the correct sign for accelerating the ionospheric electron component out of the auroral ionosphere into upward electron fluxes. A very simple model of the Hall magnetic flux suffices for demonstrating this fact.
Assume that the ion inertial Hall region has rectangular (box) shape in the plane $(x,y)$. In order to approximate the observation that the Hall magnetic field has quadrupolar shape, the Hall magnetic flux can be modelled as $$\Phi_{\rm H}(x,y)= \Phi_m\sin\left(\frac{\pi [x+\lambda_\|]}{\lambda_\|}\right)\sin\left(\frac{\pi [\lambda_i-y]}{\lambda_i}\right)$$ which accounts for an ambient antiparallel magnetic field that is directed in $\mp x$, and $\Phi_m$ is the maximum Hall magnetic flux corresponding to the highest concentration of Hall magnetic field lines pointing either in positive or negative $z$ direction. This flux is positive (directed out of the plane) in the upper left and lower right quarters of the ion inertial region, it is negative (directed into the plane) in the upper right and lower left quarters (see Figure \[figrecon-1\]).
For the velocity ${\bf V}$ we assume that in the upper quarters of the box outside the separatrices the flow is directed $-y$, in the lower quarters $+y$, while in the central parts inside the separatrices left of the [X]{}-point it is directed into $-x$, right of the [X]{}-point into $+x$. Otherwise the modulus of the velocity is assumed constant. Clearly such a model is oversimplistic, while reproducing the magnetic features of the Hall region.
Since only gradients parallel to ${\bf V}$ count in the generation of the electromotive force Eq. (\[eq:emf\]), the derivative jumps from $\nabla_x$ to $\nabla_y$ when passing from the inflow parts to the outflow part of the reconnection site, i.e. when crossing the separatrices. This outflow region is, however, narrow because the plasma is highly accelerated here. We can, therefore, in the simplified approach of our model, safely ignore it in a first discussion before commenting on its presence later on. Moreover, for simplicity we consider only the left upper part of the box located earthward of the [X]{}-point. Symmetry considerations show that the other quarters behave similarly.
Performing the differentiation, yields in the inflow region $$\label{eq:inflow}
{\cal E}_{\rm H}^{\rm in}(x,y)=\frac{\pi V\Phi_m}{\lambda_i}\sin\left(\frac{\pi [x+\lambda_\|]}{\lambda_\|}\right)\cos\left(\frac{\pi [\lambda_i-y]}{\lambda_i}\right)$$ In the outflow region one keeps $y$ constant and differentiates with respect to $x$ which, for completeness, yields $$\label{eq:outflow}
{\cal E}_{\rm H}^{\rm out}(x,y)=\frac{\pi V\Phi_m}{\lambda_\|}\cos\left(\frac{\pi [x+\lambda_\|]}{\lambda_\|}\right)\sin\left(\frac{\pi [\lambda_i-y]}{\lambda_i}\right)$$ The numerical factor in front of these expressions determines the real amplitude of the field and is of secondary importance in extracting the physical content of expressions (\[eq:inflow\]) and (\[eq:outflow\]).
Below we discuss the obvious implications of this simplified model by applying it to the more realistic elliptical Hall domain. The transfer to another more complicated geometry can be done without any restrictions as only geometric and no physical differences appear in this transfer, which avoids any unjustified mathematical complications. These do not add anything new to the implied physics except for a more precise determination of the boundaries between positive and negative electromotive potentials. Since the model is only approximate and no exact knowledge about the real geometric form of the Hall region is available, more precise mathematics is academic adding only spuriously to the inferences drawn. We intentionally refrain from it in order to avoid any exaggerated (pseudo-)interpretation.
[ ]{}
Discussion and Conclusions {#discussion-and-conclusions .unnumbered}
--------------------------
The inflow-region electromotive potential Eq. (\[eq:inflow\]) is positive whenever both signs of the trigonometric functions are positive or negative; it will be negative when the signs differ. A positive electromotive potential in the left upper quarter of the inflow region is obtained for $-\lambda_\|<x<0$ and $\lambda_i>y>\frac{1}{2}\lambda_i$ while it becomes negative when $y$ enters the interval $0<y<\frac{1}{2}\lambda_i$. Specular symmetry tells that this behaviour is the same in the entire upper and lower inflow region: Close to the poleward boundary of the ion inertial domain the electromotive force will always be positive. This is schematically demonstrated in Figure \[figrecon-2\] where the simple analytical model has formally been transferred to the elliptical shape of the ion-inertial domain. One should, however note, that basing the figure on the symmetric model does not differentiate between earthward and anti-earthward directions in the magnetotail. In the conventional view the Earth is on the left in Figure \[figrecon-2\]. Hence only the left-hand part of the figure matters for our purposes. Due to the missing ionosphere and softening of the magnetic field further downtail the right-hand part will favour the evolution of plasmoid-like structures instead.
Magnetic field lines outside the ion-inertial (Hall) region are equipotentials. The shaded zones indicated by “+” signs are domains of positive electromotive forces, which from the outside cannot be distinguished from positive space charges. [ The left-hand side in Figure \[figrecon-2\] maps down to the ionosphere, as shown in Figure \[shear\], along the non-reconnected magnetic field. From the positive electromotive potential domain in the inflow region one thus concludes that at the polar boundary of the auroral region a *downward directed electric field* will be seen which may be capable of accelerating the ionospheric electrons upward (green arrows) providing the observed downward auroral currents. Recently, the spatial auroral distribution of these electric fields has bee given [@marklund2011].]{}
[We may note that the mapping is a dynamical process which, in reality, is not instantaneous as the stationary picture suggests. Mapping down the potential into the ionosphere proceeds via kinetic Alfvén waves. That this is so can be seen from the anticipated scale of the ion inertial region. This scale transverse to the ambient magnetospheric magnetic field is the ion inertial scale $\lambda_i$ which is the transverse scale $k_\perp\lambda_i\sim 1$ of kinetic Alfvén waves. The localised electromotive force caused in the Hall region provides the source electric field for kinetic Alfvén waves which are launched along the magnetic field and transport the electric distortion caused in the ion inertial region down into the auroral ionosphere.]{}
[ ]{}
The broad inner part of the inflow region (for the moment again ignoring the interruption caused by the presence of the narrow outflow region) is an extended domain of *negative* electromotive force such that the connected ionospheric part sees an *upward* electric field that should cause the ionospheric ions to become accelerated upward, keeping the ionospheric electrons down. This is the upward current region which are constituted by the downward flowing hot magnetospheric electron component part of which comes directly from the reconnection site. This region includes the reconnection site which, however is free of Hall currents and does not give rise to the kind of Hall-induced electromotive potentials.
Of particular interest is the appearance in the outflow region of narrow domains of positive electromotive potentials located near the separatrix boundary. They result from Eq. (\[eq:outflow\]) and map down to the lower latitude ionosphere along the reconnected outflow magnetic field. Their presence implies that *the upward current region in an active aurora will always be bounded from both, the polar and the equatorial, sides by comparably narrow regions of upward ionospheric electron fluxes* corresponding to downward current flows. [This is, however, just what is regularly observed in active aurorae during substorms as shown in Figure \[aurorec-fig1a\].]{}
That our most simple analytical model reproduces this so far unexplained and thus not understood observational fact makes it highly probably that the auroral field-aligned current system is indeed created directly at the reconnection site itself in the near-Earth plasma sheet in the narrow collisionless magnetotail current layer. The vital ingredient of the mechanism that drives these currents is the presence of the Hall-magnetic field in the ion-inertial region. That the Hall field and currents should be involved in the generation of field-aligned currents in the magnetosphere had been conjectured first by @sonnerup1979.
We note, finally, the obvious possibility to make use of Eq. (\[eq:emf\]) for the purpose of an observational determination of the [electromotive force ${\cal E}(x,y)$]{} in the Hall region with the help of multi-spacecraft missions like Cluster or Themis. For this purpose it suffices to measure the plasma flow velocity and the local Hall magnetic flux.
RT thanks André Balogh, Director at ISSI, for his interest in and dicussions on this subject.
[ ]{}
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[ Elphic, R. C., Bonnell, J. W., Straneway, R. J., Kepko, L., Ergun, R. E., McFadden, J. P. Carlson, C. W., Peria, W., Catell, C. A., Klumpar, D., Shelley, E., Peterson, W., Moebius, E., Kistler, L. & Pfaff, R.: The auroral current circuit and field-aligned currents observed by FAST, Geophys. Res. Lett. 25, 2033-2036, doi: 10.1029/98GL01158, 1998.]{}
Fujimoto, M., Nakamura, M. S., Shinohara, I., Nagai, T., Mukai, T., Saito, Y., Yamamoto, T., & Kokubun, S.: Observations of earthward streaming electrons at the trailing boundary of a plasmoid, Geophys. Res. Lett. 24, 2893-2896, doi: 10.1029/97GL02821, 1997.
Marklund, G. T.: Electric fields and plasma processes in the auroral downward current region, below, within, and above the acceleration region, Space Sci. Rev. 142, 1-21, doi: 10.1007/s11214-008-9373-9, 2009.
[ Marklund, G. T., Sadeghi, S., Karlsson, T., Lindqvist, P.-A., Nilsson, H., Forsyth, C., Fazakerley, A., Lucek, E. A., & Pickett, J.: Altitude distribution of the auroral acceleration potential determined from Cluster satellite d at different heights, Phys. Rev. Lett. 106, 055002, doi: 10.1103/PhysRevLett.106.055002, 2011.]{}
Nagai, T., Shinohara, I., Fujimoto, M., Hoshino, M., Saito, Y., Machida, S., & Mukai, T.: Geotail observations of the Hall current system: Evidence of magnetic reconnection in the magnetotail, J. Geophys. Res. 106, 25929-25950, doi: 10.1029/2001JA900038, 2001.
Nakamura, R., Baumjohann, W., Asano, Y., Runov, A., Balogh, A., Owen, C. J., Fazakerley, A. N., Fujimoto, M., Klecker, B., & Rème, H.: Dynamics of thin current sheets associated with magnetotail reconnection, J. Geophys. Res. 111, A11206, doi:10.1029/2006JA011706, 2006.
: In situ detection of collisionless reconnection in the Earth’s magnetotail, [Nature]{}[ 412]{}, [414-417, doi: 10.1038/35086520]{}, [2001]{}.
Paschmann, G., Haaland, S., & Treumann, R. A.: Auroral plasma physics, Space Science Series of ISSI vol. 15, Kluwer Publ., Dordrecht 2003.
[: Onset of collisionless magnetic reconnection in thin current sheets: Three-dimensional particle simulations, [Phys. Plasmas]{} [10]{}, [3521-3527, doi: 10.1063/1.1597494]{}, [2003]{}.]{}
Sonnerup, B. U. Ö.: Magnetic field reconnection, in: Solar system plasma physics, Vol III, pp. 45-108, eds. L. T. Lanzerotti, C. F. Kennel and E. N. Parker, North-Holland, New York, 1979.
Treumann, R. A., Jaroschek, C. H., Nakamura, R., Runov, A. & Scholer, M.: The role of the Hall effect in collisionless magnetic reconnection, Adv. Space Res. 38, 101-111, doi: 10.1016/j.asr.2004.11.045, 2006.
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[^1]: Visiting the International Space Science Institute, Bern, Switzerland
[^2]: The reader should note that we do not use the term Hall reconnection as this term is occupied by prticular versions of two-fluid MHD approaches which include the flow of Hall currents in one or the other way.
|
[Positivity, Betweenness and Strictness ]{}\
[of Operator Means]{}
[Pattrawut Chansangiam]{}[^1]
**Abstract:** An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness and strictness of operator means in terms of operator monotone functions, Borel measures and certain operator equations.\
**Keywords:** operator connection; operator mean; operator monotone function.
**MSC2010:** 47A63, 47A64.
Introduction
============
According to the definition of a mean for positive real numbers in [@Toader-Toader], a mean $M$ is defined to be satisfied the following properties
- *positivity*: $s,t>0 \implies M(s,t)>0$ ;
- *betweenness*: $s {\leqslant}t \implies s {\leqslant}M(s,t) {\leqslant}t$.
A mean $M$ is said to be
- *strict at the left* if for each $a>0,b>0$, $M(a,b) = a \implies a=b$
- *strict at the right* if for each $a>0,b>0$, $M(a,b) = b \implies a=b$,
- *strict* if it is both strict at the right and the left.
A general theory of operator means was given by Kubo and Ando [@Kubo-Ando]. Let $B({\mathcal{H}})$ be the algebra of bounded linear operators on a Hilbert space ${\mathcal{H}}$. The set of positive operators on ${\mathcal{H}}$ is denoted by $B({\mathcal{H}})^+$. Denote the spectrum of an operator $X$ by $\operatorname{Sp}(X)$. For Hermitian operators $A,B \in B({\mathcal{H}})$, the partial order $A {\leqslant}B$ indicates that $B-A \in B({\mathcal{H}})^+$. The notation $A>0$ suggests that $A$ is a strictly positive operator. A *connection* is a binary operation ${\,\sigma\,}$ on $B({\mathcal{H}})^+$ such that for all positive operators $A,B,C,D$:
1. *monotonicity*: $A {\leqslant}C, B {\leqslant}D \implies A {\,\sigma\,}B {\leqslant}C {\,\sigma\,}D$
2. *transformer inequality*: $C(A {\,\sigma\,}B)C {\leqslant}(CAC) {\,\sigma\,}(CBC)$
3. *continuity from above*: for $A_n,B_n \in B({\mathcal{H}})^+$, if $A_n \downarrow A$ and $B_n \downarrow B$, then $A_n {\,\sigma\,}B_n \downarrow A {\,\sigma\,}B$. Here, $A_n \downarrow A$ indicates that $A_n$ is a decreasing sequence and $A_n$ converges strongly to $A$.
Two trivial examples are the left-trivial mean $\omega_l : (A,B) \mapsto A$ and the right-trivial mean $\omega_r: (A,B) \mapsto B$. Typical examples of a connection are the sum $(A,B) \mapsto A+B$ and the parallel sum $$\begin{aligned}
A \,:\,B = (A^{-1}+B^{-1})^{-1}, \quad A,B>0.\end{aligned}$$ In fact, the parallel sum, introduced by Anderson and Duffin [@Anderson-Duffin] for analyzing electrical networks, is a model for general connections. From the transformer inequality, every connection is *congruence invariant* in the sense that for each $A,B {\geqslant}0$ and $C>0$ we have $$\begin{aligned}
C(A {\,\sigma\,}B)C \:=\: (CAC) {\,\sigma\,}(CBC).\end{aligned}$$
A *mean* in Kubo-Ando sense is a connection $\sigma$ with fixed-point property $A {\,\sigma\,}A =A$ for all $A {\geqslant}0$. The class of Kubo-Ando means cover many well-known means in practice, e.g.
- ${\alpha}$-weighted arithmetic means: $A \triangledown_{{\alpha}} B = (1-{\alpha})A + {\alpha}B$
- ${\alpha}$-weighted geometric means: $A \#_{{\alpha}} B = A^{1/2}
({A}^{-1/2} B {A}^{-1/2})^{{\alpha}} {A}^{1/2}$
- ${\alpha}$-weighted harmonic means: $A \,!_{{\alpha}}\,
B = [(1-{\alpha})A^{-1} + {\alpha}B^{-1}]^{-1}$
- logarithmic mean: $(A,B) \mapsto A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}$ where $f: {\mathbb{R}}^+ \to {\mathbb{R}}^+$, $f(x)=(x-1)/\log{x}$, $f(0) \equiv 0$ and $f(1) \equiv 1$. Here, ${\mathbb{R}}^+=[0, \infty)$.
It is a fundamental that there are one-to-one correspondences between the following objects:
1. operator connections on $B({\mathcal{H}})^+$
2. operator monotone functions from ${\mathbb{R}}^+$ to ${\mathbb{R}}^+$
3. finite (positive) Borel measures on $[0,1]$.
Recall that a continuous function $f: {\mathbb{R}}^+ \to {\mathbb{R}}^+$ is said to be *operator monotone* if $$\begin{aligned}
A {\leqslant}B \implies f(A) {\leqslant}f(B)\end{aligned}$$ for all positive operators $A,B \in B({\mathcal{H}})$ and for all Hilbert spaces ${\mathcal{H}}$. This concept was introduced in [@Lowner]; see also [@Bhatia; @Hiai; @Hiai-Yanagi]. Every operator monotone function from ${\mathbb{R}}^+$ to ${\mathbb{R}}^+$ is always differentiable (see e.g. [@Hiai]) and concave in usual sense (see [@Hansen-Pedersen]).
A connection $\sigma$ on $B({\mathcal{H}})^+$ can be characterized via operator monotone functions as follows:
\[thm: Kubo-Ando f and sigma\] Given a connection $\sigma$, there is a unique operator monotone function $f: {\mathbb{R}}^+ \to {\mathbb{R}}^+$ satisfying $$\begin{aligned}
f(x)I = I {\,\sigma\,}(xI), \quad x {\geqslant}0.
\end{aligned}$$ Moreover, the map $\sigma \mapsto f$ is a bijection.
We call $f$ the *representing function* of $\sigma$. A connection also has a canonical characterization with respect to a Borel measure via a meaningful integral representation as follows.
\[thm: 1-1 conn and measure\] Given a finite Borel measure $\mu$ on $[0,1]$, the binary operation $$\begin{aligned}
A {\,\sigma\,}B = \int_{[0,1]} A \,!_t\, B \,d \mu(t), \quad A,B {\geqslant}0
\label{eq: int rep connection}
\end{aligned}$$ is a connection on $B({\mathcal{H}})^+$. Moreover, the map $\mu \mapsto \sigma$ is bijective, in which case the representing function of $\sigma$ is given by $$\begin{aligned}
f(x) = \int_{[0,1]} (1 \,!_t\, x) \,d \mu(t), \quad x {\geqslant}0. \label{int rep of OMF}
\end{aligned}$$
We call $\mu$ the *associated measure* of $\sigma$.
\[thm: char of mean\] Let $\sigma$ be a connection on $B({\mathcal{H}})^+$ with representing function $f$ and representing measure $\mu$. Then the following statements are equivalent.
1. $I {\,\sigma\,}I=I$ ;
2. $A {\,\sigma\,}A =A$ for all $A \in B({\mathcal{H}})^+$ ;
3. $f$ is normalized, i.e., $f(1)=1$ ;
4. $\mu$ is normalized, i.e., $\mu$ is a probability measure.
Hence every mean can be regarded as an average of weighted harmonic means. From and in Theorem \[thm: 1-1 conn and measure\], $\sigma$ and $f$ are related by $$\begin{aligned}
f(A) \:=\: I {\,\sigma\,}A, \quad A {\geqslant}0. \label{eq: f(A) = I sm A}\end{aligned}$$
In this paper, we provide various characterizations of the concepts of positivity, betweenness and strictness of operator means in terms of operator monotone functions, Borel measures and certain operator equations. It turns out that every mean satisfies the positivity property. The betweenness is a necessary and sufficient condition for a connection to be a mean. A mean is strict at the left (right) if and only if it is not the left-trivial mean (the right-trivial mean, respectively).
Positivity
==========
We say that a connection $\sigma$ satisfies the *positivity property* if $$\begin{aligned}
A,B>0 \implies A {\,\sigma\,}B >0.\end{aligned}$$ Recall that the *transpose* of a connection $\sigma$ is the connection $$\begin{aligned}
(A,B) \mapsto B {\,\sigma\,}A.\end{aligned}$$ If $f$ is the representing function of $\sigma$, then the representing function of its transpose is given by $$\begin{aligned}
g(x) = xf(\frac{1}{x}), \quad x>0
\end{aligned}$$ and $g(0)$ is defined by continuity.
\[remk: A,B>0 imply A sm B>0\] Let $\sigma$ be a connection on $B({\mathcal{H}})^+$ with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent:
1. $\sigma$ satisfies the positivity property ;
2. $I {\,\sigma\,}I >0$ ;
3. $\sigma \neq 0$ (here, $0$ is the zero connection $(A,B) \mapsto 0$) ;
4. for all $A {\geqslant}0$, $A {\,\sigma\,}A =0 \implies A=0$ (*positive definiteness*) ;
5. for all $A {\geqslant}0$, $A {\,\sigma\,}I =0 \implies A=0$ ;
6. for all $A {\geqslant}0$, $I {\,\sigma\,}A =0 \implies A=0$ ;
7. for all $A {\geqslant}0$ and $B>0$, $A {\,\sigma\,}B =0 \implies A=0$ ;
8. for all $A > 0$ and $B {\geqslant}0$, $A {\,\sigma\,}B =0 \implies B=0$ ;
9. $f \neq 0$ (here, $0$ is the function $x \mapsto 0$) ;
10. $x>0 \implies f(x)>0$ ;
11. $\mu([0,1])>0$.
The implications (1) $\Rightarrow$ (2) $\Rightarrow$ (3), (4) $\Rightarrow$ (3), (7) $\Rightarrow$ (5) $\Rightarrow$ (3), (8) $\Rightarrow$ (6) $\Rightarrow$ (3) and (10) $\Rightarrow$ (9) are clear. Using the integral representations in Theorem \[thm: 1-1 conn and measure\], it is straightforward to verify that the representing function of the zero connection $
0 \;:\; (A,B) \mapsto 0
$ is the constant function $f \equiv 0$ and its associated measure is the zero measure. Hence, we have the equivalences (3) $\Leftrightarrow$ (9) $\Leftrightarrow$ (11).
\(9) $\Rightarrow$ (10): Assume $f \neq 0$. Suppose that there is an $a>0$ such that $f(a)=0$. Then $f(x)=0$ for all $x {\leqslant}a$. The concavity of $f$ implies that $f(x) {\leqslant}0$ for all $x {\geqslant}a$. The monotonicity of $f$ forces that $f(x)=0$ for all $x {\geqslant}a$. Hence $f = 0$, a contradiction.
\(5) $\Rightarrow$ (7): Assume (5). Let $A {\geqslant}0$ and $B >0$ be such that $A {\,\sigma\,}B =0$. Then $$\begin{aligned}
0 = B^{1/2}(B^{-1/2} A B^{-1/2} {\,\sigma\,}I) B^{1/2}
\end{aligned}$$ and $B^{-1/2} A B^{-1/2} {\,\sigma\,}I =0$. Now, (5) yields $B^{-1/2} A B^{-1/2} = 0$, i.e. $A=0$.
\(6) $\Rightarrow$ (8): It is similar to (5) $\Rightarrow$ (7).
\(10) $\Rightarrow$ (1): Assume that $f(x)>0$ for all $x>0$. Since $\operatorname{Sp}(f(A))=f(\operatorname{Sp}(A))$ by spectral mapping theorem, we have $f(A)>0$ for all $A>0$. Hence, for each $A,B>0$, $$\begin{aligned}
A {\,\sigma\,}B = A^{1/2} f(A^{-1/2}BA^{-1/2})A^{1/2} >0.
\end{aligned}$$
\(10) $\Rightarrow$ (4): Assume (10). Let $A {\geqslant}0$ be such that $A {\,\sigma\,}A=0$. Then $$\begin{aligned}
A {\,\sigma\,}A &= \lim_{{\epsilon}\downarrow 0} A_{{\epsilon}} {\,\sigma\,}A_{{\epsilon}}
= \lim_{{\epsilon}\downarrow 0} A_{{\epsilon}}^{1/2} (I {\,\sigma\,}I) A_{{\epsilon}}^{1/2} \\
&= \lim_{{\epsilon}\downarrow 0} f(1) A_{{\epsilon}}
= f(1) A,
\end{aligned}$$ here $A_{{\epsilon}} \equiv A + {\epsilon}I$. Since $f(1)>0$, we have $A=0$.
\(10) $\Rightarrow$ (5): Assume (10). Let $A {\geqslant}0$ be such that $A {\,\sigma\,}I =0$. Then $g(A)=0$ where $g$ is the representing function of the transpose of $\sigma$. We see that $g(x)>0$ for $x>0$. The injectivity of functional calculus implies that $g(\ld)=0$ for all ${\lambda}\in \operatorname{Sp}(A)$. We conclude $\operatorname{Sp}(A)=0$, i.e. $A=0$.
\(10) $\Rightarrow$ (6): Assume (10). Let $A {\geqslant}0$ be such that $I \sigma A =0$. Then $f(A)=0$. By the injectivity of functional calculus, we have $f(\ld)=0$ for all ${\lambda}\in \operatorname{Sp}(A)$. The assumption (10) implies that $\operatorname{Sp}(A)=\{0\}$. Thus, $A=0$.
It is not true that $\sigma \neq 0$ implies the condition that for all $A {\geqslant}0$ and $B {\geqslant}0$, $A {\,\sigma\,}B =0 \implies A=0$. Indeed, take ${\,\sigma\,}$ to be the geometric mean and $$A= \left(\begin{array}{cc} 1 & 0 \\0 & 0 \\\end{array}\right), \quad
B= \left(\begin{array}{cc} 0 & 0 \\0 & 1 \\\end{array}\right).$$
Betweenness
===========
We say that a connection $\sigma$ satisfies the *betweenness property* if for each $A,B {\geqslant}0$, $$\begin{aligned}
A {\leqslant}B \implies A {\leqslant}A {\,\sigma\,}B {\leqslant}B.\end{aligned}$$
By Theorem \[remk: A,B>0 imply A sm B>0\], every mean enjoys the positivity property. In fact, the betweenness property is a necessary and sufficient condition for a connection to be a mean:
The following statements are equivalent for a connection $\sigma$ with representing function $f$:
1. $\sigma$ is a mean ;
2. $\sigma$ satisfies the betweenness property ;
3. for all $A {\geqslant}0$, $A {\leqslant}I \implies A {\leqslant}A {\,\sigma\,}I {\leqslant}I$ ;
4. for all $A {\geqslant}0$, $I {\leqslant}A \implies I {\leqslant}I {\,\sigma\,}A {\leqslant}A$ ;
5. for all $t{\geqslant}0$, $1 {\leqslant}t \implies 1 {\leqslant}f(t) {\leqslant}t$ ;
6. for all $t{\geqslant}0$, $t {\leqslant}1 \implies t {\leqslant}f(t) {\leqslant}1$ ;
7. for all $A,B {\geqslant}0$, $A {\leqslant}B \implies {\lVert A \rVert} {\leqslant}{\lVert A {\,\sigma\,}B \rVert} {\leqslant}{\lVert B \rVert}$ ;
8. for all $A {\geqslant}0$, $A {\leqslant}I \implies {\lVert A \rVert} {\leqslant}{\lVert A {\,\sigma\,}I \rVert} {\leqslant}1$ ;
9. for all $A {\geqslant}0$, $I {\leqslant}A \implies 1 {\leqslant}{\lVert I {\,\sigma\,}A \rVert} {\leqslant}{\lVert A \rVert}$ ;
10. the only solution $X>0$ to the equation $X {\,\sigma\,}X = I$ is $X=I$ ;
11. for all $A>0$, the only solution $X>0$ to the equation $X {\,\sigma\,}X =A$ is $X=A$.
The implications (2) $\Rightarrow$ (3), (2) $\Rightarrow$ (4), (2) $\Rightarrow$ (7) $\Rightarrow$ (8) and (11) $\Rightarrow$ (10) $\Rightarrow$ (1) are clear.
\(1) $\Rightarrow$ (2): Let $A,B {\geqslant}0$ be such that $A {\leqslant}B$. The fixed point property and the monotonicity of $\sigma$ yield $$\begin{aligned}
A = A {\,\sigma\,}A {\leqslant}A {\,\sigma\,}B {\leqslant}B {\,\sigma\,}B =B.
\end{aligned}$$
\(3) $\Rightarrow$ (1): Since $I {\leqslant}I$, we have $I {\leqslant}I {\,\sigma\,}I {\leqslant}I$, i.e. $I {\,\sigma\,}I =I$. Hence $\sigma$ is a mean by Theorem \[thm: char of mean\].
\(4) $\Rightarrow$ (1): It is similar to (3) $\Rightarrow$ (1).
\(8) $\Rightarrow$ (1): We have $1={\lVert I \rVert} {\leqslant}{\lVert I {\,\sigma\,}I \rVert} {\leqslant}1$. Hence, $$\begin{aligned}
f(1) = {\lVert f(1) I \rVert} = {\lVert I {\,\sigma\,}I \rVert} = 1.
\end{aligned}$$ Therefore, $\sigma$ is a mean by Theorem \[thm: char of mean\].
\(2) $\Rightarrow$ (5): If $t {\geqslant}1$, then $I {\leqslant}I {\,\sigma\,}(tI) {\leqslant}tI$ which is $I {\leqslant}f(t)I {\leqslant}tI$, i.e. $1 {\leqslant}f(t) {\leqslant}t$.
\(5) $\Rightarrow$ (1): We have $f(1)=1$.
\(2) $\Rightarrow$ (6) $\Rightarrow$ (1): It is similar to (2) $\Rightarrow$ (5) $\Rightarrow$ (1).
\(1) $\Rightarrow$ (11): Let $A>0$. Consider $X>0$ such that $X {\,\sigma\,}X =A$. Then by the congruence invariance of $\sigma$, we have $$\begin{aligned}
X = X^{1/2} (I {\,\sigma\,}I) X^{1/2} = X {\,\sigma\,}X = A.
\end{aligned}$$
For a connection $\sigma$ and $A,B {\geqslant}0$, the operators $A,B$ and $A {\,\sigma\,}B$ need not be comparable. The previous theorem tells us that if $\sigma$ is a mean, then the condition $0 {\leqslant}A {\leqslant}B$ guarantees the comparability between $A,B$ and $A \sigma B$.
Strictness
==========
We consider the strictness of Kubo-Ando means as that for scalar means in [@Toader-Toader]:
A mean $\sigma$ on $B({\mathcal{H}})^+$ is said to be
- *strict at the left* if for each $A>0,B>0$, $A {\,\sigma\,}B = A \implies A=B$,
- *strict at the right* if for each $A>0,B>0$, $A {\,\sigma\,}B = B \implies A=B$,
- *strict* if it is both strict at the right and the left.
The following lemmas are easy consequences of the fact that an operator monotone function $f: {\mathbb{R}}^+ \to {\mathbb{R}}^+$ is always monotone, concave and differentiable.
\[lem 1\] If $f:{\mathbb{R}}^+ \to {\mathbb{R}}^+$ is an operator monotone function such that $f$ is a constant on an interval $[a,b]$ with $a<b$, then $f$ is a constant on ${\mathbb{R}}^+$.
\[lem 2\] If $f:{\mathbb{R}}^+ \to {\mathbb{R}}^+$ is an operator monotone function such that $f$ is the identity function on an interval $[a,b]$ with $a<b$, then $f$ is the identity on ${\mathbb{R}}^+$.
\[thm: strict\_left trivial\] Let $\sigma$ be a mean with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent:
1. $\sigma$ is strict at the left ;
2. $\sigma$ is not the left-trivial mean ;
3. for all $A {\geqslant}0$, $I {\,\sigma\,}A =I \implies A=I$ ;
4. for all $A > 0$, $A {\,\sigma\,}I =A \implies A=I$ ;
5. for all $A>0$ and $B {\geqslant}0$, $A {\,\sigma\,}B =A \implies A=B$ ;
6. for all $A {\geqslant}0$, $I {\leqslant}I {\,\sigma\,}A \implies I {\leqslant}A$ ;
7. for all $A {\geqslant}0$, $I {\,\sigma\,}A {\leqslant}I \implies A {\leqslant}I$ ;
8. for all $A > 0$, $A {\leqslant}A {\,\sigma\,}I \implies A {\leqslant}I$ ;
9. for all $A > 0$, $A {\,\sigma\,}I {\leqslant}A \implies I {\leqslant}A$ ;
10. for all $A>0$ and $B {\geqslant}0$, $A {\leqslant}A {\,\sigma\,}B \implies A {\leqslant}B$ ;
11. for all $A>0$ and $B {\geqslant}0$, $A {\,\sigma\,}B {\leqslant}A \implies B {\leqslant}A$ ;
12. $f$ is not the constant function $x \mapsto 1$ ;
13. for all $x {\geqslant}0$, $f(x)=1 \implies x=1$ ;
14. for all $x {\geqslant}0$, $f(x) {\geqslant}1 \implies x {\geqslant}1$ ;
15. for all $x {\geqslant}0$, $f(x) {\leqslant}1 \implies x {\leqslant}1$ ;
16. $\mu$ is not the Dirac measure at $0$.
It is clear that $(5) \Rightarrow (1)$ and each of $(1),(4),(6)-(11)$ implies $(2)$. Also, each of $(13)-(15)$ implies $(12)$.
\(2) $\Rightarrow$ (3): Let $A {\geqslant}0$ be such that $I {\,\sigma\,}A =I$. Then $f(A)=I$. Hence, $f(\ld)=1$ for all ${\lambda}\in \operatorname{Sp}(A)$. Suppose that ${\alpha}\equiv \inf \operatorname{Sp}(A) < r(A)$ where $r(A)$ is the spectral radius of $A$. Then $f(x)=1$ for all $x \in [{\alpha}, r(A)]$. It follows that $f \equiv 1$ on ${\mathbb{R}}^+$ by Lemma \[lem 1\]. This contradicts the assumption (2). We conclude ${\alpha}=r(A)$, i.e. $\operatorname{Sp}(A)=\{{\lambda}\}$ for some ${\lambda}{\geqslant}0$. Suppose now that ${\lambda}<1$. Since $f(1)=1$, we have that $f$ is a constant on the interval $[\ld,1]$. Again, Lemma \[lem 1\] implies that $f \equiv 1$ on ${\mathbb{R}}^+$, a contradiction. Similarly, ${\lambda}>1$ gives a contradiction. Thus ${\lambda}=1$, which implies $A=I$.
\(2) $\Rightarrow$ (4): Let $A > 0$ be such that $A {\,\sigma\,}I =A$. Then $g(A)=I$ where $g$ is the representing function of the transpose of $\sigma$. Hence, $g(\ld)=\ld$ for all ${\lambda}\in \operatorname{Sp}(A)$. Suppose that ${\alpha}\equiv \inf \operatorname{Sp}(A) < r(A)$. Then $g(x)=x$ for all $x \in [{\alpha}, r(A)]$. It follows that $g(x)=x$ on ${\mathbb{R}}^+$ by Lemma \[lem 2\]. Hence, the transpose of $\sigma$ is the right-trivial mean. This contradicts the assumption (2). We conclude ${\alpha}=r(A)$, i.e. $\operatorname{Sp}(A)=\{{\lambda}\}$ for some ${\lambda}{\geqslant}0$. Suppose now that ${\lambda}<1$. Since $g(1)=1$, we have that $g(x)=x$ on the interval $[\ld,1]$. Lemma \[lem 2\] forces $g(x)=x$ on ${\mathbb{R}}^+$, a contradiction. Similarly, ${\lambda}>1$ gives a contradiction. Thus ${\lambda}=1$, which implies $A=I$.
\(3) $\Rightarrow$ (5): Use the congruence invariance of $\sigma$.
\(2) $\Rightarrow$ (6): Assume that $\sigma$ is not the left-trivial mean. Let $A {\geqslant}0$ be such that $I {\,\sigma\,}A {\leqslant}I$. Then $f(A) {\geqslant}I$. The spectral mapping theorem implies that $f(\ld) {\geqslant}1$ for all ${\lambda}\in \operatorname{Sp}(A)$. Suppose there exists a $ t \in \operatorname{Sp}(A)$ such that $t<1$. Since $f(t) {\leqslant}f(1)=1$, we have $f(t)=1$. It follows that $f(x)=1$ for $t {\leqslant}x {\leqslant}1$. By Lemma \[lem 1\], $f \equiv 1$ on ${\mathbb{R}}^+$, a contradiction. We conclude ${\lambda}{\geqslant}1$ for all ${\lambda}\in \operatorname{Sp}(A)$, i.e. $A {\geqslant}I$.
\(2) $\Rightarrow$ (7): It is similar to (2) $\Rightarrow$ (6).
\(6) $\Rightarrow$ (8): Assume (6). Let $A>0$ be such that $A {\leqslant}A {\,\sigma\,}I$. Then $$\begin{aligned}
A {\leqslant}A^{1/2} (I {\,\sigma\,}A^{-1})A^{1/2},
\end{aligned}$$ which implies $I {\leqslant}I {\,\sigma\,}A^{-1}$. By (6), we have $I {\leqslant}A^{-1}$ or $A {\leqslant}I$.
\(7) $\Rightarrow$ (9): It is similar to (6) $\Rightarrow$ (8).
\(6) $\Rightarrow$ (10): Use the congruence invariance of $\sigma$.
\(7) $\Rightarrow$ (11): Use the congruence invariance of $\sigma$.
\(2) $\Leftrightarrow$ (12) $\Leftrightarrow$ (16): Note that the representing function of the left-trivial mean is the constant function $f \equiv 1$. Its associated measure is the Dirac measure at $0$.
\(2) $\Rightarrow$ (13). Assume (2). Let $t {\geqslant}0$ be such that $f(t)=1$. Suppose that there is a ${\lambda}\neq 1$ such that $f(\ld)=1$. It follows that $f(x) =1$ for all $x$ lying between $\ld$ and $1$. Lemma \[lem 1\] implies that $f \equiv 1$ on ${\mathbb{R}}^+$, contradicting the assumption (2).
\(2) $\Rightarrow$ (14), (15). Use the argument in the proof (2) $\Rightarrow$ (13).
\[thm: strict\_right trivial\] Let $\sigma$ be a mean with representing function $f$ and associated measure $\mu$. Then the following statements are equivalent:
1. $\sigma$ is strict at the right ;
2. $\sigma$ is not the right-trivial mean ;
3. for all $A {\geqslant}0$, $A {\,\sigma\,}I =I \implies A=I$ ;
4. for all $A > 0$, $I {\,\sigma\,}A = A \implies A=I$ ;
5. for all $A {\geqslant}0$ and $B > 0$, $A {\,\sigma\,}B = B \implies A=B$ ;
6. for all $A {\geqslant}0$, $I {\leqslant}A {\,\sigma\,}I \implies I {\leqslant}A$ ;
7. for all $A {\geqslant}0$, $A {\,\sigma\,}I {\leqslant}I \implies A {\leqslant}I$ ;
8. for all $A > 0$, $A {\leqslant}I {\,\sigma\,}A \implies A {\leqslant}I$ ;
9. for all $A > 0$, $I {\,\sigma\,}A {\leqslant}A \implies I {\leqslant}A$ ;
10. for all $A {\geqslant}0$ and $B > 0$, $B {\leqslant}A {\,\sigma\,}B \implies B {\leqslant}A$ ;
11. for all $A {\geqslant}0$ and $B > 0$, $A {\,\sigma\,}B {\leqslant}B \implies A {\leqslant}B$ ;
12. $f$ is not the identity function $x \mapsto x$ ;
13. $\mu$ is not the associated measure at $1$.
Replace $\sigma$ by its transpose in the previous theorem.
We immediately get the following corollaries.
A mean is strict if and only if it is non-trivial.
\[thm: strict\_cor 2\] Let $\sigma$ be a non-trivial mean. For each $A>0$ and $B>0$, the following statements are equivalent:
1. $A = B$,
2. $A {\,\sigma\,}B =A$,
3. $A {\,\sigma\,}B = B$,
4. $B {\,\sigma\,}A = A$,
5. $B {\,\sigma\,}A = B$.
The next result is a generalization of [@Fiedler Theorem 4.2], in which the mean $\sigma$ is the geometric mean.
\[thm: strict\_cor 3\] Let $\sigma$ be a non-trivial mean. For each $A>0$ and $B>0$, the following statements are equivalent:
1. $A {\leqslant}B$,
2. $A {\leqslant}A {\,\sigma\,}B$,
3. $A {\,\sigma\,}B {\leqslant}B$,
4. $A {\leqslant}B {\,\sigma\,}A $,
5. $B {\,\sigma\,}A {\leqslant}B$.
\(i) It is not true that if $\sigma$ is not the left-trivial mean then for all $A {\geqslant}0$ and $B {\geqslant}0$, $A {\,\sigma\,}B =A \implies A=B$. Indeed, take $\sigma$ to be the geometric mean, $A=0$ and $$B= \left(\begin{array}{cc} 0 & 0 \\0 & 1 \\\end{array}\right).$$ The case of right-trivial mean is just the same.
\(ii) The assumption of invertibility of $A$ or $B$ in Corollary \[thm: strict\_cor 2\] cannot be omitted, as a counter example in (i) shows. As well, the invertibility of $A$ or $B$ in Corollary \[thm: strict\_cor 3\] cannot be omitted. Consider the geometric mean and $$A= \left(\begin{array}{cc} 1 & 0 \\0 & 0 \\\end{array}\right), \quad
B= \left(\begin{array}{cc} 0 & 0 \\0 & 1 \\\end{array}\right).$$
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M. Fielder and V. Ptak, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl. 251(1997), 1–20.
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[^1]: email: kcpattra@kmitl.ac.th; Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand.
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---
abstract: 'Previously developed “stochastic representation of deterministic interactions“ enables exact treatment of an open system without leaving its native phase space (Hilbert space) due to peculiar stochastic extension of Liouville (von Neumann) equation for its statistical operator. Can one reformulate the theory in terms of stochastic “Langevin equations” for its variables? Here it is shown that in case of classical Hamiltonian underlying dynamics the answer is principally positive, and general explicit method of constructing such equations is described.'
author:
- 'Yuriy E. Kuzovlev'
title: Designing Langevin Microdynamics in Macrocosm
---
[**I. Introduction.**]{} Any Langevin equations involve irreversibility (friction) and indeterminism (noise), as the classical equations which imitate interaction between “Brownian particle” and a fluid (see e.g. [@isi] and references therein). Both the friction and noise represent the same reversible and deterministic microscopic dynamics, but usually are presumed unambiguously (additively) distinguishable. In general, of course, such assumption is wrong, because the friction itself can essentially fluctuate, as in the case of interaction between macroscopic vibrations of a quartz crystal and its own phonon gas (see e.g. [@i0] and references therein). Therefore the question arises: how one should construct “Langevin equations” (interpreted loosely as a model replacement of underlying microscopic dynamics) to be sure they result quite accurate and thus free of artifacts?
The answer can be formulated in the framework of “the stochastic representation of deterministic interactions” [@i1; @i2; @i3; @i4; @i5; @i6], at least in two widespread situations:
i\) when the dynamics is Hamiltonian while interaction between a system of interest, “D”, and other world, “B”, is described by a bilinear contribution to Hamiltonian of “D+B” [@i1; @i2; @i3; @i5] : $$\begin{array}{c}
H=H_d+H_b+H_{int}\,\,,\,\,\, H_{int}=\sum_n D_n B_n\,\,; \label{blh}
\end{array}$$ the marks “d” and ”b” and the operators (or phase functions, in classical mechanics) $\,D_n\,$ and $\,B_n\,$ relate to “D” and “B”, respectively;
ii\) when joint evolution operator of “D+B”, $\,L\,$, has similar bilinear form [@i3; @i4] : $$\begin{array}{c}
L=L_d+L_b+L_{int}\,\,,\,\,\, L_{int}=\sum_n \Lambda ^{d}_{n}\Lambda
^{b}_{n}\,\, \label{bll}
\end{array}$$
The evolution operator is understood as those governing join statistical operator, $\,\rho\,$ , of “D+B”: $$\begin{array}{c}
d\rho/dt=L\rho\,\,\label{ev}
\end{array}$$ In Hamiltonian dynamics, $\,L=\mathcal{L}(H)\,$, where $\,\mathcal{L}(H)\,$ is quantum or classical Liouville operator, $$\begin{array}{c}
\mathcal{L}(H)\rho=\frac
{i}{\hbar}[\rho\,,H]\,\,\,\,\,\text{or}\,\,\,\,\,
\mathcal{L}(H)\rho=\left(\frac {\partial H}{\partial q}\frac
{\partial}{\partial p}-\frac {\partial H}{\partial p}\frac
{\partial}{\partial q}\right)\rho\,\,,
\end{array}$$ with $\,\{q,p\}\,$ being canonic variables. In case (\[blh\]) $\,L=\mathcal{L}(H)\,$ always has the bilinear form (\[bll\]) [@i3; @i4] (the case (\[bll\]) covers also non-canonic treatments of Hamiltonian dynamics [@i4] and, besides, essentially non-Hamiltonian and irreversible dynamics, and even Markovian probabilistic evolutions).
For simplicity, in this paper discussion of the Langevin equations will be confined by classical mechanics, moreover, starting from Sec.III, by the case (\[blh\]) only.
[**II. Characteristic functionals.**]{} The statistical operator $\,\rho\,$ from Eq.\[ev\] (density matrix, probability measure, etc.) says about current state $\,\Gamma =\Gamma_d\oplus\Gamma_b\,$ of “D+B” only. Who is interested also in its correlations with its prehistory, may consider one or another characteristic functional (CF) $$\begin{array}{c}
\text{Tr}_d\,\text{Tr}_b\,\,\rho(t,\Gamma)\left\langle
\exp\,[\,\int_{t>t^{\prime}}\sum_j v_j(t^{\prime})Q_j(\Gamma
(t^{\prime}))\,dt^{\prime}\,]\right\rangle^{\Gamma }\equiv
\end{array}$$ $$\begin{array}{c}
\equiv\,\left\langle\exp\,[\,\int_{t>t^{\prime}} \sum_j
v_j(t^{\prime})Q_j(t^{\prime})\,dt^{\prime}\,]
\right\rangle\,\,\,,\label{dcf}
\end{array}$$ where $\,Q_j(\Gamma )\,$ are some phase functions (i.e. functions of instant system’s state) and $\,v_j(t)\,$ conjugated arbitrary test functions (probe functions); Tr$_b\,$ and Tr$_d\,$ denote “traces” over phase spaces of “B” and “D”, that is integrations over $\,\Gamma_b\,$ or $\,\Gamma_d\,$; $\,\left\langle
...\right\rangle^{\Gamma }\,$ is conditional statistical average under given present state $\,\Gamma=\Gamma(t)\,$, and the right-hand side retells the left from viewpoint of exterior observers. Particularly, in case of deterministic dynamics the conditional averaging degenerates into replacing $\,\Gamma
(t^{\prime})\,$ by strictly definite function of $\,\Gamma=\Gamma(t)\,$.
In any case, if readdressing symbol $\,\rho \,$ to the whole expression under the traces in (\[dcf\]), one can write $$\begin{array}{c}
\left\langle \exp\,[\,\int_{t>t^{\prime}} \sum_j
v_j(t^{\prime})Q_j(t^{\prime})\,dt^{\prime}\,] \right\rangle
=\,\text{Tr}_d\,\text{Tr}_b\,\,\rho\,\,\,,\label{gcf}
\end{array}$$ where now, obviously, $\,\rho \,$ obeys the equation $$\begin{array}{c}
d\rho/dt=\{\sum_j v_j(t)Q_j(\Gamma) +L\}\,\rho\,\,\,\label{gev}
\end{array}$$ instead of (\[ev\]). Thus one reduces CF to slightly modified evolution equation. In fact that is a sort of famous relations between path integrals and differential equations, like the Feynman-Kac formulas [@rs; @f]. Nevertheless, we once more accented the transition from (\[dcf\]) to (\[gcf\])-(\[gev\]) (see also Sec.2 in [@i6]) because, curiously, some referees are not familiar with such possibility (by the way, some similar old examples can be found in [@bk3; @bk2]).
[**III. Stochastic representation.**]{} Consider partial probability measure of “D”’s states, $\, \rho_d \equiv
\,$Tr$\,_b\,\rho\,$, where $\,\rho \,$ satisfies the evolution equation (\[ev\]). According to [@i1; @i2; @i3], if once $\,\rho\,$ was factored, then later $\,\rho_d\,$ can be represented as the average $$\begin{array}{c}
\rho_d=\left\langle\left\langle
\widetilde{\rho}_d\,\right\rangle\right\rangle\,\, \label{sr}
\end{array}$$ of stochastic probability measure $\,\widetilde{\rho}_d\,$ which obeys the time-local differential equation $$\frac {d\widetilde{\rho}_d}{dt}=\left[\sum_n y_n(t)D_n
+\mathcal{L}\left(H_d+\sum_n
x_n(t)D_n\right)\right]\widetilde{\rho}_d\,\label{se}$$ with $\,x_n(t)\,$ and $\,y_n(t)\,$ being definite stochastic processes and $\,\left\langle\left\langle...\right\rangle\right\rangle\equiv
\langle\left\langle...\right\rangle_y\rangle_x\equiv
\left\langle\left\langle...\right\rangle_x\right\rangle_y\,$ statistical average with respect to them. Similarly, if the phase functions $\,Q_j\,$ wholly belong to “D” then their CF (\[dcf\]) can be represented, in place of (\[gcf\]) and (\[gev\]), as $$\begin{array}{c}
\left\langle \exp\,[\,\int_{t>t^{\prime}} \sum_j
v_j(t^{\prime})Q_j(t^{\prime})\,dt^{\prime}\,] \right\rangle
=\left\langle\left\langle\,\text{Tr}_d\,\,\widetilde{\rho}_d\,
\,\right\rangle\right\rangle\,\,\,,\label{qcf}
\end{array}$$ where now $\,\widetilde{\rho}_d\,$ is a solution of the stochastic equation $$\frac {d\widetilde{\rho}_d}{dt}=[\,v_j(t)Q_j+ y_n(t)D_n
+\mathcal{L}\left(H_d+
x_n(t)D_n\right)]\,\widetilde{\rho}_d\,\label{se1}$$ with the same random sources $\,x_n(t)\,$ and $\,y_n(t)\,$ (closely repeated indices imply summation).
Notice that (\[qcf\]) and (\[se1\]) again exploit the Feynman-Kac type relations, now for stochastic evolution operator $\,[\,y_n(t)D_n
+\mathcal{L}(H_d+ x_n(t)D_n)\,]\,$ in place of $\,L\,$, and that next such instants will not commented.
It is easy to see that $\,x_n(t)\,$ surrogate Hamiltonian perturbation, $\,H_d\rightarrow H_d+x_n(t)D_n\,$, of “D” by “B”. What is for $\,y_n(t)\,$, they enter (\[se\]) and (\[se1\]) like test functions conjugated with variables $\,D_n\,$. Therefore one can say that $\,y_n(t)\,$ describe observation of “D” by “B”. But any thing under observation affects the observer. Hence, in other words, $\,y_n(t)\,$ represent an opposite action of “D” onto “B”. Importantly, this passes without self-action of “D”, which is the reason for peculiarity of random processes $\,y_n(t)\,$: they are null by themselves ($\,\left\langle\left\langle
y_{n_1}(t_1)...\,y_{n_k}(t_k)\right\rangle\right\rangle =0\,$) although possess non-zero cross-correlations with $\,x_n(t)\,$ [@i1; @i2; @i3; @i4; @i6; @i5]. Such correlations are responsible for energy dissipation in “D” and similar statistical effects.
Quantitatively, full statistics of $\,x_n(t)\,$ and $\,y_n(t)\,$ is determined by separate evolution of “B” under perturbations of its Hamiltonian, $\,H_b\,\rightarrow\,H_b+f_n(t)B_n\,$, by arbitrary time-varying forces $\,f_n(t)\,$ [@i1; @i2; @i3; @i4; @i6]. In this section, let $\,\Gamma \equiv \Gamma_b\,$, and $\,B_n(t^{\prime},f,\Gamma)\,$ be values of the phase functions $\,B_n\,$ considered at time $\,t^{\prime}\,$ as functionals of the forces and functions of current “B”’s state $\,\Gamma=\Gamma(t)\,$ at time $\,t\,$. Then characteristic functional of $\,x_n(t)\,$ and $\,y_n(t)\,$ is $$\begin{array}{c}
\Xi \{u(\tau),\,f(\tau)\}\equiv
\end{array}$$ $$\begin{array}{c}
\equiv\, \left\langle\left\langle \exp\int_{t>t^{\prime}}
[\,u_n(t^{\prime})x_n(t^{\prime})+
f_n(t^{\prime})y_n(t^{\prime})]\,dt^{\prime}\right\rangle
\right\rangle=\,\,
\end{array}$$ $$\begin{array}{c}
=\text{Tr\,}_b\,\rho _b(t,f,\Gamma)\exp\left\{\int_{t>t^{\prime}}
u_n(t^{\prime})B_n(t^{\prime},f,\Gamma)dt^{\prime}\right\}
\label{cf1}
\end{array}$$ with $\,\rho _b(t,f,\Gamma)\,$ being current “B”’s distribution function. Since $\,B_n(t^{\prime}=t,f,\Gamma)\equiv B_n(\Gamma)\,$, the expression under the trace in (\[cf1\]), $$\begin{array}{c}
\widetilde{\rho}_b\,\equiv \,\rho
_b(t,f,\Gamma)\,\exp\left\{\int_{t>t^{\prime}}
u_n(t^{\prime})B_n(t^{\prime},f,\Gamma)\,dt^{\prime}\right\}\,\,\,,
\label{cdf}
\end{array}$$ satisfies the differential equation $$d\widetilde{\rho}_b/dt=\left[u_n(t)B_n\,
+\mathcal{L}\left(H_b+f_n(t)B_n\right)\right
]\widetilde{\rho}_b\,\,\,,\label{te}$$ quite similar to (\[se1\]), and CF (\[cf1\]) can be evaluated by solving this equation: $$\begin{array}{c}
\Xi\{u(\tau),\,f(\tau)\}\,=\,\text{Tr}\,_b\,\,
\widetilde{\rho}_b\,\label{cf2}
\end{array}$$ Variational differentiations of (\[cf1\]) produce the identities $$\begin{array}{c}
\left\langle\left\langle \,\prod _jx(t_j)\prod _my(\tau
_m)\,\right\rangle\right\rangle = \label{cors0}
\end{array}$$ $$=\left [ \prod _m\frac {\delta }{\delta f(\tau _m)}\,\,
\text{Tr}\,_b\,\,\rho _b(t,f,\Gamma)\prod_j B(t_j,f,\Gamma)\, \right
] _{f=0}$$ clearly explaining the peculiarity of $\,y_n(t)\,$. Besides, (\[cors0\]) shows the nullity of any cross-correlations between $\,y_n(\tau)\,$ and earlier $\,x_n(t^{\prime}\leq \tau)\,$ [@i1; @i2; @i3; @i4; @i6; @i5], which is consequence of the causality principle (none perturbation of “D” by “B” can depend on future perturbations of “B” by “D”).
[**IV. Fluctuation-dissipation relations.**]{} The phase volume conservation and generic time-reversal and time-translation symmetries of Hamiltonian mechanics result in the Onsager reciprocity relations, Kubo formulas, fluctuation-dissipation theorems [@isi] and other “fluctuation-dissipation relations” (FDR) [@bk3; @bk2; @bk1; @bk5].
In [@i6; @i5] general quantum FDR were reconsidered in terms of the stochastic representation. To exploit their classical limit, let us assume, without loss of generality, that (i) $\,f_n(-\infty)=f_n(+\infty)=0\,$, (ii) $\,B_n\,$ are chosen so that their unperturbed mean values are zeroes (i.e. $\,\left\langle\left\langle x_n(t)\right\rangle\right\rangle =0\,$), and (iii) $\,B_n\,$ possess definite time-reversal parities: $\,B_n(q,-p)=\epsilon _n B_n(q,p)\,$ with $\,\epsilon _n=\pm 1\,$. Besides, assume, with a loss of generality, that (iv) the past initial distribution function of “B” (before switching-on the “D”-“B” interaction) was the canonical one, $\,\propto\exp(-H_b/T)\,$, and (v) $\,H_b\,$ is even: $\,H_b(q,-p)=
H_b(q,p)\,$. Then the classical generating FDR [@bk3; @bk2; @bk1] yield $$\Xi\left\{u(\tau)-\frac 1T \frac {df(\tau)}{d\tau
},\,f(\tau)\right\}\,= \Xi\left\{\epsilon u(-\tau),\,\epsilon
f(-\tau)\right\}\,\label{fdr}$$ The same can be expressed [@i6] by the equalities $$\begin{array}{c}
\epsilon_n x_n(-\tau)\,\asymp \, x_n(\tau)\,\,\,,\\
\epsilon_n y_n(-\tau)\,\asymp \,
y_n(\tau)+T^{-1}\,dx_n(\tau)/d\tau\,\,, \label{xyfdr}
\end{array}$$ where symbol $\,\asymp \,$ means statistical equivalence.
For example, averaging the product of two lines of (\[xyfdr\]) taken with different arguments, it is easy to obtain such second-order relation: $$K_{jm}^{xy}(\tau )=\frac {\theta(\tau)}{T} \frac {d}{d\tau}\,
\,K_{jm}^{xx}(\tau )\,\,,\label{fdt}$$ where $\,\theta(\tau)\,$ is the Heavyside step function, $$\begin{array}{c}
K_{jm}^{xx}(\tau )\equiv \langle\langle x_j(\tau
)x_m(0)\rangle\rangle \,\,,\,\, K_{jm}^{xy}(\tau )\equiv
\langle\langle x_j(\tau )y_m(0)\rangle\rangle\,\,,
\end{array}$$ and the causality principle is accounted for as prescribed by (\[cors0\]).
[**V. Distribution function.**]{} Come back to “D” as described by the Eqs.\[sr\],\[se\],\[qcf\] and \[se1\], using $\,\Gamma\equiv \Gamma_d\equiv \{q,p\}\,$ as notation for complete set of “D”’s variables.
Equation (\[se\]) can be viewed as generating equation for CF of variables $\,D_n\,$ in the system with Hamiltonian $\,H_d+x_n(t)D_n\,$. At that, as we already mentioned, $\,y_n(t)\,$ play the role of test functions conjugated with $\,D_n\,$, while $\,x_n(t)\,$ are external forces. This picture is described by Hamilton equations and Liouville equation as follow: $$\begin{aligned}
d\Gamma(t)/dt\,=\,-[\mathcal{L}\left(H_d+x_n(t)
D_n\right)\Gamma](t)\,\,,\label{he}\\
d\breve{\rho}/dt\,=\,\mathcal{L}\left(H_d+x_n(t)D_n\right)
\breve{\rho}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\label{le0}\end{aligned}$$
Below, let $\Gamma(t)=\Gamma(t,x,\Gamma,\theta)\,$ denote solution of Eq.\[he\] with initial condition $\,\Gamma(t=\theta)=\Gamma \,$. Besides, define $\,\breve{\rho}(t,x,\Gamma)\,$ be solution of Eq.\[le0\] under condition $\,\breve{\rho}(t_0,x,\Gamma)=\rho_{d0}(\Gamma)\,$, where $\,\rho_{d0}\,$ is $\,\Gamma $’s distribution at past initial time moment, $\,t_0\,$. Formally, $\,t_0\,$ is the time when the “D”-“B” interaction was switched-on. Direct solution of (\[le0\]) reads $$\begin{array}{c}
\breve{\rho}(t,x,\Gamma)=\int\delta\{\Gamma
-\Gamma(t,x,\Gamma_0,t_0)\}\,\rho_{d0}(\Gamma_0)d\Gamma_0
\,=\\=\int\delta\{\Gamma_0
-\Gamma(t_0,x,\Gamma,t)\}\,\rho_{d0}(\Gamma_0)d\Gamma_0=\\
=\rho_{d0}(\Gamma(t_0,x,\Gamma,t)) \,\,\,, \label{df0}
\end{array}$$ where $\,\delta\{...\}\,$ means delta-function in the phase space and $\,\Gamma_0\,$ the initial state. At that, the group property of $\,\Gamma$’s transformations from one time point to another: $$\begin{array}{c}
\Gamma(t^{\prime},x,\Gamma(t,x,\Gamma_0,t_0),t)=
\Gamma(t^{\prime},x,\Gamma_0,t_0)\,\,\,,\label{gp}
\end{array}$$ and the Liouville theorem about phase volume conservation were taken into account.
In these designations, solution of Eq.\[se\] looks as $$\begin{array}{c}
\widetilde{\rho}_d\,=\,\breve{\rho}(t,x,\Gamma)
\,\exp\left\{\int_{t_0}^t y_n(t^{\prime})D_n(t^{\prime},x,\Gamma,t)\,
dt^{\prime}\right\}\,\,\label{df}
\end{array}$$ with $\,D_n(t^{\prime},x,\Gamma,t)\equiv
D_n(\Gamma(t^{\prime},x,\Gamma,t))\,$.
Since $\,y_n(t)\,$ are null by themselves and null in conjunction with any earlier $\,x_n(t^{\prime}\leq t)\,$, while $\,\breve{\rho}(t,x,\Gamma)\,$ depends on $\,x_n(t^{\prime}< t)\,$ only, and $\,\Gamma(t^{\prime},x,\Gamma,t)\,$ depend on $\,x_n(\min
(t,t^{\prime})< t^{\prime\prime}< \max(t,t^{\prime}))\,$ only, one can replace the upper integration limit in (\[df\]) by any value $\,>t\,$, in particular, by $\,\infty\,$. Then the exponent in (\[df\]) transforms into the statistically equivalent functional $$%\begin{array}{c}
S_t\{x,y,\Gamma\}\,\equiv\,\exp\left[\,\int
y_n(t^{\prime})D_n(\Gamma(t^{\prime},x,\Gamma,t))
\,dt^{\prime}\,\right]\,
%\end{array}
\label{S}$$ After this replacement, substitution of (\[df\]) to (\[sr\]), with use of identities (\[df0\]) and (\[gp\]), yields $$\begin{array}{c}
\rho_d(t,\Gamma)\,=\,\langle\langle\,\breve{\rho}(t,x,\Gamma)
\,S_t\{x,y,\Gamma\}\,\rangle\rangle\,\,=\label{rod}
\end{array}$$ $$\begin{array}{c}
=\int \langle\langle\,\delta\{\Gamma
-\Gamma(t,x,\Gamma_0,t_0)\}\,S_{t_0}\{x,y,\Gamma_0\}\,\rangle\rangle
\,\rho_{d0}(\Gamma_0)d\Gamma_0
\end{array}$$
Alternatively, by averaging directly formal operator solution of Eq.\[se\], one obtains $$\begin{array}{c}
\rho_d\,=\,\widehat{\Theta}\,\exp\left\{\int_{t_0}^t
\mathcal{L}(H_d)\,dt^{\prime}\right\}\,
\Xi\{\mathcal{L}(D),D\}\,\rho_{d0}\,\,\,,\label{ddf}
\end{array}$$ where $\,\widehat{\Theta}\,$ symbolizes chronological ordering of the following operator expression (that is ordering with respect to imaginary time argument of $\,H_d\,$ and $\,D_n\,$).
[**VI. Fluctuation statistics.**]{} Similarly to preceding section, consider Eq.\[se1\] as generating equation for joint CF of variables $\,Q_j\,$ and $\,D_n\,$.
Now, express solutions of (\[he\]) and (\[le0\]) through “D”’s state at arbitrary fixed time moment $\,\theta\,$ which is different from $\,t\,$, that is solve (\[he\]) and (\[le0\]) under initial condition $\,\Gamma(t^{\prime}=\theta)=\Gamma\,$ (thus $\,\Gamma(t^{\prime}=\theta,x,\Gamma,\theta)=\Gamma\,$). Then solution of Eq.\[se1\] can be implicitly formulated as $$\begin{array}{c}
\widetilde{\rho}_d(t,\Gamma(t,x,\Gamma,\theta))=
\,\breve{\rho}(\theta,x,\Gamma)\times\,\\\exp
\left\{\int_{t>t^{\prime}}
[v_j(t^{\prime})Q_j(t^{\prime},x,\Gamma,\theta)
+y_n(t^{\prime})D_n(t^{\prime},x,\Gamma,\theta)]\,dt^{\prime}\right\}
\label{ddf1}
\end{array}$$ with $\,Q_j(t^{\prime},x,\Gamma,\theta)\equiv $ $Q_j(\Gamma(t^{\prime},x,\Gamma,\theta))\,$. Substituting (\[ddf1\]) to (\[qcf\]) and taking into account the Liouville theorem (the phase volume conservation under arbitrary Hamiltonian evolution), at $\,t\rightarrow\infty\,$ we have
$$\begin{array}{c}
\left\langle \exp\left\{\int
\,v_j(t^{\prime})Q_j(t^{\prime})\,dt^{\prime}\right\} \right\rangle =
\left\langle\left\langle\text{Tr\,}_d\,\exp\left\{\int
v_j(t^{\prime})Q_j(t^{\prime},x,\Gamma,\theta)\,
dt^{\prime}\right\}\breve{\rho}(\theta,x,\Gamma)\,S_{\theta}\{x,y,\Gamma\}
\right\rangle\right\rangle\, \label{fcf}
\end{array}$$
In terms of various statistical moments of variables $\,Q_j\,$ (omitting their indices) $$%\begin{array}{c}
\left\langle\, Q(t_1)...\,Q(t_k)\,\right\rangle
=\text{Tr\,}_d\left\langle\left\langle
\,Q(t_1,x,\Gamma,\theta)...\,Q(t_k,x,\Gamma,\theta)\,
\breve{\rho}(\theta,x,\Gamma)\,S_{\theta}\{x,y,\Gamma\}
\,\right\rangle\right\rangle \label{moms}
%\end{array}$$ In particular, if $\,\theta\rightarrow t_0\,$ then $\,\breve{\rho}(\theta,x,\Gamma)\,$ turns into the initial distribution, $\,\rho_{d0}(\Gamma)\,$, definitively independent on $\,x_n(t)\,$: $$%\begin{array}{c}
\left\langle\, Q(t_1)...\,Q(t_k)\,\right\rangle
=\text{Tr\,}_d\,\,\rho_{d0}(\Gamma)\left\langle\left\langle
\,Q(t_1,x,\Gamma,t_0)...\,Q(t_k,x,\Gamma,t_0)\,S_{t_0}\{x,y,\Gamma\}
\,\right\rangle\right\rangle \label{moms0}
%\end{array}$$ with $\,\Gamma\,$ representing the initial state $\,\Gamma_0\,$. The same expression results from (\[moms\]) after substitution of (\[df0\]) and (\[gp\]).
Alternatively, quite similarly to (\[ddf\]), $$\begin{array}{c}
\left\langle \exp\left\{\int
\,v_j(t^{\prime})Q_j(t^{\prime})\,dt^{\prime}\right\} \right\rangle
=\text{Tr\,}_d\,\widehat{\Theta}\,\exp\{\,\int
[v_j(t^{\prime})Q_j+\mathcal{L}(H_d)]\,dt^{\prime}\}\,
\,\Xi\{\mathcal{L}(D),D\}\,\rho_{d0}
\end{array}$$ \[cdf\] The functional $\,\Xi\,$ here, defined by (\[cf1\]), at once accumulates all information about “B” which must be used when evaluating (\[fcf\])-(\[moms0\]).
[**VII. Self-interaction through environment and “scattering operator”.**]{} It is useful to emphasize rather interesting resemblance between Eq.\[moms\] or Eq.\[moms0\] and expressions for scattering amplitudes, Green functions, etc., in quantum theory of fields and many-particle systems (see e.g. [@blp; @lp]). If draw an analogy from $\,Q_j\,$ and $\,x_n(t)\,$ to electron operators and radiation field, respectively, then the averages $\,\left\langle\left\langle
Q(t_1,x,\Gamma,t_0)...Q(t_k,x,\Gamma,t_0)\,\right\rangle\right\rangle\,$ correspond to lowest-order perturbation approximation, while $\,\left\langle\left\langle
Q(t_1,x,\Gamma,t_0)...Q(t_k,x,\Gamma,t_0)\,S_{t_0}\{x,y,\Gamma\}
\,\right\rangle\right\rangle\,$ in Eq.\[moms0\] exactly summarizes all the orders of “D”’s interaction with its environment. The analogy continues in that the “complete multiple scattering operator” $\,S_{t_0}\{x,y,\Gamma\}\,$ by itself behaves like unity: $$\begin{array}{c}
\left\langle\left\langle \,S_{t_0}\{x,y,\Gamma\}
\,\right\rangle\right\rangle \,=1\,
\end{array} \label{unit}$$ This identity clearly follows from Eq.\[rod\] at $\,t\rightarrow
t_0\,$ and is easy explainable if notice that in any term of $\,\,S_{t_0}\{x,y,\Gamma\}$’s series expansion over $\,y_n(t^{\prime})\,$ and $\,x_n(t^{\prime})\,$ most late time argument belongs to some of $\,y$’s.
According to (\[cf1\]) and (\[cors0\]), separately $\,x_n(t)\,$ are nothing but noise of free unperturbed environment, like “zero, or vacuum, fluctuations”. However, along with $\,y_n(t)\,$ in $\,S_{t_0}\{x,y,\Gamma\}\,$ they represent actual noise of the environment, including its directional response to the system’s motion, in the form of both renormalization of primordial “D”’s dynamical properties and appearance of new ones: relaxation, “spectral lines broadening”, etc.
[**VIII. State-dependent noise and the fiction of friction.**]{} In [@i1; @i2; @i3] the words “Langevin equation” were addressed to objects like (\[se\]) or (\[se1\]) which emerged as stochastic extensions of the Liouville equation for probability measure of $\,\Gamma\,$. In usual sense, Langevin equations must be a stochastic extension of the Hamilton equations for $\,\Gamma\,$ themselves. Besides, one would want these equations to involve some “realistic” noises only but not auxiliary “ghost” noises like $\,y_n(t)\,$. The latter requirement means that desirable equations are certainly not literal consequence of the basic Eqs.\[se\] and \[se1\]. Instead, Langevin equations must be especially constructed as their exact statistical equivalent (or at least close approximate one).
In should be underlined that, at such target setting, a “size” of system “B” is insignificant (no matter e.g. is a Brownian particle macroscopic or as small as molecules).
With the formulated purpose, let us return to Eq.\[moms0\], choosing arbitrary functions $\,Q_j(\Gamma)\,$ as delta-functions $\,\delta\{\Gamma-\gamma\}\,$ and their index as time. Then Eq.\[moms0\] produces
$$\begin{array}{c}
W\{\gamma\}\equiv \left\langle\,\prod_t\delta\{\Gamma(t)-\gamma(t)\}
\,\right\rangle =\int\,\left\langle\left\langle
\,\prod_t\delta\{\Gamma(t,x,\Gamma_0,t_0)-\gamma(t)\}\,\exp\left\{\int
y(t)D(\gamma(t))\,dt\right\}\,\right\rangle\right\rangle\,
\rho_{d0}(\Gamma_0)d\Gamma_0 \label{pf}\\\,
\end{array}$$
which represents probability density functional for the whole system’s trajectory. Here all non-principal indices are omitted, and the delta-functions have allowed to replace $\,D(\Gamma(t,x,\Gamma_0,t_0))\,$ in the exponent by $\,D(\gamma(t))\,$.
The simplest construction of Langevin equations follows directly from careful “visual” investigation of Eq.\[pf\]. This shows that Eq.\[pf\] can be rewritten as $$\begin{array}{c}
W\{\gamma\}\,=\,\int\,\left\langle\left\langle
\,\prod_t\delta\{\Gamma(t,z,\Gamma_0,t_0)-\gamma(t)\}
\,\right\rangle\right\rangle^{\gamma}\,
\rho_{d0}(\Gamma_0)d\Gamma_0\,\,\,, \label{pf1}
\end{array}$$ with new random forces $\,z_n(t)\,$ in place of $\,x_n(t)\,$, if conditional statistics of $\,z_n(t)\,$ is defined by formulas $$\begin{array}{c}
\left\langle\left\langle\, z(t_1)...\,z(t_k)\,
\right\rangle\right\rangle^{\gamma} \,
\equiv\,\left\langle\left\langle \,x(t_1)...\,x(t_k)\,\exp\left\{\int
y(t)D(\gamma(t))\,dt\right\}\, \,\right\rangle\right\rangle
\label{momz0}
\end{array}$$ The brackets $\,\left\langle\left\langle\,
...\,\right\rangle\right\rangle^{\gamma}\,$ here have the sense of conditional averaging under given system’s trajectory $\,\gamma(t)
\,$.
At that, the role of Langevin equations governing the variables $\,\Gamma(t)\equiv \Gamma(t,z,\Gamma,t_0)\,$ and $\,Q(t)\equiv
Q(\Gamma(t,z,\Gamma,t_0))\,$ belongs to nothing but merely the Hamilton equations: $$\begin{aligned}
d\Gamma(t)/dt\,=\,-[\mathcal{L}\left(H_d+z_n(t)
D_n\right)\Gamma](t)\,\label{hez}\end{aligned}$$
Notice that in view of identity (\[unit\]) the averaging procedure defined in (\[momz0\]) automatically satisfies the normalization condition $\,\langle\langle\,1 \,\rangle\rangle^{\gamma}=1\,$. Besides, due to the above mentioned statistical peculiarities of $\,y(t)$’s, result of the averaging always agrees with the causality principle: the moments (\[momz0\]) in fact can depend on $\,\gamma(t)\,$ with $\,t<\max(t_1,...,t_k)\,$ only.
Formally, the two above expressions, (\[momz0\]) and (\[hez\]), already define what can be named “exactly equivalent Langevinian form of the stochastic representation”. It clearly emphasizes statistical nature of dissipation and friction: even if being present they still are hidden inside $\,z(t)$’s statistics, as cross-correlations of $\,x(t)$’s with $\,y(t)$’s. To see them evidently in (\[hez\]), we have to withdraw them from (\[momz0\]) in some reasonable approximation, with corresponding redefinition of noises $\,z_n(t)\,$.
[**IX. Unbiased noise and Langevin equations.**]{} With the above pointed purpose, first, assume, naturally and without loss of generality, that $\,\langle x\rangle =0\,$. Then desired dissipative contributions to (\[hez\]), together with the renormalization corrections of non-dissipative terms, can be identified among mean values of $\,z(t)$’s.
Second, consider cumulants (semi-invariants) $\,\,
\kappa_{\alpha\beta}\,\equiv\,\langle\langle\,
x_1,...,x_{\alpha},\,y_1,...,y_{\beta}\,\rangle\rangle \,$. For brevity, here the subscripts unify indices and time, and commas do emphasize that comma-separated multipliers are subject to purely irreducible correlation of $\,(\alpha +\beta)$-th order (“Malakhov’s cumulant brackets”). Then CF (\[cf1\]) can be symbolically written as
$$\Xi\{u,f\}\,=\,\exp [\,\kappa\{u,f\}\,]\,\equiv\,
\exp\left[\,\sum_{\alpha=2}^{\infty}\kappa_{\alpha 0}\frac
{u^{\alpha}}{\alpha !}\,+\,\sum_{\alpha,\beta=1}^{\infty}
\kappa_{\alpha\beta}\frac
{u^{\alpha}f^{\beta}}{\alpha!\,\beta!}\right]$$ (according to our assumption, $\,\kappa_{10}=\langle x\rangle =0\,$). Decompose it into two multipliers: $$\Xi\{u,f\}=\overline{\Xi}\{u,f\}\,\widetilde{\Xi}\{u,f\}\,\,,\,\,\,
\overline{\Xi}\{u,f\}\equiv\exp\left[\sum_{\beta=1}^{\infty}\kappa_{1
\beta}\frac {u\,f^{\beta}}{\beta!}\right]\,\,,\,\,\,
\widetilde{\Xi}\{u,f\}\equiv\exp\left[\sum_{\alpha=2}^{\infty}
\sum_{\beta=0}^{\infty}\kappa_{\alpha \beta}\frac
{u^{\alpha}f^{\beta}}{\alpha!\,\beta!}\right] \label{dec}$$ Correspondingly to this factorization of CF, both the original noises, $\,x(t)$’s and $\,y(t)$’s, divides into two components: $\,\,x=\overline{x}+\widetilde{x}\,$ and $\,y=\overline{y}+\widetilde{y}\,$, where two pairs $\,\{\widetilde{x},\widetilde{y}\}\,$ and $\,\{\overline{x},\overline{y}\}\,$ are mutually statistically independent.
It is easy to prove that for arbitrary functional $\,\Phi(\overline{x})\,$ and arbitrary function $\,f\,$ the equality holds as follows: $$%\begin{array}{c}
\langle\langle\,
\Phi(\overline{x})\,\exp(f\overline{y})\,\rangle\rangle =
\Phi\left(\overline{X}(f)\right)\,\,,\,\,\,\overline{X}(f)\equiv
\sum_{\beta=1}^{\infty}\kappa_{1\beta}\,f^{\beta}/\beta!\,\label{mv}
%\end{array}$$ This is because the pair $\,\{\overline{x},\overline{y}\}\,$, in accordance with (\[cors0\]), describes merely conditional mean value of “B”’s response to its perturbation by forces $\,f\,$. Applying the decomposition (\[dec\]) to (\[pf\]), with the help of (\[mv\]) we obtain
$$\begin{array}{c}
W\{\gamma\}=\,\int \left\langle\left\langle\,\,
\prod_t\delta\{\Gamma(t,\overline{X}(D(\gamma))+\widetilde{x},
\Gamma_0,t_0)-\gamma(t)\}\,\exp\left\{\int
\widetilde{y}(t)D(\gamma(t))\,dt\right\}\,\right\rangle\right\rangle
\,\rho_{d0}(\Gamma_0) \,d\Gamma_0\,\,\label{pf2}
\\\,
\end{array}$$
The mean response $\,\overline{X}(f)\,$, defined by (\[mv\]), with $\,f=D(\gamma)\,$, after restoration of its temporal index, reads $$\overline{X}(t,f)=\frac {\delta}{\delta
u(t)}\,\ln\Xi\{u,f\}\,|_{u=0}\,=\int\langle\langle
x(t),y(t_1)\rangle\rangle\, f(t_1)dt_1+\frac 12\int\int\langle\langle
x(t),y(t_1),y(t_2)\rangle\rangle\,f(t_1)f(t_2)dt_1dt_2\,+...\,\,,\label{mx}
\\\,\,$$ where because of (\[cors0\]) all integrals are in fact taken over $\,t_j<t\,$. It is useful to notice also that, due to the causality, Jacobian of mutual transformations between $\,\gamma\,$ and $\,\Gamma\,$ is unit.
Scanning (\[pf2\]) in comparison with (\[pf\]) and (\[pf1\]), one evidently comes to another form of the probability functional:
$$\begin{array}{c}
W\{\gamma\}\,=\,\int\,\left\langle\left\langle
\,\prod_t\delta\{\,\Gamma(t,\overline{X}(D(\gamma))+\widetilde{z}\,,
\Gamma_0,t_0)-\gamma(t)\} \,\right\rangle\right\rangle^{\gamma}\,
\rho_{d0}(\Gamma_0)d\Gamma_0\,\,\,, \label{pf3}
\\\,\,
\end{array}$$
where statistics of renormalized (in fact merely biased) noises $\,\widetilde{z}(t)\,$ is now described by
$$\begin{array}{c}
\left\langle\left\langle\,
\widetilde{z}(t_1)...\,\widetilde{z}(t_k)\,
\right\rangle\right\rangle^{\gamma} \,
\equiv\,\left\langle\left\langle
\,\widetilde{x}(t_1)...\,\widetilde{x}(t_k)\,\exp\left\{\int
\widetilde{y}(t)D(\gamma(t))\,dt\right\}
\,\right\rangle\right\rangle\,\,\label{momz1}
\\\,
\end{array}$$
At that, correspondingly to (\[pf3\]), $\,\Gamma(t)=\Gamma(t,\overline{X}(D(\Gamma))+\widetilde{z},\Gamma_0,t_0)\,$, that is stochastic Hamilton equations (\[hez\]) change to the stochastic integro-differential equations $$\begin{aligned}
d\Gamma(t)/dt\,=\,-\,[\,\mathcal{L}(H_d)\Gamma\,](t)-
(\overline{X}_n(t,D(\Gamma))+\widetilde{z}_n(t))\,
[\mathcal{L}(D_n)\,\Gamma](t)\,\,\label{hez1}\end{aligned}$$
As prescribed by (\[dec\]) and (\[momz1\]), here the noises $\,\widetilde{z}_n(t)\,$ have certainly zero mean values, while any dissipative effects of interaction with “B” are separated in $\,\overline{X}_n(t,D(\Gamma))\,$. Hence, Eqs.\[hez1\] can be by now enough surely named “Langevin equations”.
[**X. Discussion.**]{} Of course, the above result is rather trivial one. However, from the point of view of applications and practical computability, it is not quite satisfactory. The matter is that numeric modeling of noise essentially conditioned by the system it drives is generally difficult task. It would be better if the noise was reduced to unconditioned random quantities, for example, $$\begin{array}{c}
\widetilde{z}(t)=z^{(0)}(t)+
\int\,z^{(1)}(t,t_1)\,f(t_1)\,dt_1\,+\\+\,\frac
12\int\int\,z^{(2)}(t,t_1,t_2)\,f(t_1)f(t_2)\,dt_1dt_2\,+...
\,\,\,,\label{expan}
\end{array}$$ where $\,z^{(0)}(t)=\widetilde{x}(t)=x(t)\,$ is unperturbed noise, $\,z^{(1)}(t,t_1)\,$ represents stochastic linear response of “B” to its perturbation, etc., and all $\,z^{(n)}\,$ are some zero-average random functions independent on the forces. In particular, $\,z^{(1)}(t,t_1)\,$ includes fluctuations in linear friction (whose average was contained in first term of (\[mx\])).
The only situation when Eqs.\[hez1\] finalize the analysis is when the noises $\,\widetilde{z}(t)\,$ are state-independent, that is $\,z^{(n)}=0\,$ for all $\,n>0\,$. But this is unlikely realistic situation since in general it is forbidden by restrictions which follow from the phase volume conservation and microscopic reversibility. For concreteness, if “B” is equilibrium thermal bath (thermostat), these restrictions are expressed by FDR (\[fdr\]) [@bk3; @bk2; @bk1; @bk5] or equivalently (\[xyfdr\]) (notice that FDR for internally non-equilibrium baths also were considered in [@bk3; @bk2]). If noises $\,\widetilde{z}(t)\,$ are indeed state-independent, this means that $\,\kappa_{\alpha\beta}=0\,$ for all $\alpha\geq 2\,$ and $\,\beta\geq 1\,$. Then the second row from (\[xyfdr\]) clearly implies that in such case the equalities $\,\kappa_{\alpha 0}=0\,$ also should hold for all $\alpha\geq 3\,$. In other words, the noise $\,\widetilde{z}(t)\,$ can be purely state-independent only when it is purely Gaussian. Moreover, then the same FDR prescribe that $\,\kappa_{1\beta}=0\,$ for all $\,\beta\geq
2\,$, that is average response of “B” is purely linear.
Thus we come to the trite “linear Gaussian thermostat” when Eqs.\[momz1\] and \[mx\] reduce to $$\begin{array}{c}
\overline{X}_n(t,D(\Gamma))\,=\,-\,
K_{nm}^{xx}(0)D_m(\Gamma(t))/T\,\,+\,\,\label{lin}
\end{array}$$ $$+\,\frac 1T\int^{t}_{-\infty} K_{nm}^{xx}(t-t^{\prime})\,\frac
{d}{dt^{\prime}} \,D_m(\Gamma(t^{\prime}))\,dt^{\prime}\,\,\,,$$ $$\begin{array}{c}
\langle\langle\,\, \widetilde{z}_n(t_1)\,\widetilde{z}_m(t_2)\,\,
\rangle\rangle^{\gamma}\,=\,K_{nm}^{xx}(t_1-t_2)\,\\\,
\end{array}$$ Here FDR (\[fdt\]) is used, and it is taken in mind that all higher-order cumulants of $\,\widetilde{z}(t)\,$ are zeros. Discussion of more interest models will be done elsewhere.
[**XI. Example: oscillator.**]{} Consider nonlinear oscillator, assuming that “B” is “linear Gaussian thermostat” while interaction with it realizes in potential way through two statistically independent channels as follow: $$\begin{array}{c}
H_d=p^2/2m+U_0(q)\,\,,\,\,\,D_1(\Gamma)=-q\,\,,\\
\,\,\,D_2(\Gamma)=q^2/2\,\,,\,\,\,K^{xx}_{12}=K^{xx}_{21}=0
\end{array}$$
The first channel corresponds to usual thermal excitation, and the second to thermal parametric fluctuations in frequency of oscillations. The Eqs.\[hez1\] and \[lin\] yield $$\begin{array}{c}
dq(t)/dt\,=\,p(t)/m\,\,\,,\\\,\label{osc} \\
dp(t)/dt\,=\,-\,dU(q(t))/dq(t)\,+\widetilde{x}_1(t)\,+
\widetilde{x}_2(t)\,q(t)\,-\\-\,\int^t_{-\infty}
K_{11}^{xx}(t-t^{\prime})\,v(t^{\prime})\,dt^{\prime}/T\,-\\
-\,q(t)\int^{t}_{-\infty}
K_{22}^{xx}(t-t^{\prime})\,q(t^{\prime})v(t^{\prime})
\,dt^{\prime}/T\,\,,
\end{array}$$ where $\,v(t)\equiv dq(t)/dt\,$ is velocity, $\,\widetilde{x}_n(t)\,$ are mutually independent normal random processes, $\,K_{nn}^{xx}\,$ are their correlators, and $$\begin{array}{c}
U(q)\,\equiv\,
U_0(q)\,-K_{11}^{xx}(0)\,q^2/2T\,-K_{22}^{xx}(0)\,q^4/8T
\end{array}$$ is renormalized potential. Hence, correspondingly, there are two channels of friction and dissipation, and the friction channel conjugated with thermal parametric fluctuations is essentially nonlinear. Similar examples concerning thermal fluctuations in capacities of electric circuits were considered in [@bk4].
[**XII. Conclusion.**]{} For particular variant of the “stochastic representation of deterministic interactions” concerning classical Hamiltonian mechanics, we have demonstrated that by request it can be completely reformulated in terms of “Langevin equations” for internal variables of an open system. These equations are wholly housed in its own phase space and are free of the peculiar auxiliary noises $\,y_n(t)\,$, distinctive for initial “stochastic representation”. At the same time, $\,y_n(t)\,$ remain useful undercover instrument, being responsible for conditional statistical dependence of actual noise on trajectory of the system driven by it.
This Langevinian form of the theory seems more vivid, although, probably, it will occur less appropriate for practical analysis of complicated noise statistics. Besides, the original “Liouvillian form” at once covers quantum mechanics as well.
What is for its quantum Langevinian equivalent, still it remains unexplored. Notice that quantum Langevin equations for important special case of Gaussian linear thermostat were exhaustively considered in [@ef1]. Of course, more general situations also were under many considerations (see e.g. [@ef]).
But recall that the question under our principal and pragmatic interest is how much non-Gaussian non-linear generalization of quantum Langevin equations can be developed if do it wholly within native Hilbert space of an open system under consideration and with use of commutative ($\,c$-number valued) noise sources only.
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|
---
abstract: 'For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When $n\le 5$, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserving action-angle coordinates. With this coordinate system, we provide new bounds on the knotting probability of equilateral hexagons.'
address: 'Department of Mathematics and Statistics, Carleton College, Northfield, MN, 55057'
author:
- Kathleen Hake
bibliography:
- 'KnottingProbability-Hake.bib'
title: Knotting Probability of Equilateral Hexagons
---
Introduction
============
Classically a knot can be defined as a closed, non self-intersecting smooth curve embedded in Euclidean $3$-space. Two knots are considered to be equivalent if one can be smoothly deformed into another. The question of whether or not two given knots are equivalent proves to be a difficult problem. Much of theory is devoted to developing techniques to answer this question. The study of the invariance of knots has been of interest to not only mathematicians but also biologist, physicists, and computer scientists. Prominent examples of knotting appear in polymers, specifically DNA and proteins. In the early 1970’s, it was discovered that enzymes called topoisomerases causes the DNA to change its form. Type II topoisomerases bind to two segments of double-standed DNA, split one of the segments, transport the other through the break, and reseal the break. These studies suggest that the topological configuration, or the knotting, plays a role in understanding the behavior of these enzymes. Sometimes the arbitrary flexibility and lack of thickness in the classical theory of knots does not accurately depict the physical constraints of objects in nature. This inspires questions in the field of physical knot theory and models that seek to capture some of the physical properties.
A question of focus in this paper is about the statistical distribution of knot types as a function of the length. For example, what is the probability that an $n$ edged polygon is knotted? The study of knots from a probabilistic viewpoint provides an understanding of typical knots. There are many ways to model random knots. The model we will consider is that of closed polygonal curves in $\mathbb{R}^3$. A knot is realized by joining $n$ line segments. In addition, we restrict the length of the segments to be equal. We identify each $n$-sided polygonal curve with the $3n$-tuple of vertex coordinates which define it. This gives a correspondence between points in $\mathbb{R}^{3n}$ and $n$-sided polygons in $\mathbb{R}^3$. We consider the $2(n-3)$ dimensional subspace of equilateral polygons equivalent up to translations and rotations. Using techniques from symplectic geometry, we study the space of equilateral hexagons. Suppose $P$ is an equilateral hexagon. We prove that the probability $P$ is knotted is at most $\frac{14-3\pi}{192}<\frac{1}{42}$.
Background
==========
There are various ways to define a knot, all of which capture the intuitive notion of a knotted loop. We will start by defining a polygonal knot. For any two distinct points in $3$-space, $p$ and $q$, let $[p,q]$ denote the line segment joining them. For an ordered set of distinct points, $(p_1, p_2, \dots, p_n)$, the union of the segments $[p_1, p_2], [p_2,p_3],\dots, [p_{n-1},p_n],$ and $[p_n,p_1]$ is called a closed polygonal curve. If each segment intersects exactly two other segments, intersecting each only at an endpoint, then the curve is said to be simple.
A polygonal knot, $K$, is a simple, closed polygonal curve in $\mathbb{R}^3$.
If a polygonal knot, $P$, has $n$ vertices, we will call $P$ a $n$-sided polygonal knot. We label the vertices of $P$ as $v_{1}, v_{2}, \ldots, v_{n}$. We call the segments $[v_i,v_{i+1}]$ the edges of $P$ and label the edges of $P$ as $e_{1}, e_{2}, \ldots, e_{n}$, where $e_{1}=[v_1,v_2], e_2=[v_2, v_3], \dots, e_{n-1}=[v_{n-1},v_n]$, and $e_n=[v_n,v_1]$. In addition, we will select a distinguished vertex, $v_{1}$, called a root and a choice of orientation.
![The figure on the left shows a 6-sided, rooted, oriented polygonal trefoil knot. The figure on the right shows a 7-sided, rooted, oriented polygonal figure-8 knot.[]{data-label="fig:label"}](hex.pdf "fig:"){width=".35\textwidth"}![The figure on the left shows a 6-sided, rooted, oriented polygonal trefoil knot. The figure on the right shows a 7-sided, rooted, oriented polygonal figure-8 knot.[]{data-label="fig:label"}](hept.pdf "fig:"){width=".3\textwidth"}
With a distinguished vertex and orientation, we can view $P$ as a point in $\mathbb{R}^{3n}$ by listing the coordinates of the vertices starting with $v_1$ then following the orientation. Not all points in $\mathbb{R}^{3n}$ will correspond to simple polygonal curves. Therefore we define the discriminant set, in the spirit of Vassiliev [@Birman1993].
The discriminant, $\Sigma^{(n)}$, is all points in $\mathbb{R}^{3n}$ that correspond to non-embedded polygonal knots.
A polygonal knot in $\R^{3}$ fails to be embedded when two or more of the edges intersect. For an $n$-sided polygonal knot there are $\frac{1}{2}n(n-3)$ pairs of non-adjacent edges. So $\Sigma^{(n)}$ is the union of the closure of the $\frac{1}{2}n(n-3)$ real semi-algebraic cubic varieties, each piece consisting of polygons with a single intersection between non-adjacent edges[@Randell1][@Randell2]. By excluding these singular points, we are left with an open set in $\mathbb{R}^{3n}$ corresponding to embedded polygons in $\mathbb{R}^3$.
The embedding space for rooted, oriented $n$-sided polygonal knots, denoted $Geo(n)$, is defined to be $ \mathbb{R}^{3n}-\Sigma^{(n)}$.
The space $Geo(n)$ is an open $3n$-dimensional manifold. A continuous path $h:[0,1]\to Geo(n)$ is an isotopy of polygonal knots.
Two $n$-sided polygonal knots are geometrically equivalent if they lie in the same path-component of $Geo(n)$.
Path components are in bijective correspondence with the geometric knot types realizable with $n$ edges. Any polygon that is in the same path-component of the regular, planar $n$-gon is then geometrically equivalent to the unknot.
Next we will consider polygonal knots with unit length edges. Consider the function $f:Geo(n)\to \mathbb{R}^{n}$ where $(v_{1},v_{2},\ldots, v_{n})\mapsto (||v_{1}-v_{2}||,||v_{2}-v_{3}||,\ldots,||v_{n}-v_{1}||)$.
Let $f^{-1}((1,1,\ldots,1))=Equ(n)$ be the embedding space for rooted, oriented, $n$-sided equilateral knots.
Since $Equ(n)$ is the preimage of the smooth map $f$ at the regular value $(1,1,\ldots,1)$, $Equ(n)$ is a $2n$-dimensional manifold. Similar to the space of geometric knots, path-components correspond to the equilateral knot types realizable with $n$ edges. In this paper, we will focus on equilateral polygonal knots.
Every triangle is planar. A quadrilateral can be folded along a diagonal to become planar. It is also known that any pentagon can be deformed to a planar pentagon [@Randell2]. Therefore $Equ(n)$ is connected for $n\le 5$ and the case of hexagons is the first interesting example. Jorge Calvo [@Calvo] proves that $Equ(6)$ has five path-components. One component of $Equ(6)$ corresponds to the unknot, two to the right-handed trefoil and two to the left-handed trefoil. In order to distinguish between the different components, he introduces new knot invariants for equilateral hexagonal knots. First let $H=(v_{1},v_{2},\ldots,v_{6})\in Equ(6)$.
Let $H\in Equ(6)$. The curl of $H$, denoted $curl(H)$, is defined by $curl(H)=\text{sign}((v_{3}-v_{1})\times(v_{5}-v_{1})\cdot(v_{2}-v_{1}))$.
If $v_{1}, v_{3}$ and $v_{5}$ are on the $xy$-plane oriented in a counter-clockwise orientation, then $curl(H)$ denotes the sign of the $z$-coordinate of $v_{2}$. So, in a sense, some knots curl up while others curl down. We will describe a second invariant of the hexagonal knot that distinguishes its topological knot type.
Define $T_i$ to be the interior of the triangular disk spanned by $(v_{i-1}, v_{i}, v_{i+1})$.
Using a right-hand rule, we orient each $T_i$ as shown in Figure \[fig:T2\].
![In this figure triangular disk $T_2$ is shaded and the orientation from the right-hand rule is shown.[]{data-label="fig:T2"}](T2.pdf){width=".35\textwidth"}
Define $\Delta_{i}$, for $i=2,4,$ and $6$, to be the algebraic intersection number of T$_{i}$ with $H$.
![This figure shows an example of a hexagonal knot in which $\Delta_2=1$.[]{data-label="fig:D2"}](Delta2.pdf){width=".35\textwidth"}
The following Lemma distinguishes topological knot type using the algebraic intersection numbers.
([@Calvo])\[algintersection\] Let $H\in Equ(6)$. Then
1. $H$ is a right-handed trefoil iff $\Delta_{i}=1$ for all $i$,
2. $H$ is a left-handed trefoil iff $\Delta_{i}=-1$ for all $i$,
3. $H$ is an unknot iff $\Delta_{i}=0$ for some $i\in\{2,4,6\}$.
Combining the notion of curl with the appropriate intersections from Lemma \[algintersection\] we arrive at Calvo’s Geometric Knot Invariant, Joint Chirality-Curl.
([@Calvo]) Let $H\in Equ(6)$. Define Joint Chirality-Curl $$J(H)=(\Delta_{2}\Delta_{4}\Delta_{6},\Delta_{2}^2\Delta_{4}^2\Delta_{6}^2 curl(H)).$$
The Joint Chirality-Curl distinguishes between the five components of $Equ(6)$.
([@Calvo]): Let $H\in Equ(6)$. Then
$$J(H) =
\begin{cases}
(0,0) & \text{iff } $H$\text{ is unknot}\\
(1,c) & \text{iff } $H$ \text{ is right-trefoil with } $curl(H)=c$\\
(-1,c)& \text{iff } $H$\text{ is left-trefoil with }$curl(H)=c$
\end{cases}$$
The five components of $Equ(6)$ are due to the choice of a root and orientation. Consider the automorphisms $r$ and $s$ on $Equ(6)$ defined by $$r\langle v_1, v_2, v_3, v_4, v_5, v_6\rangle=\langle v_1, v_6, v_5, v_4, v_3, v_2\rangle$$ $$s\langle v_1, v_2, v_3, v_4, v_5, v_6\rangle=\langle v_2, v_3, v_4, v_5, v_6, v_1\rangle.$$
These automorphisms act on $Equ(6)$ by reversing or shifting the order of the vertices of each hexagon. They generate the dihedral group of order twelve.
([@Calvo])\[2.5Calvo\] Suppose $\Gamma$ is a subgroup of the dihedral group $\langle r,s\rangle$. Then $Geo(6)/\Gamma$ has five components if and only if $\Gamma$ is contained in the index-$2$ subgroup $\langle s^2,rs \rangle$. Otherwise, $Geo(6)/\Gamma$ has three components.
([@Calvo]) The spaces $Geo(6)/\langle s\rangle $ of non-rooted oriented embedded hexagons, and $Geo(6)/\langle r,s\rangle$ of non-rooted non-oriented embedded hexagons, each consist of three path-components.
Next we will discuss definitions and results from symplectic geometry, specifically toric symplectic manifolds [@symplectic], that apply to knot spaces.
A symplectic manifold, $M$, is an even dimensional manifold with a closed, non-degenerate $2$-form, $\omega$, called the symplectic form.
Since $\omega$ is non-degenerate, there is a canonical isomorphism between the tangent and cotangent bundles, namely
$$TM\mapsto T^*M: X\to \iota(X)\omega=\omega(X,\cdot ).$$
A symplectomorphism of a symplectic manifold $(M,\omega)$ is a diffeomorphism $\psi\in Diff(M)$ that preserves the symplectic form. The group of symplectomorphisms of $M$ is denoted $Symp(M,\omega)$.
Since $\omega$ is nondegenerate the homomorphism $T_q M\to T_q^*M: v\mapsto \iota(v)\omega_q$ is bijective. Thus there is a one-to-one correspondence between vector fields and $1$-forms via $$\chi (M)\to \Omega^1(M):X\mapsto \iota (X)\omega.$$
A vector field $X\in \chi (M)$ is called symplectic if $\iota (X)\omega$ is closed. Denote the space of symplectic vector fields by $\chi (M,\omega)$.
[@symplectic]\[symp\] Let $M$ be a closed manifold. If $t\mapsto \psi_t\in Diff(M)$ is a smooth family of diffeomorphims generated by a family of vector fields $X_t\in \chi(M)$ via $$\frac{d}{dt}\psi_t=X_t \circ \psi_t, \hspace{1cm} \psi_0=id,$$ then $\psi_t\in Symp(M,\omega)$ for every $t$ if and only if $X_t\in \chi(M,\omega)$ for every $t$.
Now consider a smooth function $H:M\to \mathbb{R}$.
The vector field $X_H:M\to TM$ determined by identity $dH=\iota(X_H)\omega$ is called the Hamiltonian vector field associated to $H$.
If $M$ is closed, then by Proposition \[symp\], the vector field $X_H$ generates a smooth $1$-parameter group of diffeomorphisms $\phi^t_H\in Diff(M)$ such that $$\frac{d}{dt} \phi^t_H=X_H\circ \phi^t_H,\hspace{1cm} \phi^0_H=id,$$ called a Hamiltonian flow associated to $H$. The identity $$dH(X_H)=(\iota(X_H)\omega)(X_H)=\omega(X_H,X_H)=0$$ shows that $X_H$ is tangent to level sets.
A useful example to consider is the unit sphere $S^2$, where $\omega$ is the standard area form. If $S^2=\{(x_1, x_2, x_3) : \sum_j x_j^2=1 \}$, then $\omega_x(u,v)= \langle x,u\times v\rangle $ for $u,v\in T_x S^2$. Consider cylindrical polar coordinates $(\theta,x_3)$ for $\theta\in [0,2\pi)$ and $x_3 \in [-1,1]$. Let $H$ be the height function $x_3$ on $S^2$. The level sets are circles at constant height. The Hamiltonian flow $\phi^t_H$ rotates each circle at constant speed and $X_H$ is the vector field $\frac{\partial }{\partial \theta}$. Thus $\phi^t(H)$ is the rotation of the sphere about its vertical axis through the angle $t$.
Consider a smooth map $[0,1]\times M\to M:(t,q)\mapsto \psi_t(q)$ such that $\psi_t\in Symp(M,\omega)$ and $\psi_0=id$. A family of such symplectomorphisms is called a symplectic isotopy of $M$. The isotopy is generated by a unique family of vector fields $X_t:M\to TM$ such that $\frac{d}{dt}\psi_t=X_t\circ\psi_t$. If all of the $1$-forms are exact then there exists a smooth family of Hamiltonian functions $H_t:M\to \mathbb{R}$ such that $\iota (X_t)\omega=dH_t$. In this case, $\psi_t$ is called a Hamiltonian isotopy.
A symplectomorphism, $\psi$, is called Hamiltonian if there exists a Hamiltonian isotopy $\psi_t\in Symp(M,\omega)$ from $\psi_0=id$ to $\psi_1=\psi$.
A Hamiltonian action of $S^1$ on $(M,\omega)$ is a $1$-parameter subgroup $\mathbb{R}\to Symp(M): t\mapsto \psi_t$ of $Symp(M)$ where $\psi_t=id$ and which is the integral of a Hamiltonian vector field $X_H$.
The Hamiltonian function $H:M\to \mathbb{R}$ in this case is called the moment map. If $k$ such symmetries commute we have an action of a torus, $T^{k}$, on $M$. Then the moment map, $\mu: M\to\R^{k}$ yields a $k$-dimensional vector of conserved quantities. If $k$ is half the dimension of $M$, then $M$ is called toric symplectic. From theorems of Atiyah[@Atiyah:1982re] and Guillemin-Sternberg[@GuillStern], the image of $\mu$ is a convex polytope, $P$, called the moment polytope. Moreover, the vertices of the moment polytope are the images under $\mu$ of the fixed point of the Hamiltonian torus action. In addition, the torus action preserves the fibers of the moment map. If we can invert $\mu$, we get a map $\alpha:P\times T^{n}\to M$ called the action-angle map.
The previous example of the unit sphere $S^2$ is a toric symplectic manifold, with circle action rotation about the $z$-axis. The moment map $H:S^2\to S^1$ is the height function, the conserved quantity as the sphere rotates. The image of $H$ is a convex polytope, namely the interval $[-1,1]$. The fibers of $H$ are horizontal circles of constant height, which are preserved under the action. Lastly the circle, $S^1$, is half the dimension of $S^2$.
The toric symplectic structure on the sphere naturally carries over to a toric symplectic structure on the product of spheres. This gives a toric symplectic structure on the space of open random walks or open polygons. We will consider the subspace of closed random walks. Let $Pol(n)$ be the $2n$ dimensional space of possibly singular polygons in $\mathbb{R}^3$ with edgelengths one. We will consider the quotient space $Pol_0(n)=Pol(n)/\text{SO}(3)$ of equilateral polygons up to translations and rotations. Jason Cantarella and Clayton Shonkwiler [@JC] describe the almost toric symplectic structure of $Pol_0(n)$. We summarize some of the important information below. To define the toric action, consider any triangulation, $T$, of an equilateral planar regular $n$-gon. Let $d_{i}$ be the lengths of the $n-3$ diagonals of the triangulation. These diagonals, along with the edges on the polygon, form $n-2$ triangles which each obey $3$ triangle inequalities. Therefore the lengths of the diagonals and the edge lengths must obey a set of $3(n-2)$ triangle inequalities, called the triangulation inequalities.
[@Kapovich][@Howardmanon][@Hitchin] The following facts are known:
- $Pol_0(n)$ is a possibly singular $(2n-6)$-dimensional symplectic manifold. The symplectic volume is equal to the standard measure.
- To any triangulation $T$ of the standard $n$-gon we can associate a Hamiltonian action of the torus $T^{n-3}$ on $Pol_0(n)$, where $\theta_i$ acts by folding the polygon around the $i^{th}$ diagonal of the triangulation.
- The moment map $\mu: Pol_0(n)\mapsto \mathbb{R}^{n-3}$ for a triangulation $T$ records the lengths $d_i$ of the $n-3$ diagonals of the triangulation.
- The inverse image $\mu^{-1}(int(P))\subset Pol_0(n)$ of the interior of the moment polytope $P$ is an toric symplectic manifold.
The moment polytope, $P_n$, is defined by the triangulation inequalities for $T$. The vertices of the moment polytope represent degenerate polygons which extremize several triangulation inequalities. Figure \[pentagon\] shows a triangulation of an equilateral pentagon and the corresponding moment polytope.
![The left image shows the fan triangulation of an equilateral pentagon, where all diagonals share a common vertex. The lengths of the diagonals, $d_1$ and $d_2$, satisfy six triangle inequalities. The figure on the left shows the moment polytope of $Pol_0(5)$ corresponding to the fan triangulation.[]{data-label="pentagon"}](pentagon.pdf "fig:"){width=".5\textwidth"}![The left image shows the fan triangulation of an equilateral pentagon, where all diagonals share a common vertex. The lengths of the diagonals, $d_1$ and $d_2$, satisfy six triangle inequalities. The figure on the left shows the moment polytope of $Pol_0(5)$ corresponding to the fan triangulation.[]{data-label="pentagon"}](pentagonpoly.pdf "fig:"){width=".55\textwidth"}
The action-angle map $\alpha:P_n\times T^{n-3} \mapsto Pol_0(n)$ for a triangulation $T$ is given by first constructing the $n-2$ triangles using the diagonal lengths, $d_i$, and edge lengths of $1$ and then joining them in $3$-space with dihedral angles given by the $\theta_i$. The polygon is the boundary of this triangulated surface. This construction only makes sense for polygons equivalent up to translations and rotations, which is why the quotient by $\text{SO}(3)$ is necessary. An example of an equilateral pentagon is shown in Figure \[pentagon2\].
![The figure shows how to construct an equilateral pentagon from the action-angle map $\alpha: P_5\times T^2\mapsto Pol_0(5)$ for the fan triangulation. A point $(d_1, d_2)$ in the moment polytope gives the information needed to construct three triangles. Then a point $(\theta_1, \theta_2)\in T^2$ gives instruction on how to attach the triangles along the diagonals. The boundary of the triangulated surface is the equilateral pentagon.[]{data-label="pentagon2"}](pent1.pdf "fig:"){width=".35\textwidth"}![The figure shows how to construct an equilateral pentagon from the action-angle map $\alpha: P_5\times T^2\mapsto Pol_0(5)$ for the fan triangulation. A point $(d_1, d_2)$ in the moment polytope gives the information needed to construct three triangles. Then a point $(\theta_1, \theta_2)\in T^2$ gives instruction on how to attach the triangles along the diagonals. The boundary of the triangulated surface is the equilateral pentagon.[]{data-label="pentagon2"}](pent2.pdf "fig:"){width=".35\textwidth"}![The figure shows how to construct an equilateral pentagon from the action-angle map $\alpha: P_5\times T^2\mapsto Pol_0(5)$ for the fan triangulation. A point $(d_1, d_2)$ in the moment polytope gives the information needed to construct three triangles. Then a point $(\theta_1, \theta_2)\in T^2$ gives instruction on how to attach the triangles along the diagonals. The boundary of the triangulated surface is the equilateral pentagon.[]{data-label="pentagon2"}](pent3.pdf "fig:"){width=".35\textwidth"}
The following theorem will be used in Section 4 when calculating the knotting probability of equilateral hexagons.
(Duistermaat-Heckman)[@DuistHeck]\[DH\] Suppose $M$ is a $2n$-dimensional toric symplectic manifold with moment polytope $P$, $T^{n}$ is the $n$-torus and $\alpha$ inverts the moment map. If we take the standard measure on the $n$-torus and the uniform measure on $\text{int}(P)$, then the map $\alpha:\text{int}(P)\times T^{n}\to M$ parametrizing a full-measure subset of $M$ in action-angles coordinates is measure-preserving. In particular, if $f:M\to \mathbb{R}$ is any integrable function then $$\int_M f(x)\text{ d}m=\int_{P\times T^n} f(d_1,\cdots,d_n,\theta_1,\cdots,\theta_n)\text{ dVol}_{\mathbb{R}^n}\wedge d\theta_1\wedge\cdots \wedge d\theta_n$$ and if $f(d_1, \cdots,d_n,\theta_1,\cdots,\theta_n)=f_d(d_1, \cdots,d_n)f_\theta(\theta_1,\cdots,\theta_n)$ then $$\int_M f(x)\text{ d}m=\int_{P} f_d(d_1,\cdots,d_n)dVol_{\mathbb{R}^n}\int_{T^n} f_\theta(\theta_1,\cdots,\theta_n)d\theta_1\wedge\cdots \wedge d\theta_n.$$\
Symplectic Structure of the space of equilateral hexagons
=========================================================
Action-Angle Coordinates
------------------------
In order to describe the action-angle coordinates on the space of equilateral hexagons, we first must consider the quotient space of $Equ(6)$.
Let $Equ_0(6)=Equ(6)/\text{SO}(3)\times\mathbb{R}^3$ be the embedding space of rooted, oriented equilateral hexagonal knots up to translations and rotations.
Let $H=(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6})\in Equ(6)$. We can translate $H$ so that $v_{1}=(0,0,0)$. Additionally we rotate $H$ so that $v_{3}$ is on the positive $x$-axis and $v_5$ on the upper-half $xy$-plane. Therefore $v_1, v_3$, and $v_5$ are on the $xy$-plane in a counter-clockwise orientation. In this section, we will consider this to be the standard position for $H\in Equ_0(6)$.
Next we can choose any triangulation of the standard planar equilateral hexagon to form our action-angle coordinates. We will use one of the triangulations that has a central triangle.
Let the $T_{135}$ triangulation be the triangulation of the regular planar equilateral hexagon which has diagonals connecting $v_{1}$ to $v_{3}$, $v_{3}$ to $v_{5}$, and $v_{5}$ to $v_{1}$, with lengths $d_{1}$, $d_{2}$, and $d_{3}$, respectively.
![This figure shows the $T_{135}$ triangulation of an equilateral hexagon.[]{data-label="fig:label"}](Ttriangulation.pdf){width=".4\textwidth"}
The lengths of the diagonals of the $T_{135}$ triangulation obey the following triangulation inequalities: $$\begin{aligned}[c]
0\le& d_1 \le2,\\
0\le& d_2 \le2,\\
0\le& d_3 \le2,\\
\end{aligned}
\quad \text{and}\quad
\begin{aligned}[c]
d_3\le& d_1+d_2,\\
d_1\le&d_3+d_2,\\
d_2\le&d_3+d_1.\\
\end{aligned}$$
The $T_{135}$ triangulation polytope, $P_6$, is the moment polytope for $Pol_0(6)$ corresponding to the $T_{135}$ triangulation and is determined by the triangulation inequalities.
![This figure shows the $T_{135}$ triangulation polytope.[]{data-label="fig:polytope"}](poly.pdf){width=".5\textwidth"}
Let $\theta_{i}$ be the dihedral angle around diagonal $d_{i}$, where the regular planar hexagon has all angles $\pi$. Then the action-angle map for $T_{135}$, $\alpha:P_6\times T^3\mapsto Pol_0(6)$ allows us to parametrize any $H\in Equ_{0}(6)$ as $H=(d_{1},d_{2}, d_{3}, \theta_{1}, \theta_{2}, \theta_{3})$. To construct an equilateral hexagonal knot in $Equ_0(6)$, first choose a point $(d_1, d_2, d_3)\in P_6$ and construct four triangles: one with lengths $d_1, d_2$, and $d_3$ and three isosceles triangles with two side lengths $1$ and third side $d_i$. The triangle with side lengths $d_1, d_2$, and $d_3$ is placed on the $xy$-plane with $v_1$ the origin and $v_3$ on the positive $x$-axis. Then a point $(\theta_1,\theta_2,\theta_3)$ in the torus $T^3$ gives instructions on how to connect the three remaining triangles.
![Given a point $(d_1,d_2,d_3)\in P_6$ four triangles are formed. Then given a triple of angles, the triangles are connected to form an equilateral hexagonal trefoil.[]{data-label="fig:label"}](trexex.pdf "fig:"){width=".6\textwidth"}![Given a point $(d_1,d_2,d_3)\in P_6$ four triangles are formed. Then given a triple of angles, the triangles are connected to form an equilateral hexagonal trefoil.[]{data-label="fig:label"}](trefexample.pdf "fig:"){width=".45\textwidth"}
For $H\in Equ_0(6)$ in standard position, the action-angle coordinates arising from the $T_{135}$ triangulation gives the following coordinates for the vertices of $H$:
$$\begin{aligned}
v_1 =& \Big(0,0,0\Big)\\
v_{2}=& \Big(\frac{d_{1}}{2}, \frac{1}{2}\sqrt{4-(d_{1})^2}\text{ cos}(\theta_{1}),\frac{1}{2}\sqrt{4-(d_{1})^2}\text{ sin}(\theta_{1})\Big),\\
v_{3}=& \Big(d_{1},0,0\Big),\\
v_{4}=&\Big(\frac{3(d_{1})^2-(d_{2})^2+(d_{3})^2}{4d_{1}}-\frac{d}{4d_{1}d_{2}}\sqrt{4-(d_{2})^2}\text{ cos}(\theta_{2}), \frac{d}{4d_{1}}-\\&\frac{(d_{1})^2+(d_{2})^2-(d_{3})^2}{4d_{1}d_{2}}\sqrt{4-(d_{2})^2}\text{ cos}(\theta_{2}), \frac{1}{2}\sqrt{4-(d_{2})^2}\text{ sin}(\theta_{2})\Big),\\
v_{5} =& \Big(\frac{(d_{1})^2-(d_{2})^2+(d_{3})^2}{2d_{1}}, \frac{d}{2d_{1}}, 0\Big),\\
v_{6} =& \Big(\frac{(d_{1})^2-(d_{2})^2+(d_{3})^2}{4d_{1}}-\frac{d}{4d_{1}d_{3}}\sqrt{4-(d_{3})^2}\text{ cos}(\theta_{3}), \frac{d}{4d_{1}}-\\&\frac{(d_{1})^2-(d_{2})^2+(d_{3})^2}{4d_{1}d_{3}}\sqrt{4-(d_{3})^2}\text{ cos}(\theta_{3}), \frac{1}{2}\sqrt{4-(d_{3})^2}\text{ sin}(\theta_{3})\Big),\end{aligned}$$
where $d=\sqrt{2(d_{1}d_{2})^2+2(d_{1}d_{3})^2+ 2(d_{2}d_{3})^2-(d_{1})^4-(d_{2})^4-(d_{3})^4}$.\
Recall that the geometric knot invariant for hexagons, Joint Chirality-Curl, distinguishes between two types of both right-handed and left-handed trefoils with $curl(H)=\text{sign}((v_{3}-v_{1})\times(v_{5}-v_{1})\cdot(v_{2}-v_{1}))$. The following two lemmas give a relation between the curl of a hexagon and the possible dihedral angles.
\[curl1\] Let $H\in Equ_0(6)$. Let $H$ be parametrized using action-angle coordinates from the $T_{135}$ triangulation. If $H$ has Joint Chirality-Curl $(\pm1,1)$, then $\theta_i\in (0,\pi)$ for $i=1,2,3$.
[*Proof:* ]{} Let $H\in Equ_{0}(6)$ be parametrized with action-angle coordinates $(d_1,d_2,d_3,\theta_1,\theta_2,\theta_3)$ arising from the $T_{135}$ triangulation. Let $H$ be in standard position. Since $v_1$, $v_3$, and $v_5$ are on the $xy$-plane oriented in a counter-clockwise direction, $curl(H)$ denotes the sign of the $z$-coordinate of $v_2$. Therefore if $curl(H)=1$, then $\theta_1\in(0,\pi)$. Suppose that $\theta_2, \theta_3\in (\pi, 2\pi)$. Then both $e_4$ and $e_5$ lie below the $xy$-plane and can not pierce $T_2$. Thus $\Delta_2=0$ and $H$ has Joint Chirality-Curl $(0,0)$. If $\theta_2\in(\pi, 2\pi)$ and $\theta_3\in(0,\pi)$, then neither $e_6$ nor $e_1$ can pierce $T_4$ and $\Delta_4=0$. Similarly if $\theta_3\in(\pi, 2\pi)$ and $\theta_2\in(0,\pi)$, then $\Delta_6=0$. Therefore if $curl(H)=1$, then $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$. [$\square$]{}
\[curl-1\] Let $H\in Equ_0(6)$. Let $H$ be parametrized using action-angle coordinates from the $T_{135}$ triangulation. If $H$ has Joint Chirality-Curl $(\pm1,-1)$, then $\theta_i\in (\pi,2\pi)$ for $i=1,2,3$.
[*Proof:* ]{} Let $H\in Equ_{0}(6)$ be parametrized with action-angle coordinates $(d_1,d_2,d_3,\theta_1,\theta_2,\theta_3)$ arising from the $T_{135}$ triangulation. Let $H$ be in standard position. If $curl(H)=-1$, then $\theta_1\in(\pi,2\pi)$. Similar to the previous argument of Lemma \[curl1\], if either of both of $\theta_2$ and $\theta_3$ are between $0$ and $\pi$, then $J(H)=(0,0)$. Therefore if $curl(H)=-1$ and $H$ is a trefoil, then $\theta_{i}\in (\pi, 2\pi)$ for all $i\in{1,2,3}$. [$\square$]{}\
Equilateral, Right-Handed, Positive Curl, Hexagonal Trefoils
------------------------------------------------------------
In this section, we will determine constraints on the values of $d_1, d_2, d_3, \theta_1, \theta_2,$ and $\theta_3$ in order to have a right-handed hexagonal trefoil with positive curl. First we will define a set of inequalities that must be satisfied in order for $H\in Equ_{0}(6)$ in standard position to have Joint Chirality-Curl $(1,1)$.
\[disc\] Let $H\in Equ_0(6)$ and parametrize $H$ with action-angle coordinates arising from the $T_{135}$ triangulation. If $J(H)=(1,1)$, then the following nine functions must be positive:
$$\begin{aligned}
f_1= & d_2\sqrt{4-(d_2)^2}\text{sin}(\theta_2)\big(d_3d-((d_1)^2-(d_2)^2+(d_3)^2)\sqrt{4-(d_3)^2}\text{cos}(\theta_3)\big)-\\& d_3\sqrt{4-(d_3)^2}\text{sin}(\theta_3)\big(d_2d-((d_1)^2+(d_2)^2-(d_3)^2)\sqrt{4-(d_2)^2}\text{cos}(\theta_2)\big),\\
g_1= & \sqrt{4-(d_2)^2}\Big(\frac{-(d_1)^2+(d_2)^2+(d_3)^2}{2d_2 d_3}\text{cos}(\theta_2)\text{sin}(\theta_3)+\text{sin}(\theta_2)\text{cos}(\theta_3)\Big)\\ &-\frac{d\text{sin}(\theta_3)}{2d_3},\\
h_1= & \sqrt{4-(d_3)^2}\Big(\frac{-(d_1)^2+(d_2)^2+(d_3)^2}{2d_2 d_3}\text{cos}(\theta_3)\text{sin}(\theta_2)+\text{sin}(\theta_3)\text{cos}(\theta_2)\Big) \\ &-\frac{d\text{sin}(\theta_2)}{2d_2},\\
f_2 = & d_3\sqrt{4-(d_3)^2}\text{sin}(\theta_3)\big(d_1d-((d_1)^2+(d_2)^2-(d_3)^2)\sqrt{4-(d_1)^2}\text{cos}(\theta_1))- \\& d_1\sqrt{4-(d_1)^2}\text{sin}(\theta_1)\big(d_3d-(-(d_1)^2+(d_2)^2+(d_3)^2)\sqrt{4-(d_3)^2}\text{cos}(\theta_3)\big),\\
g_2 = &\sqrt{4-(d_3)^2}\Big(\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3}\text{cos}(\theta_3)\text{sin}(\theta_1)+\text{sin}(\theta_3)\text{cos}(\theta_1)\Big) \\ &-\frac{d\text{sin}(\theta_1)}{2d_1},\\
h_2 = &\sqrt{4-(d_1)^2}\Big(\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3}\text{cos}(\theta_1)\text{sin}(\theta_3)+\text{sin}(\theta_1)\text{cos}(\theta_3)\Big) \\ &-\frac{d\text{sin}(\theta_3)}{2d_3},\\
f_3 = & d_1\sqrt{4-(d_1)^2}\text{sin}(\theta_1)\big(d_2d-(-(d_1)^2+(d_2)^2+(d_3)^2)\sqrt{4-(d_2)^2}\text{cos}(\theta_2)\big)-\\&d_2\sqrt{4-(d_2)^2}\text{sin}(\theta_2)\big(d_1d-((d_1)^2-(d_2)^2+(d_3)^2)\sqrt{4-(d_1)^2}\text{cos}(\theta_1)\big),\\
g_3 = &\sqrt{4-(d_1)^2}\Big(\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}\text{cos}(\theta_1)\text{sin}(\theta_2)+\text{sin}(\theta_1)\text{cos}(\theta_2)\Big)\\ & -\frac{d\text{sin}(\theta_2)}{2d_2},\\
h_3 = &\sqrt{4-(d_2)^2}\Big( \frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}\text{cos}(\theta_2)\text{sin}(\theta_1)+\text{sin}(\theta_2)\text{cos}(\theta_1)\Big) \\ & -\frac{d\text{sin}(\theta_1)}{2d_1}.\end{aligned}$$
[*Proof:* ]{}Let $H\in Equ_0(6)$ and parametrize $H$ with action-angle coordinates from the $T_{135}$ triangulation. If $curl(H)=1$, then $\theta_i\in (0,\pi)$ for all $i$ by Lemma \[curl1\]. Recall that for a hexagon, $H\in Equ_0(6)$ to be a right-handed trefoil then the algebraic intersection numbers $\Delta_i$ have to be equal to one for $i=2,4,6$. First we will consider $\Delta_4=1$. This means that the triangular disk $T_{4}$ containing $v_{3}, v_{4},$ and $v_{5}$ must be pierced by either the edge $e_{6}$ or $e_1$ so that the orientation on the edge agrees with the orientation on $T_4$ coming from a right-hand rule. If $\theta_2\in (0,\pi)$, then $e_6$ must pierce $T_4$ for $\Delta_4=1$.
In order for the line going through $v_{1}$ and $v_{6}$ to pierce $T_{4}$ the following must be positive:
$$\begin{aligned}
(v_{6}-v_{1})\times(v_{4}-v_{1})\centerdot(v_{3}-v_{1})&>0,\\
(v_{6}-v_{1})\times(v_{5}-v_{1})\centerdot(v_{4}-v_{1})&>0,\\
(v_{6}-v_{1})\times(v_{3}-v_{1})\centerdot(v_{5}-v_{1})&>0.\end{aligned}$$
{width=".5\textwidth"}
In addition, the plane containing $v_{3}, v_{4},$ and $v_{5}$ must separate $v_{1}$ from $v_{6}$. So the following must be negative $$\begin{aligned}
\big((v_{6}-v_{3})\times(v_{5}-v_{3})\centerdot(v_{4}-v_{3})\big)\big((v_{1}-v_{3})\times(v_{5}-v_{3})\centerdot(v_{4}-v_{3})\big)<0.\end{aligned}$$
Since $\theta_i\in(0,\pi)$ for all $i$, $(v_{6}-v_{1})\times(v_{3}-v_{1})\centerdot(v_{5}-v_{1})$ is always positive. Negating $(4)$ leaves three inequalities that must be satisfied so that $e_{6}$ pierces $T_4$. Letting $H$ be in standard position and evaluating the remaining three with action-angle coordinates gives three functions $f_1, g_1, h_1$ that must be positive for $H$ to have Joint Chirality-Curl $(1,1)$. Similarly, for $\Delta_2=1$ and $\Delta_6=1$ then $f_2, g_2, h_2$ and $f_3, g_3, h_3$ must be positive, respectively. Therefore if $H$ is in standard position and parametrized with action-angle coordinates form the $T_{135}$ triangulation, $f_i, g_i$, and $h_i$ must be positive for $H$ to have Joint Chirality-Curl $(1,1)$. [$\square$]{}\
Next we will prove constraints on action-angle coordinates to get a right-handed, positive curl trefoil.
\[equilateral\] Let $H\in Equ_{0}(6)$ and parametrize $H$ with with action-angle coordinates coming from the $T_{135}$ triangulation. If $H$ has Joint Chirality-Curl $(1,1)$, then the lengths of diagonals $d_i$ must be distinct.
[*Proof:* ]{}Let $H$ be in $Equ_{0}(6)$. Parametrize $H$ with action-angle coordinates coming from the $T_{135}$ triangulation, so $H=(d_1,d_2,d_3,\theta_1,\theta_2,\theta_3)$ and $H$ is in standard position. Suppose $d_1=d_2=d_3=x$, for some $x\in(0,2)$. Then $v_1=(0,0,0), v_3=(x,0,0)$, and $v_5=(\frac{x}{2}, \frac{\sqrt{3}x}{2},0)$. First we will consider the case when $x=\sqrt{3}$. When $\theta_i=0$ for all $i$, $H$ is planar but singular with vertices $v_2$, $v_4$, and $v_6$ coinciding in a single point, $(\frac{\sqrt{3}}{2},\frac{1}{2},0)$. Additionally $e_1=e_6$, $e_2=e_3$, and $e_4=e_5$. As $\theta_1$ increases from $0$ to $\pi$, $v_{2}$ traverses a circle, $c_2$, of radius $\frac{1}{2}$ centered at $(\frac{\sqrt{3}}{2},0,0)$ lying in a plane parallel to the $yz$-plane. Similarly, as $\theta_3$ increases from $0$ to $2\pi$, $v_6$ moves along a circle, $c_6$, of radius $\frac{1}{2}$ centered at the midpoint of the edge connecting $v_1$ and $v_5$. Therefore $e_1$ sweeps out a circular cone, $C_{12}$, with vertex the origin and base circle $c_2$ and $e_6$ sweeps out a circular cone, $C_{61}$, with vertex the origin and base circle $c_6$. The circles $c_2$ and $c_6$ only intersect when $\theta_1=\theta_3=0$. Therefore the respective cones only intersect in the segment from $(0,0,0)$ to $(\frac{\sqrt{3}}{2},0,0)$, corresponding to edges $e_1$ and $e_6$ coinciding when $\theta_1=\theta_3=0$. This implies that $e_2$ can not pierce $T_6$. Thus if $x=\sqrt{3}$, $H$ can not have Joint Chirality-Curl $(1,1)$.
Next we consider the case when $\sqrt{3}<x<2$. When $\theta_1, \theta_2, \theta_3=0$, $H$ is planar and embedded. Hence $H$ is unknotted. Cones $C_{12}$ and $C_{61}$, formed by edges $e_1$ and $e_6$ as $\theta_1$ and $\theta_3$ vary, do not intersect. Therefore neither $T_2$ nor $T_6$ will be pierced by $H$, so $H$ will remain unknotted.
Now consider the case when $0<x<\sqrt{3}$. If $\theta_i=\text{cos}^{-1}(\frac{\sqrt{3}x}{3\sqrt{4-x^2}})$ for all $i$, then $v_2, v_4$, and $v_6$ coincide. If $\theta_i\in(\text{cos}^{-1}(\frac{\sqrt{3}x}{3\sqrt{4-x^2}}),\pi)$ for any $i\in \{1,2,3\}$ then $H$ will be unknotted. Therefore suppose that $\theta_i\in (0,\text{cos}^{-1}(\frac{\sqrt{3}x}{3\sqrt{4-x^2}}))$ for all $i$. If $\theta_1=\theta_3=0$, then $e_2$ and $e_5$ intersect in a point on the $xy$-plane. If $1\le x<\sqrt{3}$, then the point of intersection is interior of the triangle with vertices $v_1$, $v_3$, and $v_5$. As $\theta_1$ and $\theta_3$ increase from $0$ to $\text{cos}^{-1}(\frac{\sqrt{3}x}{3\sqrt{4-x^2}})$, $e_2$ and $e_5$ continue to intersect in a point. In order for $e_2$ to pierce $T_6$, then $\theta_3>\theta_1$. Similarly $e_1$ and $e_4$ intersect when $\theta_1=\theta_2$. In order for $e_4$ to pierce $T_2$, then $\theta_1>\theta_2$. When $\theta_2=\theta_3$, $e_6$ and $e_3$ intersect. In order for $e_6$ to pierce $T_4$, then $\theta_2>\theta_3$. This implies that $\theta_3>\theta_1>\theta_2>\theta_3$, a contradiction. When $0<x<1$ and $\theta_1=\theta_3=0$, then $e_2$ and $e_5$ intersect in a point that is exterior of the triangle with vertices $v_1$, $v_3$, and $v_5$. In order for $e_2$ to pierce $T_6$, then $\theta_3<\theta_1$. In order for $e_4$ to pierce $T_2$, then $\theta_1<\theta_2$. In order for $e_6$ to pierce $T_4$, then $\theta_2<\theta_3$. This implies that $\theta_3<\theta_1<\theta_2<\theta_3$, a contradiction. Therefore when $d_1=d_2=d_3$, $H$ can not have Joint Chirality-Curl $(1,1)$. [$\square$]{}\
From Lemma \[curl1\], we know that all three dihedral angles must be in the interval $(0,\pi)$ for $curl(H)=1$. Next given any admissible triple of diagonal lengths, we prove a tighter constraints on the dihedral angles so that $H$ has Joint Chirality-Curl $(1,1)$.
\[angles\] Let $H\in Equ_{0}(6)$ and parametrize $H$ using action-angle coordinates with the $T_{135}$ triangulation. If $H$ has Joint Chirality-Curl $(1,1)$, then $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$ and $\theta_{1}+\theta_{2}<\pi$, $\theta_{1}+\theta_{3}<\pi$, and $\theta_{2}+\theta_{3}<\pi$.
[*Proof:* ]{}Let $H\in Equ_{0}(6)$ be parametrized with action-angle coordinates\
$(d_1,d_2,d_3,\theta_1,\theta_2,\theta_3)$ arising from the $T_{135}$ triangulation. If $H$ is a right-handed trefoil with $curl(H)=1$, then from Lemma \[curl1\] we know $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$. If $\theta_i\in(0,\frac{\pi}{2})$ for all $i\in\{1,2,3\}$, then clearly $\theta_{1}+\theta_{2}<\pi$, $\theta_{1}+\theta_{3}<\pi$, and $\theta_{2}+\theta_{3}<\pi$. Additionally, if $\theta_i, \theta_j\in(0,\frac{\pi}{2})$, for any two distinct $i,j\in\{1,2,3\}$, then $\theta_i+\theta_j< \pi$.
Next we will show that if $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$ and $H$ has Joint Chirality-Curl $(1,1)$, then $\theta_{1}+\theta_{2}<\pi$ and $\theta_{1}+\theta_{3}<\pi$. Towards a contradiction, suppose that $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, and $\theta_1+\theta_2= \pi$. Substituting $\theta_1=\pi-\theta_2$ into equation $g_3$ from Proposition \[disc\] and using the facts that $\text{cos}(\pi-\theta_2)=-\text{cos}(\theta_2)$ and $\text{sin}(\pi-\theta_2)=\text{sin}(\theta_2)$, we obtain the following
$$\begin{aligned}
g_3=\sqrt{4-(d_1)^2}\Big(-\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}+1\Big)\text{sin}(\theta_2)\text{cos}(\theta_2)-\frac{d\text{sin}(\theta_2)}{2d_2}.\end{aligned}$$
We make the same substitutions into equation $h_3$ to obtain $$\begin{aligned}
h_3=\sqrt{4-(d_2)^2}\Big(\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1\Big)\text{sin}(\theta_2)\text{cos}(\theta_2)-\frac{d\text{sin}(\theta_1)}{2d_1}.\end{aligned}$$
If $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1\ge0$, then $g_3$ is negative. Fix $d_1,d_2, d_3,$ and $\theta_2$, and now consider $g_3$ as a function of $\theta_1$. Since $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1\ge0$, then $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}>0$. This implies that the derivative of $g_3$ with respect to $\theta_1$, $$\sqrt{4-(d_1)^2}\Big(\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}(-\text{sin}(\theta_1))\text{sin}(\theta_2)+\text{cos}(\theta_1)\text{cos}(\theta_2)\Big),$$ is negative for $\theta_1\in(\frac{\pi}{2},\pi)$ and $\theta_2\in(0,\frac{\pi}{2})$. Since $g_3$ is negative for $\theta_1=\pi-\theta_2$ and $g_3$ is decreasing, $g_3$ is negative for all $\theta_1>\pi-\theta_2$. Next suppose $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1<0$, then $h_3$ is negative. This means that the plane, $P_2$, containing vertices $v_1, v_2$, and $v_3$ does not separate $v_4$ and $v_5$ when $\theta_1=\pi-\theta_2$. Therefore as $\theta_1$ increases to $\pi$ so that $\theta_1>\pi-\theta_2$, $P_2$ will not separate $v_4$ and $v_5$. Thus $h_3$ is negative for $\theta_1>\pi-\theta_2$. Since both $g_3$ and $h_3$ must be positive for $H$ to a right-handed trefoil with $curl(H)=1$, we have reached a contradiction. Therefore if $\theta_1\in(0,\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, and $\theta_1+\theta_2\ge \pi$, $H$ can not have Joint Chirality-Curl $(1,1)$.
Now suppose that $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, and $\theta_1+\theta_3= \pi$. Similar to the previous argument, we will substitute $\theta_1=\pi-\theta_3$ into equation $g_2$ and use the facts that $\text{cos}(\pi-\theta_3)=-\text{cos}(\theta_3)$ and $\text{sin}(\pi-\theta_3)=\text{sin}(\theta_3)$. This results in the following: $$\begin{aligned}
g_2&=\sqrt{4-(d_3)^2}\Big(\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} -1\Big)\text{cos}(\theta_3)\text{sin}(\theta_3)-\frac{d\text{sin}(\theta_1)}{2d_1}.\end{aligned}$$
Making the same substitutions into $h_2$ gives $$\begin{aligned}
h_2&=\sqrt{4-(d_1)^2}\Big(-\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} +1\Big)\text{cos}(\theta_3)\text{sin}(\theta_3)-\frac{d\text{sin}(\theta_3)}{2d_3}.\end{aligned}$$
If $\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} -1\le 0$, then $g_2$ is negative. If $\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} -1> 0$, then $h_2$ is negative. Since both equations must be positive to have Joint Chirality-Curl $(1,1)$, we have reached a contradiction. Therefore if $H$ is a right-handed trefoil with $curl(H)=1$ and $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, then $\theta_{1}+\theta_{2}<\pi$ and $\theta_{1}+\theta_{3}<\pi$.
The cases when $\theta_2\in(\frac{\pi}{2}, \pi)$, $\theta_1, \theta_3\in(0,\frac{\pi}{2})$ and $\theta_3\in(\frac{\pi}{2}, \pi)$, $\theta_1, \theta_2\in(0,\frac{\pi}{2})$ are proven in the same manner. Thus if $H$ has Joint Chirality-Curl $(1,1)$, then $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$ and $\theta_{1}+\theta_{2}<\pi$, $\theta_{1}+\theta_{3}<\pi$, and $\theta_{2}+\theta_{3}<\pi$.
[$\square$]{}\
For $H\in Equ_{0}(6)$ to be a right-handed trefoil with $curl(H)=1$, only one of the dihedral angles can be greater than $\pi/2$ with the additional condition that the sum of any two angles must be less than $\pi$. This portion of the cube $[0,2\pi]^3$ is shown in the following Figure \[torusportion\].
![The corresponding angles from Lemma \[angles\] for an equilateral hexagon to have Joint Chirality-Curl $(1,1)$.[]{data-label="torusportion"}](torusangles.pdf){width=".7\textwidth"}
When all three diagonals from the $T_{135}$ triangulation have equal length, $H$ can not have Joint Chirality-Curl $(1,1)$. So, we continue our analysis with the case where two of the diagonals have equal lengths.\
coincide when $\theta_1=\theta_2=0$. This occurs when $d_3=d_1\sqrt{4-(d_1)^2}$. If $d_3\ge d_1\sqrt{4-(d_1)^2}$, then $H$ has Joint Chirality $(0,0)$ for all $\theta_i$. Now we will consider the different ranges of $\theta_i$ for $H$ to have Joint Chirality-Curl $(1,1)$ from Lemma \[angles\].
Next we consider the case where the three diagonals of the $T_{135}$ triangulation are distinct.\
\[polytopeportionR+\] Let $H\in Equ_{0}(6)$ and parametrize $H$ using action-angle coordinates with the $T_{135}$ triangulation. Suppose $d_{1}, d_{2},$ and $d_{3}$ are distinct and let $d_{i}>d_{j}, d_{k}$. If $J(H)=(1,1)$ then $\theta_{i}\in(0,\pi)$ and $\theta_{j},\theta_{k}\in(0,\pi/2)$. Moreover, if $d_{i}>\sqrt{(d_{j})^2+(d_{k})^2}$ then $\theta_{i}\in(\pi/2,\pi)$.\
[*Proof:* ]{}Let $H\in Equ_{0}(6)$ be in standard position so that $v_1$, $v_3$, and $v_5$ are on the $xy$-plane. Suppose that the lengths of the diagonals are distinct and that $d_2>d_3>d_1$. We will show that if $H$ has Joint Chirality-Curl $(1,1)$ then $\theta_2\in (0,\pi)$ and $\theta_1,\theta_3\in(0,\frac{\pi}{2})$. Let $l_2$ be the line in the $xy$-plane perpendicular to the segment connecting $v_1$ and $v_3$ intersecting at the midpoint, $m_2$. Similarly, we define $l_4$ and $l_6$ for segments connecting $v_3$ to $v_5$ and $v_5$ to $v_1$ respectively. The three lines intersect in a unique point, $k$, the circumcenter of the triangle with vertices $v_1$, $v_3$, and $v_5$. Moreover, $l_i$ represents the orthogonal projection of $v_i$ onto the $xy$-plane as $\theta_i$ varies and $k$ is the projection of where all vertices coincide, if such point exists.
![The figures show the triangle spanned by vertices $(v_1 ,v_3 ,v_5 )$ with perpendicular bisector $l_i$. In the figure on the left, $(d_2)^2 < (d_1)^2+(d_3)^2$ and so $k$ is interior of the triangle. In the figure on the right, $(d_2)^2 >(d_1)^2+(d_3)^2$ and so $k$ is exterior of the triangle.[]{data-label="obtuse"}](circumeter.pdf "fig:"){width=".55\textwidth"}![The figures show the triangle spanned by vertices $(v_1 ,v_3 ,v_5 )$ with perpendicular bisector $l_i$. In the figure on the left, $(d_2)^2 < (d_1)^2+(d_3)^2$ and so $k$ is interior of the triangle. In the figure on the right, $(d_2)^2 >(d_1)^2+(d_3)^2$ and so $k$ is exterior of the triangle.[]{data-label="obtuse"}](circumeterobtuse.pdf "fig:"){width=".55\textwidth"}
Since $d_3>d_1$ then $l_4$ intersects the segment connecting $v_1$ to $v_5$ instead of the segment connecting $v_1$ and $v_3$. Suppose towards contradiction that $\theta_1\in(\frac{\pi}{2},\pi)$. Then the plane perpendicular to the $xy$-plane containing $l_4$ separates $e_4$ and $T_2$. Therefore $H$ can not have Joint Chirality-Curl $(1,1)$ if $\theta_1\in(\frac{\pi}{2},\pi)$. Let $\phi_1$ be the angle for $\theta_1$ where $v_2$ projects onto $k$. If $\theta_1\in (\phi_1,\frac{\pi}{2})$ then $e_4$ and $T_2$ are still separated by the plane through $l_4$. Therefore $\theta_1\in (0,\phi_1)$. Next suppose that $\theta_3\in(\frac{\pi}{2},\pi)$. Since $d_2>d_3$ the $l_2$ intersects the segment connecting $v_3$ and $v_5$. This means the plane perpendicular to the $xy$-plane containing $l_2$ separates $e_2$ and $T_6$. Therefore we have reached a contradiction and $\theta_3\in (0,\frac{\pi}{2})$. Let $\phi_3$ be the angle for $\theta_3$ for which $v_6$ projects onto $k$. In order for $e_2$ to pierce $T_6$ then $\theta_3\in (0,\phi_3)$. Let $\phi_2$ be the angle for $\theta_2$ for which $v_4$ projects onto $k$. Let $p$ be the point where $e_1$ intersects $e_4$ when $\theta_1=0$ and $\theta_2=\pi$. In order for $e_4$ to intersect $T_2$, $e_4$ must intersect the cone spanned by $e_1$. The two cones will intersect along an arc connecting $p$ to point which projects onto $k$. If $\theta_2<\phi_2$, then $e_4$ no longer intersects the cone spanned by $e_1$. Then $\theta_2\in (\phi_2,\pi)$ for $H$ to have Joint Chirality-Curl $(1,1)$. If $d_2>\sqrt{(d_1)^2+(d_3)^2}$ then the triangle with vertices $v_1$, $v_3$, and $v_5$ is obtuse, as shown in Figure \[obtuse\]. Therefore $k$ is exterior of the triangle and $\phi_2>\frac{\pi}{2}$. Hence if $d_2>\sqrt{(d_1)^2+(d_3)^2}$ then $\theta_1,\theta_3\in (0,\frac{\pi}{2})$ and $\theta_2\in (\frac{\pi}{2},\pi)$. [$\square$]{}\
The moment polytope corresponding to the $T_{135}$ triangulation is split into three equal regions, depending on the which diagonal length is largest. The function $d_1=\sqrt{(d_2)^2+(d_3)^2}$ divides the third of the polytope where $d_1>d_2, d_3$ into two regions, one for acute triangles and one for obtuse triangles, as shown in Figure \[fig:dividepolytope\].
![The figure on the left shows the portion of the moment polytope, $P_6$, where $d_1>d_2, d_3$. The figure on the left shows the portion of the moment polytope where additionally $(d_1)^2>(d_2)^2+(d_3)^2.$[]{data-label="fig:dividepolytope"}](thridpolytope.pdf "fig:"){width=".5\textwidth"}![The figure on the left shows the portion of the moment polytope, $P_6$, where $d_1>d_2, d_3$. The figure on the left shows the portion of the moment polytope where additionally $(d_1)^2>(d_2)^2+(d_3)^2.$[]{data-label="fig:dividepolytope"}](polytopeobtuse.pdf "fig:"){width=".5\textwidth"}
Equilateral, Left-Handed, Positive Curl, Hexagonal Trefoils
-----------------------------------------------------------
In this section, we will discuss constraints for $H$ to be a left-handed hexagonal trefoil with positive curl.
\[discL+\] Let $H\in Equ_0(6)$ and parametrize $H$ with action-angle coordinates arising from the $T_{135}$ triangulation. If $J(H)=(-1,1)$, then $f_i<0$, $g_i>0$, and $h_i>0$, for all $i$, where
$$\begin{aligned}
f_1= & d_2\sqrt{4-(d_2)^2}\text{sin}(\theta_2)\big(d_3d-((d_1)^2-(d_2)^2+(d_3)^2)\sqrt{4-(d_3)^2}\text{cos}(\theta_3)\big)-\\& d_3\sqrt{4-(d_3)^2}\text{sin}(\theta_3)\big(d_2d-((d_1)^2+(d_2)^2-(d_3)^2)\sqrt{4-(d_2)^2}\text{cos}(\theta_2)\big),\\
g_1= & \sqrt{4-(d_2)^2}\Big(\frac{-(d_1)^2+(d_2)^2+(d_3)^2}{2d_2 d_3}\text{cos}(\theta_2)\text{sin}(\theta_3)+\text{sin}(\theta_2)\text{cos}(\theta_3)\Big)\\ &-\frac{d\text{sin}(\theta_3)}{2d_3},\\
h_1= & \sqrt{4-(d_3)^2}\Big(\frac{-(d_1)^2+(d_2)^2+(d_3)^2}{2d_2 d_3}\text{cos}(\theta_3)\text{sin}(\theta_2)+\text{sin}(\theta_3)\text{cos}(\theta_2)\Big) \\ &-\frac{d\text{sin}(\theta_2)}{2d_2},\\
f_2 = & d_3\sqrt{4-(d_3)^2}\text{sin}(\theta_3)\big(d_1d-((d_1)^2+(d_2)^2-(d_3)^2)\sqrt{4-(d_1)^2}\text{cos}(\theta_1))- \\& d_1\sqrt{4-(d_1)^2}\text{sin}(\theta_1)\big(d_3d-(-(d_1)^2+(d_2)^2+(d_3)^2)\sqrt{4-(d_3)^2}\text{cos}(\theta_3)\big),\\
g_2 = &\sqrt{4-(d_3)^2}\Big(\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3}\text{cos}(\theta_3)\text{sin}(\theta_1)+\text{sin}(\theta_3)\text{cos}(\theta_1)\Big) \\ &-\frac{d\text{sin}(\theta_1)}{2d_1},\\
h_2 = &\sqrt{4-(d_1)^2}\Big(\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3}\text{cos}(\theta_1)\text{sin}(\theta_3)+\text{sin}(\theta_1)\text{cos}(\theta_3)\Big) \\ &-\frac{d\text{sin}(\theta_3)}{2d_3},\\
f_3 = & d_1\sqrt{4-(d_1)^2}\text{sin}(\theta_1)\big(d_2d-(-(d_1)^2+(d_2)^2+(d_3)^2)\sqrt{4-(d_2)^2}\text{cos}(\theta_2)\big)-\\&d_2\sqrt{4-(d_2)^2}\text{sin}(\theta_2)\big(d_1d-((d_1)^2-(d_2)^2+(d_3)^2)\sqrt{4-(d_1)^2}\text{cos}(\theta_1)\big),\\
g_3 = &\sqrt{4-(d_1)^2}\Big(\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}\text{cos}(\theta_1)\text{sin}(\theta_2)+\text{sin}(\theta_1)\text{cos}(\theta_2)\Big)\\ & -\frac{d\text{sin}(\theta_2)}{2d_2},\\
h_3 = &\sqrt{4-(d_2)^2}\Big( \frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}\text{cos}(\theta_2)\text{sin}(\theta_1)+\text{sin}(\theta_2)\text{cos}(\theta_1)\Big) \\ & -\frac{d\text{sin}(\theta_1)}{2d_1}.\end{aligned}$$
[*Proof:* ]{} Let $H\in Equ_0(6)$ and parametrize $H$ using action-angle coordinates for the $T_{135}$ triangulation. If $curl(H)=1$, then $\theta_i\in (0,\pi)$ for all $i$. Additionally, if $H$ has Joint Chirality-Curl $(-1,1)$ the all algebraic intersection numbers $\Delta_i$ must be negative. First we will consider that condition that $\Delta_4=-1$, meaning the algebraic intersection of $T_4$ with $H$ is $-1$. If $\theta_2\in (0,\pi)$ this means that $e_1$ must pierce $T_4$. Therefore the line going through $v_1$ and $v_2$ must pass through $T_4$ and the following three inequalities must be satisfied: $$\begin{aligned}
(v_2-v_1)\times(v_4-v_1)\centerdot(v_3-v_1)>0,\\
(v_2-v_1)\times(v_5-v_1)\centerdot(v_4-v_1)>0,\\
(v_2-v_1)\times(v_3-v_1)\centerdot(v_5-v_1)>0.\end{aligned}$$
Since $\theta_i\in(0,\pi)$, then $(v_2-v_1)\times(v_3-v_1)\centerdot(v_5-v_1)$ is always positive. Suppose $H$ is in standard position. Evaluating the remaining two expressions with action-angle coordinates from the $T_{135}$ triangulation results in $h_3>0$ and $f_3<0$.
Additionally, the plane containing $v_3, v_4$, and $v_5$ must separate $v_1$ and $v_2$. Therefore the following must be negative: $$\big( (v_2-v_3)\times(v_5-v_3)\centerdot(v_4-v_3)\big)\centerdot\big( (v_1-v_3)\times(v_5-v_3)\centerdot (v_4-v_3)\big)<0.$$ This constraint is equivalent to $g_3>0$.
Similarly, the conditions that the $\Delta_2=-1$ and $\Delta_6=-1$ are equivalent to $f_2<0, g_2>0, h_2>0$ and $f_3<0, g_3>0, h_3>0$, respectively. Therefore if $H\in Equ_0(6)$ is in standard position and has Joint Chirality-Curl $(-1,1)$, then $f_i<0$, $g_i>0$, and $h_i>0$ for all $i$. [$\square$]{}\
Next we define possible dihedral angles for an equilateral hexagon to have Joint Chirality-Curl $(-1,1)$.
\[angles(-1,1)\] Let $H\in Equ_{0}(6)$ and parametrize $H$ using action-angle coordinates with the $T_{135}$ triangulation. If $H$ has Joint Chirality-Curl $(-1,1)$, then $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$ and $\theta_{1}+\theta_{2}<\pi$, $\theta_{1}+\theta_{3}<\pi$, and $\theta_{2}+\theta_{3}<\pi$.
[*Proof:* ]{}Let $H\in Equ_{0}(6)$ be parametrized with action-angle coordinates $(d_1,d_2,d_3,\theta_1,\theta_2,\theta_3)$ arising from the $T_{135}$ triangulation. If $H$ has Joint Chirality-Curl $(-1,1)$, then from Lemma \[curl1\] we know $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$. If $\theta_i\in(0,\frac{\pi}{2})$ for all $i\in\{1,2,3\}$, then clearly $\theta_{1}+\theta_{2}<\pi$, $\theta_{1}+\theta_{3}<\pi$, and $\theta_{2}+\theta_{3}<\pi$. Additionally if $\theta_i, \theta_j\in(0,\frac{\pi}{2})$, for any two distinct $i,j\in\{1,2,3\}$, then $\theta_i+\theta_j< \pi$.
Next we will show that if $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$ and $H$ has Joint Chirality-Curl $(-1,1)$, then $\theta_{1}+\theta_{2}<\pi$ and $\theta_{1}+\theta_{3}<\pi$. Towards a contradiction, suppose that $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, and $\theta_1+\theta_2= \pi$. Substituting $\theta_1=\pi-\theta_2$ into equation $g_3$ from Proposition \[discL+\] and using the facts that $\text{cos}(\pi-\theta_2)=-\text{cos}(\theta_2)$ and $\text{sin}(\pi-\theta_2)=\text{sin}(\theta_2)$, we obtain the following $$\begin{aligned}
g_3&=\sqrt{4-(d_1)^2}\Big(-\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}+1\Big)\text{sin}(\theta_2)\text{cos}(\theta_2)-\frac{d\text{sin}(\theta_2)}{2d_2}.\end{aligned}$$
We make the same substitutions into equation $h_3$ to obtain
$$\begin{aligned}
h_3&=\sqrt{4-(d_2)^2}\Big(\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1\Big)\text{sin}(\theta_2)\text{cos}(\theta_2)-\frac{d\text{sin}(\theta_1)}{2d_1}.\end{aligned}$$
If $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1\ge0$, then $g_3$ is negative. Fix $d_1,d_2, d_3,$ and $\theta_2$, and now consider $g_3$ as a function of $\theta_1$. Since $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1\ge0$, then $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}>0$. This implies that the derivative of $g_3$ with respect to $\theta_1$, $$\sqrt{4-(d_1)^2}\Big(\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}(-\text{sin}(\theta_1))\text{sin}(\theta_2)+\text{cos}(\theta_1)\text{cos}(\theta_2)\Big),$$ is negative for $\theta_1\in(\frac{\pi}{2},\pi)$ and $\theta_2\in(0,\frac{\pi}{2})$. Since $g_3$ is negative for $\theta_1=\pi-\theta_2$ and $g_3$ is decreasing, $g_3$ is negative for all $\theta_1>\pi-\theta_2$. Next suppose $\frac{(d_1)^2+(d_2)^2-(d_3)^2}{2d_1 d_2}-1<0$, then $h_3$ is negative. Thus $h_3$ is negative for $\theta_1>\pi-\theta_2$. Since both $g_3$ and $h_3$ must be positive for $H$ to have Joint Chirality-Curl $(-1,1)$, we have reached a contradiction. Therefore if $\theta_1\in(0,\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, and $\theta_1+\theta_2\ge \pi$, $H$ can not have Joint Chirality-Curl $(-1,1)$.
Now suppose that $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, and $\theta_1+\theta_3= \pi$. Similar to the previous argument, we will substitute $\theta_1=\pi-\theta_3$ into equation $g_2$ and use the facts that $\text{cos}(\pi-\theta_3)=-\text{cos}(\theta_3)$ and $\text{sin}(\pi-\theta_3)=\text{sin}(\theta_3)$. This results in the following: $$\begin{aligned}
g_2&=\sqrt{4-(d_3)^2}\Big(\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} -1\Big)\text{cos}(\theta_3)\text{sin}(\theta_3)-\frac{d\text{sin}(\theta_1)}{2d_1}.\end{aligned}$$
Making the same substitutions into $h_2$ gives $$\begin{aligned}
h_2&=\sqrt{4-(d_1)^2}\Big(-\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} +1\Big)\text{cos}(\theta_3)\text{sin}(\theta_3)-\frac{d\text{sin}(\theta_3)}{2d_3}.\end{aligned}$$
If $\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} -1\le 0$, then $g_2$ is negative. If $\frac{(d_1)^2-(d_2)^2+(d_3)^2}{2d_1 d_3} -1> 0$, then $h_2$ is negative. Since both equations must be positive to have Joint Chirality-Curl $(-1,1)$, we have reached a contradiction. Therefore if $H$ is a left-handed trefoil with $curl(H)=1$ and $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2, \theta_3\in(0,\frac{\pi}{2})$, then $\theta_{1}+\theta_{2}<\pi$ and $\theta_{1}+\theta_{3}<\pi$.
Again the cases when $\theta_2\in(\frac{\pi}{2}, \pi)$, $\theta_1, \theta_3\in(0,\frac{\pi}{2})$ and $\theta_3\in(\frac{\pi}{2}, \pi)$, $\theta_1, \theta_2\in(0,\frac{\pi}{2})$ are proven in the same manner. Thus if $H$ has Joint Chirality-Curl $(-1,1)$, then $\theta_{i}\in (0,\pi)$ for all $i\in{1,2,3}$ and $\theta_{1}+\theta_{2}<\pi$, $\theta_{1}+\theta_{3}<\pi$, and $\theta_{2}+\theta_{3}<\pi$.
[$\square$]{}\
Next we consider the case where the three diagonals of the $T_{135}$ triangulation are distinct.\
\[polytopeportionL+\] Let $H\in Equ_{0}(6)$ and parametrize $H$ using action-angle coordinates with the $T_{135}$ triangulation. Suppose $d_{1}, d_{2},$ and $d_{3}$ are distinct and let $d_{i}>d_{j}, d_{k}$. If $J(H)=(-1,1)$ then $\theta_{i}\in(0,\pi)$ and $\theta_{j},\theta_{k}\in(0,\pi/2)$. Moreover, if $d_{i}>\sqrt{(d_{j})^2+(d_{k})^2}$ then $\theta_{i}\in(\pi/2,\pi)$.\
[*Proof:* ]{}Let $H\in Equ_{0}(6)$ be in standard position so that $v_1$, $v_3$, and $v_5$ are on the $xy$-plane. Suppose that the lengths of the diagonals are distinct and that $d_2>d_1>d_3$. We will show that if $H$ has Joint Chirality-Curl $(-1,1)$ then $\theta_2\in (0,\pi)$ and $\theta_1,\theta_3\in(0,\frac{\pi}{2})$. Let $l_2$ be perpendicular bisector to the segment connecting $v_1$ and $v_3$. Similarly, we define $l_4$ and $l_6$ to be the perpendicular bisectors to segments connecting $v_3$ to $v_5$ and $v_5$ to $v_1$, respectively. The three lines intersect in a unique point, $k$, the circumcenter of the triangle spanned by $(v_1, v_3, v_5)$. The orthogonal projection of $v_i$ onto the $xy$-plane lies on $l_i$. In addition, $k$ is the orthogonal projection of where all vertices coincide, if such point exists.
Since $d_1>d_3$ then $l_4$ intersects the segment connecting $v_1$ to $v_3$ instead of the segment connecting $v_1$ and $v_5$. Suppose towards contradiction that $\theta_3\in(\frac{\pi}{2},\pi)$. Then the plane perpendicular to the $xy$-plane containing $l_4$ separates $e_3$ and $T_6$. Therefore $H$ can not have Joint Chirality-Curl $(-1,1)$ if $\theta_3\in(\frac{\pi}{2},\pi)$.
Next suppose that $\theta_1\in(\frac{\pi}{2},\pi)$. Since $d_2>d_1$ the $l_6$ intersects the segment connecting $v_3$ and $v_5$. This means the plane perpendicular to the $xy$-plane containing $l_6$ separates $e_5$ and $T_2$. Therefore we have reached a contradiction and $\theta_1\in (0,\frac{\pi}{2})$.
Let $\psi_3$ be the angle for $\theta_3$ for which $v_6$ projects onto $k$. If $\theta_3\in(\psi_3,\pi)$, then the plane perpendicular to the $xy$-plane containing $l_4$ still separates $e_3$ and $T_6$. Thus if $H$ has Joint Chirality-Curl $(-1,1)$, then $\theta_3\in(0,\psi_3)$. Let $\psi_1$ be the angle of $\theta_1$ so that $v_1$ projects onto $k$. If $e_5$ is to intersect $T_2$, then $\theta_1\in(0,\psi_1)$. Let $\psi_2$ be the angle for $\theta_2$ for which $v_4$ projects onto $k$. Since $\theta_1\in (0,\psi_1)$ and $\theta_3\in (0,\psi_3)$ then $\theta_2\in (\psi_2,\pi)$ for $H$ to have Joint Chirality-Curl $(-1,1)$. If $d_2>\sqrt{(d_1)^2+(d_3)^2}$ then the triangle spanned by $(v_1, v_3, v_5)$ is obtuse. Therefore $k$ is exterior of the triangle spanned by $(v_1, v_3, v_5)$ and $\psi_2>\frac{\pi}{2}$. Hence if $d_2>\sqrt{(d_1)^2+(d_3)^2}$ then $\theta_1,\theta_3\in (0,\frac{\pi}{2})$ and $\theta_2\in (\frac{\pi}{2},\pi)$. [$\square$]{}\
Knotting Probability of Hexagonal Trefoils
==========================================
In this section, we will discuss the probability that a random equilateral hexagon is knotted. It has been proven that at least $\frac{1}{3}$ of hexagons with total length $2$ are unknotted[@JC]. Using action-angle coordinates and Calvo’s geometric invariant $curl$, Cantarella and Shonkwiler [@JC] prove that at least $\frac{1}{2}$ of the space of equilateral hexagons consists of unknots. In order to gain intuition on the tightness of these bounds, we performed a Monte Carlo experiment. We randomly sampled a point in the moment polytope $P_6$ and a point in the cube $[0,2\pi]^3$. We then tested whether this point satisfies the necessary constraints to have a hexagonal trefoil with Joint Chirality-Curl $(1,1)$, described in Proposition \[disc\]. Taking a sample size of $10$ million configurations, repeating this experiment multiple times, we found that on average the fraction of $(1,1)$ trefoils is $3.426005\times 10^{-5}$ with standard deviation $2.241511 \times 10^{-6}$. Since there are four types of trefoils, we estimate that the knotting probability for equilateral hexagons is about $1.370402 \times 10^{-4}$. Using the lemmas from the previous section, we improve the theoretical bound.
The probability that an equilateral hexagon is knotted is at most $\frac{14-3\pi}{192}$.
[*Proof:* ]{}Let $H \in Equ_0(6)$. We will choose the $T_{135}$ triangulation of $H$ to form our set of action-angle coordinates: $\alpha:P_6\times T^{3}\mapsto Pol_0(6)$. Since almost all of $Pol_0(6)$ is a toric symplectic manifold, Theorem \[DH\] holds for integrals over this space. First we will calculate the expected value for $H=(d_1,d_2,d_3,\theta_1,\theta_2, \theta_3)$ to have $curl=1$. Suppose that $H$ is in general position so that the lengths of the diagonals are distinct. Without loss of generality, we assume that $d_1$ is the largest of the three diagonals. The moment polytope, $P_6$, corresponding to the $T_{135}$ triangulation is $\frac{1}{2}$ of the cube $[0,2]^3$. Therefore the volume of $P_6$ is $4$. The region where one diagonal is greater than the other two divides the moment polytope into $3$ regions with equal volume of $\frac{4}{3}$. From Lemma \[polytopeportionR+\] and Lemma \[polytopeportionL+\] if $d_1>d_2,d_3$ and $curl(H)=1$, then $\theta_1\in(0,\pi)$, $\theta_2\in(0,\frac{\pi}{2})$, and $\theta_3\in(0,\frac{\pi}{2})$. Additionally, if $(d_1)^2>(d_2)^2+(d_3)^2$, then $\theta_1\in(\frac{\pi}{2},\pi)$. From Lemma \[angles\] and Lemma \[angles(-1,1)\], we know that $\theta_1+\theta_2<\pi$ and $\theta_1+\theta_3<\pi$, shown in Figure \[torussmallest\]
![The figure on the left shows the portion of the moment polytope $P_6$ where $d_1>d_2,d_3$ and $(d_1)^2>(d_2)^2+(d_3)^2$. The figure on the right shows the portion of cube $[0,2\pi]^3$ where $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2,\theta_3\in(0,\frac{\pi}{2})$, $\theta_1+\theta_2<\pi$, and $\theta_1+\theta_3<\pi$.[]{data-label="torussmallest"}](polytopeobtuse.pdf "fig:"){width=".5\textwidth"}![The figure on the left shows the portion of the moment polytope $P_6$ where $d_1>d_2,d_3$ and $(d_1)^2>(d_2)^2+(d_3)^2$. The figure on the right shows the portion of cube $[0,2\pi]^3$ where $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2,\theta_3\in(0,\frac{\pi}{2})$, $\theta_1+\theta_2<\pi$, and $\theta_1+\theta_3<\pi$.[]{data-label="torussmallest"}](torussmallestportion.pdf "fig:"){width=".45\textwidth"}
Using standard integration, we calculate that the volume of the portion of $P_6$ where $d_1>d_2$, $d_1>d_3$, and $(d_1)^2>(d_2)^2+(d_3)^2$ is equal to $\frac{2(\pi-2)}{3}$. The ratio of this volume out of the third of $P_6$ is $\frac{\pi}{2}-1$. The region where $\theta_1\in(\frac{\pi}{2},\pi)$, $\theta_2\in(0,\frac{\pi}{2})$, $\theta_3\in(0,\frac{\pi}{2})$, $\theta_1+\theta_2<\pi$ and $\theta_1+\theta_3<\pi$ is $\frac{1}{192}$ of the cube $[0,2\pi]^3$.
The portion of $P_6$ where $d_1>d_2,d_3$ and $(d_1)^2<(d_2)^2+(d_3)^2$ is $2-\frac{\pi}{2}$ of the volume of $\frac{1}{3}$ of $P_6$. The region where $\theta_1\in(0,\pi)$, $\theta_2\in(0,\frac{\pi}{2})$, $\theta_3\in(0,\frac{\pi}{2})$, $\theta_1+\theta_2<\pi$ and $\theta_1+\theta_3<\pi$, shown in Figure \[torussmall\], is $\frac{1}{48}$ of the cube $[0,2\pi]^3$.
![The figure on the left shows the portion of the moment polytope $P_6$ where $d_1>d_2,d_3$ and $(d_1)^2<(d_2)^2+(d_3)^2$. The figure on the right shows the portion of cube $[0,2\pi]^3$ where $\theta_1\in(0,\pi)$, $\theta_2,\theta_3\in(0,\frac{\pi}{2})$, $\theta_1+\theta_2<\pi$, and $\theta_1+\theta_3<\pi$.[]{data-label="torussmall"}](polytopeacute.pdf "fig:"){width=".5\textwidth"}![The figure on the left shows the portion of the moment polytope $P_6$ where $d_1>d_2,d_3$ and $(d_1)^2<(d_2)^2+(d_3)^2$. The figure on the right shows the portion of cube $[0,2\pi]^3$ where $\theta_1\in(0,\pi)$, $\theta_2,\theta_3\in(0,\frac{\pi}{2})$, $\theta_1+\theta_2<\pi$, and $\theta_1+\theta_3<\pi$.[]{data-label="torussmall"}](torussmallportion.pdf "fig:"){width=".45\textwidth"}
Therefore the expected value for $curl(H)=1$ is bounded above by $$\Big(\frac{\pi}{2}-1\Big)\Big(\frac{1}{192}\Big)+\Big(2-\frac{\pi}{2}\Big)\Big(\frac{1}{48}\Big)=\frac{7}{192}-\frac{\pi}{128}.$$ Making a similar argument for $curl(H)=-1$, we see that the knot probability is at most $$2(\frac{7}{192}-\frac{\pi}{128})=\frac{14-3\pi}{192},$$ as desired. [$\square$]{}\
Acknowledgements
================
The author would like to thank Ken Millett for his guidance on this project while at the University of California, Santa Barbara. The author would also like to thank Jorge Calvo, Jason Cantarella, and Clay Shonkwiler, whose work inspired this project.
|
---
abstract: 'We describe vacuum fluctuations and photon-field correlations in interacting quantum mechanical light-matter systems, by generalizing the application of mixed quantum-classical dynamics techniques. We employ the multi-trajectory implementation of Ehrenfest mean field theory, traditionally developed for electron-nuclear problems, to simulate the spontaneous emission of radiation in a model quantum electrodynamical cavity-bound atomic system. We investigate the performance of this approach in capturing the dynamics of spontaneous emission from the perspective of both the atomic system and the cavity photon field, through a detailed comparison with exact benchmark quantum mechanical observables and correlation functions. By properly accounting for the quantum statistics of the vacuum field, while using mixed quantum-classical (mean field) trajectories to describe the evolution, we identify a surprisingly accurate and promising route towards describing quantum effects in realistic correlated light-matter systems.'
author:
- 'Norah M. Hoffmann'
- Christian Schäfer
- Angel Rubio
- Aaron Kelly
- Heiko Appel
bibliography:
- 'HKSRA18.bib'
date:
-
-
title: 'Capturing Vacuum Fluctuations and Photon Correlations in Cavity Quantum Electrodynamics with Multi-Trajectory Ehrenfest Dynamics'
---
Introduction {#sec:intro}
============
Profound changes in the physical and chemical properties of material systems can be produced in situations where the quantum nature of light plays an important role in the interaction with the system [@Feist2015; @Schachenmayer2015; @Cirio2016]. A few notable recent examples of such effects are few-photon coherent nonlinear optics with single molecules [@maser2016], direct experimental sampling of electric-field vacuum fluctuations [@riek2015; @moskalenko2015], multiple Rabi splittings under ultra-strong vibrational coupling [@george2016], exciton-polariton condensates [@byrnes2014; @kasprzak2006], and frustrated polaritons [@Schmidt2016]. These exciting developments have been strongly driven by experimental efforts, thus exposing the immediate need for the development and improvement of theoretical approaches that can bridge the gap between quantum optics and quantum chemistry [@ruggenthaler2018].
Due to the similarity of the electron-photon and the electron-nuclear problems, simulation methods that have traditionally been of use in the quantum chemistry community, such as semiclassical and mixed quantum-classical methods, offer a potentially interesting avenue to bridge this gap. In particular, the family of trajectory-based quantum-classical methods has the advantage of providing a very intuitive, qualitative understanding of nonadiabatic molecular dynamics. Further, these techniques typically do not exhibit the pernicious exponential scaling of computational effort inherent in grid-based quantum calculations[@Thoss2004]. Available techniques in this family of exact and approximate approaches are Ehrenfest mean field dynamics, fully linearized and partially linearized path integral methods, forward-backward trajectory methods [@Hsieh2012; @he12; @SKR18], and trajectory surface-hopping algorithms [@KM13]. All these techniques have some ability to describe essential quantum mechanical effects such as tunnelling, interference, zero-point energy conservation.
Recently, Subotnik and co-workers have performed investigations of light-matter interactions where an adjusted Ehrenfest theory based method is used to simulate spontaneous emission of classical light [@CLSNS18; @CLSNS218; @LNSMCS18]. Here, in contrast with these works, we focus on the description of quantized light fields. We then generalize the well established multi-trajectory Ehrenfest method to treat quantum mechanical light-matter interactions. We highlight the possibilities and theoretical challenges of this method in comparison to the exact treatment of the quantum system, by applying this approach to investigate spontaneous emission for a model atom in an optical cavity.
The remainder of this work is divided into three sections: in Sec.\[sec:theory\] we briefly review general interacting light-matter systems, and the multi-trajectory Ehrenfest dynamics method. In this framework, we then introduce a one-dimensional model system comprising a single (two or three level) atomic system coupled to a multi-mode quantum electrodynamical (QED) cavity. In Sec.\[sec:results\] we investigate the performance of multi-trajectory Ehrenfest (MTEF) dynamics in describing the process of spontaneous emission. We conclude our results in Sec.\[sec:conc\] and discuss some prospects for future work.
Theory {#sec:theory}
======
Quantum Mechanical Light-Matter Interactions
--------------------------------------------
To begin, we describe a general coupled field-matter system using Coulomb gauge and the dipole approximation [@Faisal1987; @Flick2017a]. The total Hamiltonian for the system is [@Tokatly2013; @Flick2017; @Pellegrini2015; @Flick2015; @Craig1998] $$\hat{H} = \hat{H}_{A} + \hat{H}_{F} + \hat{H}_{AF}.
\label{G9}$$ The first term, $\hat{H}_{A}$, is the atomic Hamiltonian, which may be generally expressed in the spectral representation, $$\hat{H}_{A} = \sum_k \varepsilon_k | k \rangle \langle k |.$$ Here $\{\varepsilon_k,|k\rangle\}$ are the atomic energies and stationary states of the atomic system in absence of coupling to the cavity. The second term is the Hamiltonian of the uncoupled cavity field $\hat{H}_{F}$, $$\hat{H}_{F} = \frac{1}{2}\sum_{\alpha = 1}^{2N} \left(\hat{P}^{2}_{\alpha} + \omega^{2}_{\alpha}\hat{Q}_{\alpha}^{2} \right).
\label{G12}$$ The photon-field operators, $\hat{Q}_{\alpha}$ and $\hat{P}_{\alpha}$, obey the canonical commutation relation, $[\hat{Q}_{\alpha},\hat{P}_{\alpha'}] = \imath\hbar\delta_{\alpha,\alpha'}$, and can be expressed using creation and annihilation operators for each mode of the cavity field, $$\begin{aligned}
\hat{Q}_{\alpha} &=& \sqrt{\frac{\hbar}{2\omega_{\alpha}}}(\hat{a}^{+}_{\alpha} + \hat{a}_{\alpha}), \\
\hat{P}_{\alpha} &=& i\sqrt{\frac{\hbar\omega_{\alpha}}{2}}(\hat{a}^{+}_{\alpha} - \hat{a}_{\alpha}),\end{aligned}$$ where $\hat{a}^\dagger_\alpha$ and $\hat{a}_\alpha$ denote the usual photon creation and annihilation operators for photon mode $\alpha$. The coordinate-like operators, $\hat{Q}_{\alpha}$, are directly proportional to the electric displacement operator, while the conjugate momenta-like operators, $\hat{P}_{\alpha}$, are related to the magnetic field [@Pellegrini2015; @Flick2015]. The upper limit of the sum in Eq.(\[G12\]) is $2N$, as there are (in principle) two independent polarization degrees of freedom for each photon mode, however in the 1D cavity models presented here only a single polarization will be considered.
The final term in Eq.(\[G9\]) represents the coupling between the atom and the cavity field, $$\hat{H}_{AF} = \sum_{\alpha=1}^{2N}\Big(\omega_{\alpha}\hat{Q}_{\alpha}(\lambda_{\alpha}\cdot \hat{\mu}) + \frac{1}{2}\Big(\lambda_{\alpha} \cdot \hat{\mu} \Big)^2 \Big),$$ where we denote ${\hat{\mu}}$ as the electronic dipole moment vector of the atomic system, and ${\lambda}_{\alpha}$ as the electron-photon coupling vector [@Tokatly2013; @Flick2015]. In the case of a two-level electronic system the quadratic term in the atom-field coupling Hamiltonian simply results in a constant energy shift and hence has no effect on observables [@SRR18], and we neglect this term in the case of the three level model system. Furthermore, we note that this Hamiltonian can easily be extended to include nuclear degrees of freedom, however this has been omitted in the present work.
![Model atomic system in an electromagnetic cavity: Atom (green) trapped between two mirror-like surfaces of the cavity, supporting $2N$ photon modes with frequencies $\omega_{\alpha} = \frac{2\pi \hbar c \alpha}{L}$, where $\alpha = \{1,2, ...,2N\}$ and $L$ is the distance between the mirrors. The strength of the interactions between each mode of the cavity field and the atomic system is $\lambda_{\alpha}$.[]{data-label="AB1"}](cavity_mode_3d.pdf){width="0.5\linewidth"}
Multi-Trajectory Ehrenfest Dynamics
-----------------------------------
In this section we apply the well-known multi-trajectory Ehrenfest method, traditionally introduced to study electron-nuclear systems [@Ehrenfest1927; @Makri1987; @C.Tully1998], to coupled light-matter systems[@McLachlan1964; @C.Tully1998; @Flick2017a].
A particularly simple and instructive route to derive the MTEF mean field theory is via the quantum-classical Liouville (QCL) equation[@qcle]. This equation of motion for the density matrix is formally exact for an arbitrary quantum mechanical system that is bilinearly coupled to a harmonic environment, as is the case in the atom-field Hamiltonian studied here. The QCL equation can be written in a compact form as $$\begin{aligned}
\label{eq:qcle}
&& \frac{\partial }{\partial t}\hat{\rho}_W (X, t) =-i {\mathcal L}\hat{\rho}_W (X,t).\end{aligned}$$ It describes the time evolution of $\hat{\rho}_W(X,t)$, which is the partial Wigner transform of the density operator taken over the photon field coordinates, which are thus represented by continuous phase space variables $X=(Q,P)=(Q_1,Q_2,...,Q_{2N},P_1,P_2,...,P_{2N})$. The partial Wigner transform of the density operator, $\hat{\rho}$, is defined as $$\begin{aligned}
\label{eq:wigner}
\hat{\rho}_{W}(Q,P) = \frac{1}{(2\pi\hbar)^{2N}}\int dZ e^{i P \cdot Z} \langle Q - \frac{Z}{2} | \hat{\rho} | Q +\frac{Z}{2}\rangle. \end{aligned}$$ The QCL operator is defined as $$\label{eq:qcl_op}
i{\mathcal L} \cdot = \frac{i}{\hbar}[\hat{H}_W,\cdot] - \frac{1}{2}(\{\hat{H}_W,\cdot\}
-\{\cdot,\hat{H}_W\}),$$ where $\hat{H}_{W}$ denotes the Wigner transform of $\hat{H}$, $[\cdot,\cdot]$ is the commutator, and $\{\cdot,\cdot\}$ is the Poisson bracket in the phase space of the environmental variables.
At this point, one may arrive at MTEF equations by assuming that the total density of the system can be written as an uncorrelated product of the atomic and photonic reduced densities at all times, $$\hat{\rho}_W(X,t)=\hat{\rho}_A(t) \rho_{F,W}(X,t),$$ where the reduced density matrix of the atomic system is $$\label{eq:rdm}
\hat{\rho}_{A} (t) = \text{Tr}_F \Big( \hat{\rho}(t) \Big) = \int dX \hat{\rho}_W (X,t),$$ and the Wigner function of the cavity field is $\rho_{F,W}(X, t) = Tr_A ( \hat{\rho}_W (X,t))$. If one seeks solutions to the QCL equation of this form, the Ehrenfest mean-field equations of motion for the atomic system are obtained: $$\frac{d}{dt} \hat{\rho}_A(t) = -i\Big[ \hat{H}_A + \hat{H}_{AF,W}(X(t)), \hat{\rho}_A(t)\Big],$$ where $\hat{H}_{AF,W}$ denotes the Wigner transform of $\hat{H}_{AF}$. The evolution of the Wigner function of the photon field can be represented as a statistical ensemble of independent trajectories, with weights $w^j$, $$\rho_{F,W} (X,t) = \sum_j^{N_{traj}} w^j \delta(X-X^j(t))$$ that evolve according to Hamilton’s equations of motion, $$\frac{\partial Q_{\alpha}}{\partial t} = \frac{\partial H_{F,W}^{Eff}}{\partial P_{\alpha}},\quad \frac{\partial P_{\alpha}}{\partial t} = - \frac{\partial H_{F,W}^{Eff}}{\partial Q_{\alpha}}.$$ The effective photon field Hamiltonian is, $$H^{Eff}_{F,W} = \frac{1}{2}\sum_{\alpha}\Big( P_{\alpha}^2 + \omega_{\alpha}^2 Q_{\alpha}^2 - 2 \omega_{\alpha} \lambda_{\alpha} Q_{\alpha} \mu (t) \Big),$$ where $ \mu (t) = \text{Tr}_A(\hat{\rho}_A(0) \hat{\mu}(t))$.
The exact expression for the average value of any observable, $\langle O (t) \rangle $, can be written as $$\begin{aligned}
\langle O (t) \rangle = \text{Tr}_A \int dX \hat{O}_W(X,t) \hat{\rho}_W(X,t=0).\end{aligned}$$
We note here that for this class of systems the Ehrenfest equations of motion for the photon field coordinates correspond to a mode resolved form of Maxwell’s equations. In applying the MTEF dynamics method numerically, we use the above expressions in the following manner:
1. We first perform Monte Carlo sampling from the Wigner transform of the initial density operator of the photon field $\hat{\rho}_{F,W}(X,0)$ to generate an ensemble of initial conditions, for the trajectory ensemble $(Q_{\alpha}^j(0),P_{\alpha}^j(0))$. In this work we used uniform weights $w^j = \frac{1}{N_{traj}}$, however other importance sampling schemes could be employed as the only requirement is that the sum of the weights is normalized, $\sum_j w^j = 1$.
2. We evolve each initial condition independently according to the Ehrenfest equations of motion, producing a trajectory. In the following we refer to such a solution as an ensemble of independent trajectories.
3. Average values are constructed by summing over the entire trajectory ensemble, and normalizing the result with respect to $N_{traj}$, the total number of trajectories in the ensemble,\
$\langle O (t) \rangle = \sum_j^{N_{traj}} \text{Tr}_A \Big( \hat{O}_W(Q^j,P^j,t)\hat{\rho}_A(0)\Big)/N_{traj}$.
Here $\rho_{F,W}(X,0)$ is Wigner transform of the zero temperature vacuum state $$\rho_{F,W}(X,0)=\prod_{\alpha} \frac{1}{\pi}\exp{\left[-\frac{P^{2}_{\alpha}}{\omega_{\alpha}} - \omega_{\alpha} Q_{\alpha}^2\right]}.
\label{G20}$$
Observables and Normal Ordering
-------------------------------
Before we proceed with a discussion of our simulation results, we must note that the Wick normal ordered form for operators (denoted $:\hat{O}:$ for some operator $\hat{O}$) is used when calculating average values in this study. The reason for using the normal ordered form, in practice, is to remove the effect of vacuum fluctuations from the results, which ensures that both $\langle E \rangle = 0$ and $\langle I \rangle = 0$, irrespective of the number of photon modes in the cavity field, when the field is in the vacuum state. The effect of this operator ordering is particularly evident for the photon number operator, $$:\hat{N}_{pt}: = \frac{1}{2}\sum_{\alpha}\Big(\frac{\hat{P}^{2}_{\alpha}}{\omega_{\alpha}} + \omega_{\alpha}\hat{Q}^{2}_{\alpha} -1\Big),
\label{G5}$$ where normal ordering produces a constant shift due to the zero-point energy term.
The quantized electric field operator is defined as $$\hat{E}(r,t) = \sum_{\alpha}\sqrt{2\omega_{\alpha}} \zeta_{\alpha}(r) \hat{Q}_{\alpha}(t)$$ with $$\zeta_{\alpha}(r) = \sqrt{\frac{\hbar \omega_{\alpha}}{\epsilon_{0} L}} \sin\Big(\frac{\alpha \pi}{L} r\Big).$$ The corresponding normal-ordered electric field intensity operator is given by $$:\hat{E}^2(r,t): = :\hat{I}(r,t): = 2\sum_{\alpha} \omega_{\alpha}\zeta^{2}_{\alpha}(r)\hat{Q}_{\alpha}^{2}(t) - \sum_{\alpha}\zeta^{2}_{\alpha}(r).
\label{G15}$$ The effect of normal ordering on this quantity is shown in Fig. \[AB2\], where the intensity of the electric field is plotted in both its canonical and normal ordered forms. In addition to a constant shift with respect to the normal ordered quantity, which is identically zero, the canonical average field intensity also displays additional oscillations near the boundaries and the atomic position, corresponding to the vacuum fluctuations for this system.
[ ![Average value of the cavity electric field intensity. Wick normal ordering has been applied to the operator in the case of the red dashed line, whereas the solid black line corresponds to the original operator. The cavity field is prepared in the vacuum state, at zero temperature.[]{data-label="AB2"}](normal_vs_notnormal.pdf "fig:"){width="\columnwidth"}]{}
We also consider the second order correlation function for the photon field [@Glauber], $$:g^{2}(r_{1},r_{2},t): = \frac{ \langle : \hat{E}^{+}(r_{1},t) \hat{E}^{+}(r_{2},t)\hat{E}(r_{2},t) \hat{E}(r_{1},t) : \rangle }{ \langle : \hat{I}(r_{1},t) : \rangle \quad \langle : \hat{I}(r_{2},t) : \rangle }.
\label{G14}$$ This function is frequently used in quantum optics to discriminate between classical light and non-classical states of the photon field that exhibit photon bunching $(g^2 > 1)$ or photon anti-bunching $(g^2 < 1)$. The normal-ordered form of the numerator in $g^2$, also referred to $G^{2}(r_{1},r_{2},t)$, is $$\begin{aligned}
&:&G^{2}(r_{1},r_{2}, t): = 4\sum_{\alpha} \omega_{\alpha}^2\zeta_{\alpha}(r_{1})\zeta_{\alpha}(r_{2})\zeta_{\alpha}(r_{2})\zeta_{\alpha}(r_{1})\hat{Q}_{\alpha}^{4}(t) \nonumber \\ \nonumber
&-& \sum_{\alpha \beta}\Big(4\zeta_{\beta}(r_{1})\zeta_{\beta}(r_{2})\zeta_{\alpha}(r_{1})\zeta_{\alpha}(r_{2}) +\zeta^{2}_{\beta}(r_{2})\zeta^{2}_{\alpha}(r_{1}) \nonumber \\
& + &\zeta^{2}_{\beta}(r_{1})\zeta^{2}_{\alpha}(r_{2})\Big)\cdot2\omega_{\alpha}\hat{Q}^{2}_{\alpha}(t) . \label{G19}\end{aligned}$$
The partial Wigner transforms of the polynomial functions of the bath coordinate operators are simply polynomial functions of the continuous bath coordinates, $\Big(\hat{Q}_{\alpha}^n(t)\Big)_W = (Q_{\alpha}(t))^n$ [@Hillery]. The same is also true for the corresponding momenta, and thus the average values of the preceding operators can be easily calculated using mean-field trajectories.
Model System
------------
Following previous work[@Buzek; @Flick2017], we investigate a model atomic system in a one-dimensional electromagnetic cavity, as depicted in Fig 1. $$\begin{aligned}
\hat{H} = &\sum_{k=1}^m\epsilon_{k}\ket{k}\bra{k} + \frac{1}{2}\sum_{\alpha}^{2N}\Big(\hat{P}^{2}_{\alpha}+ \omega^{2}_{\alpha}\hat{Q}^{2}_{\alpha}\Big) \\ \nonumber +&\sum_{\alpha}^{2N}\sum_{k,l =1}^{m}\mu_{kl}\omega_{\alpha}\lambda_{\alpha}(r_A)\hat{Q}_{\alpha} \ket{k}\bra{l},
\label{G1}\end{aligned}$$ where the upper limit of the first and last summation $m$ denotes the number of atomic energy levels. In the case of a two-level atomic system, this corresponds to a special case of the spin-boson model. With the position of the atom fixed at $r_A = \frac{L}{2}$ in this study, half of the $2N$ cavity modes decouple from the atomic system by symmetry. We adopt the same parameters as in Ref. [@Flick2017; @Su1991], which are based on a 1D Hydrogen atom with a soft Coulomb potential (in atomic units): $\{\varepsilon_1,\varepsilon_2\} = \{-0.6738, -0.2798\}$, $\lambda_{\alpha}(\frac{L}{2}) = 0.0103\cdot(-1)^{\alpha}$, $L = 2.362\cdot 10^{5}$ and $\mu_{12} = 1.034$. For the three-level atom, we adopt all the same parameters for the field and the atom-field coupling as for the two-level case. The atomic energies for the three level model are $\{\varepsilon_1,\varepsilon_2,\varepsilon_3\} = \{-0.6738, -0.2798, -0.1547\}$ and as before the numerical parameters are based on the 1D soft-Coulomb Hydrogen atom. The dipole moment operator only couples adjacent states, such that, the only nonzero matrix elements are $\{\mu_{12},\mu_{23}\} = \{1.034,-2.536\}$ and their conjugates.
Results and Discussion {#sec:results}
======================
We now investigate the performance of the MTEF method in the context of cavity-bound spontaneous emission. In all calculations shown below we use 400 photon modes to represent the cavity field. We choose the atom to be initially in the excited state, and the cavity field is in the vacuum state at zero temperature. In all simulations reported here we use an ensemble of $N_{traj} = 10^4$ independent trajectories, sampled from the Wigner transform of the initial field density operator given in the previous section. This level of sampling is sufficient to converge the atomic observables to graphical accuracy, however observables and correlations functions of the photon-field would require a slightly larger trajectory ensemble for graphical convergence. All observables shown below correspond to their normal ordered forms. For our benchmark numerical treatment we solved the time-dependent Schrödinger equation by using a truncated Configuration Interaction (CI) expansion. More precisely, the photon field state-space is truncated at two excitations per photon mode, whereas for the atomic system a two and three state discrete variable representation is used in each case. Numerical convergence is checked to ensure that the CI basis that we employ is complete for the models and parameter regimes studied in this work.
Two Level Atom: One Photon Emission Process {#Se3a}
-------------------------------------------
[ ![Time-evolution of the average field intensity for the one-photon emission process, at four different time snapshots: (a) $t = 100 ~a.u.$, (b) $t = 600 ~a.u.$, (c) $t = 1200 ~a.u.$, (d) $t = 2100 ~a.u.$. (e) Zoom-in of the polariton-peak at the atomic position. Exact simulation results (black) and MTEF dynamics (green).[]{data-label="AB3"}](2-level-exact-traj-intensity.pdf "fig:"){width="\linewidth"}]{}
[ ![ Time evolution of the atomic state populations (top panel), and the total photon number (bottom panel). Top panel: Solid lines represent the atomic ground state, and dashed lines represent the excited state. Both panels: Exact simulation results (black) and MTEF (green).[]{data-label="AB4"}](2-level-population.pdf "fig:"){width="\linewidth"}]{}
[ ![Second order correlation function for the photon field, $g^{2}(r_{1},r_{2},t)$ for the two-level model, plotted at four time snapshots: (a) $t = 100 ~a.u$., (b) $t = 600 ~a.u.$, (c) $t = 1200 ~a.u.$, (d) $t = 2100 ~a.u.$. Exact (left panels), and MTEF (right panels).[]{data-label="AB5"}](2level-g2-exact.png "fig:"){width="1\linewidth"}]{}
[ ![Associated one-dimensional diagonal cuts $g^2(r_{\pm},t)$ of the second order correlation function, exact (black) and MTEF (green), plotted at four time snapshots:(a) $t = 100 ~a.u.$, (b) $t = 600 ~a.u.$, (c) $t = 1200 ~a.u.$, (d) $t = 2100 ~a.u.$.[]{data-label="AB6"}](2-diag-g2.png "fig:"){width="\linewidth"}]{}
In Fig. \[AB3\] we show the intensity of the cavity field along the axis of the cavity, at four different times. As the spontaneous emission process proceeds, a photon wavepacket with a sharp front is emitted from the atom and travels toward the boundaries where it is reflected, and then travels back to the atom (e.g. panel (c) of Fig. 3). The emitted photon is then absorbed and re-emitted by the atom, which results in the emergence of interference phenomena in the electric field. This produces a photonic wavepacket with a more complex shape (e.g. panel (d) of Fig. 3). We observe that the MTEF simulations capture the qualitative character of the spontaneous emission process extremely accurately, as well as the wavepacket propagation through the cavity. However, MTEF dynamics fails to reproduce the interference phenomena in the field due to re-emission. We do note, however, that the MTEF simulations are capable of describing the remaining field intensity at the atomic position (e.g panel (e) of Fig. 3). This feature corresponds to a bound electron-photon state, or polariton, which is an emergent hybrid state of the correlated light-matter system.
We also plot the excited state population of the atomic system, and the average value of the photon number for the field, in Fig. \[AB4\]. Again, MTEF is able to capture the qualitative behaviour of both of these quantities very nicely. However, it fails to quantitatively reproduce the correct values for the emitted photon number and atomic population transfer, as these quantities are underestimated. Furthermore, as a result of this loss in accuracy, only a part of the subsequent re-excitation and re-emission processes are captured.
In Fig. \[AB5\] we investigate the normalized second order correlation function, $g^2(r_1,r_2,t)$ for the cavity photon field. The unperturbed vacuum state, which is coherent, corresponds to $g^{2}(r_{1},r_{2},t) = 1$, given by the black background seen in Fig. \[AB5\]. The vacuum state is disturbed by the emitted wavepacket, corresponding to anti-bunched light with $g^{2}(r_{1},r_{2},t)<1$. The simplicity of the one-dimensional/one-photon process is quite clear in Fig. \[AB6\], where we show the associated one-dimensional cuts of $g^2$, along with projections of $g^2(r_1,r_2,t)$ along the positive and negative diagonals, $r_\pm = (r_1\pm r_2)/\sqrt{2}$. Here we find similar to the intensity a nice qualitative agreement between MTEF and the exact result for the first three time snapshots. However for the last time-snap-shot the exact solution shows a broader correlation than MTEF, which corresponds to the fact that MTEF is not able to accurately capture re-emission. Furthermore, as we only consider a one-photon process in this case, the correlation is symmetric in $r_{+}$ and $r_{-}$.
Three Level Atom: Two Photon Emission Process {#Se3b}
---------------------------------------------
We now investigate the three-level system, for the same observables as the previous section. The initial state for the atomic system is now the second excited state. The photonic initial state remains the zero temperature vacuum state.
[ ![Time-evolution of the average field intensity for the two-photon emission process, at time four different snapshots: (a) $t = 100 ~a.u.$, (b) $t = 600 ~a.u.$, (c) $t = 1200 ~a.u.$, (d) $t = 2100 ~a.u.$. (e) Zoom-in of the polariton-peak at the atomic position. Exact (black) and MTEF (green).[]{data-label="AB7"}](3-level-exact-traj-intensity.pdf "fig:"){width="1\linewidth"}]{}
In Fig. \[AB7\] we show the intensity of the cavity field during the two-photon emission process. Similar dynamics are observed compared with the two-level case. However, due to the additional intermediate atomic state, we now observe a double-peak feature in the emitted photonic wavepacket. This feature corresponds to the emission of two photons, as the excited atom initially drops to the first excited state emitting one photon, and then further relaxes to the ground state, emitting a second photon. The polariton peak (the central feature in the field intensity profile) is overestimated in the MTEF simulations. This overestimation is due to the incomplete relaxation of the second excited state within the Ehrenfest description.
[ ![Top panel: Time evolution of the atomic state populations: Solid line ($m=3$), dashed lines ($m=2$), and dotted line ($m=1$). Bottom panel: Total photon number as a function of time. Exact simulation results (black) and MTEF (green).[]{data-label="AB8"}](3-level-population.pdf "fig:"){width="1\linewidth"}]{}
In Fig. \[AB8\], we show the time evolution of the atomic state populations and total photon number. Again, the emitted photonic wavepacket moves through the cavity, is reflected at the mirrors, and returns to the atom. The first and second excited state are then repopulated due to stimulated absorption. A second spontaneous emission process ensues, and the emitted field again takes on a more complex profile due to interference. For the intensity, as well as the atomic population and photon number, we observe that MTEF displays qualitatively correct short-time dynamics. However it fails to describe the correct spatial structure of the (re)emitted two photon wavepacket, as well as the correct amplitude for the observables, in accordance with what was observed previously in the two-level case.
[ ![Second order correlation function for the photon field, $g^{2}(r_{1},r_{2},t)$ for the three-level model, plotted at four time snapshots: (a) $t = 100 ~a.u.$, (b) $t = 600 ~a.u.$, (c) $t = 1200 ~a.u.$, (d) $t = 2100 ~a.u.$. Exact (left panels) and MTEF (right panels).[]{data-label="AB9"}](3level-g2-exact.png "fig:"){width="0.98\linewidth"}]{}
[ ![Associated one-dimensional diagonal cuts $g^2(r_{\pm},t)$ of the second order correlation function, exact (black) and MTEF (green), plotted at four time snapshots: (a) $t = 100 ~a.u.$, (b) $t = 600 ~a.u.$, (c) $t = 1200 ~a.u.$, (d) $t = 2100 ~a.u.$.[]{data-label="AB10"}](3-diag-g2.png "fig:"){width="1\linewidth"}]{}
In Fig. \[AB9\] we show $g^2(r_1,r_2,t)$ for the two-photon emission process. The energy level spacing in the three-level truncation of the 1D soft-Coulomb hydrogen atom is uneven, such that the two emitted photons are of different frequencies. Hence, in contrast to the one-photon process, we expect to observe asymmetric features in the second order correlation function. In the exact result we observe that the vacuum state is locally disturbed by a structured, anti-bunched photon wavepacket. The fine, multi-lobed, spatial structure of the photon wavepacket is blurred into a single, rather narrow, feature in the MTEF result. However, MTEF dynamics indeed show the correct spatial asymmetry that is expected in $g^2(r_1,r_2,t)$. In the corresponding one-dimensional cuts of $g^{2}(r_{1},r_{2},t)$, shown in Fig. \[AB10\], we show in further detail the comparison of MTEF dynamics and the exact results in this more complex two-photon case.
Summary and Outlook {#sec:conc}
===================
In this work we have adapted the multi-trajectory Ehrenfest method (MTEF) to simulate correlated quantum mechanical light-matter systems. We applied this mixed quantum-classical dynamics method, which is traditonally applied to electron-nuclear dynamics problems, to two and three level model QED cavity bound atomic systems, and in order to simulate observables and correlation functions for both the atomic system, and the photon field. We find that MTEF dynamics is able to qualitatively characterize the correct dynamics for one and two photon spontaneous emission processes in a QED cavity. However, MTEF dynamics does suffer from some quantitative drawbacks. Furthermore, we also observed that MTEF dynamics simulations can, in fact, capture some quantum mechanical features such as bound polariton states and second order photon correlations. Moreover, as experimental advances drive the need for realistic descriptions of light-matter coupled systems, trajectory-based quantum-classical algorithms emerge as promising route towards treating more complex and realistic systems. In particular, combining the *ab initio* light-matter coupling methodology recently presented in Jestädt et. al. [@JRORA18] with our multi-trajectory approach, provides a computationally feasible way to simulate photon-field fluctuations and correlations in realistic three-dimensional systems. Work along these lines is already in progress. Furthermore, an alternative to the independent trajectory-based approach employed here is the conditional wavefunction approach, which allows one address nonadiabatic dynamics problems in complex systems with higher accuracy than MTEF dynamics[@Guille18], and opens up an interesting potential route for mixed quantum-classical methods in correlated light-matter systems.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Niko Säkkinen and Johannes Flick for insightful discussions and acknowledge financial support from the European Research Council (ERC-2015-AdG-694097). AK acknowledges support from the National Sciences and Engineering Research Council (NSERC) of Canada.
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---
abstract: 'We fix a monic polynomial $f(x) \in \FF_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto \omega(a)$ be the Teichmüller lift of $\FF_q$, and let $\chi:\ZZ\to \CC_p^\times$ be a finite character of $\ZZ_p$. The $L$-function associated to the polynomial $f$ and the so-called twisted character $\omega^u\times \chi$ is denoted by $L_f(\omega^u,\chi,s)$ (see Definition \[L-function\]). We prove that, when the conductor of the character is large enough, the $p$-adic Newton slopes of this $L$-function form arithmetic progressions.'
address: 'Rufei Ren, University of California, Irvine, Department of Mathematics, 340 Rowland Hall, Irvine, CA 92697'
author:
- Rufei Ren
title: 'Newton slopes for twisted Artin–Schreier–Witt Towers'
---
Introduction
============
In [@Davis-wan-xiao], Davis, Wan, and Xiao studied the $L$-function $L_f(\chi,s)$ associated to a polynomial $f(x)\in \FF_q[x]$ and a finite character $\chi:\ZZ_p\to \CC_p^\times$. They proved that
- If $\chi_1$ and $\chi_2$ are two finite characters with the same conductor $p^m\geq \frac{ap(d-1)^2}{8d}$, then $L_f(\chi_1,s)$ and $L_f(\chi_2,s)$ have the same Newton polygons.
- Let $\chi_0:\ZZ_p\to \CC_p^\times$ be a fixed character with conductor $p^{\lceil\log_p \frac{ap(d-1)^2}{8d}\rceil}.$ Then the $p$-adic Newton slopes of $L_f(\chi_1,s)$ (see Definition \[Newton slopes\]) form a disjoint union of arithmetic progressions determined by the $p$-adic Newton slopes of $L_f(\chi_0,s)$.
In [@bfz], Blache, Ferard, and Zhu studied the so-called twisted $L$-functions (see Definition \[L-function\]), whose $p$-adic Newton polygons satisfy a universal lower bound proved by C.Liu and W.Liu in [@liu-liu]. This lower bound is similar to the one given in [@Davis-wan-xiao]. Therefore, it is of interest to ask if the $p$-adic Newton slopes of the twisted $L$-functions also form arithmetic progressions. In this paper, we give an upper bound for the twisted $L$-function and prove that it coincides with the lower bound at $x=kd$ for any integer $k\geq0$. As a consequence, we prove that its $p$-adic Newton slopes indeed form arithmetic progressions.
We start with introducing some basic setups. We fix a prime number $p$. Let $\FF_q$ be the finite field of $q=p^a$ elements. Let $$\begin{aligned}
\omega:&\FF_q\to\ZZ_q\\
&\alpha\ \mapsto \omega(\alpha):=\hat \alpha\end{aligned}$$ be the Teichmüller lift of $\FF_q$. For any $u\in\{0,1,\cdots, q-1\}$, we put $$\begin{aligned}
\omega^u: &\FF_q\to\ZZ_q\\
&\alpha\ \mapsto \omega^u(\alpha)=\hat \alpha^u.\end{aligned}$$ We fix a monic polynomial $f(x) = x^d +a_{d-1}x^{d-1} + \cdots + a_0 \in \FF_q[x]$ of degree $d$ which is coprime to $p$. Set $ a_d=1$ and put $\hat a_i: = \omega( a_i)$ for $i=0, \dots, d$. The *Teichmüller lift* of the polynomial $f(x)$ is defined by $\hat f(x):=x^d + \hat a_{d-1}x^{d-1} + \cdots + \hat a_0 \in \ZZ_q[x].$
Let $u$ be an integer in the set $\{0,1,\dots,q-2\}$ and put $$u=\sum\limits_{i=0}^{a-1} u(i)p^i,$$ where $0\leq u(i)\leq p-1$ for any $0\leq i\leq a-1$.
\[L-function\] Let $\chi:\ZZ_p\to \CC_p^\times$ be a finite character with conductor $p^{m_\chi}$. The *twisted $L$-function* associated to the characters $\chi$ and $\omega^u$ is defined by $$L_f(\omega^u,\chi,s)= \prod\limits_{x \in |\GG_{m,\FF_q}|}
\frac{1}{1-\omega^u\circ{\mathrm{Norm}}_{\FF_{q^{\deg(x)}}/\FF_q} (x)\cdot\chi\Big({\mathrm{Tr}}_{\QQ_{q^{\deg(x)}}/\QQ_{p}}\big(\hat f(\hat x)\big)\Big)s^{\deg(x)}},$$ where $\GG_{m,\FF_q}$ is the one-dimensional torus over $\FF_q$ and $\deg(x)$ stands for the degree of $x$. By [@liu-wei], the $L$-function $L_f(\omega^u,\chi,s)$ is a polynomial of degree $dp^{m_\chi-1}$.
For simplicity of notations, we denote $$y_u(k):=\frac{ak(k-1)(p-1)}{2d}+\frac k d\sum\limits_{i=0}^{a-1}u(i).$$
\[Newton slopes\] We call the slopes of the line segments of the $p$-adic Newton polygon of $L_f(\omega^u, \chi, s)$ the *$p$-adic Newton slopes* of $L_f(\omega^u, \chi, s)$.
In this paper, we prove the following.
\[main\]
- The $p$-adic Newton polygon of $L_f(\omega^u,\chi, s)$ passes through the points $$\big(kd, \frac{y_u(kd)}{(p-1)p^{m_\chi-1}}\big)\quad \textrm{for any}\ 0\leq k\leq p^{m_\chi-1}.$$
- The $p$-adic Newton polygon of $L_f(\omega^u, \chi, s)$ has slopes (in increasing order) $$\bigcup_{k=1}^{p^{m_\chi-1}} \{\alpha_{k 1},\alpha_{k 2},\dots,\alpha_{k d}\},$$ where $$\frac{a(k-1)}{p^{m_{\chi}-1}}+\frac{\sum\limits_{i=0}^{a-1}u(i)}{d(p-1)p^{m_\chi-1}}\leq \alpha_{k j}\leq \frac{a(k-1)}{p^{m_{\chi}-1}}+\frac{\sum\limits_{i=0}^{a-1}u(i)}{d(p-1)p^{m_\chi-1}}+
\frac{a(d-1)}{dp^{m_\chi-1}}$$
for any $1\leq j\leq d$.
When the conductor $p^{m_\chi}$ of $L_f(\omega^u,\chi, s)$ is large enough, the $p$-adic Newton slopes of $L_f(\omega^u,\chi, s)$ have the following property.
\[strong\] Let $m_0$ be the minimal positive integer such that $p^{m_0}> \frac{adp}{8(p-1)}$ and let $0\leq \alpha_1,\alpha_2,\dots,\alpha_{p^{m_0-1}}<a$ denote the slopes of the $p$-adic Newton polygon of $L_f(\omega^u,\chi_0,s)$ for a finite character $\chi_0:\ZZ_p\to \CC_p^\times$ with $m_{\chi_0} = m_0$. Then for any finite character $\chi:\ZZ_p\to \CC_p^\times$ with $m_\chi\geq m_0$, the $p$-adic Newton polygon of $L_f(\omega^u,\chi,s)$ has slopes $$\bigcup_{i=0}^{p^{m_\chi-m_0}-1}\Big\{\frac{\alpha_1+ai}{p^{m_\chi-m_0}},\frac{\alpha_2+ai}{p^{m_\chi-m_0}},\dots, \frac{\alpha_{dp^{m_0-1}}+ai}{p^{m_\chi-m_0}}\Big\}.$$
Theorem \[strong\] says that when $m_\chi$ is large enough, the $p$-adic Newton slopes of $L_f(\omega^u,\chi, s)$ form a disjoint union of arithmetic progressions determined by the $p$-adic Newton slopes of $L_f(\omega^u,\chi_0, s)$.
This paper is inspired by the $p$-adic Newton slopes of $L_f(\chi,s)$ in arithmetic progressions (proved in [@Davis-wan-xiao]), the twisted decomposition of $$L_{f(x^{q-1})}(\chi, s):=\prod_{u=0}^{q-2} L_f(\omega^u,\chi,s)$$ in [@bfz], and the lower bound for the Newton polygon of $L_{f}(\omega^u,\chi, s)$ given in [@liu-liu]. Let $$\calC_0\to \calC_1\to\cdots\to \calC_m\to\cdots$$ be the Artin–Schreier–Witt curve tower associated to the polynomial $f(x^{q-1})$, and let $Z(\calC_m, s)$ be the zeta function of the curve $\calC_m$. It is known that $$\begin{cases}
L_{f}(\omega^u,\chi, s)&\textrm{if}\ u\neq 0\\
\frac{L_{f}(\omega^u,\chi, s)}{1-\chi(\hat f(0))}&\textrm{if}\ u= 0
\end{cases}$$ are factors of $Z(\calC_m, s)$, and the degree of $L_f(\omega^u,\chi_0,s))$ is $\frac{1}{q-1}$ of the degree of $L_f(\chi_0,s))$. Therefore, as a corollary of Theorem \[strong\], we give a more precise description of zeros of $Z(\calC_m, s)$ than the one given in [@Davis-wan-xiao].
Acknowledgments {#acknowledgments .unnumbered}
---------------
The author thanks Dennis A. Eichhorn, Karl Rubin, Daqing Wan and Liang Xiao for many valuable discussions and suggestions.
Notation
========
In this section, we introduce some notations that we will use through out the paper.
We write $v_{T}(-)$ for the $T$-adic valuation of elements in $\CC_p[T]$ and $v_{p}(-)$ for the $p$-adic valuation of elements in $\CC_p$.
\[Newton polygon\] Given a set $S:=\big\{(k,d_k)\;\big|\;0\leq k\leq n\big\}$. The *Newton polygon* of $S$, denoted by $\operatorname{NP}(S)$, is the lower convex hull of points in $S$. We call $n$ *the length* of $\operatorname{NP}(S)$.
For a power series $F(s)=\sum\limits_{i=0}^n \mathbbm{u}_i(T)s^i$, we put $$\operatorname{NP}_T(F):=\operatorname{NP}\Big(\Big\{(k,v_T(\mathbbm u_k))\;\big|\;0\leq k\leq n\Big\}\Big),$$ where $n\in \ZZ_{\geq 0}\cup\infty$.
\[definition of R\] For a Newton polygon $\operatorname{NP}$, we write $R(\operatorname{NP})$ for the multiset of slopes in $\operatorname{NP}$.
It has an inverse, denoted by $R^{-1}$, mapping a multiset $\SS$ to the lower convex whose slopes coincide with this multiset.
\[properties for NP\]
- Let $\SS_1$ and $\SS_2$ be two multisets in $\QQ$. We denote by $$\SS_1\uplus\SS_2$$ the union of $\SS_1$ and $\SS_2$ as multisets.
- For any two Newton polygons $\operatorname{NP}_1$ and $\operatorname{NP}_2$, we write $$\operatorname{NP}_1\oplus\operatorname{NP}_2$$ for the Newton polygon whose slopes are the union of the slopes of $\operatorname{NP}_1$ and $\operatorname{NP}_2$.
- We denote by $\operatorname{NP}(i)$ the height of $\operatorname{NP}$ at $x=i$.
- For any $t\in \QQ$, we denote $t+\operatorname{NP}$ be the Newton polygon such that $$(t+\operatorname{NP})(k)=\operatorname{NP}(k)+t\quad\textrm{for any}\ 0\leq k\leq n,$$ where $n$ is the length of $\operatorname{NP}$.
Let $\operatorname{NP}_1$ and $\operatorname{NP}_2$ be two polygons of same length $n$. If $$\operatorname{NP}_1(k)\geq NP_2(k)$$ holds for any $0\leq k\leq n$, then we call that $\operatorname{NP}_1$ is above $\operatorname{NP}_2$ and denote this by $\operatorname{NP}_1\geq \operatorname{NP}_2$.
\[sum inequality\] If $\Big\{\operatorname{NP}_{1,i}\;\big|\;1\leq i\leq m\Big\}$ and $\Big\{\operatorname{NP}_{2,i}\;\big|\;1\leq i\leq m\Big\}$ are two sets of Newton polygons such that for any $0\leq i\leq m$,
- $\operatorname{NP}_{1,i}$ and $\operatorname{NP}_{2,i}$ have the same length, and
- $\operatorname{NP}_{1,i}\geq\operatorname{NP}_{2,i}$,
then $$\bigoplus\limits_{i=1}^m \operatorname{NP}_{1,i}\geq \bigoplus\limits_{i=1}^m \operatorname{NP}_{2,i}.$$
It follows directly from the definition of “$\oplus$”.
For any positive integer $\ell$, the sum $$S_{f,\omega^u,\chi}(\ell,s)=\sum_{x\in\FF^\times_{q^\ell}}\omega^u\circ{\mathrm{Norm}}_{\FF_{q^\ell}/\FF_q} (x)\cdot(1+T)^{\Big({\mathrm{Tr}}_{\QQ_{q^{\ell}}/\QQ_{p}}\big( \hat f(\hat x)\big)\Big)}\in \ZZ_q[\![T]\!]$$ is called a twisted $T$-adic exponential sum of $f(x)$.
The $L$-function $L_f(\omega^u,T,s)$ (see Definition \[L-function\]) satisfies $$L_f(\omega^u,T,s)= \exp(\sum_{\ell=1}^{\infty}S_{f,\omega^u,\chi}(\ell,s)\frac{s^\ell}{\ell}).$$ It is easy to check $$\label{specialization of L}
L_f(\omega^u,T,s)\big|_{T=\chi(1)-1}=L_f(\omega^u,\chi,s).$$
\[core lemma\] If we put $g(x):=f(x^{q-1})$, then $$\label{core eq}
L_g(\omega^0,T,s)=\prod_{u=0}^{q-2} L_f(\omega^u,T,s).$$
Put $$(1+T)^{\Big({\mathrm{Tr}}_{\QQ_{q^{\ell}}/\QQ_{p}}\big(\hat f(\hat x)\big)\Big)}:=\sum_{n=0}^\infty b_{\ell, n}(T)\hat x^n\in \ZZ_p[\![T]\!][\![\hat x]\!],$$ where $b_{\ell, n}(T)\in \ZZ_p[\![T]\!]$ for all $n\geq 0$ and $\ell\geq 1$.
Notice that for each $u\in\{0,1,\dots,q-2\}$, we have $$S_{f,\omega^u,\chi}(\ell,s)=(q^\ell-1)\sum_{n=1}^\infty b_{\ell, (q^\ell-1)(n-\frac{u}{q-1})}(T).$$ Therefore, by taking the sum of $u$ over the set $\{0,1,\dots, q-2\}$, we get $$\begin{aligned}
\sum_{u=0}^{q-2}S_{f,\omega^u,T}(\ell,s)=&(q^\ell-1)\sum_{u=0}^{q-2}\sum_{n=1}^\infty b_{\ell, (q^\ell-1)(n-\frac{u}{q-1})}(T)\\=&(q^\ell-1)\sum_{n=1}^\infty b_{\ell, \frac{n(q^\ell-1)}{q-1}}(T).
\end{aligned}$$
On the other hand, by definition, it is easy to check that $$S_{g,\omega^0,T}(\ell,s)=(q^\ell-1)\sum_{n=1}^\infty b_{\ell, \frac{n(q^\ell-1)}{q-1}}(T).$$Therefore, we have $$S_{g,\omega^0,T}(\ell,s)=\sum_{u=0}^{q-2}S_{f,\omega^u,T}(\ell,s),$$ for all $\ell\geq 1$, which implies $$L_g(\omega^0,T,s)=\prod_{u=0}^{q-2} L_f(\omega^u,T,s).\qedhere$$
The *characteristic power series* of $f$ is given by $$C_f(\omega^u,T,s):=\prod_{i=0}^{\infty}L_f(\omega^u,T,q^is),$$ which is known as a $p$-adic entire power series.
By Lemma \[core lemma\], we know that $$C_g(\omega^0,T,s)=\prod_{u=0}^{q-2} C_f(\omega^u,T,s).$$
We denote by $\operatorname{NP}_f(L,\omega^u,T)$ (resp. $\operatorname{NP}_f(L,\omega^u,\chi)$) the $T$-adic Newton polygon (resp. $p$-adic Newton polygon) of $L_f(\omega^u,T,s)$ (resp. $L_f(\omega^u,\chi,s)$).
Similarly, we write $\operatorname{NP}_f(C,\omega^u,T)$ and $\operatorname{NP}_f(C,\omega^u,\chi)$ for the $T$-adic Newton polygon (resp. $p$-adic Newton polygon) of $C_f(\omega^u,T,s)$ and $C_f(\omega^u,\chi,s)$ respectively.
The T-adic Dwork’s Trace Formula
================================
In this section, we recall properties of the $L$-function associated to a $T$-adic exponential sum as considered by Liu and Wan in [@liu-wan]. Its specializations to appropriate values of $T$ interpolate the $L$-functions considered above.
We first recall that the *Artin–Hasse exponential series* is defined by $$\label{Artin-Hasse}
E(\pi) = \exp\big( \sum_{i=0}^\infty \frac{\pi^{p^i}}{p^i} \big) = \prod\limits_{p \nmid i,\ i \geq 1} \big( 1-\pi^i\big)^{-\mu(i)/i} \in 1+ \pi + \pi^2 \ZZ_p[\![ \pi ]\!].$$ Setting $T = E(\pi) -1$ defines an isomorphism $\ZZ_p\llbracket \pi \rrbracket \cong \ZZ_p\llbracket T\rrbracket$.
For our given polynomial $f(x) = \sum\limits_{i=0}^d a_i x^i \in \ZZ_q[x]$, we put $$\label{E:Ef(x)}
E_f(x) := \prod\limits_{i=0}^d E(a_i \pi x^i) \in \ZZ_q[\![\pi]\!] [\![ x ]\!].$$
We follow the notation of [@liu-liu]. Set $$C_u:=\{v\in \ZZ_{\geq 0}\;\big|\; v\equiv u\}$$ and $$\textbf{B}_u:=\Big\{\sum_{v\in C_u} b_vT^{\frac{v}{d(q-1)}}x^{\frac{v}{(q-1)}}\;\big|\; b_v\in \ZZ_q[\![T^{\frac{v}{d(q-1)}}]\!]\ \textrm{and}\ v_T(b_v)\to \infty \Big\}.$$
- For two integers $n$ and $m$, we denote by $n\%m$ the residue class of $n$ modulo $m$ in $\{0,1,\dots,m-1\}$.
- Recall $u\in \{0,1,\dots,q-2\}$. We write $b_u|a$ for the minimal positive integer such that $u^{p^{b_u}}\equiv u \pmod q$.
- Denote $$u_i:=(up^i)\% (q-1)\ \textrm{for}\ i=0,\dots, b_u-1$$ and put $$\widetilde{\textbf{B}}_u=\bigoplus_{i=0}^{b_u-1} \textbf{B}_{u_i}$$ to be the total Banach space associated to $u$.
- Choose a permutation $(i_1,i_2,\dots,i_{b_u})$ of $\{1,2,\dots,b_u\}$ such that the sequence $\{u_{i_n}\}$ is non-decreasing. Put $$\biguplus\limits_{i=0}^{b_{u}-1} C_{u_i}:=\big(c_{u,n}\big)_{n\in \ZZ_{\geq 0}}$$ to be a non-decreasing sequence.
It is easy to check that $$\label{c}
c_{u,n}=(q-1)\lfloor\frac{n}{b_u}\rfloor+u_{i_{(n\%b_u)}}.$$
Let $\psi_p$ denote the operator on $\widetilde{\textbf{B}}_u$ given by $$\psi_p\Big(\sum_{n\geq 0}^\infty d_n(T) x^n\Big): = \sum_{n\geq 0}^\infty d_{pn}( T) x^n,$$ and let $\psi$ be the composite linear operator $$\label{E:psi}
\psi := \sigma\circ\psi_p \circ E_f(x): \widetilde{\textbf{B}}_u \longrightarrow \widetilde{\textbf{B}}_u,$$ where $\sigma$ is the Frobenius automorphism of $\ZZ_q$, and $E_f(x)$ acts on $g\in \widetilde{\textbf{B}}_u$ by $$E_f(x)(g):=E_f(x) \cdot g.$$
By Dwork’s trace formula, we have
The characteristic power series $C_f(\omega^u,T,s)$ satisfies $$C_f(\omega^u,T,s)=\det\Big(1-\psi^as\;\big|\;\textbf{B}_u/\ZZ_q[\![T^{\frac{1}{d(q-1)}}]\!]\Big).$$
By [@liu-liu Lemma 4.2], we have $$\label{ll}
C_f(\omega^u,T,s)^b=\det\Big(1-\psi^as\;\big|\;\widetilde{\textbf{B}}_u/\ZZ_q[\![T^{\frac{1}{d(q-1)}}]\!]\Big).$$
We write $$B=\big(T^{\frac{v}{d(q-1)}}x^{\frac{c_{u,n}}{(q-1)}}\big)_{n\in \ZZ_{\geq 0}}$$ for a basis of $\widetilde{\textbf{B}}_u$ over $\ZZ_q[\![T^{\frac{v}{d(q-1)}}]\!]$ and denote by $N$ the standard matrix of $\psi$ associated to the basis $B$.
It is not hard to check that $N$ is an infinite dimensional nuclear matrix of the form $$\label{E:explicit N}
N=\begin{pmatrix}
*T^{\frac{(p-1)c_{u,0}}{d(q-1)}}&*T^{\frac{(p-1)c_{u,0}}{d(q-1)}}&\cdots& *T^{\frac{(p-1)c_{u,0}}{d(q-1)}}&\cdots\\
*T^{\frac{(p-1)c_{u,1}}{d(q-1)}}&*T^{\frac{(p-1)c_{u,1}}{d(q-1)}}&\cdots& *T^{\frac{(p-1)c_{u,1}}{d(q-1)}}&\cdots\\
\vdots & \vdots & \ddots&\vdots&\\
*T^{\frac{(p-1)c_{u,n}}{d(q-1)}}&*T^{\frac{(p-1)c_{u,n}}{d(q-1)}}&\cdots& *T^{\frac{(p-1)c_{u,n}}{d(q-1)}}&\cdots\\
\vdots & \vdots & \ddots & \vdots & \ddots
\end{pmatrix}.$$
By [@ren-wan-xiao-yu Corollary 3.9], we know $$\label{dwork}
\det\Big(1-\psi^as\;\big|\;\widetilde{\textbf{B}}_u/\ZZ_q[\![T^{\frac{1}{d(q-1)}}]\!]\Big)= \det\big(I-s \sigma^{a-1}(N) \cdots \sigma(N) N \big).$$
For a matrix $M$, we write $$\left[ \begin{array}{cccccccccc}
m_1 & m_2 &\cdots&m_{k} \\
n_1 & n_2 &\cdots&n_{k} \end{array} \right]_M$$ for the $k\times k$-submatrix formed by elements whose row indices belong to $\{m_1,\dots,m_{k}\}$ and whose column indices belong to $\{n_1,\dots,n_{k}\}$.
\[lower bound\] Let $(t_{ij})_{j\in \ZZ_{\geq 0}}$ be $n$ non-decreasing sequences, and let $M_1,M_2,\dots,M_n$ be $n$ nuclear matrices such that $$M_i={\mathrm{Diag}}(T^{t_{i1}},T^{t_{i2}},\dots)\cdot M_i'\quad \textrm{for any}\ 1\leq i\leq n,$$ where $M_i'$ are infinite matrix whose entries belong to $\ZZ_q[\![T^{\frac{1}{d(q-1)}}]\!]$. Then the $T$-adic Newton polygon $$\operatorname{NP}_T\big(\det(1-M_{n} \cdots M_2 M_1s)\big)\geq \operatorname{NP}\Big(\Big\{(k,\sum_{i=1}^n\sum\limits_{j=1}^{k}(t_{ij}))\;|\; k\geq 0\Big\}\Big).$$
Put $$\det(I-sM_{n} \cdots M_2 M_1):=\sum\limits_{k=0}^\infty (-1)^k \mathbbm r_k(T) s^k .$$ From the definition of characteristic power series, we get $$\label{E:expression of char power series}
\begin{split}
\mathbbm r_k(T)& =\sum\limits_{0\leq m_1<m_2<\cdots<m_{k}<\infty}\det \left[ \begin{array}{cccccccccc}
m_1 & m_2 &\cdots&m_{k} \\
m_1 & m_2 &\cdots&m_{k} \end{array} \right]_{M_{n} \cdots M_2 M_1} \\
&=\sum_{\substack{0\leq m_{1,1}<m_{1,2}<\cdots<m_{1,k}<\infty \\ \cdots\\0\leq m_{n,1}<m_{n,2}<\cdots<m_{n,k}<\infty}} \det\bigg(\prod\limits_{i=1}^{n}\left[ \begin{array}{cccccccccc}
m_{i+1,1} & m_{i+1,2} &\cdots&m_{i+1,k} \\
m_{i,1} & m_{i,2} &\cdots&m_{i,k} \end{array} \right]_{M_i}\bigg) \\
&=\sum_{\substack{0\leq m_{1,1}<m_{1,2}<\cdots<m_{1,k}<\infty \\ \cdots\\0\leq m_{n,1}<m_{n,2}<\cdots<m_{n,k}<\infty}} \prod\limits_{i=1}^{n}\bigg(
\det \left[ \begin{array}{cccccccccc}
m_{i+1,1} & m_{i+1,2} &\cdots&m_{i+1,k} \\
m_{i,1} & m_{i,2} &\cdots&m_{i,k} \end{array} \right]_{M_i}\bigg).
\end{split}$$ Here and after, we set $m_{n+1,i}=m_{1,i}$ for all $1\leq i\leq k$. Since $$v_T\bigg(\det \left[ \begin{array}{cccccccccc}
m_{i+1,1} & m_{i+1,2} &\cdots&m_{i+1,k} \\
m_{i,1} & m_{i,2} &\cdots&m_{i,k} \end{array} \right]_{M_i}\bigg)\geq
\sum_{j=1}^{k}t_{ij},$$ we complete the proof.
\[definition of HP\] The *Hodge polygon* of $C_f(\omega^u,T,s)$, denoted by ${\mathrm{HP}}(d,\omega^u,T)$, is the lower convex hull of set $$\Big\{\Big(k, \frac{a(p-1)}{db_u(q-1)}\sum\limits_{j=0}^{kb_u-1}c_{u,j}\Big)\;\big|\; k\geq 0\Big\}.$$
Each point in $\Big\{(k, \frac{a(p-1)}{db_u(q-1)}\sum\limits_{j=0}^{kb_u-1}c_{u,j})\Big\}$ is a vertex of ${\mathrm{HP}}(d,\omega^u,T).$
It follows that sequence $\Big(\frac{a(p-1)}{db_u(q-1)}\sum\limits_{j=(k-1)b_u}^{kb_u-1}c_{u,j}\Big)_{k\in \ZZ_{\geq 0}}$ is strictly increasing in $k$.
Recall $$u=\sum\limits_{j=0}^{a-1}u(j)p^j\quad \textrm{and}\quad y_u(k)=\frac{ak(k-1)(p-1)}{2d}+\frac{k\sum\limits_{j=0}^{a-1}{u(j)}}{d}.$$
\[computation\] We have $$y_u(k)=\frac{a(p-1)}{db_u(q-1)}\sum\limits_{j=1}^{kb_u}c_{u,j}.$$
From , we know $$\label{sum of c}
\begin{split}
\sum\limits_{j=0}^{kb_u-1}c_{u,j}=&\sum\limits_{j=0}^{k-1}\sum\limits_{\ell=0}^{b_u-1}c_{u,jb_u+\ell}\\=&\sum\limits_{j=0}^{k-1}\Big[jb_u(q-1)+\sum_{i=0}^{b_u-1}u_i\Big]\\
=&\frac{k(k-1)b_u(q-1)}{2}+k\sum_{i=0}^{b_u-1}u_i.
\end{split}$$ Since $$\label{sum of u}\begin{split}
\sum_{i=0}^{b_u-1}u_i=\frac{b_u}{a}(\sum\limits_{i=0}^{a-1}up^i\%(q-1))
= \frac{b_u}{a}(\sum\limits_{j=0}^{a-1}u(j)\frac{q-1}{p-1}),
\end{split}$$ we know $$\begin{split}
\frac{a(p-1)}{db_u(q-1)}\sum\limits_{j=1}^{kb_u}c_{u,j}=&\frac{a(p-1)}{db_u(q-1)}\Big(\frac{k(k-1)b_u(q-1)}{2}+\frac{k(q-1)b_u\sum\limits_{i=0}^{b_u-1}u_i}{a(p-1)}\Big)\\=&\frac{ak(k-1)(p-1)}{2d}+\frac{k\sum\limits_{j=0}^{a-1}u(j)}{d}\\=&y_u(k).\qedhere
\end{split}$$
\[passing some certain points\] The Hodge polygon ${\mathrm{HP}}(d,\omega^u,T)$ passes through the points $$\Big(k, y_u(k)\Big)\quad \textrm{for any}\ k\geq 0.$$
\[HP\] The polygons $\operatorname{NP}_f(C,\omega^u,T)$ and ${\mathrm{HP}}(d,\omega^u,T)$ satisfy $$\operatorname{NP}_f(C,\omega^u,T)\geq {\mathrm{HP}}(d,\omega^u,T).$$
Since matrix $N$ is nuclear as in , its conjugates $\sigma^i(N)$ are also nuclear matrices of the same form. Therefore, applying Lemma \[lower bound\] to the product of these matrices yields $$\operatorname{NP}_T\Big(\det\big(I-s \sigma^{a-1}(N) \cdots \sigma(N) N \big)\Big)\geq \operatorname{NP}\Big(\Big\{\big(k, \frac{a(p-1)}{d(q-1)}\sum\limits_{j=1}^kc_{u,j}\big)\;\big|\;k\geq 0\Big\}\Big).$$
From and , we have $$\begin{aligned}
\operatorname{NP}_f(C,\omega^u,T)\geq& \operatorname{NP}\Big(\Big\{\big(k, \frac{a(p-1)}{d(q-1)}\sum\limits_{j=1}^kc_{u,j}\big)\;\big|\;k\geq 0\Big\}\Big)\\=&{\mathrm{HP}}(d,\omega^u,T).\qedhere
\end{aligned}$$
\[HP for chi\] For any character $\chi:\ZZ_p\to \CC_p^\times$ with conductor $p^{m_\chi}$, we have $$\operatorname{NP}_f(C,\omega^u,\chi)\geq \frac{1}{p^{m_\chi-1}(p-1)}{\mathrm{HP}}(d,\omega^u,T).$$
It simply follows $$\begin{split}
\operatorname{NP}_f(C,\omega^u,\chi)\geq& \frac{1}{p^{m_\chi-1}(p-1)}\operatorname{NP}_f(C,\omega^u,T)\\ \geq&\frac{1}{p^{m_\chi-1}(p-1)}{\mathrm{HP}}(d,\omega^u,T).\qedhere
\end{split}$$
Proof of Theorem \[main\] and Theorem \[strong\]
================================================
In this section, we prove the main theorems.
\[coincide for chi1\]
- The Newton polygon $\operatorname{NP}_f(C,\omega^u,T)$ passes through the points $$\Big(kd,y_u(kd)\Big)\quad \textrm{for all}\ k\geq 0.$$
- If we write $$\label{ru}
C_f(\omega^u,T,s)=\sum\limits_{k=0}^\infty r_{u,k}(T)s^k,$$ then for any $k\geq 0$ and $0\leq u\leq q-2$, the leading term of $r_{u,kd}$ is of the form $$*T^{y_u(kd)},$$ where $*$ represents for a $p$-adic unit.
\[def of UP\] We denote by ${\mathrm{UP}}(d,\omega^u,T)$ the lower convex hull of the points in $$\Big\{(kd,y_u(kd))\;\big|\;k\geq 0\Big\}.$$
\[upper T\] The polygon ${\mathrm{UP}}(d,\omega^u,T)$ forms an upper bound of $\operatorname{NP}_f(C,\omega^u,T)$.
This follows directly from Proposition \[coincide for chi1\] (a).
\[upper chi\] Any finite character $\chi:\ZZ_p\to \CC_p^\times$ with conductor $p^{m_\chi}$ satisfies $$\operatorname{NP}_f(C,\omega^u,\chi)\leq\frac{1}{(p-1)p^{m_\chi-1}}{\mathrm{UP}}(d,\omega^u,T).$$
It follows from Theorem \[coincide for chi1\] (b).
We will give the proof of Proposition \[coincide for chi1\] later.
\[technical lemma\] Let $\operatorname{NP}_1, \operatorname{NP}_2,\dots,\operatorname{NP}_n$ be $n$ Newton polygons. Assume for each $1\leq i\leq n$ there is a rational number $c$ and a vertex $(k_i, y_i)$ of $\operatorname{NP}_i$ such that all segments of $\operatorname{NP}_i$ before this point have slopes strictly less than $c$, while all segments after that point have slopes greater than $c$. Then $\bigoplus\limits_{i=1}^n \operatorname{NP}_i$ passes though the point $$\Big(\sum_{i=1}^{n}k_i, \sum_{i=1}^{n}y_i\Big).$$
The proof follows from the definition of direct sum “$\oplus$” of polygons.
\[specialization\] Any finite character $\chi$ with conductor $p^{m_\chi}$ satisfies $$\label{ineq}
(p-1)p^{m_\chi}\operatorname{NP}_g(C,\omega^0,\chi)\geq \operatorname{NP}_g(C,\omega^0,T).$$
It is enough to show each monomial $aT^i\in \ZZ_q[T]$ satisfy $$v_p(a(\chi(1)-1)^i)\geq v_T(aT^i),$$ which follows $$\begin{split}
(p-1)p^{m_\chi}v_p(a(\chi(1)-1)^i)=&(p-1)p^{m_\chi}(v_p(a)+iv_p(\chi(1)-1))\\=
&(p-1)p^{m_\chi}v_p(a)+i\\\geq&v_T(aT^i).\qedhere
\end{split}$$
Proof of (a). Fix a finite character $\chi_1:\ZZ_p\to \CC_p^\times$ with conductor $p$. By Lemma \[specialization\], we have $$(p-1)\operatorname{NP}_g(C,\omega^0,\chi_1)\geq \operatorname{NP}_g(C,\omega^0,T).$$
By [@Davis-wan-xiao Proposition 3.2], the $p$-adic Newton polygon $\operatorname{NP}_g(C,\omega^0,\chi_1)$ passes through the points $$\Big(kd(q-1), \frac{ak\big((q-1)kd-1\big)(p-1)}{2}\Big)\quad \textrm{for any}\ k\geq 0.$$ Hence, we know that $\operatorname{NP}_g(C,\omega^0,T)$ is not above point $$\Big(kd(q-1), \frac{ak((q-1)kd-1)(p-1)}{2}\Big)\quad \textrm{for any}\ k\geq 0.$$
On the other hand, by Definition \[definition of HP\] and Lemma \[computation\], we have
- For any $0\leq u\leq q-2$ and $k\geq 0$, the point $$\Big(kd,y_u(kd)\Big)$$ is a vertex of ${\mathrm{HP}}(d,\omega^u,T)$.
- All segments of ${\mathrm{HP}}(d,\omega^u,T)$ before this point have slopes strictly less than $ak(p-1)$, while all segments after this point have slopes greater than $ak(p-1)$.
By checking the conditions in Lemma \[technical lemma\], we prove $\bigoplus\limits_{u=0}^{q-2}{\mathrm{HP}}(d,\omega^u,T)$ passes through the points $$\Big(kd(q-1), \sum\limits_{u=0}^{q-2}y_u(kd)\Big).$$
Combining it with Proposition \[HP\] yields that $\operatorname{NP}_g(C,\omega^0,T)$ is not above the points $$\Big(kd(q-1), \sum\limits_{u=0}^{q-2}y_u(kd)\Big)\quad \textrm{for any}\ k\geq 0.$$ Thus, $$\begin{split}
\label{fake inequality}\sum\limits_{u=0}^{q-2}\Big(\frac{ak(kd-1)(p-1)}{2}+k\sum\limits_{i=0}^{a-1}u(i)\Big)=&\sum\limits_{u=0}^{q-2}y_u(kd)\\\leq& \frac{ak((q-1)kd-1)(p-1)}{2}.
\end{split}$$ Now we show that is actually an equality.
Consider $$\begin{split}
\sum\limits_{u=0}^{q-2}\sum\limits_{i=0}^{a-1}u(i)=&-a(p-1)+\sum\limits_{i=0}^{a-1}\sum\limits_{u=0}^{q-1}u(i)\\=&-a(p-1)+\sum\limits_{i=0}^{a-1}q\frac{p-1}{2}\\
=&\frac{aq(p-1)}{2}-a(p-1).
\end{split}$$ Then we simplify the left-hand side of by $$\begin{split}
\sum\limits_{u=0}^{q-2}\Big(\frac{ak(kd-1)(p-1)}{2}+k\sum\limits_{i=0}^{a-1}u(i)\Big)=&(p-1)\Big(\frac{aqk}{2}-ak+\frac{(q-1)ak(kd-1)}{2}\Big)\\
=&\frac{ak((q-1)kd-1)(p-1)}{2},
\end{split}$$ which is equal to its right-hand side. It implies for any $u\in \{0,1,\dots,q-2\}$, the Newton polygon $\operatorname{NP}_f(C,\omega^u,T)$ passes through the points $$\Big(kd,y_u(kd)\Big)\quad \textrm{for any}\ k\geq 0.$$
Proof of (b). From (a), we are able to write $$\label{expression of ruk}
r_{u,kd}(T):=\sum\limits_{i=y_u(kd)}^\infty r_{u,kd,i}T^i,$$ where $r_{u,kd,i}$ belongs to $\calO_{\CC_p}$.
Put $C_g(\omega^0,T,s)=\sum\limits_{n=0}^{\infty}w_n(T)s^{n}.$ From [@Davis-wan-xiao], we know that the leading term of $w_{kd}(T)$ has the form $$*_kT^{\frac{ak((q-1)kd-1)(p-1)}{2}},$$ where $*_k$ is a $p$-adic unit. It is easy to show that $$\prod\limits_{u=0}^{q-2} r_{u,kd,y(kd)}=*_k ,$$ which implies that $r_{u,kd,i}$ are all $p$-adic units.
Now we are ready to prove our main theorems of this paper.
\(a) From , we obtain $$\begin{split}
C_f(\omega^u,T,s)|_{T=\chi(1)-1}=&C_f(\omega^u,\chi,s)\\
=&\sum\limits_{k=0}^\infty r_{u,k}(\chi(1)-1)s^k.
\end{split}$$ Therefore, by Proposition \[coincide for chi1\] (b), the Newton polygon $(p-1)p^{m_\chi-1}\operatorname{NP}_f(C,\omega^u,\chi)$ is not above point $$\Big(kd,y_u(kd)\Big)\quad \textrm{for all}\ k\geq 0.$$
On the other hand, the Hodge polygon ${\mathrm{HP}}(d,\omega^u,T)$ forms a lower bound of $(p-1)p^{m_\chi-1}\operatorname{NP}_f(C,\omega^u,\chi)$ and for all $k\geq 0$ the points $$\Big(kd,y_u(kd)\Big)$$ are also vertices of ${\mathrm{HP}}(d,\omega^u,T)$.
Therefore, the points $$\Big(kd,y_u(kd)\Big)$$ are forced to be the vertices of $(p-1)p^{m_\chi-1}\operatorname{NP}_f(C,\omega^u,\chi)$.
A simple argument about the relation between roots of a power series and its $p$-adic Newton polygon completes the proof.
\(b) Since the slopes of segments of ${\mathrm{HP}}(d,\omega^u,T)$ between $x=d(k-1)$ and $x=dk$ are in the interval $$\Big[ a(k-1)(p-1)+\frac{1}{d}\sum\limits_{i=0}^{a-1}u(i), a(k-1)(p-1)+ \frac{a}{d}(d-1)(p-1)+\frac{1}{d}\sum\limits_{i=0}^{a-1}u(i)
\Big],$$ by simply applying (a), we know that the slopes of segments of $(p-1)p^{m_\chi-1}\operatorname{NP}_f(C,\omega^u,\chi)$ between $x=d(k-1)$ and $x=dk$ also in this interval, which completes the proof of (b).
Recall ${\mathrm{UP}}(d,\omega^u,T)$ is the upper bound of $\operatorname{NP}_f(C,\omega^u,T)$ defined in Notation \[def of UP\].
\[distance\] The vertical distance between points in ${\mathrm{UP}}(d,\omega^u,T)$ and $\operatorname{NP}_f(C,\omega^u,T)$ is bounded above by $\frac{ad(p-1)}{8}$.
By Corollary \[upper T\] and Proposition \[HP\], we know $${\mathrm{UP}}(d,\omega^u,T)\geq \operatorname{NP}_f(C,\omega^u,T)\geq {\mathrm{HP}}(d,\omega^u,T).$$ By Corollary \[passing some certain points\], the polygon ${\mathrm{HP}}(d,\omega^u,T)$ above the parabola $P$ defined by $$P(x):= \frac{ax(x-1)(p-1)}{2d}+\frac{x\sum\limits_{i=0}^{a-1}u(i)}{d}.$$ Since all vertices $(kd,y_u(kd))$ of ${\mathrm{UP}}(d,\omega^u,T)$ coincide with the parabola $P$, by simple calculation, the maximal vertical distance of ${\mathrm{UP}}(d,\omega^u,T)$ and $P$ is equal to $$\max\limits_{k\in \ZZ_{\geq 0}}\Big\{\frac{P(d(k+1))+P(d(k))}{2}-P(d(k+1/2))\Big\}
=\max\limits_{k\in \ZZ_{\geq 0}} \Big\{\frac{ad(p-1)}{8}\Big\}=\frac{ad(p-1)}{8}.\qedhere$$
\[independent\] Let $\chi$ be a finite character with conductor $p^{m_\chi}> \frac{adp}{8}$. Then the Newton polygon $p^{m_\chi}\operatorname{NP}_f(C,\omega^u,\chi)$ is independent of $\chi$.
Recall in we denote $$C_f(\omega^u,T,s)=\sum\limits_{k=0}^\infty r_{u,k}(T)s^k.$$ By Proposition \[HP\] and Corollary \[passing some certain points\], we are able to write $r_{u,k}(T)$ of the form $$r_{u,k}(T)=\sum\limits_{j=y_u(k)}^\infty r_{u,k,i}T^j.$$ Assume that $i(k)$ is the smallest integer such that
- $i(k)\leq {\mathrm{UP}}(d,\omega^u,T)(k)$, where ${\mathrm{UP}}(d,\omega^u,T)(k)$ is the height of ${\mathrm{UP}}(d,\omega^u,T)$ at $x=k$.
- The corresponding coefficient $r_{u,k,i(k)}$ is a $p$-adic unit.
If such $i(k)$ does not exist, we simply put $i(k)=\infty$.
Then we will show that for any $\chi$ satisfying $$\label{condition for chi}
p^{m_\chi}> \frac{adp}{8},$$ the Newton polygon $p^{m_\chi-1}(p-1)\operatorname{NP}_f(C,\omega^u,\chi)$ is same as $\operatorname{NP}\Big(\Big\{(k,i(k))\;\big|\;k\geq 0\Big\}\Big).$
Since $C_f(\omega^u,T,s)\in \ZZ_p[\![T]\!],$ for any $\ell< i(k)$ we have $$\label{ineq1}\begin{split}
v_p\big(r_{u,k,\ell}(\chi(1)-1)^\ell\big)\geq& 1+(\frac{\ell}{p^{m_\chi-1}(p-1)})\\
=&\frac{p^{m_\chi-1}(p-1)+\ell-i(k)+i(k)}{p^{m_\chi-1}(p-1)}.
\end{split}$$ By Lemma \[distance\] and the definition of $i(k)$, we know that $$\label{aaa}
i(k)-\ell\leq \frac{ad(p-1)}{8}.$$ It follows from the inequalities , and that $$\begin{split}
v_p\Big(r_{u,k,\ell}\cdot(\chi(1)-1)^\ell\Big)\geq& \frac{p^{m_\chi-1}(p-1)-\frac{ad(p-1)}{8}+i(k)}{p^{m_\chi-1}(p-1)}\\
>& \frac{i(k)}{{p^{m_\chi-1}(p-1)}}\\
=&v_p\Big(r_{u,k,i(k)}\cdot(\chi(1)-1)^{i(k)}\Big).
\end{split}$$ The inequality above implies that $v_p\big(u_{u,k}\cdot(\chi(1)-1)\big)$ is
- either equal to $\frac{i(k)}{{p^{m_\chi-1}(p-1)}}$
- or greater than $\frac{1}{(p-1)p^{m_\chi-1}}{\mathrm{UP}}(d,\omega^u,T)(k)$.
Then this proposition follows directly from Corollary \[upper chi\].
For a Newton polygon $\operatorname{NP}$ and a rational number $t$ recall the definition of Newton polygon $t+\operatorname{NP}$ in Notation \[properties for NP\] (d).
Let $\chi:\ZZ_p\to \CC_p^\times$ be a finite character with conductor $p^{m_\chi}$. Then we have $$\Big\{\alpha\in R(\operatorname{NP}_f(C,\omega^u,\chi))\;\big|\;\alpha< ak\Big\}=\biguplus_{i=0}^{k-1}R\Big(ai+\operatorname{NP}(L_f(\omega^u,\chi,q^{i}s))\Big),$$ where $R$ is defined in Definition \[definition of R\].
Since $$C_f(\omega^u,\chi,s)=\prod_{i=0}^{\infty}L_f(\omega^u,\chi,q^is)\quad\textrm{and}\quad R\big(\operatorname{NP}_f(\omega^u,\chi,s)\big)\subset [0,a),$$ we know that $$\begin{split}
\Big\{\alpha\in R(\operatorname{NP}_f(C,\omega^u,\chi))\;\big|\;\alpha< ak\Big\}=&\biguplus_{i=0}^{k-1} \Big\{p\textrm{-adic Newton slopes of}\ L_f(\omega^u,\chi,q^{i}s)\Big\}\\=&\biguplus_{i=0}^{k-1}R\Big(ai+\operatorname{NP}(L,\omega^u,\chi)\Big).\qedhere
\end{split}$$
By Proposition \[independent\], we have $$\begin{split}
&R\big(\operatorname{NP}(L_f(\omega^u,\chi,s))\big)\\
=&\Big\{\alpha\in R\big(\operatorname{NP}_f(C,\omega^u,\chi)\big)\;\Big|\;\alpha< a\Big\}\\
=&\Big\{\frac{\alpha}{p^{m_\chi}}\;\Big|\; \alpha\in p^{m_\chi}R\big(\operatorname{NP}_f(C,\omega^u,\chi)\big)\ \textrm{and}\ \alpha< ap^{m_\chi}\Big\}\\
=&\Big\{\frac{\alpha}{p^{m_\chi}}\;\Big|\; \alpha\in p^{m_0}R\big(\operatorname{NP}_f(C,\omega^u,\chi_0)\big)\ \textrm{and}\ \alpha< ap^{m_\chi}\Big\}\\
=&\Big\{\frac{\alpha p^{m_0}}{p^{m_\chi}}\;\Big|\; \alpha\in R\big(\operatorname{NP}_f(C,\omega^u,\chi_0)\big)\ \textrm{and}\ \alpha< ap^{m_\chi-m_0}\Big\}\\
=&\biguplus_{i=0}^{p^{m_\chi-m_0}-1}R\Big(\frac{1}{p^{m_\chi-m_0}}\big(ai+\operatorname{NP}(L,\omega^u,\chi)\big)\Big)\\
=&\bigcup_{i=0}^{p^{m_\chi-m_0}-1}\Big\{\frac{\alpha_1+ai}{p^{m_\chi-m_0}},\frac{\alpha_2+ai}{p^{m_\chi-m_0}},\dots, \frac{\alpha_{dp^{m_0-1}}+ai}{p^{m_\chi-m_0}}\Big\}.\qedhere
\end{split}$$
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---
abstract: 'Spider silk is a remarkable example of bio-material with superior mechanical characteristics. Its multilevel structural organization of dragline and viscid silk leads to unusual and tunable properties, extensively studied from a quasi-static point of view. In this study, inspired by the Nephila spider orb web architecture, we propose a novel design for mechanical metamaterials based on its periodic repetition. We demonstrate that spider-web metamaterial structure plays an important role in the dynamic response and wave attenuation mechanisms. The capability of the resulting structure to inhibit elastic wave propagation in sub-wavelength frequency ranges is assessed and parametric studies are performed to derive optimal configurations and constituent mechanical properties. The results show promise for the design of innovative lightweight structures for tunable vibration damping and impact protection, or the protection of large scale infrastructure such as suspended bridges.'
author:
- Marco Miniaci
- Anastasiia Krushynska
- 'Alexander B. Movchan'
- Federico Bosia
- 'Nicola M. Pugno'
title: 'Spider-Web Inspired Mechanical Metamaterials'
---
Many natural materials display outstanding properties that can be attributed to their complex structural design, developed in the course of millions of years of evolution [@Gao2003; @Aizenberg2005; @Kamat2000]. Particularly fascinating are spider silks, which exhibit unrivaled strength and toughness when compared to most materials [@Vollrath1992; @Gosline1999; @Boutry2009; @Meyer2014]. Previous studies have revealed that mechanical performance of spider webs is not only due to the remarkable properties of the silk material, but also to an optimized architecture that is adapted to different functions [@Cranford2012; @Zaera2014].
Structural behaviour of orb spider webs has been extensively analyzed under quasi-static [@Boutry2009; @Cranford2012; @Aoyanagi2010] and dynamic [@Alam2007; @Ko2004] loading conditions. However, the spider web structure has yet to be exploited for the design of mechanical metamaterials. The latter include phononic crystals and elastic metamaterials, and are usually periodic composites capable of inhibiting the propagation of elastic or acoustic waves in specific frequency ranges called band gaps. This unique ability opens a wide range of application opportunities, such as seismic wave insulation [@Miniaci2016a], environmental noise reduction [@MartinezSala1995], sub-wavelength imaging and focusing [@Bigoni2013], acoustic cloaking [@Farhat2008], and even thermal control [@Maldovan2013]. In phononic crystals, band gaps are induced due to Bragg scattering from periodic inhomogeneities [@Brillouin1946], while in elastic metamaterials, sub-wavelength band gaps can be generated due to localized resonances [@Liu2000]. The latter are commonly achieved by employing heavy constituents [@Liu2000; @Pennec2010; @Hussein2014; @Krushynska2014]. However, recently it has been found that hierarchically organized [@Miniaci2016b] or lattice-type structures can also generate sub-wavelength band gaps [@LimBertoldi2015; @WangBertoldi2015]. From this perspective, a spider-web inspired, lattice-based elastic metamaterial seems to be a promising alternative to control low-frequency wave propagation.
In this letter, we design a novel metamaterial inspired by the Nephila orb web architecture and analyze the dynamics of elastic waves propagating in the considered structures. The capability of inducing local resonance band gaps is studied by highlighting the importance of the structural topology as well as the material properties of single constituents. We find that the band gaps are associated with either ring-shaped resonators or constituent parts of the bearing frame and can be tuned to desired frequencies by varying material parameters of the constituents or the number of the ring resonators. Overall, our analysis shows that the spider web-inspired architectures allow to extend the effective attenuation frequency ranges compared to a simple lattice [@Martinsson2003] and simultaneously provide lightweight yet robust structures.
We consider a spider web-inspired metamaterial in the form of an infinite in-plane lattice modeled by periodically repeating representative unit cells in a square array. The primary framework of the unit cell is a square frame with supporting radial ligaments (Fig. \[models\]a). The ligaments intersect the frame at right-angle junctions acting as ‘hinge’ joints (square junctions in Fig. \[models\]a). The secondary framework is defined by a set of equidistant circumferential ligaments (or ring resonators) attached to the radial ligaments by hinge joints, further called ‘connectors’ to distinguish them from the joints in the first framework (Fig. \[models\]b). The geometry of the metamaterial is completely defined by 5 parameters: unit cell pitch $a$, size of square joints $b$, thickness of radial and circumferential ligaments $c$, number of ring resonators $N$, and radius of a ring resonator $R_N$. We initially consider $a = 1 [m]$, $b = 0.04 \cdot a$, $c = 0.01 \cdot a$, $N=7$, and $R_N = 0.1 \cdot a \cdot (N+1)/2$. The material properties of the primary and secondary frameworks correspond to the parameters of dragline ($E_{d} = 12$ GPa, $\nu_{d} = 0.4$, $\rho_{d} = 1200$ kg$/$m$^3$) and viscid ($E_{v} = 1.2$ GPa, $\nu_{v} = 0.4$, $\rho_{v} = 1200$ kg$/$m$^3$) silks of the Nephila orb spider web [@Zaera2014], respectively. Material properties of the connectors can assume dragline or viscid silk values, as specified below.
The propagation of elastic waves in infinite and finite-size lattices is investigated numerically by using the Finite Element (FE) commercial package COMSOL Multiphysics [@COMSOL]. Unit cells are discretized by triangular linear elements of size $a/20$. Wave dispersion in the infinite lattice is studied by applying the Bloch conditions [@Hussein2014] at the unit cell boundaries and performing frequency modal analysis for wavenumbers along the 3 high-symmetry directions $\Gamma-X-M$ within the first irreducible Brillouin zone [@Miniaci2015].
First, we investigate the propagation of small-amplitude elastic waves in an infinite structure formed by the primary framework unit cell (Fig. \[models\]a), called “regular lattice” metamaterial. Figure \[DispersionCurves\]a shows the band structure for the regular lattice as a function of reduced wave vector $k^* = [k_x a/\pi; k_y a/\pi]$, where it appears that there are no band gaps in the considered frequency range, up to 400 Hz. However, the structure is characterized by several localized modes at various frequencies represented by (almost) flat bands. Consideration of the corresponding vibration patterns reveals that the motions are localized within the radial ligaments.
Next, the circumferential elements are introduced to analyze the wave dynamics in a spider-web inspired metamaterial formed by the unit cell shown in Fig. \[models\]b. Here, we explore three possibilities: 1) the circumferential ligaments have the same mechanical properties as the radial ligaments (dragline silk); 2) the circumferential ligaments have the mechanical properties of viscid silk, while connectors of radial and circumferential ligaments have the properties of dragline silk, and 3) both the circumferential ligaments and the connectors have the properties of viscid silk. This analysis allows to evaluate the influence of material parameters on the wave attenuation performance of the spider-web topology.
The band structure of the spider-web metamaterial made of the dragline silk material is shown in Figure \[DispersionCurves\]b and exhibits a complete band gap between the $10^{th}$ and $11^{th}$ bands at frequencies from $346.5$ to $367.4$ Hz, which is shaded in light gray. Vibration patterns at the band gap bounds (points C and D in Fig. \[DispersionCurves\]b) reveal that the whole unit cell is involved in the motion. As the band gap bounds are formed by non-flat curves, the band gap is not due to local resonance effects. At the same time, it cannot be induced by Bragg scattering, since it is located at least twice below the frequencies at which a half-wavelength of either longitudinal (2314 Hz) or shear (945 Hz) waves in the silk material is equal to the unit cell size, and the bounds are shifted from the edge points ($\Gamma$, $X$, and $M$) of the irreducible Brillouin zone. Further analysis of the band gap origin is beyond the scope of this letter, since we are mainly focusing on a spider-web inspired structure with *different* mechanical properties for radial and circumferential ligaments [@Cranford2012; @Zaera2014]. Another remarkable feature of the band structure in Fig. \[DispersionCurves\]b is the smaller number of localized modes compared to Fig. \[DispersionCurves\]a, which may be explained with the elimination of local resonances due to the coupling between motions in radial and circumferential ligaments.
The assignment of the material parameters of viscid silk to ring resonators leads to the appearance of two band gaps in the corresponding band structures (Fig. \[DispersionCurves\]c-d) regardless of the mechanical properties of the connectors joining radial and circumferential ligaments. Due to the more compliant properties of the resonators, the band gaps are located at lower frequencies compared to the case in Fig. \[DispersionCurves\]b. Moreover, it is obvious that these are so-called hybridization band gaps [@Sainidou2002] induced by local resonances, since the lower bounds are formed by flat curves representing localized motions (see modes relative to points E-H in Fig. \[DispersionCurves\]c-d), and the reduced Bloch wave vector $k^*$ has a $\pi$ change inside each band gap. In the case when the ring resonators and the connectors have the same mechanical properties, the band gaps are shifted to lower frequencies due to a more compliant behavior of the connectors (Fig. \[DispersionCurves\]d).
Another peculiarity of the band structures in Fig. \[DispersionCurves\]c-d in comparison to Fig. \[DispersionCurves\]a-b is a larger number of almost flat pass bands, many of which correspond to localized modes. In the cases when the ring resonators are characterized by a low stiffness the localized modes are associated to a family of standing waves mostly dominated by high inertia of the resonators (see modes relative to points E-H in Fig. \[DispersionCurves\]c-d). If the connectors between radial and circumferential ligaments have the same material properties as the resonators (which is the closest configuration to a real spider web), it appears that in the low-frequency regime, frequencies of standing waves may be associated with the ring resonators only. The natural frequencies $\omega_n$ for non-axisymmetric in-plane flexural vibrations of these resonators can be expressed [@Tim; @Love] in closed form: $$\omega_n = k ~ \frac{n (n^2-1)}{R^2 \sqrt{n^2 + 1}}, ~ n > 1. \label{omega}$$ Here $R$ stands for the radius of a ring resonator, and $k$ is a dimensional constant that depends on the elastic modulus of the ring resonator, the mass density, and its cross-section. Examples of vibrational modes for several values of $n$ are shown in the Supplementary Material. However, solution (\[omega\]) cannot be applied to a spider-web lattice system, since the dynamic response of the latter is governed by the entire structure and not the individual decoupled resonators (see the Supplementary Material for details).
Wave propagation in the regular and spider-web lattice metamaterials can be better understood by analyzing the transformations of individual modes in band structures of Fig. \[DispersionCurves\] for varying geometrical and mechanical parameters of the unit cell. For example, the frequency range of the lowest (localized) mode is the same for the regular and spider-web-type metamaterials, except for the case in Fig. \[DispersionCurves\]b, whereas the second mode (second curve from the bottom) is shifted towards higher frequencies, as circumferential ligaments are introduced. These features can be explained by examining the vibration modes of the corresponding systems, a detailed analysis of which is given in the Supplementary Material.
Further, we investigate the evolution of the band structure for the spider-web inspired metamaterials by varying the mechanical properties and number of ring resonators. The parametric study is performed for metamaterials with the ring resonators with intermediate stiffness values between those of dragline and viscid silks. The overall band structure resembles that shown in Figs. \[DispersionCurves\]c-d. Here, we focus our attention only on the band gap frequencies. Figure \[ParametricAnalysis\]a shows band gap frequencies versus ratios $E_{rr}/E_{rl}$, where $E_{rr}$ and $E_{rl}$ are the stiffnesses of the ring resonators and radial ligaments, respectively. It appears that for $E_{rr}/E_{rl} = 0.6$ the first band gap is maximized, and the second one is closed. In general, as the stiffness of the ring resonators increases, inhibited frequency ranges are translated towards higher frequencies, except one narrow band gap located slightly above 150 Hz, whose frequencies are independent of the mechanical parameters of the resonators.
Next, we investigate the wave dispersion in structures with a varying number of ring-shaped resonators. Figure \[ParametricAnalysis\]b shows dispersion diagrams for the unit cells with 1,3,5, and 7 stiff ring resonators (dragline silk). In all cases, the unit cells with a reduced number of resonators are obtained by removing the most internal resonators from the original geometry shown (Fig. \[models\]b). Stiff rings are chosen here, since the corresponding band structures have fewer pass bands, thus facilitating the analysis. In the considered frequency range, the number of localized modes decreases with the increase of the number of the ring resonators, since the structure becomes stiffer. This results in an increase in the number of band gaps, which are also shifted to lower frequencies, since the wave attenuation is enhanced due to increased number of resonators. Interestingly, the unit cells with 3-7 ring resonators possess the first (narrow) band gap at the same frequencies, slightly above 150 Hz. Examination of the corresponding vibration forms reveals that this band gap is induced by local resonances in the external square frame. Thus the gap frequencies are independent of the number of the ring resonators, as well as of their mechanical properties (Fig. \[ParametricAnalysis\]a).
Having demonstrated that infinite spider-web inspired lattice materials are characterized by locally resonant band gaps, we now investigate how the transmission in finite-size structures is affected. The analysis of finite-size structure is performed in the frequency domain for a model comprising 25 unit cells placed in a square array with traction-free boundary conditions. The structure is excited at the central point by applying harmonic in-plane displacement at a frequency of 186 Hz (within a band gap) at an angle of $\pi/4$ with respect to the horizontal axis. Figure \[FrequencyAnalysis\] presents frequency-domain responses in terms of in-plane displacements for two structures formed by the regular and spider-web lattice unit cells with viscid silk ring resonators. A scale factor of 2500 is applied to highlight the dynamics of the structure. Maximum and minimum values of displacements are shown in red and dark blue, respectively. As can be easily seen, all of the regular-lattice structure is involved in the motion (Fig. \[FrequencyAnalysis\]a), while the spider-web inspired system is capable of strongly attenuating vibrations after a few unit cells (Fig. \[FrequencyAnalysis\]b). A similar behavior can be observed for other excitation frequencies within the predicted band gaps. These results confirm the predictions derived from the wave dispersion analysis. Figure \[FrequencyAnalysis\]b suggests an important application, namely the generation of a defect mode in a cluster with highly localized vibrations around its center to obtain efficient wave attenuation effects in desired frequency ranges.
In summary, we have numerically studied the propagation characteristics of elastic waves in regular and spider web-inspired beam lattices, based on the Nephila orb web architecture. Our results indicate that the spider-web inspired lattices possess locally resonant band gaps induced by ring-shaped resonators and parts of the bearing frame. The band gaps can be easily tuned in a wide range of frequencies by varying the mechanical properties or the number of the resonators. Interestingly, despite the fact that the ring resonators are responsible for the generation of band gaps, their eigenfrequencies cannot be directly used to predict the band gap bounds, since the overall structure plays an important role in their formation. We have found that mechanical properties of the connectors between ring resonators and the lattice frame also influence the inhibited frequency ranges. Though lattice systems with locally resonant band gaps have already been reported [@WangBertoldi2015; @LimBertoldi2015], this study shows that spider-web inspired lattice metamaterials possess more parameters to tune the band gaps to desired frequencies and are easier to manipulate/manufacture compared to hierarchically organized lattice-type structures. In the proposed metamaterial configuration, the role of the ring resonators is three-fold: (1) they govern the local resonance mechanisms responsible for inducing sub-wavelength band gaps, (2) they contribute to the formation of band gaps due to local resonance in the bearing frame and (2) they are parts of the structural lattice coupled to the bearing frame to enhanced wave attenuation performance with significant reductions of the wave energy transferred through the web architecture. Thus, results from this study can lead to new ideas for the design of lightweight and robust bio-inspired metamaterial structures with tunable properties. This work also suggests a new functionality for spider webs and new applications for the corresponding metamaterials and metastructures, e.g. for earthquake protection of suspended bridges.
**Acknowledgements**
M.M. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement N. 658483. A.K. has received funding from the European Union’s Seventh Framework programme for research and innovation under the Marie Sk[ł]{}odowska-Curie grant agreement N. 609402-2020 researchers: Train to Move (T2M). N.M.P. is supported by the European Research Council in the following active projects: ERC StG Ideas 2011 BIHSNAM no. 279985; ERC PoC 2015 SILKENE no. 693670; as well as by the European Commission under the Graphene Flagship (WP ‘14 Polymer Nanocomposites’, no. 696656). F.B. is supported by BIHSNAM.
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[max width=0.8]{}
No. ImageNet fsPASCAL fsCOCO FSS FSS(test set)
----- ---------- ---------- -------- ----- ---------------
66.45%
71.34%
79.30%
80.12%
81.97%
82.66%
: In Table 5 of the paper, models trained on ImageNet, FSS, and fsCOCO were compared and the test set results on FSS and fsCOCO were presented. Here we show more experiment settings that also include training on fsPASCAL, with their test results evaluated on the FSS test set. Similar to Table 5, All learning rates are initially set to $10^{-4}$ except the models trained without using ImageNet pre-trained weights, which are set to $10^{-3}$.[]{data-label="diffdatasets"}
|
---
abstract: 'We provide strong evidence that the relaxation dynamics of one-dimensional, metallic Fermi systems resulting out of an abrupt amplitude change of the two-particle interaction has aspects which are universal in the Luttinger liquid sense: The leading long-time behavior of certain observables is described by universal functions of the equilibrium Luttinger liquid parameter and the renormalized velocity. We analytically derive those functions for the Tomonaga-Luttinger model and verify our hypothesis of universality by considering spinless lattice fermions within the framework of the density matrix renormalization group.'
author:
- 'C. Karrasch'
- 'J. Rentrop'
- 'D. Schuricht'
- 'V. Meden'
title: Luttinger liquid universality in the time evolution after an interaction quench
---
The equilibrium low-energy physics of a large class of one-dimensional (1d), correlated, metallic Fermi systems is described by the Luttinger liquid (LL) phenomenology [@Giamarchi03; @Schoenhammer05]. The Tomonaga-Luttinger (TL) model is the effective low-energy fixed point model of the LL universality class and thus plays the same role as the free Fermi gas in Fermi liquid theory. This universality relies on the renormalization group (RG) irrelevance of contributions such as the momentum dependence of the two-particle interaction [@Meden99] or the curvature of the single-particle dispersion [@Imambekov11] which are present in microscopic models but ignored in the TL model. For a model falling into the LL universality class it is not necessary to explicitely compute thermodynamic observables and correlation functions if one is interested in the low-energy limit. One only needs to determine two numbers – the LL parameter $K$ and the renormalized velocity $v$ of the excitations – which fully characterize the low-energy physics of a spinless LL (on which we focus). Those depend on the band structure and filling as well as the amplitude and range of the two-particle interaction of the microscopic model at hand and can be extracted from the ground-state energy [@Haldane80] or ‘simple’ response functions [@llparam]. Thereafter, correlation functions at long length scales or thermodynamic quantities at low energies can be obtained by plugging in $K$ and $v$ into analytic expressions derived within the exactly solvable TL model [@Giamarchi03; @Schoenhammer05].
The recent progress in experimentally controlling isolated many-body states, in particular in cold atomic gases [@Bloch08], led to numerous theoretical studies on the dynamics of closed quantum systems resulting out of an abrupt change of the amplitude $U$ of the two-particle interaction [@Polkovnikov11]. One assumes that the system is prepared in a canonical thermal state or, at temperature $T=0$ on which we focus, the ground state of an initial Hamiltonian $H_{\rm i}$ with a given $U_{\rm i}$; often $U_{\rm i}=0$ is considered. At time $t=0$ the interaction is quenched to $U_{\rm f}$ and the time evolution is performed with the final Hamiltonian $H_{\rm f}$. Fundamental questions discussed are [@Polkovnikov11]: (a) Do some observables become stationary at large times? (b) How can they be classified (locality)? (c) Is it possible to compute their steady-state expectation values using an appropriate density matrix $\rho_{\rm st}$? It was conjectured that in models with many integrals of motion, e.g. those which are solvable by Bethe ansatz (‘integrable’) [@Faribault09; @Mossel10; @Gritsev10], $\rho_{\rm st}$ is not of thermal but rather of generalized (‘Gibbs’) canonical form [@Rigol07]. This has been confirmed for models which can be mapped to effective noninteracting ones [@Rigol07; @Calabrese07; @Eckstein08; @Kollar08; @Rossini09; @Calabrese11; @Barthel08], in particular the TL model [@Cazalilla06] and a variant of the latter [@Kennes10], but a general prove is lacking. For generic models one generally expects $\rho_{\rm st}$ to be thermal.
Here we address questions about the *relaxation dynamics towards the steady state*. Considering several models which in equilibrium fall into the LL class we ask: (A) [*Can the time evolution be characterized as universal in the LL sense?*]{} As model-dependent high energy processes matter at short to intermediate times one can only expect to find LL universality on *large time scales* – but even then it is not obvious whether the notion of RG irrelevance can be transferred from equilibrium to the nonequilibrium dynamics [@Mitra11; @Rentrop12]. (B) [*If LL universality is found, does it hold independently of the number of integrals of motion and thus independently of the expected nature of the steady state*]{} (generalized canonical versus thermal)? To answer those questions we proceed in steps. We compute the time evolution of the $Z$[*-factor*]{} jump of the momentum distribution function $n(k,t)$ at the Fermi momentum $k_{\rm F}$ and the [*kinetic energy per length*]{} $e_{\rm kin}(t)$ for the spinless TL model with arbitrary momentum dependent interactions $g_{2/4}(k)$ using bosonization [@Giamarchi03; @Schoenhammer05]. $e_{\rm kin}(t)$ is defined as the expectation value of the initial Hamiltonian $H_{\rm i}$ and thus describes how excitations die out. The quench is performed out of the noninteracting ground state. We analytically show that for any continous $g_{2/4}(k)$ the long-time dynamics are given by [@footnote0]
\[asymptotics\] $$\begin{aligned}
Z(t) & \sim t^{-\gamma_{\rm st}(K)}~, \label{asymptotics1} \\
\left|de_{\rm kin}(t)/dt
\right| &\sim \epsilon(K,v)t^{-3} ~. \label{asymptotics2}\end{aligned}$$
The decay of $Z(t)$ is governed by a universal exponent $\gamma_{\rm st}(K)$ that depends on the equilibrium LL parameter $K$ only; $e_{\rm kin}(t)$ features an asymptotic power law with an interaction-independent exponent but universal prefactor $\epsilon(K,v)$ determined by $K$ as well as by the renormalized velocity $v$. In equilibrium, this is the characteristic $K$- and $v$-dependence of correlation functions or of thermodynamic quantities, which *a posteriori* motivates to consider $Z(t)$ and $e_{\rm kin}(t)$ as representative examples. The notion of LL universality can now be defined in analogy to equilibrium: *The quench dynamics is universal if Eqs. (\[asymptotics\]) describe the long-time relaxation for any model falling into the equilibrium LL universality class* if the corresponding values for $K$ and $v$ are plugged in. To investigate this we compute $Z(t)$ for a 1d lattice of spinless fermions with nearest-neighbor hopping and interaction $\Delta$ [@Manmana07] as well as an extension of the latter including a next-to-nearest-neighbor interaction $\Delta_2$. We use the numerical time-dependent density-matrix RG (DMRG) [@Schollwoeck11; @white; @tdmrg]. The model with $\Delta_2=0$ has many conserved quantities, is Bethe ansatz integrable, and thus $K$ as well as $v$ are known analytically [@Haldane80; @Qin97]. The $\Delta$-$\Delta_2$–model, however, is believed to be not exactly solvable. For $\Delta_2>0$ we extract $K$ and $v$ from equilibrium quantities (e.g. the small momentum density response function) using DMRG [@footnote2; @llparam; @LLpaper]. Our data for the $Z$-factor agrees with Eq. (\[asymptotics1\]) for any interaction strength, filling factor, and irrespective of the integrability of the model. The results for $e_{\rm kin}(t)$ are consistent with Eq. (\[asymptotics2\]), but on the time scales accessible by DMRG the asymptotic behavior is still masked by oscillatory terms of higher order in $t^{-1}$. To unambiguously determine the prefactor of the $t^{-3}$-decay of the energy we resort to a numerical trick. Instead of performing the time evolution with $\exp{(-i H_{\rm f} t)}$ we apply the imaginary time analogue $\exp{(- H_{\rm f} \tau)}$. In this case the [*total*]{} energy per length $e(\tau)$ – which is no longer conserved – is the natural observable. For the TL model we show that the asymptotics is completely analogous to Eq. (\[asymptotics2\]) with $t \to \tau$ and $\epsilon(K,v)$ replaced by a different function $\epsilon_{\rm it}(K,v)$. For the lattice model, the $\tau^{-3}$-decay manifests over several orders of magnitude, and the prefactor agrees with the TL prediction.
This altogether provides strong evidence that questions (A) and (B) can be answered by ‘yes’. We conjecture that the universality of the quench dynamics also holds for other models falling into the equilibrium LL class.
![(Color online) Time evolution of the $Z$-factor out of the noninteracting ground state of a 1d metallic Fermi system after switching on two-particle terms at time $t=0$. Dashed lines show the *universal asymptotic power law* $t^{-\gamma_{\rm st}(K)}$ with an exponent determined by the equilibrium LL parameter $K$. (a) TL model. The plots displays box-like two-particle interactions $g(k)$ of strength $g=g(0)$; the asymptotics are universal for any $g(k)$. Time is given in units of $(v_{\rm F}k_{\rm c})^{-1}$. (b, c, Inset) Spinless lattice fermions of Eq. (\[Hlatticemodel\]) at filling $\nu$ featuring nearest ($\Delta$) and next-nearest ($\Delta_2$) neighbor interactions. []{data-label="fig:zfac"}](zfac.eps){width="0.95\linewidth"}
{width="0.46\linewidth"} {width="0.46\linewidth"}
*The TL model* — After bosonizing [@Giamarchi03; @Schoenhammer05] the density of left and right moving fermions with a linear dispersion the Hamiltonian of the TL model is quadratic in operators $b_n^{(\dag)}$ which obey bosonic commutation relations: $$\begin{aligned}
H = \sum_{n >0 } && \left[ k_n \left( v_{\rm F} + \frac{g_4(k_n)}{2 \pi} \right)
\left( b_n^\dag b_n^{} + b_{-n}^\dag b_{-n}^{} \right) \right. \nonumber \\
&& \left. + k_n \frac{g_2(k_n)}{2 \pi}
\left( b_n^\dag b_{-n}^\dag + b_{-n}^{} b_{n}^{} \right) \right] ~,
\label{HTL}\end{aligned}$$ where $k_n=2 \pi n / L$, $n \in {\mathbb Z}$, $L$ denotes the chain length, and $v_{\rm F}$ is the Fermi velocity. The two coupling functions (potentials) $g_{2/4}$ determine the strength of the scattering of fermions on different branches ($g_2$) and the same branch ($g_4$). Usually the $k$-dependence of $g_{2/4}$ is neglected and integrals are regularized in the ultraviolet through an ad hoc procedure [@Giamarchi03; @Schoenhammer05]. As the momentum dependence is RG irrelevant this is justified in equilibrium if all energy scales are sent to zero [@Meden99]. For the quench dynamics – even at asymptotic times – it is, however, not clear if the same reasoning holds and we thus keep the full $k$-dependence and consider coupling [*functions.*]{} In fact, it was recently shown that the momentum dependence indeed affects the long-time dynamics of certain observables [@Rentrop12]. For the system to be a LL in equilibrium we require that $0 < g_{2/4}(0) < \infty $ (repulsive interactions) and that $g_{2/4}(k)$ decay on a scale $k_{\rm c}$. The Hamiltonian of Eq. (\[HTL\]) can be diagonalized to $H = \sum_{n \neq 0} \omega(k_n) \, \alpha_n^\dag \alpha_n^{}
+ E_{\rm gs}$ by introducing new modes $\alpha_n = c(k_n) b_n + s(k_n) b^\dag_{-n}$ with $$\begin{aligned}
&& s^2(k) = \frac{1}{2} \left[ \frac{1+ \hat g_4(k)}{W(k)} -1 \right] = c^2(k) -1
~,\label{manydefs} \\ &&
\omega(k) = v_{\rm F} |k| \, W(k) = v_{\rm F} |k|\sqrt{(1+ \hat g_4(k))^2 - \hat g_2^2(k)} \nonumber ~, \end{aligned}$$ where $\hat g_{2/4}= g_{2/4}/(2 \pi v_{\rm F})$, and $E_{\rm gs}$ denoting the ground state energy. The LL parameter and the renormalized velocity read $$\begin{aligned}
K= \sqrt{\frac{1+\hat g_4(0) -\hat g_2(0)}{1+\hat g_4(0) +\hat g_2(0)}}~ , \; \; \; \; v=v_{\rm F} W(0)~ .
\label{Kv}\end{aligned}$$ As our initial state we take the noninteracting ground state $\left| E_{\rm gs}^{0} \right\rangle$ which is given by the vacuum $\left| \mbox{vac}(b) \right>$ with respect to the $b_n$. Expectation values of the time-evolved state $ \left| \Psi(t) \right\rangle
= \exp(-i H t) \left| E_{\rm gs}^{0} \right\rangle$ can be computed straightforwardly using the simple time dependence of the eigenmode operators $\alpha_n^{(\dag)}$ and their linear dependence on the $b_n^{(\dag)}$ [@Rentrop12].
After bosonizing the fermionic field operator [@Giamarchi03; @Schoenhammer05] the $Z$-factor $Z(t)=\lim_{k\nearrow k_{\rm F}}n(k,t)-
\lim_{k\searrow k_{\rm F}}n(k,t)$ is easily obtained (taking $L \to \infty$) [@Cazalilla06; @Rentrop12; @Dora11]: $$\begin{aligned}
Z(t) = \exp{ \left\{ - \int_0^\infty \!\!\! dk \, \frac{4 s^2(k) c^2(k)}{k}
\left( 1- \cos{\left[2 \omega(k) t\right]} \right) \right\} } .
\nonumber\end{aligned}$$ Independent of the form of $g_{2/4}(k)$ (even for potentials with a discontinous jump to zero at $k_{\rm c}$) the large-time behavior is given by Eq. (\[asymptotics1\]) with $\gamma_{\rm st} = (K^2 + K^{-2}-2)/4$; it manifests on the (nonuniversal) scale $(v_{\rm F}k_{\rm c})^{-1}$. Figure \[fig:zfac\](a) shows $Z(t)$ obtained by numerically performing the integral for a simple box shaped potential $\hat g_2(k) = \hat g_4(k) = g
\Theta(k_{\rm c} - |k|)/2$ of varying amplitude $g$. The asymptotic power-law is modulated by oscillations which decay faster than $t^{- \gamma_{\rm st}}$.
The kinetic energy per length $e_{\rm kin}(t)$ reads ($L \to \infty$) $$\begin{aligned}
e_{\rm kin}(t) = \frac{ v_{\rm F}}{2 \pi} \!\! \int_0^\infty \!\!\!\! dk k 4 s^2(k) c^2(k)
\left\{ 1- \cos{[2 \omega(k) t]}
\right\} .
\label{EkintTD}\end{aligned}$$ The steady-state value is obtained by dropping the oscillatory term which averages out for $t\to\infty$. For [*continuous*]{} coupling functions $g_{2/4}(k)$ of range $k_{\rm c}$ asymptotic analysis yields Eq. (\[asymptotics2\]) as the leading term in the long-time limit;[@footnote4] the coefficient is given by $\epsilon(K,v)=\gamma_{\rm st}(K) v_{\rm F}/(4 \pi v^2)$. Figure \[fig:ekin\](a) shows the derivative of $e_{\rm kin}$ for $\hat g_2(k) = \hat g_4(k) = g(k)$, a Gaussian potential $g(k)=g \exp(-[k/k_{\rm c}]^2/2) /2$ as well as a quartic potential $g(k) = g/(1+[k/k_{\rm c}]^4)/2$ and varying interaction strengths. As either $g(0)$ or the lowest nonvanishing Taylor expansion order of $g(k)-g(0)$ increases, the amplitude of an oscillatory term which decays faster than the leading one becomes stronger. The (nonuniversal) scale on which the asymptotic $t^{-3}$-behavior dominates thus heavily depends on the strength and type of potential at hand \[compare the inset and the main part of Figure \[fig:ekin\](a)\].
*Microscopic lattice model* — As a next step we provide strong evidence that Eqs. (\[asymptotics\]) describe the long-time relaxation dynamics of any model which *in equilibrium* falls into the LL universality class. To this end, we consider spinless lattice fermions, $$\begin{aligned}
\mbox{} \hspace{-.3cm} H \! = \! \sum_{j} \! \left[ \frac{1}{2} c_j^{\dag}
c_{j+1}^{\phantom{\dagger}} + \mbox{H.c.} + \Delta n_j n_{j+1} + \Delta_2 n_j n_{j+2}
\right] ,
\label{Hlatticemodel}\end{aligned}$$ with $n_j = c_j^\dag c_j-1/2$. We study the quench dynamics using an infinite-system DMRG algorithm [@Schollwoeck11; @tebd]. We determine $|E_{\rm gs}^0\rangle$ by applying an imaginary time evolution $\exp(-\tau H|_{\Delta=\Delta_2=0})$ to a random initial matrix product state with a fixed matrix dimension $\chi$ until the energy has converged to typically $8-10$ relative digits. Operators $\exp(\sim H)$ are factorized by a second or fourth order Trotter decomposition. Thereafter, we compute the real time evolution $|\Psi(t)\rangle=\exp(-itH)|E_{\rm gs}^0\rangle$ in presence of the two-particle terms $\Delta$ and $\Delta_2$. $\chi$ is dynamically increased in order to maintain a fixed discarded weight. We carefully ensure that the latter is chosen small enough (and that the initial $\chi$ is large enough) to obtain numerically exact results.
![(Color online) Imaginary time evolution $\exp(-\tau H)|E_{\rm gs}^0\rangle\stackrel{\tau\to\infty}
{\longrightarrow}|E_{\rm gs}\rangle$ towards the *interacting* ground state $|E_{\rm gs}\rangle$. The main part shows DMRG data for the $\tau$-derivative of the total energy in the lattice model. The $\tau^{-3}$-decay predicted by bosonization manifests over several orders of magnitude. The prefactor agrees with the TL formula $\epsilon_{\rm it}(K,v)$ for all parameters (this is illustrated in the insets). The imaginary-time energy dynamics are thus universal in the LL sense.[]{data-label="fig:imag"}](imag.eps){width="0.95\linewidth"}
The time evolution of the momentum distribution function $ n(k,t) =
\sum_j e^{ikj} \big\langle\Psi(t)\big| c_j^\dagger c_j^{\phantom{\dagger}}
\big|\Psi(t)\big\rangle$ and the corresponding $Z$-factor can be computed straightforwardly; the $j$-sum is carried out up to $\sim10000$ sites. Results for $Z(t)$ are shown in Figure \[fig:zfac\](b,c). Its long-time asymptotics indeed shows a power-law decay $t^{-\gamma_{\rm st}(K)}$, and the exponent agrees to the one predicted by the TL formula (dashed lines) if the latter is evaluated for $K$ corresponding to the microscopic parameters under consideration. We take $K$ from the Bethe ansatz ($\Delta_2=0$) [@Haldane80; @Qin97] or the equilibrium density response ($\Delta_2>0$) [@llparam; @LLpaper]. The agreement with the TL model result holds for any interaction strength [@footnote2], for any filling factor $\nu$, and irrespective of the integrability of the model. This strongly indicates that the asymptotic dynamics of the $Z$-factor is indeed universal in the LL sense.
The time derivative of the kinetic energy (the expectation value of $H|_{\Delta=\Delta_2=0}$) per length (site) is shown in Figure \[fig:ekin\](b). Its magnitude decays as $t^{-3}$, and the ratio between prefactors (where the factor $v_{\rm F}$ drops out) at different interaction strengths is consistent with the TL formula (see the dashed lines). However, oscillations have not died out completely, and a pure power law cannot be identified unambigously. To further support that this is merely because the time scales reachable in our DMRG calculation are too small – remind that for the TL model the scale where $de_{\rm kin}/dt$ is governed by Eq. (\[asymptotics2\]) strongly depends on the strength and type of the potential in contrast to $Z(t)$ where it is always $(v_{\rm F}k_{\rm c})^{-1}$ – and that the energy relaxation is indeed universal, we consider an imaginary time evolution, $\left| \Psi(\tau) \right\rangle
= \exp{(- H \tau)}\left| E_{\rm gs}^{0} \right\rangle/\left\langle E_{\rm gs}^{0} \right|
\exp{(- 2 H \tau)} \left| E_{\rm gs}^{0} \right\rangle^{1/2}$. For $\tau\to\infty$, $\left| \Psi(\tau) \right\rangle$ approaches the ground state $ \left| E_{\rm gs}
\right\rangle$ of the interacting Hamiltonian. The total energy is the natural observable to compute in this academic scenario. Its asymptotic behavior within the TL model is completely analogous to Eq. (\[asymptotics2\]) with $t \to \tau$ and $\epsilon(K,v) \to \epsilon_{\rm it}(K,v) =
\mbox{Li}_2([K+K^{-1}-2]/[K+K^{-1}+2])/(8 \pi v)$, where $ \mbox{Li}_2$ denotes the dilogarithm [@footnote3]. For the lattice model, one can easily access large imaginary times using DMRG, and the $\tau^{-3}$-decay manifests over several orders of magnitude. This is illustrated in Fig. \[fig:imag\]. The prefactor (shown in the insets) agrees with $\epsilon_{\rm it}(K,v)$ (the latter depends on $K$ and $v$ only; thus, one does not need to consider ratios) for all interactions and fillings. The dynamics of the total energy at large $\tau$ is universal.
*Conclusion* — We have obtained exact expressions for the time evolution of the $Z$-factor and of the kinetic energy after an interaction quench within the Tomonaga-Luttinger model. For any continous two-particle potential, their long-time asymptotes $Z\sim t^{-\gamma_{\rm st}}$, $de_{\rm kin}/dt\sim \epsilon(K,v)/t^3$ are universal functions of the LL parameter $K$ and the renormalized velocity $v$. We studied a similar scenario for spinless lattice fermions using DMRG; for large times, $Z(t)$ and $e_{\rm kin}(t)$ are described by the above expressions. This provides strong evidence that the relaxation dynamics after an interaction quench within any model that falls into the equilibrium LL universality class has aspects which are universal in the LL sense.
*Acknowledgments* — We thank S. Kehrein, D. M. Kennes, J. E. Moore, and K. Schönhammer for fruitful discussions. This work was supported by the DFG via KA3360-1/1 (CK), the Emmy-Noether program (DS), and FOR 912 (VM).
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This holds as long as the interaction is smaller than a critical scale beyond which $\omega(k)$ of Eq. (\[manydefs\]) develops stationary points [@Rentrop12]. This scale depends on the form of the $g_{2/4}(k)$.
This result can straightforwardly be obtained by writing $\left| \mbox{vac}(b) \right>= \prod_{n >0 }
\sum_{l \geq 0} a_l^{(n)} \left| l \right>_{n} \left| l \right>_{-n}$, where $\left| l \right>_{m}$ is the $l$-th oscillator eigenstate with respect to the eigenmode given by $\alpha_m^{(\dag)}$ and applying $\exp(- \tau H)$. The expansion coefficients are $a_l^{(n)}=(-1)^l [s(k_n)/c(k_n)]^l/c(k_n)$.
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abstract: 'A general semiclassical method in phase space based on the final value representation of the Wigner function is considered that bypasses caustics and the need to root-search for classical trajectories. We demonstrate its potential by applying the method to the Kerr Hamiltonian, for which the exact quantum evolution is punctuated by a sequence of intricate revival patterns. The structure of such revival patterns, lying far beyond the Ehrenfest time, is semiclassically reproduced and revealed as a consequence of constructive and destructive interferences of classical trajectories.'
author:
- 'Gabriel M. Lando'
- 'Raúl O. Vallejos'
- 'Gert-Ludwig Ingold'
- 'Alfredo M. Ozorio de Almeida'
title: 'Quantum revival patterns from classical phase-space trajectories'
---
Introduction
============
The semiclassical approximation to quantum mechanics is valuable not only when an exact solution is out of reach, but it also sheds light on a quantum system’s classical backbone in both chaotic and integrable scenarios [@Gutzwiller]. It was not until the seminal work of Tomsovic and Heller [@HelTom], however, that semiclassical approximations were shown to describe intricate phenomena in non-trivial time regimes. Indeed, such approximations remained valid for longer than the previously established threshold for accuracy, the Ehrenfest time, at which classical structure finer than a quantum cell starts to develop [@Gutzwiller; @Maia; @Roman]. Motivated by this success, semiclassical methods have since been applied for systems with ever-increasing complexity, testing the limit of what one would consider to be exclusively quantum [@Zi; @Suarez; @Sepulveda].
Among the class of intrinsically quantum phenomena, the so called quantum revival patterns, also known as fractional revivals [@Robinett; @Berry], are a formidable example: They are characterized by shifted and superposed replicas of the initial distribution. Reproducing this phenomenon using the standard semiclassical methods initiated by van Vleck [@Van; @Vleck; @Gutzwiller] is problematic mainly because the semiclassical propagator diverges at regions known as caustics, which proliferate in the time interval required for a revival pattern to form. Even though it is possible to use the more sophisticated uniform approximations [@Ber76], which override infinities, they are valid only locally until the next caustic is met. Such approximations are also unable to amend a standard difficulty in semiclassical propagation, known as the root-search problem, which is the need to seek and select relevant classical trajectories based on boundary conditions.
Initial value representations (IVR) [@Mil70] have been tailored to deal with such difficulties. The Herman-Kluk propagator [@HerKluk; @Kay1; @FGrossmann], for instance, is an integral over classical trajectories defined by their initial values, requiring no root-search. As an off-shot, the caustic singularities are replaced by zeros and a workable approximation for quantum evolution is then achieved [@Mil01; @Mil12].
However, in several applications such as the semiclassical treatment of decoherence [@Zurek], it is desirable to employ methods that have been developed based on the Weyl representation of quantum mechanics. Here, one evolves directly the Wigner function [@Wigner; @Rios; @Dittrich] or its Fourier transform, the chord function, by the Weyl transform of the semiclassical propagator [@Ber89; @Report; @Brodier]. A mere change of variables to initial or final values of the trajectories results in divergence-free phase space evolution [@IVRFVR]. Thus, the desirable features of previous IVR techniques follow naturally, without any need to substitute the standard semiclassical propagators that are derived directly from path integrals [@Gutzwiller].
Whether or not an initial (or final) value representation will be able to remain accurate and unshaken by caustics, which usually spoil standard propagation, must be checked numerically. Since the time evolution under the action of quadratic Hamiltonians is semiclassically exact, such systems are not suited as a testbed. The difficulty is then that higher-order Hamiltonians, which generate non-linear classical motion, do not generally have exactly solvable quantum evolutions. How can one then be certain about features in the semiclassical evolution without an exact result with which to compare them? Indeed, one finds that a vast literature has grown that is based on the convenience of these methods without ever addressing this basic dilemma.
Here, we propose the square of the simple harmonic oscillator, the 4th-order Kerr system, as the ideal benchmark test for a semiclassical method for systems with one degree of freedom. It is singled out by the following properties: (a) One can calculate exactly its intricate quantum evolution, which displays a periodic structure of the aforementioned quantum revival patterns; (b) classical trajectories can be obtained analytically, thus avoiding integration errors; (c) the caustic structure is so complex that it must reveal any shortcoming in the method. Also, since semiclassical methods are only exact for quadratic Hamiltonians, the semiclassical treatment of Kerr propagation is still only an approximation, regardless of the exactness of its classical trajectories. This is the stringent test to which we submit the recently proposed phase space final value representation (FVR) [@IVRFVR].
The Kerr system is also of significant experimental interest. The optical Kerr effect can be emulated, e.g., in a Bose-Einstein condensate confined by a three-dimensional optical lattice [@greiner02]. A three-dimensional circuit quantum electrodynamic architecture was also used to engineer an artificial Kerr medium in order to observe fractional revivals of a coherent state [@kirchmair13]. Due to the extremely weak nonlinearities of most materials, however, collapses and revivals due to the Kerr effect have not yet been observed in optical media.
Direct use of the semiclassical propagator by Toscano *et al.* for the Kerr system did allow for accurate evaluation of the autocorrelation function of a coherent state for long times [@toscano09], but the wave functions were only evaluated far from caustics. Tomsovic *et al.* used sophisticated complex time-dependent semiclassical propagators to accurately calculate autocorrelation functions of Gaussian many-body states of Bose-Hubbard systems with their Kerr-like Hamiltonian beyond the Ehrenfest time [@tomsovic18]. The many-body context was also the motivation of the cruder semiclassical approximation for the Wigner function propagator in [@mathew18], which nonetheless did detect full revivals for the Kerr system through the annihilation operator’s expectation value for an initial coherent state. Moreover, the Herman-Kluk approximation was applied to the $0$-dimensional Bose-Hubbard model: By lifting the wave function into phase space, simple interferences were visually captured for an initial coherent state placed very close to the origin in a Kerr-like system [@grossman]. However, none of the previous explorations attain the high degree of detailed verification of a semiclassical method as we here exhibit for the Kerr evolution.
The text is organized as follows: In Sec. II we introduce the Kerr system, presenting its exact classical and quantum evolutions. This is followed by a description of the FVR method in Sec. III and its results in Sec. IV. We discuss the semiclassical mechanism for revival production in Sec. V and finish the paper with the final remarks of Sec. VI. Movies of quantum and semiclassical evolution of the Wigner function for a coherent state are provided as Supplemental Material.
The Kerr System
===============
The 4th-order Kerr Hamiltonian which we consider here is essentially the square of the Hamiltonian of a simple harmonic oscillator. With an appropriate choice of units for position, momentum, and energy, we can always bring the Kerr Hamiltonian into the form $$H(q, p) = (p^2+q^2)^2\,.
\label{eq:HKerr}$$
Viewing (\[eq:HKerr\]) as a classical Hamiltonian, one finds that $$\omega = 4 (p^2+q^2)$$ is conserved for each orbit, playing the role of an angular frequency. The resulting Hamilton equations of motion can then be solved analytically. Since orbits with a larger radius have higher angular velocities, the initial classical distribution will both revolve around the origin and stretch into a thin filament, as can be seen in the evolution of a coherent state displayed in the left column of Fig. \[fig:wigner\_cl\_qu\_sc\] for four different times $t$.
Quantum mechanically, $q$ and $p$ in (\[eq:HKerr\]) become operators satisfying the commutation relation $[\hat q,\hat p] = i$, where we adopt $\hbar=1$. Introducing the number operator $\hat n$ of the harmonic oscillator, we can express the Kerr Hamiltonian as $$H = (2\hat n+1)^2
\label{eq:HKerr_n}$$ with its Fock eigenstates $\vert n\rangle$.
It has long been noticed [@Stoler1] that under the action of the Kerr Hamiltonian (\[eq:HKerr\_n\]) fractional revivals occur in the quantum evolution of a coherent state $$\vert \alpha \rangle = e^{-\frac{\vert \alpha \vert^2}{2}}
\sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} \vert n \rangle \,
\label{coher}$$ where $\alpha = \left( \langle\hat{q}\rangle + i \langle\hat{p}\rangle
\right)/\sqrt{2}$. Following [@Aver1], we choose times $t=(2a/b)
T_\text{rev}$, where the integers $a$ and $b$ are mutually prime and the time for a complete revival is given by $$T_\text{rev}= \frac{\pi}{4} \, . \label{eq:rev}$$ By choosing sufficiently large integers, any given time can be approximated to the desired accuracy. It can be shown [@Aver1] that for times $t$ of the form introduced here, the evolved coherent state $\vert\alpha(t)\rangle$ can be expressed as a superposition of $b$ or $b/2$ coherent states when $b$ is odd or even, respectively. In particular, for $a=1$ and $b=2m$, one finds a fractional revival pattern of order $m$.
The classical evolution of the Wigner function for a coherent state is obtained by solving the Liouville equation or, equivalently, by propagating the initial Wigner function using the classical equations of motion. The quantum evolution is given by time-evolving under and taking its well-known Wigner transform , to be introduced in the next section. In Fig. \[fig:wigner\_cl\_qu\_sc\] we provide the classical (left column) and quantum (right column) exact Wigner functions for the evolution of an initial coherent state at four distinct times. We see that for $t_1$ the classical backbone is clearly visible in the quantum Wigner function, together with the typical interference patterns. Such clear analogies between classical and quantum behavior are expected up to the Ehrenfest time, $$T_E = \frac{2 \pi}{\omega_c}, \label{eq:ehren}$$ in which the initial distribution’s center, moving with angular velocity $\omega_c$, has performed a full revolution around the origin [@Roman]. For $t_2$, for instance, we have already exceeded $T_E$ and the superposition of multiple interferences masks the relationship to the underlying classical structure. The following panels for $t_3$ and $t_4$ display fractional revival patterns. Their time values of $t_3 = T_\text{rev}/5$ and $t_4 = T_\text{rev}/2$ by far surpass the Ehrenfest time: Not only has the classical filament become quite thin, but the gap between different windings has narrowed to $\mathcal{O}(\hbar^{1/2})$.
![The classical and quantum Wigner functions of a coherent state initially centered at the phase-space point $(q, p) = (5, 0)$ are displayed for four different times, with the largest one exceeding $ 6 T_E$, the Ehrenfest time for this case being given by as $T_E \approx 0.063$. The times $t_3 = T_\text{rev}/5$ and $t_4 = T_\text{rev}/2$ correspond to fractional revivals, where $T_\text{rev}$ is given by . The gray squares in the classical plots represent regions of area $\hbar$.[]{data-label="fig:wigner_cl_qu_sc"}](fig_1_new){width="\linewidth"}
Evolution of the Wigner Function
================================
Our focus is the analysis of the intricate features of the evolved state $|\psi(t)\rangle$, best examined within a full representation in phase space with coordinates $(q,p)$, the position and the momentum, respectively. Among these, the Weyl representation, whose main object is the Wigner function $$W (q, p, t) = \frac{1}{\pi \hbar} \! \int \! d\tilde q\langle q+\tilde q | \psi(t) \rangle\langle\psi (t) | q-\tilde q\rangle
e^{-2i\tilde q p/ \hbar}\,, \label{wig}$$ is equivalent to, e.g, the position or momentum representations of quantum mechanics. In particular, position marginal distributions are given by $$\vert \langle q|\psi(t)\rangle \vert^2 = \int dp \, W (q,p,t) \, ,
\label{eq:marginal}$$ and analogously for momentum. The squared autocorrelation is also obtainable from the Wigner function as $$A^2 (t) = 2 \pi \int dq dp \,W(q,p,t) W (q,p,0) \, .
\label{eq:autoc}$$ Hence, the portrait of the evolved Wigner function is sufficient for testing any semiclassical method.
The divergences of standard semiclassical methods can be traced to the amplitude in the propagator as caustics are approached. A change of variables in the several alternative methods proposed in [@IVRFVR] suppress these divergences, with the further advantage that the FVR preserves its semiclassical form for purely quadratic Hamiltonians.
![A final chord $\xi'$, with center $\text{x}'$, is evolved backwards to form an initial chord $\xi(\xi', \text{x}', t)$, centered at $\text{x}(\xi',\text{x}',t)$. By considering the endpoints of $\xi'$ and $\xi$, such evolution is described by the circuit $\eta_-' \mapsto \eta_- \mapsto \eta_+ \mapsto \eta_+'$. The first and last propagations are performed along classical trajectories (dashed lines), while the middle one is a reflection around $\text{x}$, given by $\eta_+ = 2\text{x} - \eta_-$. The shaded areas $R$ and $R'$ represent the Wigner function at initial and final times, respectively.[]{data-label="fig:quad"}](quad.pdf){width=".7\linewidth"}
The time evolved Wigner function within the FVR approach depends on pairs of trajectories with initial values $\eta_\pm$ and final values $\eta_\pm'$, as depicted in Fig. \[fig:quad\]. The initial center and chord are given by $\text{x} = (\eta_+ + \eta_-)/2$ and $\xi = \eta_+ - \eta_-$, respectively; the final center and chord, $\text{x}'$ and $\xi'$, are defined accordingly. Then, following [@IVRFVR], $$W(\text{x}',t) =\!\! \int \! \frac{d\xi_p'd\xi_q'}{(2\pi)} \left\vert \det \frac{d \xi }{d \xi'} \! \right\vert^\frac{1}{2} \! \! \! \exp \left\{ \! i \! \left[ \tilde{S}_{\text{x}'}(\xi) \! + \! \frac{\tilde{\sigma} \pi}{2}\right] \! \right\} \! \chi \left( \xi \right) \, , \label{fvr}$$
![Density plot for the determinant in in the $(\xi_q',\xi_p')$-plane at times $t=0.013$ and $t=0.071$ for the final Wigner function evaluated at $(q,p)=(5,2)$. The caustic submanifolds at which the original root-search based propagator diverges are displayed as solid black curves. The initial state is the same as for Fig. \[fig:wigner\_cl\_qu\_sc\].[]{data-label="fig:caustics"}](fig_3){width="\linewidth"}
$\!\!$which is expressed in terms of functions that we now describe. The argument $\xi$ of both $\chi$ and the action $ \tilde{S}_{\text{x}'}$ is defined in the present FVR as the backward classical propagation of the integration variables $\xi' = (\xi'_q, \xi'_p)$. The initial state, on which absolutely no restriction is placed, enters through the symplectic Fourier transform of its Wigner function at $t=0$, $$\chi \left( \xi \right) = \int \frac{dy}{(2\pi)} \, \exp \left( -i y \cdot J \xi \right) W(y,0)\,,$$ $J$ being the symplectic matrix. Note that, for coherent states, both $\chi(\xi)$ and $W(y,0)$ are Gaussians. The exponential in (\[fvr\]) arises from a semiclassical approximation of an evolved phase space reflection [@IVRFVR; @Brodier]. The choice of final instead of initial values as integration variables is just a matter of choice when classical trajectories are exact, but preferable when only numerical solutions are available [@footnote2]. Considering that the initial chord $\xi$ is determined by the pair of trajectories propagated backward from $\eta_\pm'$, the first term in the phase of (\[fvr\]) can be written as $$\tilde{S}_{\text{x}'}(\eta'_\pm, \eta_\pm) = \eta_+ \cdot J \eta_- + t \Delta H(\eta_\pm) - \oint_{\mathcal{C}(\eta_\pm',\eta_\pm)} p \, dq \, . \label{sss}$$ Here, $\Delta H(\eta_\pm) = H(\eta_+) - H(\eta_-)$ is the energy difference between the endpoints of $\xi$ and $\mathcal{C} (\eta_\pm',\eta_\pm) $ is the closed contour given by $\eta_-' \mapsto \eta_- \mapsto \eta_+ \mapsto \eta_+' \mapsto \eta_-'$, detailed in Fig. \[fig:quad\]. The term $\tilde{\sigma}$ in (\[fvr\]) counts the zeros of the determinant in the prefactor up to time $t$ and is related to the Maslov index [@Littlejohn; @OAI]. The regions defined by this determinant’s kernel are exactly the caustic submanifolds, which form a complex web in the Kerr system, as can be seen in Fig. \[fig:caustics\].
Semiclassical Evolution for the Kerr Hamiltonian
================================================
Semiclassical propagation depends entirely on classical trajectories and must not only reproduce the interferences between the classical whirl and itself, but also eventually cancel them in large phase-space regions at fractional revival patterns (cf. panels for $t_3$ and $t_4$ in the right column of Fig. \[fig:wigner\_cl\_qu\_sc\]). Fig. \[fig:wigner\_sc\] shows that, despite the complex caustic structure and large interference-free regions, the FVR method successfully reproduces fractional revival patterns that occur for times much longer than the Ehrenfest time.
![The semiclassical Wigner functions obtained from the FVR in Eq. (left column) for the same time values as in Fig. \[fig:wigner\_cl\_qu\_sc\]. We repeat the exact quantum equivalents for comparison (right column). Notice how the FVR is able to transform the thin classical spirals in Fig. \[fig:wigner\_cl\_qu\_sc\] into pentagonal and cat states. For more details about the time evolution see the movies in the Supplemental Material.[]{data-label="fig:wigner_sc"}](fig_4_new){width="\linewidth"}
It should be recalled that standard semiclassical methods based on root-search are limited to initial states that are either initial coherent states [@HelTom; @Littlejohn] or approximate WKB states [@Gutzwiller; @Maia]. A shifted first excited Fock state lies outside both these classes, but as Fig. \[fig:fock\] demonstrates, our FVR approximation captures its full time evolution just as easily as that of initial coherent states.
![(a) Semiclassical (left) and quantum (right) Wigner functions for the triangular revival at $t=\pi/12$. The initial state is a displaced $n=1$ Fock state centered at $(q,p)=(5,0)$. (b) Position marginal distributions obtained from for the post-normalized Wigner functions displayed in (a).[]{data-label="fig:fock"}](fig_5){width="\linewidth"}
The squared autocorrelation obtained from the semiclassical and quantum Wigner functions using is displayed in Fig. \[fig:autocorr\]. It again affirms the FVR accuracy. Note that due to the fine semiclassical undulations inside the revived coherent states (see Fig. \[fig:wigner\_sc\]), the Wigner function loses a small fraction of its normalization – a homogeneous loss that can be corrected by simple post-normalization. This loss is due to a subset of trajectories that do not contribute to the mechanism described in the following section.
![Comparison of the time dependence of the squared autocorrelation function for the quantum result (line) and the semiclassical result after post-normalization (points) for the initial state used in Fig. \[fig:wigner\_cl\_qu\_sc\]. Beyond half the revival time $t=\pi/8 \approx 0.393$, $A^2 (t)$ is seen to continue symmetrically. In order to make the fine structure more visible, we have left out part of the time interval.[]{data-label="fig:autocorr"}](fig_7){width="\linewidth"}
Semiclassical Mechanism for Revival Patterns
============================================
Some light can be shed on how the classical trajectories combine to create revivial patterns. We start with the full revival at $T_\text{rev} = \pi/4$. Here, the final Wigner function is equal to the initial one, implying that the final chords $\xi'$ must be backwards evolved near themselves, i.e. $\xi' \approx \xi$. Since the orbits are circles, this condition fixes contributing $\xi'$ chords as the ones whose endpoints $\eta_+'$ and $\eta_-'$ perform an integer number of complete revolutions around the origin, finishing near their initial values $\eta_+$ and $\eta_-$. In terms of the orbits’ angular frequencies, $$\begin{aligned}
\omega_\pm T_\text{rev} = 4|\eta_\pm|^2 T_\text{rev} = 2 \pi j_\pm \, , \label{eq:times}\end{aligned}$$ where we define the winding numbers $j_\pm$ for $\eta_\pm$. The pairs of orbits whose radii lie in between the successive “time-quantized" values define long chords with rapidly oscillating phases that cancel out: Their absence is responsible for the fine undulations in the revived coherent states and affects the Wigner function’s normalization.
Substituting the variables in in Eq. , $$\tilde{S}_{\text{x}'}(\eta'_\pm, \eta_\pm) - \eta_+ \cdot J \eta_- = \left( \frac{\pi^2}{4 T_\text{rev}} \right)(j_+^2 - j_-^2) \, ,$$ and since the winding numbers for this case are exactly the Maslov contributions, the final phase \[modulo a symplectic Fourier transform\] is finally given by $$\tilde{S}_{\text{x}'}(\eta'_\pm, \eta_\pm) + \frac{\tilde{\sigma} \pi}{2} = \pi (j_+-j_-) [1 + (j_+ + j_-)] \, . \label{eq:phase}$$ Since $j_+ - j_-$ and $j_+ + j_-$ have the same parity, the phase in is always an even multiple of $\pi$. The relevant final chords are, therefore, selected such that the final Wigner function is localized exactly over the initial one, reproducing the complete revival as expected.
For fractional revivals with times $t = \pi/\beta$, the only difference is that the relevant final chords might perform fractional revolutions around the origin. We can express this condition as $$\begin{aligned}
\frac{\omega_\pm \pi}{\beta} = \frac{4|\eta_\pm|^2 \pi}{\beta} = 2 \pi (j_\pm + \alpha) \, , \label{eq:times2}\end{aligned}$$ where $\alpha$ is a rational number that reflects the positions of the final coherent states and thus depends on $\beta$. In contrast to the revived coherent states, the interference patterns appearing near the origin for the cat state revival ($\beta = 8$) are due exclusively to long chords, typically spanning the diameter of the classical spiral.
Conclusion
==========
The ability to reproduce the complete kaleidoscopic evolution for the quantum Kerr system on the basis of classical phase space trajectories is indisputable evidence for our FVR method. The undulations visible in the semiclassical Wigner functions play no role in extracting typical quantum mechanical quantities obtained by integration, a process which filters out residual classical fine structures. Preliminary results evince that our FVR method is not restricted to exact classical systems, demonstrating the generality of this approach [@new].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Jonas Bucher and Fabricio Toscano for stimulating discussions. Partial financial support from CNPq and the National Institute for Science and Technology: Quantum Information is gratefully acknowledged.
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|
DESY 07-117
7.0[**Next-to-leading order corrections in\
exclusive meson production**]{} 4.0M. Diehl and W. Kugler\
*Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany* 5.0**Abstract**\
3.0
Introduction {#sec:intro}
============
Generalized parton distributions (GPDs) have developed into a versatile tool to quantify important aspects of hadron structure in QCD. In particular they contain unique information on the transverse spatial distribution of partons [@Burkardt:2002hr] and on spin-orbit effects and orbital angular momentum inside the nucleon [@Ji:1996ek; @Burkardt:2005km]. Deeply virtual Compton scattering is widely recognized as the process providing the theoretically cleanest access to GPDs, with a wealth of observables calculable in the large $Q^2$ limit [@Belitsky:2001ns] and with the calculation of the hard-scattering subprocess now pushed to next-to-next-to-leading order (NNLO) accuracy in $\alpha_s$ [@Kumericki:2007sa]. A quantitative theoretical description of exclusive meson production remains a challenge. It would offer the possibility to obtain important complementary information, difficult to obtain from Compton scattering alone. Perhaps most importantly, vector meson production is directly sensitive to gluon distributions, which in the Compton process are $\alpha_s$ suppressed relative to quark distributions and only accessible through scaling violation (just as in the well-known case of inclusive deep inelastic scattering). Given in addition the large number of channels that can be studied and the wealth of high-quality data in a wide range of kinematics from collider to fixed-target energies [@Aharon:2007bq; @Hadjidakis:2004jc], it should be worthwhile to try and push the theory description of exclusive meson production as far as possible.
In this work we study the exclusive production of light mesons at large photon virtuality $Q^2$ within the framework of collinear factorization [@Collins:1996fb]. In Bjorken kinematics, the process amplitude can be approximated by the convolution of hard-scattering kernels with generalized parton distributions and the quark-antiquark distribution amplitude of the produced meson. The hard-scattering kernels have been calculated to $O(\alpha_s^2)$, i.e.to next-to-leading order (NLO) accuracy [@Melic:1998qr; @Belitsky:2001nq; @Ivanov:2004zv]. The aim of the present paper is to investigate in some detail the size of the NLO corrections compared with the leading-order (LO) results, on which phenomenological studies have so far relied.
The collinear factorization approach provides an approximation of the leading helicity amplitudes for meson production in the Bjorken limit, up to relative corrections of order $1/Q^2$. These power corrections cannot be calculated systematically (and in fact the derivation [@Collins:1996fb] of the factorization theorem suggests that these corrections do not all factorize into hard-scattering kernels and nonperturbative quantities pertaining to either the nucleon or the produced meson). One particular source of power corrections can however readily be identified, namely the effect of the transverse momentum of partons entering the hard-scattering subprocess, which in the collinear approximation is neglected in the calculation of the hard-scattering kernel. A number of approaches include these $k_T$ effects, in particular the studies in [@Vanderhaeghen:1999xj; @Goloskokov:2006hr] based on the modified hard-scattering formalism of Sterman et al. [@Botts:1989kf], and calculations like [@Frankfurt:1995jw] which are based on the color dipole formulation. In the work by Martin, Ryskin and Teubner [@Martin:1996bp], parton-hadron duality is used to model the meson formation and thus the transverse momentum of the hadronizing quarks is included in the calculation, whereas the transverse momentum of gluons in the proton is treated within high-energy $k_T$ factorization. The studies just quoted agree in that transverse momentum effects result in substantial power corrections to the collinear approximation for $Q^2$ up to several ${\operatorname{GeV}}^2$. Unfortunately, the calculation of full NLO corrections in $\alpha_s$ remains not only a practical but also a conceptual challenge in all of these approaches, so that the perturbative stability of their results cannot be investigated at present. (The approach of Sterman et al.takes partial account of radiative corrections, resumming a certain class of them into Sudakov form factors.)
A consistent simultaneous treatment of radiative and power corrections being out of reach at this time, a possible strategy is to study the NLO corrections in the collinear approximation and in particular to identify kinematical regions where these corrections are moderate or small. There one can then use with greater confidence formulations incorporating power corrections. In this spirit the present investigation should be understood. We will study both the cross section for meson production from an unpolarized target and the transverse target polarization asymmetry. This asymmetry is one of the few observables sensitive to the nucleon helicity-flip distributions (in particular for gluons) and hence to the spin-orbit and orbital angular momentum effects mentioned above. We will in particular see whether corrections tend to cancel in this polarization asymmetry, as is often assumed.
In the bulk of this paper we concentrate on the production of vector mesons. In Sect. \[sec:kernels\] we set up our notation and recall important properties of the hard-scattering kernels at NLO, as well as giving a one-variable representation of these kernels after Gegenbauer expansion of the meson distribution amplitude. In Sect. \[sec:models\] we specify the model of the generalized parton distributions $H$ and $E$ we use for our numerical studies. The size of radiative corrections involving convolutions with distributions $H$ is then studied in Sects. \[sec:small-x\] and \[sec:large-x\] for small and large $x_B$, respectively, and the convolutions involving distributions $E$ are quantified in Sect. \[sec:E-convolutions\]. In Sect. \[sec:cross\] we then look at the NLO corrections at the level of the observable cross section and polarization asymmetry. A brief study of exclusive pion production in Sect. \[sec:pseudo\] complements our work, and in Sect. \[sec:sum\] we summarize our main findings. A number of more lengthy formulae is collected in appendices.
Hard-scattering kernels {#sec:kernels}
=======================
In the main part of this paper we are concerned with exclusive production of a vector meson $$\label{vector-prod}
\gamma^*(q) + p(p) \to V(q') + p(p')$$ in the limit of large $Q^2 = -q^2$ at fixed Bjorken variable $x_B =
Q^2 /(2 p\cdot q)$ and fixed $t = (p-p')^2$. To leading order in $1/Q$, the amplitude for longitudinal polarization of photon and meson can be written as $$\begin{aligned}
\label{meson_amp_NLO}
\mathcal{M} &=
\frac{2\pi\sqrt{4\pi\alpha}}{\xi{\mskip 1.5mu}Q N_c}\, Q_V f_V
\int_0^1 dz\, \phi_V(z)
\int_{-1}^1 dx\, \biggl\{ T_g(z,x,\xi) {\mskip 1.5mu}F^g(x,\xi,t)
\nonumber \\
&\qquad
+ \frac{1}{n_f} \Bigl[ T_a({\bar{z}},x,\xi) - T_a(z,-x,\xi) \Bigr] {\mskip 1.5mu}F^{S}(x,\xi,t)
+ T_b(z,x,\xi) {\mskip 1.5mu}F^{S}(x,\xi,t)
\phantom{\biggl[ \biggr]}
\nonumber \\
&\qquad
+ e_{V}^{(3)}\, \Bigl[ T_a({\bar{z}},x,\xi) - T_a(z,-x,\xi) \Bigr] \,
\Bigl[ F^{u(+)}(x,\xi,t) - F^{d(+)}(x,\xi,t) \Bigr]
\phantom{\biggl[ \biggr]}
\nonumber \\
&\qquad
+ e_{V}^{(8)}\, \Bigl[ T_a({\bar{z}},x,\xi) - T_a(z,-x,\xi) \Bigr] \,
\Bigl[ F^{u(+)}(x,\xi,t) + F^{d(+)}(x,\xi,t)
- 2 F^{s(+)}(x,\xi,t) \Bigr]
\biggr\}\end{aligned}$$ with ${\bar{z}}= 1-z$, $N_c=3$, and the electromagnetic fine structure constant $\alpha$. Throughout this paper we work with $n_f=3$ active quark flavors. The proton matrix elements $F$ are parameterized by generalized parton distributions, $$\label{GPD-def}
F^{q,g}(x,\xi,t) =
\frac{1}{(p+p')\cdot n} \left[
H^{q,g}(x,\xi,t)\, \bar u(p') {\mskip 1.5mu}\slashed{n}{\mskip 1.5mu}u(p)
+ E^{q,g}(x,\xi,t)\, \bar u(p')\,
\frac{i\sigma^{\alpha\beta}n_{\alpha} (p'-p)_\beta}{2 m_p}\, u(p)
\right]$$ for quarks and gluons, where we use the conventions of [@Diehl:2003ny]. Here $n$ is a light-like auxiliary vector, $\xi
= x_B /(2-x_B)$ is the skewness variable, and $m_p$ denotes the nucleon mass. We have further introduced the combination $$\label{def-Fplus}
F^{q(+)}(x,\xi,t) = F^q(x,\xi,t) - F^q(-x,\xi,t)$$ with positive charge conjugation parity. In we have arranged the terms containing quark distributions into the flavor singlet $$\begin{aligned}
\label{def-FS}
F^{S} &= F^{u(+)} + F^{d(+)} + F^{s(+)}\end{aligned}$$ and the flavor triplet and octet combinations, $F^{u(+)} - F^{d(+)}$ and $F^{u(+)} + F^{d(+)} - 2F^{s(+)}$. The factors $$\begin{aligned}
\label{charge_factors}
Q_\rho &= \tfrac{1}{\sqrt{2}} \,,
&
Q_\omega &= \tfrac{1}{3\sqrt{2}} \,,
&
Q_\phi &= -\tfrac{1}{3}\end{aligned}$$ and $$\begin{aligned}
\label{quark_charge_factors}
e_{\rho}^{(3)} &= e_{\rho}^{(8)} = e_{\omega}^{(8)} = \tfrac{1}{6} \,,
&
e_{\omega}^{(3)} &= \tfrac{3}{2} \,,
&
e_{\phi}^{(3)} &= 0 \,,
&
e_{\phi}^{(8)} &= -\tfrac{1}{3}\end{aligned}$$ correspond to a respective flavor content $$\tfrac{1}{\sqrt{2}} \bigl( |u\bar{u}\rangle-|d\bar{d}\rangle \bigr) \,,
\qquad
\tfrac{1}{\sqrt{2}} \bigl( |u\bar{u}\rangle+|d\bar{d}\rangle \bigr) \,,
\qquad
|s\bar{s}\rangle$$ of the $\rho$, $\omega$ and $\phi$. The meson distribution amplitudes $\phi_V(z)$ are normalized as $\int_0^1 dz\, \phi_V(z) = 1$, and the decay constants have the values $f_\rho = 209 {\operatorname{MeV}}$, $f_\omega = 187
{\operatorname{MeV}}$, $f_\phi = 221 {\operatorname{MeV}}$ [@Beneke:2003zv]. We finally have hard-scattering kernels in , where $T_g$ goes with gluon and $T_a$, $T_b$ go with quark distributions in the proton. In the graphs for $T_a$ quark lines connect the proton and meson side, whereas in the graphs for $T_b$ the proton and meson side are only connected by gluon lines. $T_b$ thus starts at order $\alpha_s^2$ and only goes with the quark singlet distribution $F^S$. Example graphs for the three kernels at NLO are shown in Fig. \[fig:graphs\]. We will refer to $T_g$, $T_a$, $T_b$ as the gluon, the quark non-singlet, and the pure quark singlet kernel, respectively.
![\[fig:graphs\] Example graphs for the hard-scattering kernels $T_a$, $T_b$ and $T_g$ at order $\alpha_s^2$.](plots/mesons.eps){width="\textwidth"}
For better legibility we have not displayed the dependence on the renormalization and factorization scales in . The renormalization scale $\mu_R$ appears as argument of $\alpha_s$ and through explicit logarithms in the hard-scattering kernels $T$. The kernels further contain logarithms of the respective factorization scales $\mu_{DA}$ and $\mu_{GPD}$ for the meson distribution amplitude and the generalized parton distributions. The NLO kernels in [@Belitsky:2001nq; @Ivanov:2004zv] are given for a common factorization scale $\mu_F = \mu_{DA} = \mu_{GPD}$. We can restore the individual logarithms of $\mu_{DA}$ and $\mu_{GPD}$ from the requirement that within the calculated precision the process amplitude must be independent of these scales. As an example consider the term $$\begin{aligned}
\label{scale-derivative}
& \frac{{\mathrm{d}}}{{\mathrm{d}}\ln\mu_{DA}^{2} \rule{0pt}{0.9em}}\, \int_0^1 dz\,
\phi_V(z; \mu_{DA})\,
T_a({\bar{z}},x,\xi;{\mskip 1.5mu}\alpha_s(\mu_R),\mu_R, \mu_{GPD},\mu_{DA},Q)
\nonumber \\[0.3em]
&\qquad
= \int_0^1 dz\,
\biggl[ \frac{{\mathrm{d}}}{{\mathrm{d}}\ln\mu_{DA}^{2} \rule{0pt}{0.9em}}\,
\phi_V(z; \mu_{DA}) \biggr]\,
T_a({\bar{z}},x,\xi;{\mskip 1.5mu}\alpha_s(\mu_R),\mu_R, \mu_{GPD},\mu_{DA},Q)
\nonumber \\[0.2em]
&\qquad
+ \int_0^1 dz\, \phi_V(z; \mu_{DA})\,
\biggl[ \frac{{\mathrm{d}}}{{\mathrm{d}}\ln\mu_{DA}^{2} \rule{0pt}{0.9em}}\,
T_a({\bar{z}},x,\xi;{\mskip 1.5mu}\alpha_s(\mu_R),\mu_R, \mu_{GPD},\mu_{DA},Q)
\biggr] \,,\end{aligned}$$ where the scale dependence of $\phi_V(z; \mu_{DA})$ is given by the ERBL evolution equation [@Efremov:1980qk]. At leading order this gives a term ${\mathrm{d}}/{\mathrm{d}}(\ln\mu_{DA}^{2}{\mskip 1.5mu})\, \phi_V(z; \mu_{DA})$ of order $\alpha_s$, whose convolution with the $O(\alpha_s)$ part of $T_a$ must cancel against the contribution from explicit logarithms of $\mu_{DA}$ in the $O(\alpha_s^2)$ part of $T_a$. An analogous argument holds for the dependence on $\mu_{GPD}$, with the complication that the gluon and quark singlet distributions mix under evolution. More precisely, the convolution of ${\mathrm{d}}/{\mathrm{d}}(\ln\mu_{GPD}^{2}{\mskip 1.5mu})\, F^S(x,\xi,t; \mu_{GPD})$ with the $O(\alpha_s)$ part of $T_a$ cancels at $O(\alpha_s^2)$ against the contributions from logarithms of $\mu_{GPD}$ in $T_a$ and in $T_g$. Likewise, the convolution of ${\mathrm{d}}/{\mathrm{d}}(\ln\mu_{GPD}^{2}{\mskip 1.5mu})\,
F^g(x,\xi,t; \mu_{GPD})$ with the Born term of $T_g$ cancels at $O(\alpha_s^2)$ against the contributions from logarithms of $\mu_{GPD}$ in $T_g$ and in the pure singlet kernel $T_b$. We have explicitly checked that the scale dependence of the hard-scattering kernels given in [@Ivanov:2004zv] cancels in the process amplitude as just described, using the LO evolution equations for GPDs given in App. \[app:evolution\].
Separating the $\mu_{DA}$ and $\mu_{GPD}$ dependence, we can write the kernels as $$\begin{aligned}
\label{orig_kernels}
T_g(z,x,\xi) &=
- \alpha_s\,
\frac{\xi}{(\xi-x-i\epsilon) (\xi+x-i\epsilon)}\, \frac{1}{z{\bar{z}}}
\biggl[1 + \frac{\alpha_s}{4\pi}\;
\mathcal{I}_g\Bigl( z,\frac{\xi-x}{2\xi} \Bigr)
\biggr] \,,
\nonumber \\
T_b(z,x,\xi) &=
\phantom{-}C_F\, \frac{\alpha_s^2}{8\pi}\,
\frac{1}{z{\bar{z}}}\; \mathcal{I}_b\Bigl( z,\frac{\xi-x}{2\xi} \Bigr) \,,
\nonumber \\
T_a({\bar{z}},x,\xi) &=
- C_F{\mskip 1.5mu}\alpha_s\, \frac{\xi}{\xi-x-i\epsilon}\, \frac{1}{{\bar{z}}}\,
\biggl[ 1 + \frac{\alpha_s}{4\pi}\;
\mathcal{I}_a\Bigl( {\bar{z}},\frac{\xi-x}{2\xi} \Bigr)
\biggr]\end{aligned}$$ with $$\begin{aligned}
\label{I_gluon}
\mathcal{I}_g(z,y) &= \left[{\mskip 1.5mu}2 C_A \left(\frac{{\bar{y}}}{y}+\frac{y}{{\bar{y}}}\right)
\bigl( y\ln y + {\bar{y}}\ln{\bar{y}}\bigr)
- C_F \left( \frac{y}{{\bar{y}}} \ln y + \frac{{\bar{y}}}{y} \ln{\bar{y}}\right)
\right] \ln\frac{Q^2}{\mu^2_{GPD}}
\nonumber \\
&\quad
+ \beta_0 \ln\frac{\mu^2_{R}}{\mu_{GPD}^2}
+ C_F{\mskip 1.5mu}\bigl( 3 + 2 z \ln{\bar{z}}+ 2 {\bar{z}}\ln z \bigr)
\ln\frac{Q^2}{\mu^2_{DA}}
+ \mathcal{K}_g(z,y) \,,
\nonumber \\
\mathcal{I}_b(z,y) &=
2 ({\bar{y}}-y) \left( \frac{\ln y}{\bar{y}}+\frac{\ln\bar{y}}{y} \right)
\ln\frac{Q^2}{\mu_{GPD}^2}
+ \mathcal{K}_b(z,y)
\intertext{and}
\label{I_a}
\mathcal{I}_a(v,u) &=
\beta_0{\mskip 1.5mu}\biggl( \frac{5}{3} - \ln(vu)
- \ln\frac{Q^2}{\mu^2_R} \biggr)
+ C_F\, \bigl( 3 + 2\ln u \bigr) \ln\frac{Q^2}{\mu^2_{GPD}}
+ C_F\, \bigl( 3 + 2\ln v \bigr) \ln\frac{Q^2}{\mu^2_{DA}}
\nonumber \\
&\quad
+ \mathcal{K}_a(v,u) \,,
\phantom{\biggl[ \biggr]}\end{aligned}$$ where ${\bar{y}}= 1-y$ and we use the standard notation $$\begin{aligned}
\label{color_factors}
C_F &= \frac{N_c^2-1}{2N_c} \,,
&
C_A &= N_c \,,
&
\beta_0 &= \frac{11}{3}{\mskip 1.5mu}N_c - \frac{2}{3}{\mskip 1.5mu}n_f \,.\end{aligned}$$ The functions $\mathcal{K}_g$, $\mathcal{K}_b$ and $\mathcal{K}_a$ are independent of $Q^2$ and the renormalization and factorization scales. They contain factors $C_F$ or $C_A$ but not $\beta_0$. Their expressions can be found in [@Ivanov:2004zv], taking into account that the kernels $T_g$ and $T_b$ here are denoted by $T_g$ and $T_{(+)}$ there, and that $$\begin{aligned}
\label{dima-trans}
T(v,u) \,\bigg|_{\text{\protect\cite{Ivanov:2004zv}}}
&= \frac{C_F{\mskip 1.5mu}\alpha_s}{4 v u}
\biggl[ 1 + \frac{\alpha_s}{4\pi}\;
\mathcal{I}_a(v,u) \biggr]_{\text{here}} \,,
&
y \,\big|_{\text{\protect\cite{Ivanov:2004zv}}}
&= - y \,\big|_{\text{here}} \,.\end{aligned}$$ Note that the pure singlet kernel $T_b$ does not contain logarithms of $\mu_{DA}$ and $\mu_{R}$ at $O(\alpha_s^2)$, since there is no Born level contribution against which they could cancel in the scale dependence of the process amplitude. There is however a logarithm of $\mu_{GPD}$, since the corresponding derivative of the Born level convolution of $T_g$ with $F^g$ contains a term going with the quark singlet distribution $F^S$, as already mentioned after .
The kernels in have singularities for real-valued arguments. One readily finds that $x/\xi = (\hat{s} - \hat{u}) /Q^2$, where $\hat{s}$ and $\hat{u}$ are the Mandelstam variables for the parton-level subprocess $\gamma^* q\to (q\bar{q}){\mskip 1.5mu}q$ or $\gamma^*
g\to (q\bar{q}){\mskip 1.5mu}g$. The prescriptions $\hat{s}+i\epsilon$ for the $\hat{s}$-channel and $\hat{u}+i\epsilon$ for the $\hat{u}$-channel singularities thus instruct us to take $x+i\epsilon$ for $x>0$ and $x-i\epsilon$ for $x<0$. Correspondingly, the second argument $(\xi -
x)/(2\xi)$ of $\mathcal{I}_g$, $\mathcal{I}_b$ and $\mathcal{I}_a$ must be taken with $-i\epsilon$ for $x>0$ and $+i\epsilon$ for $x<0$. In $T_a(z,-x,\xi)$ the second argument of $\mathcal{I}_a$ is $(\xi +
x)/(2\xi)$, which has to be taken with $-i\epsilon$ for $x<0$ and $+i\epsilon$ for $x>0$. We remark that, as it is written, the $i\epsilon$ prescription in [@Ivanov:2004zv] for the gluon and the pure singlet kernel is correct for $x>0$ but incorrect for $x<0$. Likewise, the prescription given in [@Belitsky:2001nq; @Ivanov:2004zv] for the quark non-singlet kernel is correct for $x>0$ if the corresponding argument is $(\xi - x)/(2\xi)$ and for $x<0$ if the argument is $(\xi +
x)/(2\xi)$, but incorrect in the other cases.[^1]
Gegenbauer expansion {#sec:gegenbauer}
--------------------
Let us expand the meson distribution amplitude on Gegenbauer polynomials, $$\label{gegen-phi}
\phi_V(z;\mu) = 6 z(1-z) \sum_{n=0}^\infty
a_n(\mu)\, C_n^{3/2}(2z-1) \,,$$ where $a_0 = 1$ according to the normalization condition $\int_0^1
dz\, \phi_V(z) = 1$. To leading order, the Gegenbauer coefficients evolve as $$\label{DA-evol}
a_n(\mu) = a_n(\mu_0)\,
\left( \frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}
\right)^{\gamma_n/\beta_0}$$ with anomalous dimensions $$\begin{aligned}
\label{anom-dim}
\gamma_0 &= 0 \,, & \gamma_2 &= \tfrac{25}{6}\, C_F \,, &
\gamma_4 &= \tfrac{91}{15}\, C_F \,,\end{aligned}$$ where $\alpha_s(\mu)$ is the running coupling at one-loop accuracy. One has $\gamma_n \approx 4 C_F \ln(n+1)$ within at most $6\%$ for all $n$. For $V=\rho, \omega, \phi$ only coefficients $a_n$ with even $n$ are nonzero due to charge conjugation invariance, and in all subsequent expressions of this paper we consider $n$ to be even. Calculations of the distribution amplitudes in models or on the lattice typically give values for the first or the first two nonvanishing moments, see e.g.[@Ball:1996tb; @Bakulev:2005cp; @Braun:2006dg], so that a truncated version of the expansion is very often used in phenomenological studies. Convolution with individual terms in also allows us to reduce the hard-scattering kernels for meson production to functions of a single longitudinal variable. More precisely, we can rewrite the process amplitude as $$\begin{aligned}
\label{gegen-amp}
\mathcal{M} =
\frac{2\pi\sqrt{4\pi\alpha}}{\xi{\mskip 1.5mu}Q N_c}\, Q_V f_V
\sum_{n=0}^\infty a_n^{}{\mskip 1.5mu}\biggl[ \mathcal{F}^g_n + \mathcal{F}^{S(a)}_n + \mathcal{F}^{S(b)}_n
+ e_{V}^{(3)}\, \mathcal{F}^{(3)}_{n\phantom{V}}
+ e_{V}^{(8)}\, \mathcal{F}^{(8)}_{n\phantom{V}} \biggr]\end{aligned}$$ with convolutions in $x$ $$\begin{aligned}
\label{F-def}
\mathcal{F}^g_n
&= \int_{-1}^1 dx\, T_{g,n}(x,\xi){\mskip 1.5mu}F^g(x,\xi,t) \,,
\qquad\qquad\qquad
\mathcal{F}^{S(b)}_n
= \int_{-1}^1 dx\, T_{b,n}(x,\xi){\mskip 1.5mu}F^{S}(x,\xi,t) \,,
\nonumber\\
\mathcal{F}^{S(a)}_n
&= \int_{-1}^1 dx\, \Bigl[ T_{a,n}(x,\xi) - T_{a,n}(-x,\xi) \Bigr] {\mskip 1.5mu}\frac{1}{n_f}\, F^{S}(x,\xi,t) \,,
\nonumber \\
\mathcal{F}^{(3)}_n
&= \int_{-1}^1 dx\, \Bigl[ T_{a,n}(x,\xi) - T_{a,n}(-x,\xi) \Bigr] \,
\Bigl[ F^{u(+)}(x,\xi,t) - F^{d(+)}(x,\xi,t) \Bigr] \,,
\displaybreak
\nonumber \\
\mathcal{F}^{(8)}_n
&= \int_{-1}^1 dx\, \Bigl[ T_{a,n}(x,\xi) - T_{a,n}(-x,\xi) \Bigr] \,
\Bigl[ F^{u(+)}(x,\xi,t) + F^{d(+)}(x,\xi,t)
- 2 F^{s(+)}(x,\xi,t) \Bigr] \,,\end{aligned}$$ which depend on $\xi$ and $t$, and logarithmically on $Q^2$ and on the factorization and renormalization scales. At order $\alpha_s^2$ the dependence on $\mu_R$ and on $\mu_{DA}$ cancels in each separate convolution, while the dependence on $\mu_{GPD}$ cancels in $\mathcal{F}^{\smash{(3)}}_{n\phantom{i}}$ and $\mathcal{F}^{\smash{(8)}}_{n\phantom{i}}$ and in the sum $\mathcal{F}^{\smash[b]{g}}_n + \mathcal{F}^{\smash{S(a)}}_n +
\mathcal{F}^{\smash{S(b)}}_n$ as discussed after . In analogy to we define convolutions $\mathcal{H}$ and $\mathcal{E}$ for the individual distributions $H$ and $E$ in . The kernels $T_{g,n}$, $T_{a,n}$, $T_{b,n}$ are obtained from $T_g$, $T_a$, $T_b$ by multiplying with $6 z(1-z)\, C_{\smash{n}}^{3/2}(2z -
1)$ and integrating over $z$. For $n=0$ we find $$\begin{aligned}
\label{gegen-kernels}
T_{g,n}(x,\xi) &= - 3 \alpha_s\,
\frac{2\xi}{(\xi-x-i\epsilon) (\xi+x-i\epsilon)}
\biggl[ 1 + \frac{\alpha_s}{4\pi}\,
t_{g,n}\left( \frac{\xi-x}{2\xi} \right) \biggr] \,,
\nonumber \\
T_{b,n}(x,\xi) &= \phantom{-}3 C_F\, \frac{\alpha_s^2}{4\pi}\,
t_{b,n}\left( \frac{\xi-x}{2\xi} \right) \,,
\nonumber \\
T_{a,n}(x,\xi) &= - 3 C_F\, \alpha_s\,
\frac{\xi}{\xi-x-i\epsilon}\,
\biggl[ 1 + \frac{\alpha_s}{4\pi}\,
t_{a,n}\left( \frac{\xi-x}{2\xi} \right) \biggr]\end{aligned}$$ with $$\begin{aligned}
\label{kernels_asy}
t_{g,0}(y) &=
\biggl[ 2C_A{\mskip 1.5mu}(y^2+{\bar{y}}^2) - C_F{\mskip 1.5mu}y \biggr] \frac{\ln y}{{\bar{y}}}\,
\ln\frac{Q^2}{\mu^2_{GPD}}
+ \frac{\beta_0}{2}{\mskip 1.5mu}\ln\frac{\mu_R^2}{\mu^2_{GPD}}
\nonumber \\
&\quad + C_F \biggl[
- \frac{5}{2}
+ \left(\frac{1}{{\bar{y}}}+1-4y\right) \ln y
- \frac{y}{2}\, \frac{\ln^2 y}{{\bar{y}}}
\nonumber \\
& \qquad\qquad
- 2 ({\bar{y}}-y) {\operatorname{Li}}_2{\bar{y}}- 4y{\bar{y}}\biggl(3{\operatorname{Li}}_3{\bar{y}}- \ln y\, {\operatorname{Li}}_2 y
- \frac{\pi^2}{6} \ln y \biggr)
\biggr]
\nonumber \\
&\quad + C_A \left[
- \left( \frac{6}{{\bar{y}}} - 8y \right) \ln y
+ \left( \frac{1}{{\bar{y}}} - 2y \right) \ln^2 y + 2({\bar{y}}-y){\operatorname{Li}}_2{\bar{y}}\right]
+ \{y\to{\bar{y}}\}\,, \phantom{\biggl[ \biggr]}
\nonumber \\
t_{b,0}(y) &=
2 ({\bar{y}}-y)\, \frac{\ln y}{{\bar{y}}}\,
\biggl[ \ln\frac{Q^2}{\mu_{GPD}^2} - 3 \biggr]
+ ({\bar{y}}-y)\, \frac{\ln^2 y}{{\bar{y}}} + 4 {\operatorname{Li}}_2{\bar{y}}- \{y\to{\bar{y}}\} \,,
\nonumber \\
t_{a,0}(y) &=
\beta_0 \left[ \frac{19}{6} - \ln y
- \ln\frac{Q^2}{\mu^2_R} \right]
+ C_F \biggl[
\left( 3 + 2 \ln y \right){\mskip 1.5mu}\ln\frac{Q^2}{\mu^2_{GPD}}
- \frac{77}{6} - \left( \frac{1}{{\bar{y}}} - 3 \right) \ln y + \ln^2 y
\biggr]
\nonumber \\
&\quad
+ \left( 2 C_F - C_A \right)
\biggl\{ - \frac{1}{3} - 4(2-3y) \ln{\bar{y}}+ 2(1-6y) \ln y + 4(1-3y){\mskip 1.5mu}\bigl( {\operatorname{Li}}_2 y - {\operatorname{Li}}_2{\bar{y}}\bigr)
\nonumber \\
&\qquad\qquad
+ 2(1-6y{\bar{y}}) \left[ 3 \bigl( {\operatorname{Li}}_3{\bar{y}}+ {\operatorname{Li}}_3 y \bigr)
- \ln y\, {\operatorname{Li}}_2 y - \ln{\bar{y}}\, {\operatorname{Li}}_2{\bar{y}}- \frac{\pi^2}{6}\, \bigl( \ln y + \ln{\bar{y}}\bigr)
\right]
\biggr\} \,.\end{aligned}$$ The corresponding kernels for $n=2$ and $n=4$ are given in App. \[app:kernels\]. The $i\epsilon$ prescription to be used in is the same as specified at the end of the previous subsection. This implies that in $t_{g,n}(y)$, $t_{b,n}(y)$, $t_{a,n}(y)$ and $t_{a,n}({\bar{y}})$ one has to take $\ln(y -i\epsilon)$, ${\operatorname{Li}}_2({\bar{y}}+
i\epsilon)$ and ${\operatorname{Li}}_3({\bar{y}}+ i\epsilon)$ for $y<0$. For the gluon and pure singlet kernel, which dominate in process amplitudes at small $\xi$, we have in particular $$\begin{aligned}
\label{im_gluon}
\frac{1}{\pi} {\operatorname{Im}}t_{g,0}(y) &=
- \biggl[ 2C_A{\mskip 1.5mu}(y^2+{\bar{y}}^2) - C_F{\mskip 1.5mu}y {\mskip 1.5mu}\biggr] \frac{1}{{\bar{y}}}\,
\ln\frac{Q^2}{\mu_{GPD}^2}
\nonumber \\
&\quad
- C_F \biggl[ 1-4y + \frac{1 - y \ln(-y)}{{\bar{y}}}
+ 2({\bar{y}}-y) \ln{\bar{y}}+ 2y{\bar{y}}\left(
\ln^2{\bar{y}}+ 2 {\operatorname{Li}}_2 y + \frac{\pi^2}{3} \right) \biggr]
\nonumber \\
&\quad
+ 2C_A \biggl[ \frac{3}{{\bar{y}}} - 4y
- \left( \frac{1}{{\bar{y}}} - 2y \right) \ln(-y)
+ ({\bar{y}}-y) \ln{\bar{y}}\biggr] \,,
\nonumber \\
\frac{1}{\pi} {\operatorname{Im}}t_{b,0}(y) &=
2\, \frac{{\bar{y}}-y}{{\bar{y}}}\, \biggl[ 3 - \ln(-y)
- \ln\frac{Q^2}{\mu_{GPD}^2} \biggr] + 4 \ln{\bar{y}}\end{aligned}$$ in the region $y<0$. In the limit $y\to 0$ all three expressions in contain singular terms proportional to $\ln y$ and $\ln^2 y$. For the convolution we should however consider $(y {\bar{y}})^{-1}{\mskip 1.5mu}t_{g,n}(y)$, $y^{-1}{\mskip 1.5mu}t_{a,n}(y)$ and ${\bar{y}}^{-1}{\mskip 1.5mu}t_{a,n}({\bar{y}})$ according to . With the appropriate $i\epsilon$ prescription, these kernels contain terms which for $y\to 0$ go like $(y-i\epsilon)^{-1} \ln^m(y-i\epsilon)$, where $m=0,1,2$.
Model for the unpolarized GPDs {#sec:models}
==============================
It is difficult to study the impact of NLO corrections at the level of the hard-scattering kernels given in the previous subsection, especially since they are not smooth functions but distributions with singularities at $y=0$. We will therefore use model GPDs to investigate the radiative corrections at the level of the convolution integrals . The aim of this work is not a systematic improvement of existing models, nor a detailed exploration of model uncertainties on observables in exclusive meson production. We do however require that the models we use are consistent with known theoretical requirements and basic phenomenological constraints.
For $H^q$ and $H^g$ we adopt the widely used ansatz of [@Musatov:1999xp; @Goeke:2001tz] based on double distributions, where a $\xi$ dependence is generated according to $$\begin{aligned}
\label{dd-models}
H^{q(+)}(x,\xi,t) &=
\int_{-1}^1 d\beta \int_{-1+|\beta|}^{1-|\beta|} d\alpha\;
\delta(x-\beta-\xi\alpha)\, h^{(2)}(\beta,\alpha)\,
H^{q(+)}(\beta,0,t) \,,
\nonumber \\
H^g(x,\xi,t) &=
\int_{-1}^1 d\beta \int_{-1+|\beta|}^{1-|\beta|} d\alpha\;
\delta(x-\beta-\xi\alpha)\, h^{(2)}(\beta,\alpha)\,
H^g(\beta,0,t)\end{aligned}$$ with $$\label{profile}
h^{(b)}(\beta,\alpha) = \frac{\Gamma(2b+2)}{2^{2b+1}\Gamma^2(b+1)}\,
\frac{[ (1-|\beta|)^2- \alpha^2 ]^b}{(1-|\beta|)^{2b+1}} \,.$$ The distributions at zero skewness are taken as $$\begin{aligned}
\label{t-dep-h}
H^{q(+)}(x,0,t) &=
q_v(x) \exp\bigl[ t f_{q_v}(x) \bigr]
+ 2{\mskip 1.5mu}\bar{q}(x) \exp\bigl[ t f_{\bar{q}}(x) \bigr] \,,
\nonumber \\[0.3em]
H^g(x,0,t) &= x g(x) \exp\bigl[ t f_{g}(x) \bigr]\end{aligned}$$ for $x>0$, with the values for $x<0$ following from the symmetry properties of the distributions. Here $q_v(x) = q(x) - \bar{q}(x)$, $\bar{q}(x)$ and $g(x)$ are the usual unpolarized densities for valence quarks, antiquarks and gluons, for which we take the CTEQ6M parameterization [@Pumplin:2002vw]. This parameterization has an identical strange and antistrange sea, so that $s_v(x)=0$. The ansatz is taken at a starting scale $\mu_0$ and then evolved with the LO evolution equations given in App. \[app:evolution\]. For the studies in Sects. \[sec:small-x\] and \[sec:large-x\] we take $\mu_0 = 1.3{\operatorname{GeV}}$, which is the starting scale of evolution for the CTEQ6M densities. In Sects. \[sec:E-convolutions\] and \[sec:cross\] we will instead take $\mu_0 = 2{\operatorname{GeV}}$, since this will allow us to use the results for the $t$ dependence of valence distributions obtained in [@Diehl:2004cx].
For the $t$ dependence in the ansatz we follow the modeling strategy of [@Goeke:2001tz] and take an exponential behavior in $t$ with an $x$ dependent slope. For valence quarks we take the slope functions $$\label{DFJK4-f}
f_{q_v}(x) = \alpha'_v (1-x)^3 \ln\frac{1}{x} + B_{q_v} (1-x)^3
+ A_{q_v} x (1-x)^2$$ with parameters $\alpha'_v = 0.9 {\operatorname{GeV}}^{-2}$ and $$\begin{aligned}
\label{DFJK-params}
A_{u_v} &= 1.26 {\operatorname{GeV}}^{-2} \,, & B_{u_v} &= 0.59 {\operatorname{GeV}}^{-2} \,,
\nonumber \\
A_{d_v} &= 3.82 {\operatorname{GeV}}^{-2} \,, & B_{d_v} &= 0.32 {\operatorname{GeV}}^{-2} \,,\end{aligned}$$ from [@Diehl:2004cx]. We recall the sum rule $$\begin{aligned}
\label{F1-sum-rule}
F_1^q(t) &= \int_{-1}^1 dx\, H^q(x,0,t)
= \int_0^1 dx\, q_v(x) \exp\bigl[ t f_{q_v}(x) \bigr] \,,\end{aligned}$$ from which one obtains the electromagnetic Dirac form factors of proton and neutron by appropriate quark flavor combinations. Together with the CTEQ6M distributions at $\mu_0 = 2{\operatorname{GeV}}$, the ansatz in and gives a good description of the data for these form factors. For gluons we take a slightly simpler form than and set $$\label{gluon-profile}
f_g(x) = \alpha'_g (1-x)^2 \ln\frac{1}{x} + B_g (1-x)^2 \,.$$ For the parameters we take $$\begin{aligned}
\label{g-prof-param}
\alpha'_g &= 0.164 {\operatorname{GeV}}^{-2} \,, &
B_g &= 1.2 {\operatorname{GeV}}^{-2}\end{aligned}$$ so as to match recent H1 data on ${J\mskip -2mu/\mskip -0.5mu\Psi}$ photoproduction, whose $t$ dependence is well fitted by [@Aktas:2005xu] $$\label{h1-jpsi}
\frac{d\sigma}{dt} \propto \exp\biggl[
\left( b_0 + 4\alpha'_g \ln\frac{W_{\gamma p}}{W_0} \right) t
\,\biggr]$$ with central values $b_0 = 4.63 {\operatorname{GeV}}^{-2}$ and $\alpha'_g = 0.164
{\operatorname{GeV}}^{-2}$ for $W_{0} = 90 {\operatorname{GeV}}$. To connect with we have used the approximate relation $d\sigma/dt \propto |H^g(\xi,\xi,t)|^2$, which is obtained when only keeping the imaginary part of the tree-level amplitude, where $2 \xi =
(M_{{J\mskip -2mu/\mskip -0.5mu\Psi}} /W_{\gamma p})^2$ in terms of the $\gamma p$ c.m. energy. With the ansatz one approximately has $H^g(\xi,\xi,t) \propto \exp\bigl[ t f_g(2\xi) \bigr]$ for the $t$ dependence of the GPD [@Goloskokov:2006hr].
Whereas information on valence quark GPDs can be obtained from the sum rules and information on gluon GPDs from ${J\mskip -2mu/\mskip -0.5mu\Psi}$ production, almost nothing is so far known about the $t$ dependence of GPDs for antiquarks. As a simple ansatz we shall take their slope functions equal to those in the valence sector, $$\begin{aligned}
f_{\bar{u}} &= f_{u_v} \,, &
f_{\smash{\bar{d}}} &= f_{d_v} \,, &
f_{\bar{s}} &= f_{d_v} \,,\end{aligned}$$ bearing in mind that it remains an outstanding task to develop more realistic models.
Nucleon helicity-flip distributions {#sec:e-model}
-----------------------------------
The nucleon helicity-flip distributions $E^q$ and $E^g$ are less-well known than their counterparts $H^q$ and $H^g$, because their values at $\xi=0$ and $t=0$ cannot be measured in inclusive processes and are thus subject to considerable uncertainty.
The model described in this subsection refers to a scale of $\mu_0
=2{\operatorname{GeV}}$. We make a double distribution based ansatz $$\begin{aligned}
\label{dd-models-e}
E^{q(+)}(x,\xi,t) &=
\int_{-1}^1 d\beta \int_{-1+|\beta|}^{1-|\beta|} d\alpha\;
\delta(x-\beta-\xi\alpha)\, h^{(2)}(\beta,\alpha)\,
E^{q(+)}(\beta,0,t) \,,
\nonumber \\
E^g(x,\xi,t) &=
\int_{-1}^1 d\beta \int_{-1+|\beta|}^{1-|\beta|} d\alpha\;
\delta(x-\beta-\xi\alpha)\, h^{(2)}(\beta,\alpha)\,
E^g(\beta,0,t)\end{aligned}$$ as in , and for $x>0$ set $$\begin{aligned}
\label{t-dep-e}
E^{q(+)}(x,0,t) &=
e_{q_v}(x) \exp\bigl[ t{\mskip 1.5mu}g_{q_v}(x) \bigr]
+ 2 e_{\bar{q}}(x) \exp\bigl[ t{\mskip 1.5mu}g_{\bar{q}}(x) \bigr] \,,
\nonumber \\[0.3em]
E^g(x,0,t) &= x e_g(x) \exp\bigl[ t{\mskip 1.5mu}g_{g}(x) \bigr] \,,\end{aligned}$$ with the corresponding values for $x<0$ determined by the symmetry properties of the distributions. For the forward limit of the valence distribution we take $$\label{e-val-model}
e_{q_v}(x) = \kappa_q \, N(\alpha_v,\beta_{q_v})\,
x^{-\alpha_v}{\mskip 1.5mu}(1-x)^{\beta_{q_v}} \,,$$ whose normalization factor $$\label{norm-fact}
N(\alpha,\beta) =
\frac{\Gamma(2-\alpha+\beta)}{\Gamma(1-\alpha)\, \Gamma(1+\beta)}$$ ensures the sum rules $$\begin{aligned}
\label{E-kappa-sum}
\kappa_q &= \int_{-1}^1 dx\, E^q(x,0,0)
= \int_0^1 dx\, e_{q_v}(x) \,,\end{aligned}$$ where $\kappa_u \approx 1.67$ and $\kappa_d \approx -2.03$ are the contributions of $u$ and $d$ quarks to the anomalous magnetic moment of the proton. For the functions controlling the $t$ dependence we take the same form as in , $$\label{DFJK4-g}
g_{q_v}(x) = \alpha'_v (1-x)^3 \ln\frac{1}{x} + D_{q_v} (1-x)^3
+ C_{q_v} x (1-x)^2 \,.$$ With the parameters $\alpha_v^{} = 0.55$, $\alpha'_v = 0.9 {\operatorname{GeV}}^{-2}$ and $$\begin{aligned}
\label{DFJK4-E-par}
\beta_u &= 3.99 \,, &
C_{u_v \,} &= 1.22 {\operatorname{GeV}}^{-2} \,, & D_{u_v} &= 0.38 {\operatorname{GeV}}^{-2} \,,
\nonumber \\
\beta_d &= 5.59 \,, &
C_{d_v} &= 2.59 {\operatorname{GeV}}^{-2} \,, & D_{d_v} &= -0.75 {\operatorname{GeV}}^{-2} \,,\end{aligned}$$ from [@Diehl:2004cx] one obtains a good fit to the electromagnetic Pauli form factors of proton and neutron via the generalization of the sum rule to finite $t$.
For the forward limit of the distributions of antiquarks and gluons we make the same simple ansatz as in , $$\begin{aligned}
\label{eg-model}
e_{\bar{q}}(x) &= k_{\bar{q}} \,
x^{-\alpha_{\bar{q}}}{\mskip 1.5mu}(1-x)^{\beta_{\bar{q}}} \,,
&
e_g(x) &= k_g \, x^{-\alpha_g}{\mskip 1.5mu}(1-x)^{\beta_g} \,,\end{aligned}$$ and for the $t$ dependence in the gluon sector we set $$\label{e-g-profile}
g_g(x) = \alpha'_g (1-x)^2 \ln\frac{1}{x} + D_g (1-x)^2 \,,$$ in analogy to the form we used for $H^g$. We presently have not no phenomenological information on these distributions, but two theoretical constraints. There is a condition that ensures positive semidefinite densities of partons in the transverse plane [@Burkardt:2003ck], which with our ansatz for the GPDs reads [@Diehl:2004cx] $$\begin{aligned}
\label{pos-cond}
\left[ \frac{e_{\bar{q}}(x)}{\bar{q}(x)} \right]^2 & \le
8{\mskip 1.5mu}\mathrm{e}{\mskip 1.5mu}m_p^2 \,
\left[ \frac{g_{\bar{q}}(x)}{f_{\bar{q}}(x)} \right]^3
\bigl[ f_{\bar{q}}(x) - g_{\bar{q}}(x) \bigr] \,,
\nonumber \\
\left[ \frac{e_g(x)}{g(x)} \right]^2 & \le
8{\mskip 1.5mu}\mathrm{e}{\mskip 1.5mu}m_p^2 \,
\left[ \frac{g_g(x)}{f_g(x)} \right]^3
\bigl[ f_g(x) - g_g(x) \bigr]\end{aligned}$$ if we neglect for simplicity the polarized antiquark and gluon distributions compared with the unpolarized ones. On the other hand we have the sum rule $$\begin{aligned}
\label{zero-sum-rule}
0 &= \int_{0}^1 dx\, E^g(x,0,0) + \sum_q \int_{-1}^1 dx\, x E^q(x,0,0)
\nonumber \\
&= \int_{0}^1 dx\, x e^g(x)
+ \sum_q \int_{0}^1 dx\, x \bigl[ e_{q_v}(x) + 2 e_{\bar{q}}(x)
\bigr]\end{aligned}$$ following from the conservation of the energy-momentum tensor. For the parameters in we take $$\begin{aligned}
\alpha'_g &= 0.164 {\operatorname{GeV}}^{-2} \,, &
D_g &= 1.08 {\operatorname{GeV}}^{-2} \,,\end{aligned}$$ with $\alpha'_g$ as in and $D_g$ slightly smaller than its counterpart $B_g$ for $H^g$, so that the positivity condition can be fulfilled. Assuming a similar small-$x$ behavior of the distributions for proton helicity-flip and non-flip, we take in the values $\alpha_{\bar{q}} = 1.25$ and $\alpha_g = 1.10$, which we obtain when fitting the CTEQ6M distributions to a power law in the $x$ range from $10^{-4}$ to $10^{-3}$.
Since it turns out that the transverse target polarization asymmetry in $\rho$ production is very sensitive to the details of the helicity-flip distributions, we will explore two model scenarios in our numerical studies:
1. a scenario where the sea quark distributions $e_{\bar{q}}$ behave similarly to the valence distributions $e_{q_v}$. For the $t$ dependence we then take $g_{\bar{u}}(x) = g_{u_v}(x)$ and $g_{\smash{\bar{d}}}(x) = g_{d_v}(x)$. The parameters $k_{\bar{q}}$ in are taken such that second moments at $t=0$ fulfill $$\label{val-scen}
\frac{\int_0^1 dx\, x e_{\bar{q}}(x)}{\int_0^1 dx\, x e_{q_v}(x)}
= \frac{\int_0^1 dx\, x \bar{q}(x)}{\int_0^1 dx\, x q_v(x)}$$ for $q = u,d$, where the ratio on the r.h.s. is taken from the CTEQ6M parameterization at $\mu=2 {\operatorname{GeV}}$. Its value is $0.095$ for $u$ and $0.30$ for $d$ quarks. This fixes the values of $k_{\bar{q}}\,
N^{-1}(\alpha_{\bar{q}} -1, \beta_{\bar{q}})$ with $N$ given in . For the strange distribution we set $e_s = e_{\bar{s}} =0$, and $k_g\,
N^{-1}(\alpha_g -1, \beta_g)$ is then fixed by the sum rule .
The powers $\beta_{\bar{q}}$ and $\beta_g$ controlling the large-$x$ behavior are finally taken to have the smallest values for which the positivity condition holds in the range $x<0.9$ (for higher $x$ even the unpolarized densities are so uncertain that we do not insist on the positivity conditions to be fulfilled).
2. a scenario where $e_{\bar{q}}$ behaves similarly to the gluon distribution $e_g$ . The $t$ dependence is now modeled by taking $g_{\bar{q}}(x) = g_g(x)$ for $q = u,d,s$. For the second moments we impose $$\label{glu-scen}
\frac{\int_0^1 dx\, x e_{\bar{q}}(x)}{\int_0^1 dx\, x e_g(x)}
= \frac{\int_0^1 dx\, x \bar{q}(x)}{\int_0^1 dx\, x g(x)}$$ for the three light quark flavors, where with the CTEQ6M distributions the r.h.s. is equal to $0.064$, $0.083$, $0.036$ for $u$, $d$, $s$, respectively. We now have a nonzero $e_{s} = e_{\bar{s}}$. The values of $k_{\bar{q}}\, N^{-1}(\alpha_{\bar{q}} -1, \beta_{\bar{q}})$ and $k_g\, N^{-1}(\alpha_g -1, \beta_g)$ are taken to fulfill both and , and the powers $\beta_{\bar{q}}$, $\beta_g$ are set to the minimal values for which positivity holds in the range $x<0.9$.
The parameters resulting from this modeling procedure are collected in Table \[tab:params\], and the distributions at $\xi=0$ and $t=0$ for model 1 are shown in Fig. \[E-model\].
-------------------------- ---------------------- --------------------- ---------------------- --------------------- ---------------------- -- --------------------- --------------------- --------------------- ---------------------
$u_v$ $d_v$ $\bar{u}$ $\bar{d}$ $g$ $\bar{u}$ $\bar{d}$ $\bar{s}$ $g$
$\alpha_a$ ${\phantom{-}}0.55$ ${\phantom{-}}0.55$ ${\phantom{-}}1.25$ ${\phantom{-}}1.25$ ${\phantom{-}}1.10$ ${\phantom{-}}1.25$ ${\phantom{-}}1.25$ ${\phantom{-}}1.25$ ${\phantom{-}}1.10$
$\beta_a$ ${\phantom{-}}3.99$ ${\phantom{-}}5.59$ ${\phantom{-}}9.6$ ${\phantom{-}}9.2$ ${\phantom{-}}6.7$ ${\phantom{-}}7.6$ ${\phantom{-}}6.5$ ${\phantom{-}}5.5$ ${\phantom{-}}2.5$
$k_a$ ${\phantom{-}}1.71$ $-2.36$ ${\phantom{-}}0.06$ $-0.18$ ${\phantom{-}}0.26$ $-0.0016$ $-0.0018$ $-0.0007$ $-0.017$
$\int_0^1 dx\, x e_a(x)$ ${\phantom{-}}0.138$ $-0.130$ ${\phantom{-}}0.013$ $-0.039$ ${\phantom{-}}0.044$ $-0.0004$ $-0.0005$ $-0.0002$ $-0.0059$
-------------------------- ---------------------- --------------------- ---------------------- --------------------- ---------------------- -- --------------------- --------------------- --------------------- ---------------------
: \[tab:params\] Parameters in the ansatz for different parton species $a$ in the two models described in the text. The values for valence quarks apply to both models, with normalization parameters given by $k_{q_v} = \kappa_q{\mskip 1.5mu}N(\alpha_{v}, \beta_{q_v})$ according to . The last line gives the second Mellin moment at $\mu= 2{\operatorname{GeV}}$ in the forward limit.
We find that in model 2, both sea quark and gluon distributions are nearly zero (so that we do not attach importance to the unrealistically small value of $\beta_g$ obtained with our above procedure). Their smallness can be traced back to the small value of the flavor singlet integral $$\label{e-singlet-int}
\int_0^1 dx\, x \bigl[ e_{u_v}(x) + e_{d_v}(x) \bigr] = 0.008$$ in the valence sector of our ansatz. In model 2, the distributions $e_{\bar{q}}$ and $e_g$ have the same sign as a consequence of and due to the sum rule can only be tiny. Somewhat larger distributions for sea quarks and gluons are obtained in model 1, where they have opposite sign because of .
![\[E-model\] The forward limits $e_a(x)$ of the nucleon helicity-flip distributions at $\mu= 2{\operatorname{GeV}}$ for different parton species $a$ in model 1.](plots/E_plot1.ps "fig:"){width="\plotwidth"} ![\[E-model\] The forward limits $e_a(x)$ of the nucleon helicity-flip distributions at $\mu= 2{\operatorname{GeV}}$ for different parton species $a$ in model 1.](plots/E_plot2.ps "fig:"){width="\plotwidth"}
We note that the parameters we have taken for the valence part of $E^q$ are by no means precisely determined by a fit to the Pauli form factors: alternative fits in [@Diehl:2004cx] gave a similarly good description of the form factor data, with some variation of the resulting value of the integral in . Nevertheless, any model where $e_{u_v}$ and $e_{d_v}$ have similar shapes and no zeroes in $x$ will yield rather small values of this integral, given the strong cancellation between $u$ and $d$ quark contributions in the moment $\int_0^1 dx\, \bigl[
e_{u_v}(x) + e_{d_v}(x) \bigr] = \kappa_u + \kappa_d \approx -0.36$. It would be interesting to explore how much the integral and as a consequence the sea quark and gluon distributions can vary in realistic models, but such an investigation is beyond the scope of this work.
We end this section by quoting the values for the total angular momentum carried by quarks and antiquarks of a given flavor in our model, given by $$J_{q} = \frac{1}{2} \int_{-1}^1 dx\, x
\bigl[ H^q(x,0,0) + E^q(x,0,0) \bigr]$$ according to Ji’s sum rule [@Ji:1996ek]. With the parameters in Table \[tab:params\] and the CTEQ6M distributions we find $$\begin{aligned}
J_u &= 0.25 \,, & J_d &= -0.01 \,,
& & \text{(model 1)}
\nonumber \\
J_u &= 0.24 \,, & J_d &= {\phantom{-}}0.03
& & \text{(model 2)}\end{aligned}$$ at the scale $\mu= 2{\operatorname{GeV}}$ of our model. We note that this is in rather good agreement with the results of recent lattice calculations, with $J_u = 0.214(16)$ and $J_d = -0.001(16)$ reported in [@Hagler:2007xi], and $J_u = 0.33(2)$ and $J_d = -0.02(2)$ in [@Schierholz:2007]. Let us reiterate that with just two sets of model parameters we cannot exhaust the range of possible scenarios but only provide two representatives that are consistent with presently known constraints. As just discussed, the relative smallness of sea quark and gluon distributions compared with the nucleon helicity conserving case should however be typical of a rather wide class of models.
Vector meson production at small $x_B$ {#sec:small-x}
======================================
We now study numerically the importance of NLO corrections in vector meson production. Here and in the following sections we use the two-loop strong coupling for $n_f=3$ flavors with a QCD scale parameter $\Lambda^{(3)} = 226 {\operatorname{MeV}}$. This value corresponds to $\Lambda^{(4)} = 326 {\operatorname{MeV}}$, $\Lambda^{(5)} = 372 {\operatorname{MeV}}$ and to $\alpha_{\smash{s}}^{(5)}(M_Z) = 0.118$ when matching at $m_c=1.3{\operatorname{GeV}}$ and $m_b=4.5{\operatorname{GeV}}$, which are the values used in the CTEQ6M parton analysis [@Pumplin:2002vw]. We also take $n_f=3$ fixed in the evolution and the hard-scattering kernels. Taking $n_f=4$ with massless charm or $n_f=5$ with massless charm and bottom would not be a good approximation for the rather moderate values of $Q^2$ we will discuss for fixed-target kinematics. On the other hand, taking $n_f=3$ and neglecting charm altogether is admittedly not a good approximation for the larger $Q^2$ relevant in collider kinematics. However, with $\alpha_{\smash{s}}^{(3)} = 0.164$ compared to $\alpha_{\smash{s}}^{(5)} = 0.178$ at $\mu=10{\operatorname{GeV}}$ we expect that this inaccuracy will not affect the conclusions at high $Q^2$ we shall draw from our studies.
We have performed the evolution of the GPDs at LO using the momentum-space evolution code of [@Vinnikov:2006xw]. As explained in Sect. \[sec:kernels\], taking LO evolution together with the NLO hard-scattering kernels is sufficient to obtain scale independence of the process amplitude up to uncalculated corrections of order $\alpha_s^3$. With the input scale of evolution not taken too small, NLO evolution effects should be rather moderate at the $Q^2$ values relevant in fixed-target kinematics, whereas our general conclusions for high $Q^2$ and small $x_B$ will again not depend on this level of detail. We note that the NLO kernels in momentum space are available in the literature [@Belitsky:1999hf], but their considerable length makes it difficult to implement them in a fast numerical evaluation. For including NLO effects in the evolution it should be more efficient to use the Mellin space approach recently followed for deeply virtual Compton scattering in [@Kumericki:2007sa].
Here and in the following section we consider the convolutions of hard-scattering kernels with GPDs at $t=0$. For nonzero $\xi = x_B
/(2-x_B)$ this should be understood in the sense of an analytic continuation, since the physical region for meson production is $-t
\ge 4 m_p^2{\mskip 1.5mu}\xi^2/(1-\xi^2)$ in Bjorken kinematics. To explore the importance of NLO corrections we do not see this as a shortcoming.
![\[LO\_NLO\_comparison\_small\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 4 {\operatorname{GeV}}$. The scales are set to $\mu_R = \mu_{GPD} = \mu_{DA} = Q$. The NLO terms are for Gegenbauer index $n=0$ unless specified explicitly. Here and in the following plots the label “NLO” denotes the $O(\alpha_s^2)$ part of the convolutions, whereas the sum of $O(\alpha_s)$ and $O(\alpha_s^2)$ terms is labeled by “LO+NLO”.](plots/gluon_real_small_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[LO\_NLO\_comparison\_small\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 4 {\operatorname{GeV}}$. The scales are set to $\mu_R = \mu_{GPD} = \mu_{DA} = Q$. The NLO terms are for Gegenbauer index $n=0$ unless specified explicitly. Here and in the following plots the label “NLO” denotes the $O(\alpha_s^2)$ part of the convolutions, whereas the sum of $O(\alpha_s)$ and $O(\alpha_s^2)$ terms is labeled by “LO+NLO”.](plots/gluon_imag_small_x_LO_NLO.ps "fig:"){width="\plotwidth"}\
![\[LO\_NLO\_comparison\_small\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 4 {\operatorname{GeV}}$. The scales are set to $\mu_R = \mu_{GPD} = \mu_{DA} = Q$. The NLO terms are for Gegenbauer index $n=0$ unless specified explicitly. Here and in the following plots the label “NLO” denotes the $O(\alpha_s^2)$ part of the convolutions, whereas the sum of $O(\alpha_s)$ and $O(\alpha_s^2)$ terms is labeled by “LO+NLO”.](plots/qs_s_real_small_x_LO_NLO_muF_4GeV.ps "fig:"){width="\plotwidth"} ![\[LO\_NLO\_comparison\_small\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 4 {\operatorname{GeV}}$. The scales are set to $\mu_R = \mu_{GPD} = \mu_{DA} = Q$. The NLO terms are for Gegenbauer index $n=0$ unless specified explicitly. Here and in the following plots the label “NLO” denotes the $O(\alpha_s^2)$ part of the convolutions, whereas the sum of $O(\alpha_s)$ and $O(\alpha_s^2)$ terms is labeled by “LO+NLO”.](plots/qs_s_imag_small_x_LO_NLO_muF_4GeV.ps "fig:"){width="\plotwidth"}\
![\[LO\_NLO\_comparison\_small\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 4 {\operatorname{GeV}}$. The scales are set to $\mu_R = \mu_{GPD} = \mu_{DA} = Q$. The NLO terms are for Gegenbauer index $n=0$ unless specified explicitly. Here and in the following plots the label “NLO” denotes the $O(\alpha_s^2)$ part of the convolutions, whereas the sum of $O(\alpha_s)$ and $O(\alpha_s^2)$ terms is labeled by “LO+NLO”.](plots/g_s_qs_sum_real_small_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[LO\_NLO\_comparison\_small\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 4 {\operatorname{GeV}}$. The scales are set to $\mu_R = \mu_{GPD} = \mu_{DA} = Q$. The NLO terms are for Gegenbauer index $n=0$ unless specified explicitly. Here and in the following plots the label “NLO” denotes the $O(\alpha_s^2)$ part of the convolutions, whereas the sum of $O(\alpha_s)$ and $O(\alpha_s^2)$ terms is labeled by “LO+NLO”.](plots/g_s_qs_sum_imag_small_x_LO_NLO.ps "fig:"){width="\plotwidth"}
![\[small\_x\_nonsinglet\_LO\_NLO\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 4 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns3_real_small_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[small\_x\_nonsinglet\_LO\_NLO\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 4 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns3_imag_small_x_LO_NLO.ps "fig:"){width="\plotwidth"}\
![\[small\_x\_nonsinglet\_LO\_NLO\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 4 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns8_real_small_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[small\_x\_nonsinglet\_LO\_NLO\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 4 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns8_imag_small_x_LO_NLO.ps "fig:"){width="\plotwidth"}
![\[imag\_diff\_Q2\] Dependence on the common scale $\mu =
\mu_R = \mu_{GPD}$ for the sum of convolutions in the gluon and quark singlet sector.](plots/diff_mu_all_scales_imag_xi_10-2_Q2_4.ps "fig:"){width="\plotwidth"} ![\[imag\_diff\_Q2\] Dependence on the common scale $\mu =
\mu_R = \mu_{GPD}$ for the sum of convolutions in the gluon and quark singlet sector.](plots/diff_mu_all_scales_imag_xi_10-2.ps "fig:"){width="\plotwidth"}\
![\[imag\_diff\_Q2\] Dependence on the common scale $\mu =
\mu_R = \mu_{GPD}$ for the sum of convolutions in the gluon and quark singlet sector.](plots/diff_mu_all_scales_imag_xi_10-3_Q2_4.ps "fig:"){width="\plotwidth"} ![\[imag\_diff\_Q2\] Dependence on the common scale $\mu =
\mu_R = \mu_{GPD}$ for the sum of convolutions in the gluon and quark singlet sector.](plots/diff_mu_all_scales_imag_xi_10-3.ps "fig:"){width="\plotwidth"}\
![\[imag\_diff\_Q2\] Dependence on the common scale $\mu =
\mu_R = \mu_{GPD}$ for the sum of convolutions in the gluon and quark singlet sector.](plots/diff_mu_all_scales_imag_xi_10-4_Q2_4.ps "fig:"){width="\plotwidth"} ![\[imag\_diff\_Q2\] Dependence on the common scale $\mu =
\mu_R = \mu_{GPD}$ for the sum of convolutions in the gluon and quark singlet sector.](plots/diff_mu_all_scales_imag_xi_10-4.ps "fig:"){width="\plotwidth"}
Let us start our discussion with the gluon and quark singlet sector. Here and in following we shall always present the convolutions for Gegenbauer index $n=0$ unless indicated otherwise. In Fig. \[LO\_NLO\_comparison\_small\_x\] we show the LO and NLO pieces of the convolutions for the scale choice $\mu_R = \mu_{GPD} =
\mu_{DA} = Q$. The size of corrections at small $x_B$ is dramatic: we have large NLO corrections with opposite sign compared to the LO term for $\mathcal{H}^g$, and a similarly large NLO contribution from $\mathcal{H}^{S(b)}$ with sign opposite to the LO result for $\mathcal{H}^{S(a)}$. In the sum of gluon and quark singlet terms, the NLO corrections drastically reduce the LO result or even lead to a change of sign between LO and the sum of LO and NLO results. We also observe that for higher Gegenbauer index the NLO corrections tend to be even more important. Note that the LO term of the convolutions is the same for all $n$ as can be seen from and . The size of NLO corrections in $\mathcal{H}^{S(a)}$ is comparatively moderate, at least for lower Gegenbauer moments. The same is seen for the quark non-singlet convolutions in Fig. \[small\_x\_nonsinglet\_LO\_NLO\]. Of course, the gluon and quark singlet terms will dominate meson production at small $x_B$ in those channels where it is allowed by the meson quantum numbers.
In Fig. \[imag\_diff\_Q2\] we explore the influence of the scale choice by varying $\mu_R = \mu_{GPD}$ simultaneously. For $x_B=
2\times 10^{-3}$ we find an indication for the onset of perturbative stability at $Q= 7{\operatorname{GeV}}$ but not yet at $Q= 4{\operatorname{GeV}}$. For $x_B= 2\times
10^{-2}$ the situation is less severe, with moderate corrections in a wide $\mu$ range already at $Q= 4{\operatorname{GeV}}$. In contrast, when going down to $x_B= 2\times 10^{-4}$ we find very large corrections even at $Q=
7{\operatorname{GeV}}$. We have checked that the conclusions in the respective kinematics do not change when we vary $\mu_{GPD}$ while keeping $\mu_R
= Q$ fixed.
![\[cs\_rho\_zeus\] Cross section for $\gamma^* p\to \rho{\mskip 1.5mu}p$ with longitudinal photon polarization. Bands correspond to the range $Q/2 < \mu < 2Q$ and solid lines to $\mu=Q$. We also show the power-law behavior $\sigma \propto W^{0.88}$ (with arbitrary normalization) obtained from a fit to data in the range $0.001
\protect{\raisebox{-4pt}{ $\,\stackrel{\textstyle <}{\sim}\,$}}x_B \protect{\raisebox{-4pt}{ $\,\stackrel{\textstyle <}{\sim}\,$}}0.005$ [@ZEUS:2001].](plots/rho_ZEUS_cs.ps){width="\plotwidth"}
Figure \[cs\_rho\_zeus\] shows how the perturbative instability we observed in the convolutions affects the longitudinal cross section for $\rho$ production. Here we have taken the asymptotic form of the meson distribution amplitude, i.e. set $a_n=0$ for $n\ge 2$. In the NLO result for the cross section we have squared the coherent sum of LO and NLO terms in the process amplitude,[^2] i.e. we have taken $|\mathcal{M}_{\mathrm{LO}} +
\mathcal{M}_{\mathrm{NLO}}|^2$. We see that the NLO corrections severely decrease the LO result. As a consequence of the cancellations between LO and NLO contributions, the scale dependence of the cross section does not decrease. We also show in the figure the power-law behavior $\sigma \propto W^{0.88}$ obtained from a fit to data in the range $0.001 \protect{\raisebox{-4pt}{ $\,\stackrel{\textstyle <}{\sim}\,$}}x_B \protect{\raisebox{-4pt}{ $\,\stackrel{\textstyle <}{\sim}\,$}}0.005$ [@ZEUS:2001]. As observed in [@Goloskokov:2006hr], a double distribution model with the CTEQ6M distributions as input lead to a rather good description of this experimentally observed energy dependence if the cross section is evaluated at LO. With the strong cancellations from the $O(\alpha_s^2)$ corrections, one obtains an NLO result whose energy behavior is much too weak.
![\[fig:kernels\] The factorization scale dependent and independent terms of ${\operatorname{Im}}t_{g,n}$ and ${\operatorname{Im}}t_{b,n}$ as specified in , shown for $n=0$ and $y<0$.](plots/gluon_imag_single_terms.ps "fig:"){width="\plotwidth"} ![\[fig:kernels\] The factorization scale dependent and independent terms of ${\operatorname{Im}}t_{g,n}$ and ${\operatorname{Im}}t_{b,n}$ as specified in , shown for $n=0$ and $y<0$.](plots/singlet_imag_single_terms.ps "fig:"){width="\plotwidth"}
Let us discuss how the huge size of corrections can be understood at an analytical level, following the line of argument given in [@Ivanov:2004zv; @Ivanov:2004vd]. Using and we can approximate the hard-scattering kernels for large negative $y$ as $$\begin{aligned}
\label{asy-g}
\frac{1}{\pi}\, {\operatorname{Im}}\mathcal{I}_g(z,y)
&= 4 C_A \biggl[ \ln(z{\bar{z}}) + \ln \frac{Q^2}{\mu_{GPD}^2} \biggr] {\mskip 1.5mu}y
+ O(1) \,,
&
{\operatorname{Re}}\mathcal{I}_g(z,y)
&= O(1) \,,
\nonumber \\[0.2em]
\frac{1}{\pi} {\operatorname{Im}}\mathcal{I}_b(z,y) &= 4 \biggl[
1 - \ln(z{\bar{z}}) - \ln \frac{Q^2}{\mu_{GPD}^2} \biggr]
+ O\bigl( y^{-1} \bigr) \,,
&
{\operatorname{Re}}\mathcal{I}_b(z,y) &= O\bigl( y^{-1} \bigr) \,,\end{aligned}$$ where here and in the following the order of corrections is given up to powers of $\ln{\bar{y}}$. The quark non-singlet kernel is subleading compared with the pure singlet one, $$\begin{aligned}
\label{asy-a}
\frac{\mathcal{I}_a({\bar{z}}, y)}{y} \sim
\frac{\mathcal{I}_a(z, {\bar{y}})}{{\bar{y}}} \sim
O\bigl( y^{-1} \bigr) \,,\end{aligned}$$ where we have divided $\mathcal{I}_a({\bar{z}}, y)$ by $y$ corresponding to the prefactor in the complete kernel . From we readily obtain $$\begin{aligned}
\label{tgb-approx}
\frac{1}{\pi} {\operatorname{Im}}t_{g,n}(y) &= - 4 C_A
\biggl[ c_n - \ln\frac{Q^2}{\mu^2_{GPD}} \biggr] {\mskip 1.5mu}y
+ O(1) \,,
\nonumber \\
\frac{1}{\pi} {\operatorname{Im}}t_{b,n}(y) &=
4 \biggl[ c_n+1 - \ln\frac{Q^2}{\mu^2_{GPD}} \biggr]
+ O\bigl( y^{-1} \bigr)\end{aligned}$$ with constants $$\begin{aligned}
c_{0} &= 2 \,, &
c_{2} &= \frac{11}{3} \approx 3.7 \,, &
c_{4} &= \frac{137}{30} \approx 4.6 \,, &
c_{n} &= - \int_0^1 dz\, \ln(z{\bar{z}})\, C_{n}^{3/2}(2z-1)\end{aligned}$$ that increase with the Gegenbauer index $n$. In Fig. \[fig:kernels\] we show for the case $n=0$ that these approximations become very good for increasing $|y|$, where we have decomposed the exact kernels as $$\begin{aligned}
\label{kernels-CG}
{\operatorname{Im}}t_{g,n}^{}(y) &= {\operatorname{Im}}t_{g,n}^{C}(y)
+ {\operatorname{Im}}t_{g,n}^{G}(y){\mskip 1.5mu}\ln\frac{Q^2}{\mu^2_{GPD}} \,,
\nonumber \\
{\operatorname{Im}}t_{b,n}^{}(y) &= {\operatorname{Im}}t_{b,n}^{C}(y)
+ {\operatorname{Im}}t_{b,n}^{G}(y){\mskip 1.5mu}\ln\frac{Q^2}{\mu^2_{GPD}} \,.\end{aligned}$$
Let us now rewrite the convolutions of kernels and GPDs in terms of the variable $\omega = x/\xi$, $$\begin{aligned}
\label{small-x-con}
& {\operatorname{Im}}{\mskip 1.5mu}\Bigl[ \mathcal{H}^g_n + \mathcal{H}^{S(b)}_n \Bigr]
= - 6 \alpha_s\,
\biggl[ \pi H^g(\xi,\xi,t)
+ \frac{\alpha_s}{4\pi} {\operatorname{Im}}\int_0^{1/\xi} d\omega
\nonumber \\[0.3em]
& \qquad \times \biggl\{
\frac{2}{1-\omega-i\epsilon}\;
t_{g,n}\biggl( \frac{1-\omega-i\epsilon}{2} \biggr)\,
\frac{H^g(\omega\xi,\xi,t)}{1+\omega}
- C_F\, t_{b,n}\biggl( \frac{1-\omega-i\epsilon}{2} \biggr)\,
\xi H^S(\omega\xi,\xi,t) \biggr\}
\biggr] \,.\end{aligned}$$ For $\omega \ge \omega_0$ with some $\omega_0 \gg 1$ we can use the approximation of the hard-scattering kernels, and further approximate $1+\omega \approx \omega$ in the first term on the second line. This gives $$\begin{aligned}
\label{small-x-app}
&\hspace{-1.4em} - \frac{1}{6 \pi \alpha_s}
{\operatorname{Im}}{\mskip 1.5mu}\Bigl[ \mathcal{H}^g_n + \mathcal{H}^{S(b)}_n \Bigr]
\,\approx\,
H^g(\xi,\xi,t) + \frac{\alpha_s}{\pi}\,
\int_0^{\omega_0} d\omega\, \ldots
\nonumber \\[0.3em]
& - \frac{\alpha_s}{\pi}\,
\int_{\omega_0}^{1/\xi} d\omega\,
\biggl\{
C_A \biggl[ c_n - \ln\frac{Q^2}{\mu^2_{GPD}} \biggr]\,
\frac{H^g(\omega\xi,\xi,t)}{\omega}
+ C_F \biggl[ c_n+1 - \ln\frac{Q^2}{\mu^2_{GPD}} \biggr]\,
\xi H^S(\omega\xi,\xi,t)
\biggr\} \,,\end{aligned}$$ where the integral over $\omega$ on the first line is to be taken with the unapproximated integrand from . It grows with $\xi$ like $H^g(\omega\xi,\xi,t)$ or $\xi H^S(\omega\xi,\xi,t)$ but lacks the enhancement due to the upper limit $1/\xi$ of the integral on the second line. Restricting our discussion to $t=0$ for simplicity, we can for sufficiently large $\omega$ neglect the effect of skewness in the GPDs and then have $$\begin{aligned}
H^g(\omega\xi,\xi,0) &\approx \omega\xi{\mskip 1.5mu}g(\omega\xi) \,,
&
H^S(\omega\xi,\xi,0) &\approx S(\omega\xi)
= \sum\nolimits_q\, \bigl[ q(\omega\xi) + \bar{q}(\omega\xi) \bigr] \,,\end{aligned}$$ where $S(x)$ is the usual quark singlet distribution. In a very rough approximation one may treat $x g(x)$ and $x S(x)$ as constant at small $x$. In one then has loop integrals $\int d\omega
/\omega$ for both the gluon and the quark term, which generate large logarithms $\ln(\omega_0{\mskip 1.5mu}\xi)$ for $1/\xi \gg \omega_0$. These logarithms are of BFKL type and correspond to graphs with $t$-channel gluon exchange in the hard-scattering kernel, such as those for $T_b$ and $T_g$ in Fig. \[fig:graphs\].
In a phenomenologically more realistic approximation one has $x g(x)
\approx a{\mskip 1.5mu}x^{-\lambda}$ at small $x$ and a similar behavior with different values of $a$ and $\lambda$ for $x S(x)$. This gives $$\label{small-x-pow-approx}
\int_{\omega_0}^{1/\xi} d\omega\,
\frac{H^g(\omega\xi,\xi,0)}{\omega} \approx
a{\mskip 1.5mu}\xi^{-\lambda} \int_{\omega_0}^{1/\xi}
d\omega\, \omega^{-\lambda-1}
\approx
\frac{a}{\lambda}\, \bigl( \omega_0{\mskip 1.5mu}\xi \bigr)^{-\lambda}$$ for $1/\xi \gg \omega_0$, when the bulk of the integral comes from the region where the small-$x$ approximation of the gluon density is valid. With $\lambda$ being rather small for the gluon distribution in a wide range of the factorization scale, the term has the same power behavior $\xi^{-\lambda}$ as the Born term $H^g(\xi,\xi,0)$ in but is numerically enhanced by $1/\lambda$. A contribution analogous to is obtained from the quark singlet term in and comes with a similar enhancement.
Concerning the choice of factorization scale, it is clear that the size of the corrections in is decreased if $\mu_{GPD}$ is taken smaller than $Q$. It is also clear that no scale choice can eliminate both the gluon and quark singlet contribution in this expression. To make at least the gluon term for $n=0$ disappear one needs $\mu_{GPD}^2 = e^{-2}{\mskip 1.5mu}Q^2 \approx 0.14\, Q^2$. For a wide range of $Q^2$ this is outside the perturbative region or at least so low that the quark singlet distribution has a rather small power $\lambda$ and can thus give important corrections. We note that previous analyses of vector meson production at small $x_B$ have argued for a factorization scale well below $Q^2$, based on different estimates of the typical virtualities in the leading-order graphs [@Frankfurt:1995jw; @Ryskin:1995hz]. We also note that the $\mu_R$ dependent term $\beta_0 \ln(\mu_R^2 /\mu^2_{GPD})$ in the gluon kernel does not appear in the approximation which dominates the convolutions at small $x_B$. The choice of $\mu_R$ can thus not cure the huge NLO corrections we have discussed.
Vector meson production at moderate to large $x_B$ {#sec:large-x}
==================================================
Let us now investigate the NLO corrections in typical fixed-target kinematics, as it is accessible at HERMES, JLab and COMPASS. We take again $t=0$ and for definiteness present estimates at $Q^2 =4 {\operatorname{GeV}}^2$. For larger $Q^2$, which will in particular be accessible with the JLab energy upgrade to $12{\operatorname{GeV}}$, the corrections are in general smaller.
![\[LO\_NLO\_comparison\_large\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/gluon_real_large_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[LO\_NLO\_comparison\_large\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/gluon_imag_large_x_LO_NLO.ps "fig:"){width="\plotwidth"}\
![\[LO\_NLO\_comparison\_large\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/qs_s_real_large_x_LO_NLO_muF_2GeV.ps "fig:"){width="\plotwidth"} ![\[LO\_NLO\_comparison\_large\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/qs_s_imag_large_x_LO_NLO_muF_2GeV.ps "fig:"){width="\plotwidth"}\
![\[LO\_NLO\_comparison\_large\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/g_s_qs_sum_real_large_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[LO\_NLO\_comparison\_large\_x\] LO and NLO terms of the convolutions in the gluon and quark singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/g_s_qs_sum_imag_large_x_LO_NLO.ps "fig:"){width="\plotwidth"}
In Fig. \[LO\_NLO\_comparison\_large\_x\] we compare the LO and NLO parts of the convolution integrals. In the gluon sector we find no simple picture, with relative corrections that are typically moderate but become large for ${\operatorname{Re}}\mathcal{H}^g$ at smaller $x_B$ and for ${\operatorname{Im}}\mathcal{H}^g$ at larger $x_B$. For the quark singlet the situation is similar to the one in the small-$x_B$ region, i.e. we have rather large NLO corrections from $\mathcal{H}^{S(b)}$ with sign opposite to the LO part of $\mathcal{H}^{S(a)}$, whereas the NLO corrections in $\mathcal{H}^{S(a)}$ are smaller. Adding gluon and quark singlet contributions, we find that for $n=0$ the NLO corrections are of reasonable size for the imaginary part. For the real part at lower $x_B$, the corrections are however large and of opposite sign compared to the Born term. We note that the convolutions $\mathcal{H}$ satisfy a dispersion relation in $1/x_B$ for fixed $Q^2$ and $t$ [@Anikin:2007yh]. In this representation their real parts at a given $x_B$ are sensitive to the imaginary part at smaller values of $x_B$, where the NLO corrections rapidly increase as we have seen in the previous section. Turning to the quark non-singlet convolutions, we see in Fig. \[ns\_LO\_NLO\_comparison\_large\_x\] that for $n=0$ the NLO corrections are comparatively moderate for the imaginary part and larger for the real part.
Going from $n=0$ to higher Gegenbauer indices $n=2$ and $n=4$, the NLO corrections become larger, as we see in Figs. \[LO\_NLO\_comparison\_large\_x\] and \[ns\_LO\_NLO\_comparison\_large\_x\] and already observed at small $x_B$. Generically this is not unexpected, since the $z$ dependent kernels contain logarithms $\ln z$ and $\ln{\bar{z}}$ which enhance the endpoint regions of the $z$ integration, and those endpoint regions are more prominent for higher Gegenbauer polynomials in the expansion . Note that according to phenomenological estimates or lattice calculations the coefficients $a_n$ of these polynomials are clearly smaller than $a_0$, so that increasing corrections to $\mathcal{H}_n$ for higher $n$ do not affect the sum $\sum_n a_n \mathcal{H}_n$ as much. We note that in the modified hard-scattering approach of Sterman et al. [@Botts:1989kf], which goes beyond the collinear approximation used in the present work, the endpoint regions in $z$ are suppressed by radiative corrections that are resummed into Sudakov form factors. As just discussed, we do not observe such a suppression in the fixed-order results analyzed here, where various positive and negative corrections compete with each other—only some of them related to the Sudakov factor. How the situation will be at higher orders is an important question, which goes beyond the scope of the present work.
![\[ns\_LO\_NLO\_comparison\_large\_x\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns3_real_large_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[ns\_LO\_NLO\_comparison\_large\_x\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns3_imag_large_x_LO_NLO.ps "fig:"){width="\plotwidth"}\
![\[ns\_LO\_NLO\_comparison\_large\_x\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns8_real_large_x_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[ns\_LO\_NLO\_comparison\_large\_x\] LO terms and the sum of LO and NLO terms of the convolutions in the quark non-singlet sector at $Q = 2 {\operatorname{GeV}}$, with $\mu_R = \mu_{GPD} = \mu_{DA} = Q$.](plots/ns8_imag_large_x_LO_NLO.ps "fig:"){width="\plotwidth"}
Let us now take a closer look at the $\mu_R$ dependence of the corrections. As we explained in Sect. \[sec:kernels\], the pure quark singlet kernel $T_b$ is independent of this scale at $O(\alpha_s^2)$. According to the gluon kernel $T_g$ depends on $\mu_R^2$ only through $\beta_0 \ln(\mu_R^2 /\mu^2_{GPD})$, which originates from graphs with gluon propagator corrections such as the one shown in Fig. \[fig:beta-graphs\]. The $\mu_{GPD}$ dependence of this term is connected with the contribution proportional to $\beta_0$ in the evolution kernel $V^{gg}$ for the gluon GPD, given in . As already pointed out in Sect. \[sec:small-x\], the term $\beta_0 \ln(\mu_R^2 /\mu^2_{GPD})$ does not contribute to the large-$|y|$ behavior of ${\operatorname{Im}}t_{g,n}(y)$ and is hence not relevant for the huge NLO corrections at small $x_B$.
![\[fig:beta-graphs\] Example graphs giving rise to terms proportional to $\beta_0$ in the hard-scattering kernels $T_a$ and $T_g$.](plots/mesons-beta.eps){width="66.00000%"}
For the kernel $T_{a}$ the situation is more involved. The general structure of its convolution with the quark singlet distribution $H^S$ can be written as $$\begin{aligned}
\label{F-terms}
\mathcal{H}^{S(a)}_{n\phantom{P}} &=
\beta_0^{}\, \biggl( \mathcal{H}^{S(a)}_{n, \beta}
+ \mathcal{H}^{S(a)}_{n, R}\, \ln\frac{Q^2}{\mu^2_{R}}
\biggr)
+ \mathcal{H}^{S(a)}_{n, C}
+ \mathcal{H}^{S(a)}_{n, G} \ln\frac{Q^2}{\mu^2_{GPD}}
+ \mathcal{H}^{S(a)}_{n, D} \ln\frac{Q^2}{\mu^2_{DA}}\end{aligned}$$ with an analogous decomposition for the convolutions $\mathcal{H}^{(3)}_n$ and $\mathcal{H}^{(8)}_n$. The terms proportional to $\beta_0$ originate from graphs with gluon propagator corrections such as in Fig. \[fig:beta-graphs\], whereas the terms with subscripts $C, G, D$ do not contain $\beta_0$. In Fig. \[NLO\_single\_terms\] we show the corresponding contributions for $n=0$. We see that terms multiplying $\ln(Q^2 /\mu^2_{GPD})$ are rather small, whereas those going with $\ln(Q^2 /\mu^2_{DA})$ are of course absent for $n=0$. The term $\mathcal{H}_{0, R}$ is clearly smaller than $\mathcal{H}_{0, \beta}$ and has opposite sign. $\mathcal{H}_{0, C}$ has also the opposite sign compared to $\mathcal{H}_{0, \beta}$ but is similar in magnitude. We note that $|\mathcal{H}_{n, \beta}|$ increases with $n$, as can be seen from and .
![\[NLO\_single\_terms\] Individual terms in the convolutions of the quark non-singlet kernel for $n=0$. The quark distributions are evaluated at $\mu_{GPD}= 2 {\operatorname{GeV}}$ and the running coupling in the kernels at $\mu_R = 2{\operatorname{GeV}}$.](plots/quarksinglet_real_large_x_single_terms_muF_2GeV.ps "fig:"){width="\plotwidth"} ![\[NLO\_single\_terms\] Individual terms in the convolutions of the quark non-singlet kernel for $n=0$. The quark distributions are evaluated at $\mu_{GPD}= 2 {\operatorname{GeV}}$ and the running coupling in the kernels at $\mu_R = 2{\operatorname{GeV}}$.](plots/quarksinglet_imag_large_x_single_terms_muF_2GeV.ps "fig:"){width="\plotwidth"} ![\[NLO\_single\_terms\] Individual terms in the convolutions of the quark non-singlet kernel for $n=0$. The quark distributions are evaluated at $\mu_{GPD}= 2 {\operatorname{GeV}}$ and the running coupling in the kernels at $\mu_R = 2{\operatorname{GeV}}$.](plots/ns3_real_large_x_single_terms_muF_2GeV.ps "fig:"){width="\plotwidth"} ![\[NLO\_single\_terms\] Individual terms in the convolutions of the quark non-singlet kernel for $n=0$. The quark distributions are evaluated at $\mu_{GPD}= 2 {\operatorname{GeV}}$ and the running coupling in the kernels at $\mu_R = 2{\operatorname{GeV}}$.](plots/ns3_imag_large_x_single_terms_muF_2GeV.ps "fig:"){width="\plotwidth"}
Let us briefly comment on the BLM scale setting prescription [@Brodsky:1982gc], which has been discussed in the context of exclusive meson production in [@Belitsky:2001nq; @Anikin:2004jb]. This prescription aims at including the corrections from graphs like those of Fig. \[fig:beta-graphs\] in the argument of the running coupling, and for the case at hand takes $\mu_R$ such that the contribution from $\mathcal{H}_{n, \beta}$ cancels against the one from $\mathcal{H}_{n, R}{\mskip 1.5mu}\ln(Q^2 /\mu^2_{R})$ in . As is evident from Fig. \[NLO\_single\_terms\], this requires $\mu_R^2$ to be substantially lower than $Q^2$. For most of experimentally accessible kinematics, the resulting $\mu_R$ is in fact far below the region where perturbation theory can be applied. In such a situation, the perturbative running of $\alpha_s$ is often modified such that the coupling saturates for decreasing $\mu_R$. We note that in the context of our NLO analysis, the logarithm $\beta_0
\ln(Q^2 /\mu^2_{R})$ in the hard-scattering kernel is intimately related with the perturbative running of $\alpha_s(\mu_R)$, so that keeping one while modifying the other is not obviously consistent.
We also remark that if $\mathcal{H}_{n, \beta}$ and $\mathcal{H}_{n,
R}{\mskip 1.5mu}\ln(Q^2 /\mu^2_{R})$ are made to cancel by the BLM scale choice, one is left with a relatively large correction from $\mathcal{H}_{n,
C}$. For scale choices where $\mu_R^2$ is closer to $Q^2$, one instead has a partial cancellation between $\mathcal{H}_{n, C}$ and $\mathcal{H}_{n, \beta}$. A more detailed analysis for the similar case of the electromagnetic form factor is given in [@Bakulev:2000uh], which also discusses the issue of Sudakov-type corrections we raised above.
![\[diff\_mur\_results\] Dependence of the convolutions on $\mu_R$. The factorization scale is held fixed at $\mu_{GPD} = Q$.](plots/diff_mur_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_mur\_results\] Dependence of the convolutions on $\mu_R$. The factorization scale is held fixed at $\mu_{GPD} = Q$.](plots/diff_mur_imag_xi_10-1.ps "fig:"){width="\plotwidth"}\
![\[diff\_mur\_results\] Dependence of the convolutions on $\mu_R$. The factorization scale is held fixed at $\mu_{GPD} = Q$.](plots/ns3_diff_mur_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_mur\_results\] Dependence of the convolutions on $\mu_R$. The factorization scale is held fixed at $\mu_{GPD} = Q$.](plots/ns3_diff_mur_imag_xi_10-1.ps "fig:"){width="\plotwidth"}\
![\[diff\_mur\_results\] Dependence of the convolutions on $\mu_R$. The factorization scale is held fixed at $\mu_{GPD} = Q$.](plots/ns8_diff_mur_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_mur\_results\] Dependence of the convolutions on $\mu_R$. The factorization scale is held fixed at $\mu_{GPD} = Q$.](plots/ns8_diff_mur_imag_xi_10-1.ps "fig:"){width="\plotwidth"}
![\[diff\_muf\_results\] Dependence of the convolutions on $\mu_{GPD}$. The renormalization scale is held fixed at $\mu_R =
Q$.](plots/diff_muf_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_muf\_results\] Dependence of the convolutions on $\mu_{GPD}$. The renormalization scale is held fixed at $\mu_R =
Q$.](plots/diff_muf_imag_xi_10-1.ps "fig:"){width="\plotwidth"}\
![\[diff\_muf\_results\] Dependence of the convolutions on $\mu_{GPD}$. The renormalization scale is held fixed at $\mu_R =
Q$.](plots/ns3_diff_muf_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_muf\_results\] Dependence of the convolutions on $\mu_{GPD}$. The renormalization scale is held fixed at $\mu_R =
Q$.](plots/ns3_diff_muf_imag_xi_10-1.ps "fig:"){width="\plotwidth"}\
![\[diff\_muf\_results\] Dependence of the convolutions on $\mu_{GPD}$. The renormalization scale is held fixed at $\mu_R =
Q$.](plots/ns8_diff_muf_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_muf\_results\] Dependence of the convolutions on $\mu_{GPD}$. The renormalization scale is held fixed at $\mu_R =
Q$.](plots/ns8_diff_muf_imag_xi_10-1.ps "fig:"){width="\plotwidth"}
![\[diff\_mu\_results\] Dependence of the convolutions on the common scale $\mu = \mu_R = \mu_{GPD} = \mu_{DA}$.](plots/diff_mu_all_scales_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_mu\_results\] Dependence of the convolutions on the common scale $\mu = \mu_R = \mu_{GPD} = \mu_{DA}$.](plots/diff_mu_all_scales_imag_xi_10-1.ps "fig:"){width="\plotwidth"}\
![\[diff\_mu\_results\] Dependence of the convolutions on the common scale $\mu = \mu_R = \mu_{GPD} = \mu_{DA}$.](plots/ns3_diff_mu_all_scales_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_mu\_results\] Dependence of the convolutions on the common scale $\mu = \mu_R = \mu_{GPD} = \mu_{DA}$.](plots/ns3_diff_mu_all_scales_imag_xi_10-1.ps "fig:"){width="\plotwidth"}\
![\[diff\_mu\_results\] Dependence of the convolutions on the common scale $\mu = \mu_R = \mu_{GPD} = \mu_{DA}$.](plots/ns8_diff_mu_all_scales_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[diff\_mu\_results\] Dependence of the convolutions on the common scale $\mu = \mu_R = \mu_{GPD} = \mu_{DA}$.](plots/ns8_diff_mu_all_scales_imag_xi_10-1.ps "fig:"){width="\plotwidth"}
Figure \[diff\_mur\_results\] shows the dependence of the convolutions on $\mu_R$ at fixed $\mu_{GPD} = Q$. Within the $\mu_R$ range shown we generally find a moderate scale dependence, both at LO and at NLO. An exception is the region $\mu_R {\raisebox{-4pt}{ $\,\stackrel{\textstyle <}{\sim}\,$}}2{\operatorname{GeV}}$, where the growth of the LO results simply reflects the growth of $\alpha_s(\mu_R)$. Note that with the parameters specified at beginning of Sect. \[sec:small-x\] we have $\alpha_{\smash{s}}^{(3)}(2 {\operatorname{GeV}}) =
0.30$ and $\alpha_{\smash{s}}^{(3)}(1 {\operatorname{GeV}}) = 0.51$. The NLO results further contain explicit logarithms $\ln(Q^2 /\mu_R^2)$, which in some cases can cause corrections to grow out of control, especially for the real parts of convolutions. We note that for ${\operatorname{Re}}\mathcal{H}^{(8)}$ the NLO correction is is unusually large compared with the LO term. This is because of a nearby zero in $x_B$, as is seen in Fig. \[ns\_LO\_NLO\_comparison\_large\_x\], and should not be a reason of particular concern.
The variation of the convolutions with $\mu_{GPD}$ at fixed $\mu_R =
Q$ is shown in Fig. \[diff\_muf\_results\]. We again find a rather moderate scale dependence, except when $\mu_{GPD}$ becomes small. The dependence on a single scale $\mu = \mu_R = \mu_{GPD}$ is shown in Fig. \[diff\_mu\_results\]. Note that in many cases the individual variation of $\mu_R$ decreases the amplitude in absolute size whereas the variation of $\mu_{GPD}$ increases it, with both tendencies partially canceling when the scales are set equal. Again we find that the scale dependence becomes quite drastic below $2{\operatorname{GeV}}$.
We finally discuss the dependence on $\mu_{DA}$ for Gegenbauer indices $n>0$. According to the convolutions $\mathcal{H}_n(\mu_{DA})$ appear multiplied by $a_n(\mu_{DA})$ in the process amplitude, where the scale dependence of both factors partially cancels. In Fig. \[a2\_diff\_muda\_results\] we therefore plot convolutions multiplied with $a_2(\mu_{DA}) /a_2(\mu_0) = \bigl[
\alpha_s(\mu_{DA}) / \alpha_s(\mu_0) \bigr]{}^{\gamma_2/\beta_0}$ following the relation . The corresponding plots for $n=4$ and for convolutions in the quark non-singlet sector look very similar. We find that the dependence on $\mu_{DA}$ is slightly decreased when going to NLO.
![\[a2\_diff\_muda\_results\] Dependence on the factorization scale $\mu_{DA}$ of the convolutions in the gluon and quark singlet sector multiplied by the scale dependence $a_{2}(\mu_{DA})
/a_{2}(\mu_0)$ of the corresponding Gegenbauer coefficient. The reference scale for $a_{2}$ is taken as $\mu_0 = 2{\operatorname{GeV}}$, and the other scales are set to $\mu_{GPD} = \mu_{R} = Q = 2{\operatorname{GeV}}$.](plots/a2_diff_mufda_real_xi_10-1.ps "fig:"){width="\plotwidth"} ![\[a2\_diff\_muda\_results\] Dependence on the factorization scale $\mu_{DA}$ of the convolutions in the gluon and quark singlet sector multiplied by the scale dependence $a_{2}(\mu_{DA})
/a_{2}(\mu_0)$ of the corresponding Gegenbauer coefficient. The reference scale for $a_{2}$ is taken as $\mu_0 = 2{\operatorname{GeV}}$, and the other scales are set to $\mu_{GPD} = \mu_{R} = Q = 2{\operatorname{GeV}}$.](plots/a2_diff_mufda_imag_xi_10-1.ps "fig:"){width="\plotwidth"}
Proton helicity flip amplitudes {#sec:E-convolutions}
===============================
We now turn to the convolutions of the hard-scattering kernels with the GPDs describing proton helicity flip. In this section we take $t=
-0.4{\operatorname{GeV}}^2$, which is the value for which will present estimates for observables in the next section.
![\[E-gs-mod-1\] LO and NLO terms of the convolutions in the gluon and quark singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4 {\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/E_gluon_real_LO_NLO_large_x.ps "fig:"){width="\plotwidth"} ![\[E-gs-mod-1\] LO and NLO terms of the convolutions in the gluon and quark singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4 {\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/E_gluon_imag_LO_NLO_large_x.ps "fig:"){width="\plotwidth"}\
![\[E-gs-mod-1\] LO and NLO terms of the convolutions in the gluon and quark singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4 {\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/E_quarksinglet_real_LO_NLO_large_x.ps "fig:"){width="\plotwidth"} ![\[E-gs-mod-1\] LO and NLO terms of the convolutions in the gluon and quark singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4 {\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/E_quarksinglet_imag_LO_NLO_large_x.ps "fig:"){width="\plotwidth"}\
![\[E-gs-mod-1\] LO and NLO terms of the convolutions in the gluon and quark singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4 {\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/E_g_s_qs_sum_real_LO_NLO_large_x.ps "fig:"){width="\plotwidth"} ![\[E-gs-mod-1\] LO and NLO terms of the convolutions in the gluon and quark singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4 {\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/E_g_s_qs_sum_imag_LO_NLO_large_x.ps "fig:"){width="\plotwidth"}
![\[E-gs-mod-2\] As Fig. \[E-gs-mod-1\] but for model 2.](plots/E_gluon_real_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"} ![\[E-gs-mod-2\] As Fig. \[E-gs-mod-1\] but for model 2.](plots/E_gluon_imag_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"}\
![\[E-gs-mod-2\] As Fig. \[E-gs-mod-1\] but for model 2.](plots/E_quarksinglet_real_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"} ![\[E-gs-mod-2\] As Fig. \[E-gs-mod-1\] but for model 2.](plots/E_quarksinglet_imag_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"}\
![\[E-gs-mod-2\] As Fig. \[E-gs-mod-1\] but for model 2.](plots/E_g_s_qs_sum_real_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"} ![\[E-gs-mod-2\] As Fig. \[E-gs-mod-1\] but for model 2.](plots/E_g_s_qs_sum_imag_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"}
![\[E-ns-mod-1\] LO and NLO terms of the convolutions in the quark non-singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4
{\operatorname{GeV}}^2$, with $\mu_R = \mu_{GPD} = Q$.](plots/E_nonsinglet_real_LO_NLO_large_x.ps "fig:"){width="\plotwidth"} ![\[E-ns-mod-1\] LO and NLO terms of the convolutions in the quark non-singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4
{\operatorname{GeV}}^2$, with $\mu_R = \mu_{GPD} = Q$.](plots/E_nonsinglet_imag_LO_NLO_large_x.ps "fig:"){width="\plotwidth"}\
![\[E-ns-mod-1\] LO and NLO terms of the convolutions in the quark non-singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4
{\operatorname{GeV}}^2$, with $\mu_R = \mu_{GPD} = Q$.](plots/E_d-s_real_LO_NLO_large_x.ps "fig:"){width="\plotwidth"} ![\[E-ns-mod-1\] LO and NLO terms of the convolutions in the quark non-singlet sector for model 1 at $Q = 2 {\operatorname{GeV}}$ and $t= -0.4
{\operatorname{GeV}}^2$, with $\mu_R = \mu_{GPD} = Q$.](plots/E_d-s_imag_LO_NLO_large_x.ps "fig:"){width="\plotwidth"}
![\[E-ns-mod-2\] As Fig. \[E-ns-mod-1\] but for model 2.](plots/E_nonsinglet_real_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"} ![\[E-ns-mod-2\] As Fig. \[E-ns-mod-1\] but for model 2.](plots/E_nonsinglet_imag_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"}\
![\[E-ns-mod-2\] As Fig. \[E-ns-mod-1\] but for model 2.](plots/E_d-s_real_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"} ![\[E-ns-mod-2\] As Fig. \[E-ns-mod-1\] but for model 2.](plots/E_d-s_imag_LO_NLO_large_x_model_2.ps "fig:"){width="\plotwidth"}
In Figs. \[E-gs-mod-1\] and \[E-gs-mod-2\] we compare the LO and NLO terms of the convolutions in the gluon and quark singlet sector for the two models described in Sect. \[sec:e-model\]. For model 1 the individual corrections for gluon and quark convolutions look quite similar to those we saw for $\mathcal{H}$ in the previous section. The sum of gluon and quark singlet contributions at LO is however very small in this model because of cancellations, so that the NLO term dominates in a wide kinematical region. In model 2 the gluon contributions are nearly absent, so that the quark singlet contribution dominates in this sector. We note that, contrary to the individual terms, the sum of gluon and quark singlet contributions comes out to be rather similar in the two models and is small compared with the flavor non-singlet contributions shown in Figs. \[E-ns-mod-1\] and \[E-ns-mod-2\]. According to our discussion at the end of Sect. \[sec:e-model\] this has its origin in the sum rule for the second moment of $E$ at $t=0$, so that we expect a small net contribution from gluons and the quark singlet in large class of models for $E$.
As shown in Figs. \[E-ns-mod-1\] and \[E-ns-mod-2\], the NLO corrections to the quark non-singlet convolutions are relatively moderate but not small, similarly to the case of $\mathcal{H}$. The size of the convolutions is quite different in the two models, indicating the important role played at intermediate $x_B$ by sea quarks in model 1. Let us recall that with the double distribution ansatz the GPDs at $x\sim \xi$ are sensitive to forward parton distributions with momentum fractions well below $\xi$, as discussed in Sect. 4.3.3 of [@Diehl:2003ny].
Cross sections and asymmetries {#sec:cross}
==============================
Having studied in detail the building blocks of the scattering amplitude for vector meson production, we now combine them to observables. We recall that to leading order in $1/Q$ there are just two of these: the unpolarized $\gamma^* p$ cross section and the asymmetry for a transversely polarized target, both referring to longitudinal polarization of virtual photon and produced meson. The $ep$ cross section in the leading $1/Q$ approximation can be written as $$\label{AUT-def}
\frac{d\sigma(ep\to epV)}{dt\, dQ^2\, dy\, d\phi\, d\phi_S}
= \frac{\alpha}{4\pi^3}\, \frac{1-x_B}{Q^2}\,
\frac{1-y}{y}\, \frac{d\sigma_L}{dt}
\Bigl[ 1 + S_T \sin(\phi-\phi_S){\mskip 1.5mu}A_{UT} \Bigr]$$ where $y$ is the usual inelasticity variable for deep inelastic scattering and $S_T$ denotes the transverse component of the target polarization. $\phi$ is the azimuthal angle between lepton plane and hadron plane, and $\phi_S$ is the azimuthal angle between lepton plane and target spin vector, both defined according to the Trento convention [@Bacchetta:2004jz]. The $\gamma^* p$ cross section $d\sigma_L/dt$ and the polarization asymmetry $A_{UT}$ depend on $x_B$, $Q^2$ and $t$. To leading order in $1/Q$ they are given by $$\begin{aligned}
\label{sigmaL}
\frac{d\sigma_L}{dt} &=
\frac{\pi^2}{9}\, \frac{\alpha}{Q^6}\,
\frac{(2-x_B)^2}{1-x_B}\, f_V^2\,
\Bigl[{\mskip 1.5mu}(1-\xi^2)\, | \mathcal{H}_V^{} |^2
- \bigl({\mskip 1.5mu}t/(4m_p^2) + \xi^2 {\mskip 1.5mu}\bigr)\, | \mathcal{E}_V^{} |^2
- 2\xi^2 {\operatorname{Re}}\bigl( \mathcal{E}_V^*{\mskip 1.5mu}\mathcal{H}_V^{} \bigr)
{\mskip 1.5mu}\Bigr]
\intertext{and}
\label{AUT}
A_{UT} &= \frac{\sqrt{t_0-t\rule{0pt}{1.8ex}}}{m_p}\,
\frac{\sqrt{1-\xi^2}\,
{\operatorname{Im}}\bigl( \mathcal{E}_V^*{\mskip 1.5mu}\mathcal{H}_V^{} \bigr)}{\rule{0pt}{1em}
(1-\xi^2)\, | \mathcal{H}_V^{} |^2
- \bigl({\mskip 1.5mu}t/(4m_p^2) + \xi^2 {\mskip 1.5mu}\bigr)\, | \mathcal{E}_V^{} |^2
- 2\xi^2 {\operatorname{Re}}\bigl( \mathcal{E}_V^*{\mskip 1.5mu}\mathcal{H}_V^{} \bigr)} \,,\end{aligned}$$ where $t_{0} = -4 m_p^2{\mskip 1.5mu}\xi^2 /(1-\xi^2)$. Here we have combined the convolutions into $$\label{F_V-def}
\mathcal{F}_V = Q_V \sum_{n=0}^\infty a_n^{}{\mskip 1.5mu}\biggl[ \mathcal{F}^g_n + \mathcal{H}^{S(a)}_n + \mathcal{F}^{S(b)}_n
+ e_{V}^{(3)}\, \mathcal{F}^{(3)}_{n\phantom{V}}
+ e_{V}^{(8)}\, \mathcal{F}^{(8)}_{n\phantom{V}} \biggr] \,,$$ with analogous combinations for $\mathcal{H}_V$ and $\mathcal{E}_V$. In the remainder of this section we take the asymptotic form of the meson distribution amplitude, i.e. we set $a_n=0$ for $n\ge 2$. As long as $\mathcal{E}_V$ is not much larger than $\mathcal{H}_V$, the cross section is dominated by the term with $|\mathcal{H}_V|^2$ in a wide range of kinematics, where the prefactors $\xi^2$ and $t/(4m_p^2)$ of the other terms are small. The asymmetry is then approximately given by $$\label{asy-approx}
A_{UT} \approx \frac{\sqrt{t_0-t\rule{0pt}{1.8ex}}}{m_p} \;
\frac{{\operatorname{Im}}\bigl( \mathcal{E}_V^{*}{\mskip 1.5mu}\mathcal{H}_V^{} \bigr)}{| \mathcal{H}_V^{} |^2}
= \frac{\sqrt{t_0-t\rule{0pt}{1.8ex}}}{m_p} \;
\biggl| \frac{\mathcal{E}_V}{\mathcal{H}_V}\, \biggr|
\sin\delta_V \,,$$ where $\delta_V = \arg (\mathcal{H}_V /\mathcal{E}_V)$ is the relative phase between $\mathcal{H}_V$ and $\mathcal{E}_V$.
Figure \[rho\_re\_im\] shows the real and imaginary parts of the convolutions appearing in and . $\mathcal{H}_\rho$ is dominated by the gluon and quark singlet part, and in line with our discussion in Sect. \[sec:large-x\] we find rather moderate corrections for the imaginary part but a very unstable real part in a wide range of $x_B$. As for $\mathcal{E}_\rho$, its real part is very small and subject to large relative corrections, whereas its imaginary part is much larger and receives corrections of order $100\%$ . As we see in Fig. \[E-ns-mod-1\], the individual flavor non-singlet combinations $\mathcal{E}^{(3)}$ and $\mathcal{E}^{(8)}$ are less affected by corrections, but they have opposite sign and partially cancel in the sum relevant for $\rho$ production. The rather small but unstable contribution from the gluon and quark singlet terms is hence important in this channel and largely responsible for the NLO corrections seen in Fig. \[rho\_re\_im\]. As a further consequence of the cancellations just mentioned, the size of $\mathcal{E}_\rho$ is tiny compared with $\mathcal{H}_\rho$. The quark flavor combination relevant for $\rho$ production is $2u + d$, where in our model the flavor combinations add for $H$ but largely cancel for $E$.
![\[omega\_re\_im\] As Fig. \[rho\_re\_im\] but for $\omega$ production.](plots/asym_single_terms_H.ps "fig:"){width="\plotwidth"} ![\[omega\_re\_im\] As Fig. \[rho\_re\_im\] but for $\omega$ production.](plots/asym_single_terms_E.ps "fig:"){width="\plotwidth"}
![\[omega\_re\_im\] As Fig. \[rho\_re\_im\] but for $\omega$ production.](plots/omega_asym_single_terms_H.ps "fig:"){width="\plotwidth"} ![\[omega\_re\_im\] As Fig. \[rho\_re\_im\] but for $\omega$ production.](plots/omega_asym_single_terms_E.ps "fig:"){width="\plotwidth"}
For $\omega$ production we see in Fig. \[omega\_re\_im\] that at smaller $x_B$ the convolution $\mathcal{H}_\omega$ is about $1/3$ of $\mathcal{H}_\rho$, which follows from the dominance of the gluon contribution as seen in and , whereas at larger $x_B$ differences between the two channels appear. $\mathcal{E}_\omega$ is much bigger than $\mathcal{E}_\rho$ since in the combination $2u - d$ the contributions of $u$ and $d$ quarks add for $E$, and the size of its radiative corrections reflects the one of $\mathcal{E}^{(3)}$ in Fig. \[E-ns-mod-1\]. We note that the dominance of the imaginary over the real part in $\mathcal{E}_\omega$ and $\mathcal{E}_\rho$ is less pronounced in model 2, as can be anticipated by comparing Figs. \[E-ns-mod-1\] and \[E-ns-mod-2\].
The cross section $d\sigma_L/dt$ for $\rho$ production is dominated by $({\operatorname{Im}}\mathcal{H}_\rho)^2$, except for contributions from $({\operatorname{Re}}\mathcal{H}_\rho)^2$ at small $x_B$ for NLO and at large $x_B$ for LO. Given the size of corrections to ${\operatorname{Im}}\mathcal{H}_\rho$ in Fig. \[rho\_re\_im\] we thus have quite substantial NLO effects in the cross section at $Q^2= 4{\operatorname{GeV}}^2$, as shown in Fig. \[cs\_rho\]. For $Q^2= 9{\operatorname{GeV}}^2$ and $x_B >0.1$ the relative corrections decrease. The plot has been calculated with model 1 for $E$, but since its contribution to $d\sigma_L/dt$ is negligible the corresponding curves for model 2 look very similar. To obtain an estimate of scale uncertainties, we show bands corresponding to $\mu= \mu_R =\mu_{GPD}$ between $2{\operatorname{GeV}}$ and $2Q$. Given our discussion in the previous section, we do not consider it meaningful to go to scales below $2{\operatorname{GeV}}$, so that the bands in the figure are strongly asymmetric. For $Q^2=4 {\operatorname{GeV}}^2$ they go only in one direction, and the band of the LO result does not provide an estimate for the size of the NLO corrections, which turn out to go in the other direction.
We have a very peculiar situation for the polarization asymmetry $A_{UT}$ in $\rho$ production, which as shown in Figs. \[rho\_asym\_1\] and \[rho\_asym\_2\] is very small in both models 1 and 2 due to the cancellations in $\mathcal{E}_\rho$ discussed above. $A_{UT}$ changes quite dramatically from LO to NLO in a wide range of kinematics, clearly because of the NLO corrections in the numerator. A closer look at Fig. \[rho\_re\_im\] reveals that the large perturbative corrections in ${\operatorname{Im}}\bigl( \mathcal{E}_\rho^*{\mskip 1.5mu}\mathcal{H}_\rho^{} \bigr)$ are mainly due to the large corrections to both ${\operatorname{Re}}\mathcal{H}_\rho$ and ${\operatorname{Re}}\mathcal{E}_\rho$. These hardly affect the unpolarized cross section, which is strongly dominated by ${\operatorname{Im}}\mathcal{H}_\rho$. At higher $Q^2$ the instability of $A_{UT}$ is less pronounced, and in model 2 we even have quite small corrections. We note that the bands from the scale variation at LO order are extremely narrow in Figs. \[rho\_asym\_1\] and \[rho\_asym\_2\]. This is because the scale variation of $\alpha_s(\mu_R)$ cancels in the ratio $A_{UT}$ at LO and because in the kinematics we are looking at, the $\mu_{GPD}$ dependence of $\mathcal{H}_\rho$ and $\mathcal{E}_\rho$ is rather weak. In this situation, the scale uncertainty of the LO result does obviously not provide a good estimate for the size of higher-order corrections. Let us finally remark that at $t= -0.4{\operatorname{GeV}}^2$ the asymmetry $A_{UT}$ must go to zero as $x_B$ tends to $0.484$ because of the prefactor $\sqrt{t_0-t\rule{0pt}{1.8ex}}$ in .
The cross section for $\omega$ production is shown in Fig. \[cs\_omega\] and shows a similar pattern of NLO corrections to the one in $\rho$ production, reflecting the similar pattern of corrections we have seen for ${\operatorname{Im}}\mathcal{H}_\rho$ and ${\operatorname{Im}}\mathcal{H}_\omega$. As a result the ratio of cross sections $d\sigma_L/dt$ in the two channels is quite stable under radiative corrections, as seen in Fig. \[omega\_rho\_ratio\]. The target polarization asymmetry, shown in Fig. \[omega\_asym\] for model 1, changes however drastically between LO and NLO at small to intermediate $x_B$. This is because ${\operatorname{Im}}\mathcal{E}_\omega$ then dominates over ${\operatorname{Re}}\mathcal{E}_\omega$, so that its product with the unstable convolution ${\operatorname{Re}}\mathcal{H}_\omega$ controls the numerator of the asymmetry. The absolute size of $A_{UT}$ can be large in this channel since $|\mathcal{E}_\omega| \sim |\mathcal{H}_\omega|$ in our model. According to Fig. \[omega\_re\_im\], the relative phase $\delta_\omega$ is close to zero at LO for $x_B {\raisebox{-4pt}{ $\,\stackrel{\textstyle <}{\sim}\,$}}0.3$, so that the factor $\sin\delta_\omega$ in makes $A_{UT}$ small and prone to large radiative corrections.
Let us finally take a look at $\phi$ production. At LO this channel is strongly dominated by gluon exchange, since in our models strange quark distributions are small for $H$ and even more so for $E$. At NLO we have further contributions from the pure singlet terms $\mathcal{H}^{S(b)}$ and $\mathcal{E}^{S(b)}$, which are not negligible. We see in Fig. \[cs\_phi\] that the NLO corrections to the cross section are large at small $x_B$ and slowly decrease with $x_B$. Except for the region of small $x_B$, this pattern is quite different from the one in $\rho$ production, so that the cross section ratio for the two channels receives important corrections at larger $x_B$ as we see in Fig. \[phi\_rho\_ratio\]. The asymmetry $A_{UT}$ is essentially zero at LO, because in our model the relative phase $\delta_\phi$ between $\mathcal{H}_\phi$ and $\mathcal{E}_\phi$ is very close to zero. This changes at NLO, where in model 1 we obtain a small to moderate $A_{UT}$, as shown in Fig. \[phi\_asym\].
![\[rho\_asym\_2\] As Fig. \[rho\_asym\_1\] but for model 2.](plots/rho_cs_Q2_4.ps "fig:"){width="\plotwidth"} ![\[rho\_asym\_2\] As Fig. \[rho\_asym\_1\] but for model 2.](plots/rho_cs_Q2_9.ps "fig:"){width="\plotwidth"}
![\[rho\_asym\_2\] As Fig. \[rho\_asym\_1\] but for model 2.](plots/rho_asym_Q2_4.ps "fig:"){width="\plotwidth"} ![\[rho\_asym\_2\] As Fig. \[rho\_asym\_1\] but for model 2.](plots/rho_asym_Q2_9.ps "fig:"){width="\plotwidth"}
![\[rho\_asym\_2\] As Fig. \[rho\_asym\_1\] but for model 2.](plots/rho_asym_Q2_4_Modell_2.ps "fig:"){width="\plotwidth"} ![\[rho\_asym\_2\] As Fig. \[rho\_asym\_1\] but for model 2.](plots/rho_asym_Q2_9_Modell_2.ps "fig:"){width="\plotwidth"}
![\[omega\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \omega{\mskip 1.5mu}p$.](plots/omega_cs_Q2_4.ps "fig:"){width="\plotwidth"} ![\[omega\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \omega{\mskip 1.5mu}p$.](plots/omega_cs_Q2_9.ps "fig:"){width="\plotwidth"}
![\[omega\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \omega{\mskip 1.5mu}p$.](plots/ratio_omega_rho_Q2_4.ps "fig:"){width="\plotwidth"} ![\[omega\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \omega{\mskip 1.5mu}p$.](plots/ratio_omega_rho_Q2_9.ps "fig:"){width="\plotwidth"}
![\[omega\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \omega{\mskip 1.5mu}p$.](plots/omega_asym_Q2_4.ps "fig:"){width="\plotwidth"} ![\[omega\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \omega{\mskip 1.5mu}p$.](plots/omega_asym_Q2_9.ps "fig:"){width="\plotwidth"}
![\[phi\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \phi{\mskip 1.5mu}p$.](plots/phi_cs_Q2_4.ps "fig:"){width="\plotwidth"} ![\[phi\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \phi{\mskip 1.5mu}p$.](plots/phi_cs_Q2_9.ps "fig:"){width="\plotwidth"}
![\[phi\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \phi{\mskip 1.5mu}p$.](plots/ratio_phi_rho_Q2_4.ps "fig:"){width="\plotwidth"} ![\[phi\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \phi{\mskip 1.5mu}p$.](plots/ratio_phi_rho_Q2_9.ps "fig:"){width="\plotwidth"}
![\[phi\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \phi{\mskip 1.5mu}p$.](plots/phi_asym_Q2_4.ps "fig:"){width="\plotwidth"} ![\[phi\_asym\] As Fig. \[rho\_asym\_1\] but for $\gamma^* p\to \phi{\mskip 1.5mu}p$.](plots/phi_asym_Q2_9.ps "fig:"){width="\plotwidth"}
Pseudoscalar meson production {#sec:pseudo}
=============================
Having studied in detail the production of vector mesons, let us finally take a look at pseudoscalar production. We will only consider $\gamma^* p\to \pi^+ n$, which was already studied at NLO in [@Belitsky:2001nq]. Gluon distributions do not contribute in this channel.
In the collinear approximation the amplitude for this process can be written as $$\begin{aligned}
\label{pion_amp_NLO}
\mathcal{M} &=
\frac{4\pi\sqrt{4\pi\alpha}}{\xi{\mskip 1.5mu}Q N_c}\, f_\pi
\int_0^1 dz\, \phi_\pi(z)
\int_{-1}^1 dx\,
\Bigl[ e_u{\mskip 1.5mu}T_a({\bar{z}},x,\xi) - e_d{\mskip 1.5mu}T_a(z,-x,\xi) \Bigr] \,
\Bigl[ \widetilde{F}^{u}(x,\xi,t) - \widetilde{F}^{d}(x,\xi,t) \Bigr]
\nonumber \\
&= \frac{4\pi\sqrt{4\pi\alpha}}{\xi{\mskip 1.5mu}Q N_c}\, f_\pi
\sum_{n=0}^\infty a_n^{}{\mskip 1.5mu}\widetilde{\mathcal{F}}^{{\mskip 1.5mu}\pi}_n\end{aligned}$$ with $e_u = 2/3$, $e_d = -1/3$ and $f_\pi = 131 {\operatorname{MeV}}$. $\phi_\pi(z)$ is the twist-two distribution amplitude of the pion and has a Gegenbauer decomposition as in . The convolutions $\widetilde{\mathcal{F}}^{{\mskip 1.5mu}\pi}_n$ are defined as $$\label{Ftilde-def}
\widetilde{\mathcal{F}}^{{\mskip 1.5mu}\pi}_n =
\int_{-1}^1 dx\,
\Bigl[ e_u{\mskip 1.5mu}T_{a,n}(x,\xi) - e_d{\mskip 1.5mu}T_a(-x,\xi) \Bigr] \,
\Bigl[ \widetilde{F}^{u}(x,\xi,t) - \widetilde{F}^{d}(x,\xi,t) \Bigr] \,,$$ and the kernels $T_a(\bar{z},x,\xi)$ and $T_{a,n}(x,\xi)$ are the same as in Sect. \[sec:kernels\]. The matrix elements $\widetilde{F}^q$ are the counterparts of $F^q$ for polarized quarks and given by $$\widetilde{F}^q(x,\xi,t) =
\frac{1}{(p+p')\cdot n} \left[
\widetilde{H}^q(x,\xi,t)\, \bar u(p') {\mskip 1.5mu}\slashed{n}
\gamma_5{\mskip 1.5mu}u(p)
+ \widetilde{E}^q(x,\xi,t)\, \bar u(p')\,
\frac{(p'-p)\cdot n}{2 m_p}\, \gamma_5{\mskip 1.5mu}u(p) \right]$$ in terms of the generalized parton distributions $\widetilde{H}$ and $\widetilde{E}$, where as in the unpolarized case we use the conventions of [@Diehl:2003ny]. Since the hard-scattering kernel in is neither even nor odd in $x$, the convolution involves both the charge-conjugation even and odd combinations $$\begin{aligned}
\label{def-Ftilde-pm}
\widetilde{F}^{q(+)}(x,\xi,t) &=
\widetilde{F}^q(x,\xi,t) + \widetilde{F}^q(-x,\xi,t) \,,
&
\widetilde{F}^{q(-)}(x,\xi,t) &=
\widetilde{F}^q(x,\xi,t) - \widetilde{F}^q(-x,\xi,t) \,.\end{aligned}$$
We model the distributions $\widetilde{H}$ in close analogy to the unpolarized case and set $$\begin{aligned}
\label{dd-models-pol}
\widetilde{H}^{q(+)}(x,\xi,t) &=
\int_{-1}^1 d\beta \int_{-1+|\beta|}^{1-|\beta|} d\alpha\;
\delta(x-\beta-\xi\alpha)\, h^{(2)}(\beta,\alpha)\,
\widetilde{H}^{q(+)}(\beta,0,t) \,,
\nonumber \\
\widetilde{H}^{q(-)}(x,\xi,t) &=
\int_{-1}^1 d\beta \int_{-1+|\beta|}^{1-|\beta|} d\alpha\;
\delta(x-\beta-\xi\alpha)\, h^{(2)}(\beta,\alpha)\,
\widetilde{H}^{q(-)}(\beta,0,t) \,,\end{aligned}$$ with $h^{(2)}(\beta,\alpha)$ as in and $$\begin{aligned}
\widetilde{H}^{q(+)}(x,0,t)
&= \Delta q_v(x) \exp\bigl[ t f_{q_v}(x) \bigr]
+ 2 \Delta \bar{q}(x) \exp\bigl[ t f_{\bar{q}}(x) \bigr] \,,
\nonumber \\
\widetilde{H}^{q(-)}(x,0,t)
&= \Delta q_v(x) \exp\bigl[ t f_{q_v}(x) \bigr]\end{aligned}$$ for $x>0$. The values for $x<0$ are determined by the symmetry properties following from . For the polarized valence and antiquark densities $\Delta q_v$ and $\Delta \bar{q}$ we use the NLO parameterization from [@Bluemlein:2002be] at $\mu=
2{\operatorname{GeV}}$, and for the $t$ dependence we take the same functions $f_{q_v}(x)$ as in , and furthermore set $f_{\bar{q}}(x) = f_{q_v}(x)$. As was shown in [@Diehl:2004cx], this gives a good description of the isovector axial form factor via the sum rule $$F_A(t) = \int_0^1 dx\, \bigl[ \widetilde{H}^{u(+)}(x,0,t)
- \widetilde{H}^{d(+)}(x,0,t) \bigl] \,.$$ For the nucleon helicity-flip distribution $\widetilde{E}$ we take a pion exchange ansatz $$\label{Etilde-model}
\widetilde{E}^{u}(x,\xi,t) = -\widetilde{E}^{d}(x,\xi,t) =
\frac{\theta\bigl( \xi-|x| \bigr)}{2\xi}\,
\phi_\pi\left(\frac{x+\xi}{2\xi}\right)
\frac{2m_p^2\, g_A}{m_\pi^2 -t}\,
\frac{\Lambda^2-m_\pi^2}{\Lambda^2-t} ,$$ with the nucleon axial charge $g_A \approx 1.26$ and a cutoff parameter $\Lambda = 0.8 {\operatorname{GeV}}$ [@Koepf:1995yh] to suppress large off-shellness of the exchanged pion in the $t$ channel.
Results {#sec:pi-results}
-------
![\[Htilde\] The convolution $\widetilde{\mathcal{H}}^{{\mskip 1.5mu}\pi}_{{\mskip 1.5mu}0}$ defined as in , evaluated at $Q= 2{\operatorname{GeV}}$ and $t=
-0.4{\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/H_tilde_pion_real_LO_NLO.ps "fig:"){width="\plotwidth"} ![\[Htilde\] The convolution $\widetilde{\mathcal{H}}^{{\mskip 1.5mu}\pi}_{{\mskip 1.5mu}0}$ defined as in , evaluated at $Q= 2{\operatorname{GeV}}$ and $t=
-0.4{\operatorname{GeV}}^2$. The scales are set to $\mu_R = \mu_{GPD} = Q$.](plots/H_tilde_pion_imag_LO_NLO.ps "fig:"){width="\plotwidth"}
The convolution $\widetilde{\mathcal{H}}^{{\mskip 1.5mu}\pi}_{n}$ at LO and NLO is shown in Fig. \[Htilde\] for $n=0$. We find moderate corrections for the imaginary part and larger ones for the real part. For $\widetilde{\mathcal{E}}^{{\mskip 1.5mu}\pi}_{n}$ we can easily give the analytic form of the NLO result. The scale dependent terms admit a closed expression, $$\begin{aligned}
\label{E-tilde-res-gen}
\sum_{n} a_n^{}{\mskip 1.5mu}\widetilde{\mathcal{E}}^{{\mskip 1.5mu}\pi}_n
\,\propto\,
\sum_{m,n} a_m^{} & a_n^{}{\mskip 1.5mu}\biggl\{ 1 + \frac{\alpha_s}{4\pi}
\nonumber \\
& \; \times \biggl[\,
\beta_0{\mskip 1.5mu}\biggl( \frac{14}{3} + \frac{\gamma_m+\gamma_n}{2C_F}
- \ln\frac{Q^2}{\mu_R^2} \biggr)
- \gamma_m \ln\frac{Q^2}{\mu^2_{GPD}}
- \gamma_n \ln\frac{Q^2}{\mu^2_{DA}} + \ldots \,\biggr]
\biggr\} \,,\end{aligned}$$ where the $\ldots$ denote contributions which depend neither on $Q^2$ and the scales nor on $\beta_0$. Including these terms we can write $$\begin{aligned}
\label{Etilde-res}
\sum_{n} & a_n^{}{\mskip 1.5mu}\widetilde{\mathcal{E}}^{{\mskip 1.5mu}\pi}_n
\,\propto\,
(1 + a_2 + a_4)^2 + \frac{\alpha_s(\mu_R)}{\pi}
\biggl[
\frac{79}{12} + 25.0{\mskip 1.5mu}a_2 + 32.8{\mskip 1.5mu}a_4
+ 53.4{\mskip 1.5mu}a_2{\mskip 1.5mu}a_4 + 21.4{\mskip 1.5mu}a_2^2 + 32.6{\mskip 1.5mu}a_4^2
\nonumber \\
& - \frac{9}{4}\, (1+a_2+a_4)^2{\mskip 1.5mu}\ln\frac{Q^2}{\mu_R^2}
- (1+a_2+a_4)
\biggl( \frac{25}{18}\, a_2 + \frac{91}{45}\, a_4 \biggl)
\biggl( \ln\frac{Q^2}{\mu^2_{GPD}} + \ln\frac{Q^2}{\mu^2_{DA}}
\biggr) \biggr]
+ \ldots \,,\end{aligned}$$ where we have set $n_f=3$ in $\beta_0$ and where we approximated numerically the coefficients written with a decimal point. Here the $\ldots$ denote terms with higher Gegenbauer coefficients. Note that these coefficients appear twice, once for the produced pion and once for the pion exchange ansatz of the distribution $\widetilde{E}$. Up to a global factor, the expression also gives the NLO result for the electromagnetic pion form factor $F_\pi(Q^2)$ at large spacelike momentum transfer $Q^2$, and it agrees with the result in the detailed study [@Melic:1998qr]. Let us first discuss the case $m=n=0$ relevant for the asymptotic form of the pion distribution amplitude, where the convolution has no dependence on $\mu_{GPD}$ and $\mu_{DA}$. We then have the rather large coefficient $79/12 \approx 6.6$ in square brackets, so that with the scale choice $\mu_R = Q$ there are quite large NLO corrections. The corrections are zero for $\mu_R^2 = e^{-79/27}{\mskip 1.5mu}Q^2 \approx
0.05\, Q^2$, which is outside the perturbative region for most cases relevant in practice. The BLM scale for this case is yet smaller: with we reproduce the well-known result $\mu_R^2 = e^{-14/3}{\mskip 1.5mu}Q^2 \approx 0.01{\mskip 1.5mu}Q^2$ [@Belitsky:2001nq]. The coefficient of $\alpha_s/\pi$ in is then $-47/12 \approx -3.9$ and thus again rather large, but of course the scale $\mu_R$ is outside the perturbative region for all experimentally relevant kinematics. We finally see in that for higher Gegenbauer moments the correction terms are larger than for $m=n=0$. The reason for this is the same which we discussed in Sect. \[sec:large-x\] for the convolutions $\mathcal{H}$. In we also see that the BLM scale becomes smaller for higher $m$ and $n$.
The observables for exclusive pion production at leading order in $1/Q$ are the same as for vector meson production, and the $ep$ cross section is given as in . The cross section for a longitudinal photon and the transverse target asymmetry are now respectively given by $$\begin{aligned}
\label{pi-cs}
\frac{d\sigma_L}{dt} &=
\frac{\pi^2}{9}\, \frac{\alpha}{Q^6}\,
\frac{(2-x_B)^2}{1-x_B}\, (2 f_\pi)^2\,
\Bigl[{\mskip 1.5mu}(1-\xi^2)\, | \widetilde{\mathcal{H}}_\pi^{} |^2
- \xi^2{\mskip 1.5mu}t/(4m_p^2)\, | \widetilde{\mathcal{E}}_\pi^{} |^2
- 2\xi^2 {\operatorname{Re}}\bigl( \widetilde{\mathcal{E}}_\pi^*{\mskip 1.5mu}\widetilde{\mathcal{H}}_\pi^{} \bigr)
{\mskip 1.5mu}\Bigr]
\intertext{and}
A_{UT} &= - \frac{\sqrt{t_0-t\rule{0pt}{1.8ex}}}{m_p}\,
\frac{\xi{\mskip 1.5mu}\sqrt{1-\xi^2}\,
{\operatorname{Im}}\bigl( \widetilde{\mathcal{E}}_\pi^*{\mskip 1.5mu}\widetilde{\mathcal{H}}_\pi^{} \bigr)}{\rule{0pt}{1.05em}
(1-\xi^2)\, | \widetilde{\mathcal{H}}_\pi^{} |^2
- \xi^2{\mskip 1.5mu}t/(4m_p^2)\, | {\mskip 1.5mu}\widetilde{\mathcal{E}}_\pi^{} |^2
- 2\xi^2 {\operatorname{Re}}\bigl( \widetilde{\mathcal{E}}_\pi^*{\mskip 1.5mu}\widetilde{\mathcal{H}}_\pi^{} \bigr)} \,,\end{aligned}$$ with $$\begin{aligned}
\widetilde{\mathcal{H}}_\pi &= \sum_{n=0}^\infty a_n^{}{\mskip 1.5mu}\widetilde{\mathcal{H}}^{{\mskip 1.5mu}\pi}_n \,,
&
\widetilde{\mathcal{E}}_\pi &= \sum_{n=0}^\infty a_n^{}{\mskip 1.5mu}\widetilde{\mathcal{E}}^{{\mskip 1.5mu}\pi}_n \,.\end{aligned}$$ For numerical estimates we take the asymptotic pion distribution amplitude in the following, setting $a_n=0$ for $n\ge 2$. We note that the recent lattice study [@Braun:2006dg] obtained a rather moderate value $a_2(\mu_0) = 0.201(114)$ at $\mu_0 = 2{\operatorname{GeV}}$.
In Fig. \[cs\_pi\_HE\] we show the separate contributions from the terms with $|\widetilde{\mathcal{H}}_\pi|^2$ and with $|\widetilde{\mathcal{E}}_\pi|^2$ in , as well as the full result. We see that at the value of $t$ chosen here, the contribution from $|\widetilde{\mathcal{H}}_\pi|^2$ is more important, mainly because of the suppression factor $(\Lambda^2-m_\pi^2)
/(\Lambda^2-t)$ in our model for $\widetilde{E}$. The square of this factor is $0.36$ at $t=-0.4{\operatorname{GeV}}^2$.
We compare the LO and NLO results for the cross section in Fig. \[cs\_pi\] and find that the NLO corrections are quite large, even at $Q^2= 9{\operatorname{GeV}}^2$. In contrast, the corrections for the beam spin asymmetry are very small as seen in Fig. \[pi\_asym\], in line with the findings reported in [@Belitsky:2001nq]. Note that with our model $\widetilde{\mathcal{E}}_\pi$ is purely real, so that at intermediate $x_B$ the large relative NLO corrections in ${\operatorname{Re}}\widetilde{\mathcal{H}}_\pi$ do not affect the numerator of $A_{UT}$ in . Approximating the asymmetry as $$A_{UT} \approx
- \frac{\sqrt{t_0-t\rule{0pt}{1.8ex}}}{m_p} \;
\frac{\xi {\operatorname{Im}}\bigl( \widetilde{\mathcal{E}}_\pi^*{\mskip 1.5mu}\widetilde{\mathcal{H}}_\pi^{} \bigr)}{\rule{0pt}{1.05em}
| \widetilde{\mathcal{H}}_\pi^{} |^2}
= - \frac{\sqrt{t_0-t\rule{0pt}{1.8ex}}}{m_p} \;
\biggl| \frac{\xi{\mskip 1.5mu}\widetilde{\mathcal{E}}_\pi}{\rule{0pt}{1.05em}
\widetilde{\mathcal{H}}_\pi}\, \biggr|
\sin\delta_\pi$$ with $\delta_\pi = \arg( \widetilde{\mathcal{H}}_\pi
/\widetilde{\mathcal{E}}_\pi)$, we can understand why only small corrections are seen in this case: the relative phase $\delta_\pi$ is well different from zero, and the NLO corrections increase both $|\widetilde{\mathcal{H}}_\pi|$ and $|\widetilde{\mathcal{E}}_\pi|$.
![\[pi\_asym\] The transverse target spin asymmetry for $\pi^+$ production, as defined in . The meaning of the bands and solid lines is as in Fig. \[cs\_pi\].](plots/Contrib_pion.ps){width="\plotwidth"}
![\[pi\_asym\] The transverse target spin asymmetry for $\pi^+$ production, as defined in . The meaning of the bands and solid lines is as in Fig. \[cs\_pi\].](plots/pion_cs_Q2_4.ps "fig:"){width="\plotwidth"} ![\[pi\_asym\] The transverse target spin asymmetry for $\pi^+$ production, as defined in . The meaning of the bands and solid lines is as in Fig. \[cs\_pi\].](plots/pion_cs_Q2_9.ps "fig:"){width="\plotwidth"}
![\[pi\_asym\] The transverse target spin asymmetry for $\pi^+$ production, as defined in . The meaning of the bands and solid lines is as in Fig. \[cs\_pi\].](plots/pion_asym_Q2_4.ps "fig:"){width="\plotwidth"} ![\[pi\_asym\] The transverse target spin asymmetry for $\pi^+$ production, as defined in . The meaning of the bands and solid lines is as in Fig. \[cs\_pi\].](plots/pion_asym_Q2_9.ps "fig:"){width="\plotwidth"}
Summary {#sec:sum}
=======
In this work we have analyzed the NLO corrections for exclusive meson production at large $Q^2$ in the collinear factorization approach. Using the Gegenbauer expansion of meson distribution amplitudes, we have rewritten the hard-scattering kernels of [@Ivanov:2004zv] into functions depending on only one variable, and we have separated the explicit logarithms in the factorization scale for the meson distribution amplitude and the generalized parton distributions.
For vector meson production at small $x_B$ we find huge NLO corrections even for $Q^2$ well above $10{\operatorname{GeV}}^2$, in agreement with the results obtained in [@Ivanov:2004zv]. The corrections have opposite sign compared to the Born term and can be traced back to BFKL type logarithms in the hard-scattering kernels, which appear with rather large numerical prefactors in this process. We conclude at this stage that a quantitative control of radiative corrections at small $x_B$ will require resummation of these logarithms. First steps in this direction have been reported in [@Dima:2007]. If successful, such a resummation in combination with a dispersion relation [@Anikin:2007yh] may also be useful for stabilizing the real part of the amplitude, where we find very large NLO corrections even at $x_B\sim 0.1$.
At intermediate to large $x_B$, typical of fixed-target experiments, we have investigated the production of $\rho^0$, $\omega$, $\phi$ and of $\pi^+$. We find NLO corrections to the longitudinal cross sections of up to $100\%$, which somewhat decrease in size when going from $Q^2 = 4{\operatorname{GeV}}^2$ to $9{\operatorname{GeV}}^2$. Note that the meson production cross section depends *quadratically* on generalized parton distributions—the increased sensitivity to these basic quantities comes with an increased sensitivity to higher-order corrections. We generally find that uncertainties on the cross section due to the choice of renormalization and factorization scales are not too large at LO and do not significantly decrease when going to NLO. For scales below $4{\operatorname{GeV}}^2$, however, NLO corrections often grow out of control. The cross section ratio for $\omega$ to $\rho$ production turns out to be very stable under corrections, but less so the one for $\phi$ to $\rho$. For the transverse target polarization asymmetry $A_{UT}$ in $\pi^+$ production we find quite small NLO effects, confirming the results in [@Belitsky:2001nq]. For vector meson production this is however not the case. With the models we have used for the nucleon helicity-flip distributions $E$, the numerator of the asymmetry in this channel is dominated by the product $({\operatorname{Im}}\mathcal{E}_V) ({\operatorname{Re}}\mathcal{H}_V)$ in a wide range of kinematics and therefore suffers from the perturbative instability we find for ${\operatorname{Re}}\mathcal{H}_V$ at small to intermediate $x_B$, even if the corrections to ${\operatorname{Im}}\mathcal{E}_V$ are not too large. It is often assumed that corrections tend to cancel in asymmetries. The examples we have studied show that this may hold in specific cases but not in others, and that special care is needed for observables like $A_{UT}$ that depend on the relative phase between amplitudes.
We should recall that in the kinematics we studied, one must expect that our leading-twist results receive power corrections that cannot be neglected when comparing with data. They will certainly affect the cross sections and will not always cancel in cross section ratios. An example is the transverse target polarization asymmetry in $\pi^+$ production. The phenomenological estimates in [@Vanderhaeghen:1999xj] found that the convolution $\widetilde{\mathcal{H}}_\pi$ is decreased by effects of transverse parton momentum in the hard scattering, whereas $\widetilde{\mathcal{E}}_\pi$ is increased by the soft overlap mechanism that has been extensively studied in the context of the pion form factor. Together, these corrections may significantly increase leading-twist estimates for $A_{UT}$.
From our numerical studies we must conclude that a precise quantitative interpretation of exclusive meson production requires large $Q^2$, say above $10{\operatorname{GeV}}^2$. In addition it would be highly valuable to have a consistent scheme for combining radiative with power corrections, at least in parts. Nevertheless, we find that valuable information on generalized parton distributions can be obtained also from data at lower $Q^2$. In particular, a large measured asymmetry $A_{UT}$ in vector meson production would give valuable constraints on the size of the proton helicity-flip distribution $E^g$ for gluons, which are most difficult to obtain in deeply virtual Compton scattering or from lattice QCD calculations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We gratefully acknowledge discussions with L. Favart, H. Fischer, P. Kroll, A. Rostomyan and A. Schäfer. Special thanks are due to D. Yu. Ivanov for numerous conversations and advice. This work is supported by the Helmholtz Association, contract number VH-NG-004.
Polylogarithms {#app:polylog}
==============
We collect here some properties of the polylogarithms that appear in the hard-scattering kernels for meson production. Their definitions are $$\begin{aligned}
{\operatorname{Li}}_2 z &= - \int_0^1 \frac{dt}{t}\, \ln(1- z t) \,,
&
{\operatorname{Li}}_3 z &= \int_0^1 \frac{dt}{t}\, {\operatorname{Li}}_2(z t) \,,\end{aligned}$$ from which one readily obtains for the imaginary parts $$\begin{aligned}
{\operatorname{Im}}{\mskip 1.5mu}\Bigl[ {\operatorname{Li}}_2({\bar{y}}+ i\epsilon) \Bigr] &=
\pi{\mskip 1.5mu}\theta(-y){\mskip 1.5mu}\ln{\bar{y}}\,,
&
{\operatorname{Im}}{\mskip 1.5mu}\Bigl[ {\operatorname{Li}}_3({\bar{y}}+ i\epsilon) \Bigr] &=
\frac{\pi}{2}\, \theta(-y){\mskip 1.5mu}\ln^2{\bar{y}}\,.\end{aligned}$$ The limiting behavior for $y\to -\infty$ can be obtained from the expansions $$\begin{aligned}
\label{poly-asy1}
{\operatorname{Li}}_2 y &= -\frac{\pi^2}{6} - \frac{1}{2}{\mskip 1.5mu}\ln^2(-y)
- \sum_{n=1}^\infty \frac{y^{-n}}{n^2} \,,
&
{\operatorname{Li}}_3 y &= -\frac{\pi^2}{6}{\mskip 1.5mu}\ln(-y) - \frac{1}{6}{\mskip 1.5mu}\ln^3(-y)
+ \sum_{n=1}^\infty \frac{y^{-n}}{n^3} \,,\end{aligned}$$ which are valid for $y< -1$, and from $$\begin{aligned}
\label{poly-asy2}
{\operatorname{Re}}{\mskip 1.5mu}\Bigl[ {\operatorname{Li}}_2{\bar{y}}\Bigr] &=
\frac{\pi^2}{3}
- \frac{1}{2}{\mskip 1.5mu}\ln^2{\bar{y}}- \sum_{n=1}^\infty \frac{{\bar{y}}^{-n}}{n^2} \,,
&
{\operatorname{Re}}{\mskip 1.5mu}\Bigl[ {\operatorname{Li}}_3{\bar{y}}\Bigr] &=
\frac{\pi^2}{3}{\mskip 1.5mu}\ln{\bar{y}}- \frac{1}{6}{\mskip 1.5mu}\ln^3{\bar{y}}+ \sum_{n=1}^\infty \frac{{\bar{y}}^{-n}}{n^3} \,,\end{aligned}$$ which holds for $y < 0$. A useful relation finally is $${\operatorname{Li}}_2 y + {\operatorname{Li}}_2 {\bar{y}}= \frac{\pi^2}{6} - (\ln y)\, (\ln{\bar{y}}) \,.$$ A wealth of further information can be found in [@Devoto:1983tc].
Hard-scattering kernels for higher Gegenbauer moments {#app:kernels}
=====================================================
In this appendix we give the analogs of the hard-scattering kernels in for Gegenbauer index $n=2$ and $n=4$. For the gluon kernel we find $$\begin{aligned}
\label{app:gluon}
t_{g,2}(y) &=
\biggl[ 2C_A{\mskip 1.5mu}(y^2+{\bar{y}}^2) - C_F{\mskip 1.5mu}y \biggr] \frac{\ln y}{{\bar{y}}}\,
\ln\frac{Q^2}{\mu^2_{GPD}}
+ \frac{\beta_0}{2}{\mskip 1.5mu}\ln\frac{\mu_R^2}{\mu^2_{GPD}}
- \frac{25}{12}\, C_F \ln\frac{Q^2}{\mu^2_{DA}}
\nonumber \\
&\quad
+ C_F \biggl[
\frac{35}{36} (5-54y{\bar{y}})
- \frac{y}{2}\, \frac{\ln^2 y}{{\bar{y}}}
- 7({\bar{y}}-y){\mskip 1.5mu}(1-30y{\bar{y}}) {\operatorname{Li}}_2{\bar{y}}\nonumber \\
&\qquad\qquad
+ \left( \frac{1}{{\bar{y}}} - \frac{3}{2}
-\frac{392}{3}y+525y^2-420y^3 \right) \ln y
\biggr]
\nonumber \\
&\quad
+ C_A \biggl[
- \frac{15}{4}\, (1-4y{\bar{y}})
+ \left( \frac1{\bar{y}}-2y \right) \ln^2 y
+ ({\bar{y}}-y){\mskip 1.5mu}(7-60y{\bar{y}}) {\operatorname{Li}}_2{\bar{y}}\nonumber \\
&\qquad\qquad
- \left( \frac{23}{3{\bar{y}}}+\frac{5}{6}-58y+150y^2-120y^3 \right) \ln y
\biggr]
\nonumber \\
&\quad
+ 6y{\bar{y}}{\mskip 1.5mu}\Bigl[ 5(1-4y{\bar{y}}){\mskip 1.5mu}C_A - 14 (1-5y{\bar{y}}){\mskip 1.5mu}C_F \Bigr]
\left( 3{\operatorname{Li}}_3{\bar{y}}- \ln y\, {\operatorname{Li}}_2 y - \frac{\pi^2}{6} \ln y \right)
+ \{y\to{\bar{y}}\}\,,
\nonumber \\
t_{g,4}(y) &=
\biggl[ 2C_A{\mskip 1.5mu}(y^2+{\bar{y}}^2) - C_F{\mskip 1.5mu}y \biggr] \frac{\ln y}{{\bar{y}}}\,
\ln\frac{Q^2}{\mu^2_{GPD}}
+ \frac{\beta_0}{2}{\mskip 1.5mu}\ln\frac{\mu_R^2}{\mu^2_{GPD}}
- \frac{91}{30}\, C_F \ln\frac{Q^2}{\mu^2_{DA}}
\nonumber \\
&\quad
+ C_F \biggl[
\frac{27287}{1800}-595y{\bar{y}}+2520(y{\bar{y}})^2
- \frac{y}{2}\, \frac{\ln^2 y}{{\bar{y}}}
+ 16({\bar{y}}-y) \Bigl( 1-105y{\bar{y}}+630(y{\bar{y}})^2 \Bigr) {\operatorname{Li}}_2{\bar{y}}\nonumber \\
&\qquad\qquad
+ \left( \frac{1}{{\bar{y}}} - \frac{5}{2}
-\frac{11596}{15}y+9660y^2-34160y^3+45360y^4-20160y^5 \right) \ln y
{\mskip 1.5mu}\biggr]
\nonumber \\
&\quad
+ C_A \biggl[
- \frac{35}{16}\, (1-4y{\bar{y}}) (5-72y{\bar{y}})
+ \left( \frac1{\bar{y}}-2y \right) \ln^2 y
+ 2 ({\bar{y}}-y) \Bigl( 8-315y{\bar{y}}+1260(y{\bar{y}})^2 \Bigr) {\operatorname{Li}}_2{\bar{y}}\nonumber \\
&\qquad\qquad
- \left( \frac{257}{30{\bar{y}}}+\frac{77}{60}-\frac{1741}{5}y
+2940y^2-8960y^3+11340y^4-5040y^5 \right) \ln y
\biggr]
\nonumber \\
&\quad
+ 30y{\bar{y}}{\mskip 1.5mu}\Bigl[ 7(1-4y{\bar{y}})(1-6y{\bar{y}}){\mskip 1.5mu}C_A
\nonumber \\
&\qquad\qquad
- 16{\mskip 1.5mu}\Bigl( 1-14y{\bar{y}}+42(y{\bar{y}})^2 \Bigr){\mskip 1.5mu}C_F \Bigr]
\left( 3{\operatorname{Li}}_3{\bar{y}}- \ln y\, {\operatorname{Li}}_2 y - \frac{\pi^2}{6} \ln y \right)
+ \{y\to{\bar{y}}\} \,, \phantom{\biggl[ \biggr]}\end{aligned}$$ and for the pure singlet kernel $$\begin{aligned}
\label{app:singlet}
t_{b,2}(y) &=
2({\bar{y}}-y) \, \frac{\ln y}{{\bar{y}}}\,
\biggl[ \ln\frac{Q^2}{\mu_{GPD}^2} - \frac{23}{6} \biggr]
+ ({\bar{y}}-y)\, \frac{\ln^2 y}{{\bar{y}}}
- \frac{15}{2} ({\bar{y}}-y)
\nonumber \\
&\quad
+ 2(7-60y{\bar{y}}) {\operatorname{Li}}_2{\bar{y}}- \left( \frac{5}{3}-90y+120y^2\right) \ln y
\nonumber \\
&\quad
+ 60 ({\bar{y}}-y){\mskip 1.5mu}y{\bar{y}}\left[ 3 {\operatorname{Li}}_3{\bar{y}}+ \biggl( {\operatorname{Li}}_2{\bar{y}}+ \ln^2{\bar{y}}- \frac{\pi^2}{3} \biggl){\mskip 1.5mu}\ln y
\right]
- \{y\to{\bar{y}}\} \,,
\nonumber \\
t_{b,4}(y) &=
2({\bar{y}}-y) \, \frac{\ln y}{{\bar{y}}}\,
\biggl[ \ln\frac{Q^2}{\mu_{GPD}^2} - \frac{257}{60} \biggr]
+ ({\bar{y}}-y)\, \frac{\ln^2 y}{{\bar{y}}}
- \frac{35}{8}\, ({\bar{y}}-y) (5-72y{\bar{y}})
\nonumber \\
&\quad
+ 4{\mskip 1.5mu}\Bigl( 8-315y{\bar{y}}+1260(y{\bar{y}})^2 \Bigr) {\operatorname{Li}}_2{\bar{y}}- \left( \frac{77}{30}-665y+4550y^2-8820y^3+5040y^4 \right) \ln y
\nonumber \\
&\quad
+ 420 ({\bar{y}}-y){\mskip 1.5mu}y{\bar{y}}{\mskip 1.5mu}(1-6y{\bar{y}}) \left[ 3 {\operatorname{Li}}_3{\bar{y}}+ \biggl( {\operatorname{Li}}_2{\bar{y}}+ \ln^2{\bar{y}}- \frac{\pi^2}{3} \biggl){\mskip 1.5mu}\ln y
\right]
- \{y\to{\bar{y}}\} \,.\end{aligned}$$ The quark non-singlet kernel reads $$\begin{aligned}
\label{app:non-sing}
t_{a,2}(y) &=
\beta_0 \left[ \frac{21}{4} - \ln y
- \ln\frac{Q^2}{\mu^2_R} \right]
\nonumber \\
&\quad
+ C_F \biggl[
\left( 3 + 2 \ln y \right){\mskip 1.5mu}\ln\frac{Q^2}{\mu^2_{GPD}}
- \frac{25}{6}\, \ln\frac{Q^2}{\mu^2_{DA}}
- \frac{1019}{72} - \left( \frac{1}{{\bar{y}}} + \frac{7}{6} \right) \ln y
+ \ln^2 y
\biggr]
\nonumber \\
&\quad
+ \left( 2 C_F - C_A \right)
\biggl\{ \frac{401}{12}-255y+270y^2
- \left( \frac{299}{3}-867y+1830y^2-1080y^3 \right) \ln{\bar{y}}\nonumber \\
&\qquad
+ \left( \frac{56}{3}-357y+1290y^2-1080y^3 \right) \ln y
+ 2{\mskip 1.5mu}\bigl( 22-291y+780y^2-540y^3 \bigr)\,
\bigl( {\operatorname{Li}}_2 y - {\operatorname{Li}}_2{\bar{y}}\bigr)
\nonumber \\
&\qquad
+ 12{\mskip 1.5mu}(1-21y+106y^2-175y^3+90y^4) \phantom{\biggl[ \biggr]}
\nonumber \\
&\qquad\quad
\times \biggl[ 3 \bigl( {\operatorname{Li}}_3{\bar{y}}+ {\operatorname{Li}}_3 y \bigr)
- \ln y\, {\operatorname{Li}}_2 y - \ln{\bar{y}}\, {\operatorname{Li}}_2{\bar{y}}- \frac{\pi^2}{6}\, \bigl( \ln y + \ln{\bar{y}}\bigr) \biggr]
\biggr\} \,,
\nonumber \\
t_{a,4}(y) &=
\beta_0 \left[ \frac{31}{5} - \ln y
- \ln\frac{Q^2}{\mu^2_R} \right]
\nonumber\\
&\quad
+ C_F \biggl[
\left( 3 + 2 \ln y \right){\mskip 1.5mu}\ln\frac{Q^2}{\mu^2_{GPD}}
- \frac{91}{15}\, \ln\frac{Q^2}{\mu^2_{DA}}
- \frac{10213}{900} - \left( \frac{1}{{\bar{y}}} + \frac{46}{15} \right) \ln y
+ \ln^2 y
\biggr]
\nonumber \\
&\quad
+ \left( 2 C_F - C_A \right)
\biggl\{ \frac{4903}{40}-\frac{5775}{2}y+\frac{57085}{4}y^2
-23310y^3+11970y^4
\nonumber \\
&\qquad
- \left( \frac{21109}{60}-\frac{41451}{5}y+\frac{103285}{2}y^2
-125020y^3+129150y^4-47880y^5 \right) \ln{\bar{y}}\nonumber \\
&\qquad
+ \left( \frac{2899}{60}-\frac{11001}{5}y+\frac{45535}{2}y^2
-78400y^3+105210y^4-47880y^5 \right) \ln y
\nonumber \\
&\qquad
+ \bigl( 137-4506y+35280y^2-100380y^3+117180y^4-47880y^5 \bigr)\,
\bigl( {\operatorname{Li}}_2 y - {\operatorname{Li}}_2{\bar{y}}\bigr) \phantom{\biggl[ \biggr]}
\nonumber \\
&\qquad
+ 30{\mskip 1.5mu}\bigl( 1-48y+580y^2-2590y^3+5166y^4-4704y^5+1596y^6 \bigr)
\phantom{\biggl[ \biggr]}
\nonumber \\
&\qquad\quad
\times \biggl[ 3 \bigl( {\operatorname{Li}}_3{\bar{y}}+ {\operatorname{Li}}_3 y \bigr)
- \ln y\, {\operatorname{Li}}_2 y - \ln{\bar{y}}\, {\operatorname{Li}}_2{\bar{y}}- \frac{\pi^2}{6}\, \bigl( \ln y + \ln{\bar{y}}\bigr) \biggr]
\biggr\} \,.\end{aligned}$$ Using , and the representation $$\gamma_n = (-1)^{n+1}\,
2 C_F \int_0^1 dz\, (1-z){\mskip 1.5mu}(3 + 2 \ln z)\, C_n^{3/2}(2z-1)$$ of the anomalous dimensions, we can give a closed form for the scale dependent terms for all even $n$, $$\begin{aligned}
\label{higher-kern}
t_{g,n}(y) &=
\biggl[ 2C_A{\mskip 1.5mu}(y^2+{\bar{y}}^2) - C_F{\mskip 1.5mu}y \biggr] \frac{\ln y}{{\bar{y}}}\,
\ln\frac{Q^2}{\mu^2_{GPD}}
+ \frac{\beta_0}{2}{\mskip 1.5mu}\ln\frac{\mu_R^2}{\mu^2_{GPD}}
- \frac{\gamma_n}{2}{\mskip 1.5mu}\ln\frac{Q^2}{\mu^2_{DA}} + \{y\to{\bar{y}}\}
+ \ldots \,,
\nonumber \\
t_{b,n}(y) &=
2({\bar{y}}-y) \, \frac{\ln y}{{\bar{y}}}\, \ln\frac{Q^2}{\mu_{GPD}^2} - \{y\to{\bar{y}}\}
+ \ldots \,,
\nonumber \\
t_{a,n}(y) &= \beta_0 \biggl[ \frac{19}{6} + \frac{\gamma_n}{2 C_F}
- \ln y - \ln\frac{Q^2}{\mu_{R}^2} \biggr]
+ C_F (3 + 2\ln y) \ln\frac{Q^2}{\mu_{GPD}^2}
- \gamma_n \ln\frac{Q^2}{\mu_{DA}^2} + \ldots \,,\end{aligned}$$ where the terms denoted by $\ldots$ are independent of $Q^2$ and the scales and do not involve $\beta_0$. From the scale dependence of the Gegenbauer coefficients of the meson distribution amplitude we can readily reconstruct their evolution equation $$\mu^2 \frac{{\mathrm{d}}}{{\mathrm{d}}\mu^2}\, a_n(\mu)
= - \frac{\alpha_s(\mu)}{4\pi}\, \gamma_n{\mskip 1.5mu}a_n(\mu) +
O(\alpha_s^2) \,.$$ With and we see that the $\mu_{DA}$ dependence of the process amplitude cancels up to terms of order $\alpha_s^3$, as it must be.
Evolution kernels {#app:evolution}
=================
For definiteness we give here the LO evolution kernels for GPDs, which we have used to check the scale invariance of the NLO amplitude for meson production as explained in Sect. \[sec:kernels\]. The non-singlet evolution equation reads $$\label{gpd_ns_evol}
\mu^2 \frac{{\mathrm{d}}}{{\mathrm{d}}\mu^2}\, F^{NS}(x,\xi,t) =
\int_{-1}^1 \frac{dy}{|\xi|}\,
V^{NS}\biggl( \frac{x}{\xi},\frac{y}{\xi} \biggr)
F^{NS}(y,\xi,t) \,,$$ where $F^{NS}$ can be a flavor non-singlet combination such as $F^{u(+)} - F^{d(+)}$, or the charge-conjugation odd combination $F^{q(-)}(x,\xi,t) = F^q(x,\xi,t) + F^q(-x,\xi,t)$ for a single quark flavor. In the gluon and quark singlet sector we have a matrix equation $$\label{gpd_s_evol}
\mu^2\frac{{\mathrm{d}}}{{\mathrm{d}}\mu^2}\,
\begin{pmatrix}
F^S(x,\xi,t)\; \\[1.8ex]
F^g(x,\xi,t)
\end{pmatrix}
= \int_{-1}^1 \frac{dy}{|\xi|}
\begin{pmatrix}
V^{qq}\left( \frac{x}{\xi},\frac{y}{\xi} \right) &
\xi^{-1}{\mskip 1.5mu}V^{qg}\left( \frac{x}{\xi},\frac{y}{\xi} \right)
\\[1.2ex]
\xi V^{gq}\left( \frac{x}{\xi},\frac{y}{\xi} \right) \; &
V^{gg}\left( \frac{x}{\xi},\frac{y}{\xi} \right)
\end{pmatrix}
\;
\begin{pmatrix}
F^S(y,\xi,t)\; \\[1.8ex]
F^g(y,\xi,t)
\end{pmatrix}$$ with $F^{S}$ defined in . At $O(\alpha_s)$ one has $V^{NS}(x,y) = V^{qq}(x,y)$ and $$\begin{aligned}
\label{evol-kernels}
V^{qq}(x,y)
&= \frac{\alpha_s}{4\pi}\, C_F
\left[ \rho(x,y)\, \frac{1+x}{1+y} \left(1+\frac{2}{y-x}\right)
+ \{x\rightarrow -x,y\rightarrow -y\} \right]_+ \,,
\nonumber \\
V^{qg}(x,y) &= -\frac{\alpha_s}{4\pi}\, 2 T_F{\mskip 1.5mu}n_f
\left[ \rho(x,y)\, \frac{1+x}{(1+y)^2}\, (1-2x+y-xy)
- \{x\rightarrow -x,y\rightarrow -y\} \right] \,,
\nonumber \\
V^{gq}(x,y) &= \frac{\alpha_s}{4\pi}\, C_F
\left[ \rho(x,y)\, \left( (2-x) (1+x)^2
- \frac{(1+x)^2}{1+y} \right)
- \{x\rightarrow -x,y\rightarrow -y\} \right] \,,
\nonumber \\
V^{gg}(x,y) &= \frac{\alpha_s}{4\pi}\, C_A
\left[ \rho(x,y)\, \frac{(1+x)^2}{(1+y)^2}
\left(2+\frac{2}{y-x}\right)
+ \{x\rightarrow -x,y\rightarrow -y\} \right]_+
\nonumber \\
&+ \frac{\alpha_s}{4\pi}\, C_A
\left[ \rho(x,y)\, \frac{(1+x)^2}{(1+y)^2}\, (1-2x+2y-xy)
+ \{x\rightarrow -x,y\rightarrow -y\} \right]
\nonumber \\
&+ \frac{\alpha_s}{4\pi}\,
\left( \beta_0 - \frac{14}{3}\, C_A \right) \delta(x-y)\end{aligned}$$ with $T_F = 1/2$ and the remaining constants as given in . The plus-prescription appearing in $V^{qq}$ and $V^{gg}$ is defined by $$\label{plus_description}
\bigl[ f(x,y) \bigr]_+ = f(x,y) - \delta(x-y) \int dz\,f(z,y) \,,$$ and the function $\rho(x,y)$ specifies the support as $$\label{rho_function}
\rho(x,y) = \theta\left(\frac{1+x}{1+y}\right)
\theta\left(1-\frac{1+x}{1+y}\right) {\operatorname{sgn}}(1+y)
= \theta(y-x)\, \theta(x+1) - \theta(x-y)\, \theta(-x-1) \,.$$
The evolution equations for polarized GPDs read as in and , with the unpolarized matrix elements $F$ and kernels $V$ replaced by their polarized counterparts $\widetilde{F}$ and $\widetilde{V}$. With $\widetilde{F}^{q(+)}$ and $\widetilde{F}^{q(-)}$ defined in above, $\widetilde{F}^{NS}$ can be either a flavor non-singlet combination like $\widetilde{F}^{u(+)} -
\widetilde{F}^{d(+)}$ or a charge-conjugation odd combination $\widetilde{F}^{q(-)}$, whereas the flavor singlet combination is given by $$\widetilde{F}^{S} = \widetilde{F}^{u(+)} +
\widetilde{F}^{d(+)} + \widetilde{F}^{s(+)} \,.$$ To $O(\alpha_s)$ the polarized evolution kernels are $$\widetilde{V}^{NS}(x,y) = \widetilde{V}^{qq}(x,y) = V^{qq}(x,y)$$ and $$\begin{aligned}
\label{pol-kernels}
\widetilde{V}^{qg}(x,y) &= -\frac{\alpha_s}{4\pi}\, 2 T_f{\mskip 1.5mu}n_f
\left[ \rho(x,y)\, \frac{1+x}{(1+y)^2}
- \{x\rightarrow -x,y\rightarrow -y\} \right] \,,
\nonumber \\
\widetilde{V}^{gq}(x,y) &= \frac{\alpha_s}{4\pi}\, C_F
\left[ \rho(x,y)\, \frac{(1+x)^2}{1+y}
- \{x\rightarrow -x,y\rightarrow -y\} \right] \,,
\nonumber \\
\widetilde{V}^{gg}(x,y) &= \frac{\alpha_s}{4\pi}\, C_A
\left[ \rho(x,y)\, \frac{(1+x)^2}{(1+y)^2}
\left( 2+\frac{2}{y-x} \right)
+ \{x\rightarrow -x,y\rightarrow -y\} \right]_+
\nonumber \\
&+ \frac{\alpha_s}{4\pi}\,
\left( \beta_0 - \frac{14}{3}\, C_A \right) \delta(x-y) \,.\end{aligned}$$
The kernels given here agree with those in [@Blumlein:1999sc] if one takes into account that any contribution to $V^{gq}(x,y)$ which is even in $y$ at fixed $x$ will drop out in the convolution . Taking the limit $\xi\rightarrow 0$ as $$\label{kernel-forward}
\lim_{\xi\rightarrow 0^+}
\frac{1}{\xi}
\begin{pmatrix}
\phantom{\frac{1}{\xi}}{\mskip 1.5mu}V^{qq}\left( \frac{z}{\xi},\frac{1}{\xi} \right) &
\frac{1}{\xi}{\mskip 1.5mu}V^{qg}\left( \frac{z}{\xi},\frac{1}{\xi} \right) \\[1.2ex]
\frac{\xi}{z}{\mskip 1.5mu}V^{gq}\left( \frac{z}{\xi},\frac{1}{\xi} \right) &
\frac{1}{z}{\mskip 1.5mu}V^{gg}\left( \frac{z}{\xi},\frac{1}{\xi} \right)
\end{pmatrix}
= \begin{pmatrix}
P^{qq}(z) & P^{qg}(z) \\[1.8ex]
P^{gq}(z) & P^{gg}(z)
\end{pmatrix}$$ one obtains the usual DGLAP evolution kernels from , and in analogy one recovers the polarized DGLAP kernels from . The factors $\frac{1}{z}$ in front of $V^{gq}$ and $V^{gg}$ reflect the different forward limits of the quark and gluon GPDs.
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[^1]: We thank Dima Ivanov for discussions on this point. The numerical results in [@Ivanov:2004zv] were obtained with the correct prescription.
[^2]: We thus keep terms of $O(\alpha_s^3)$ in the cross section, although the accuracy of the NLO calculation is only up to $O(\alpha_s^2)$. This should not be seen as a problem, as it will not make a considerable difference in situations where perturbative corrections are moderate, whereas in situations where NLO corrections are huge we would neither trust the cross section with or without the partially included $O(\alpha_s^3)$ terms.
|
---
abstract: 'Consistent Hashing functions are widely used for load balancing across a variety of applications. However, the original presentation and typical implementations of Consistent Hashing rely on randomised allocation of hash codes to keys which results in a flawed and approximately-uniform allocation of keys to hash codes. We analyse the desired properties and present an algorithm that perfectly achieves them without resorting to any random distributions. The algorithm is simple and adds to our understanding of what is necessary to create a consistent hash function.'
author:
- 'Matthew Sackman\'
bibliography:
- '../../library.bib'
title: Perfect Consistent Hashing
---
Introduction
============
A hash function is a function that deterministically and uniformly maps keys of unbounded size to members of a finite set of hash codes. It is [*deterministic*]{} in that in the absence of changes to the set of hash codes, the same key is always mapped to the same hash code. It is [*uniform*]{} in that each hash code is equally likely to be generated. If $h_C(\kappa)$ is the hash function $h$ with the set of hash codes $C$, applied to the key $\kappa$, then $\forall \sigma
\in C, \forall \kappa.\: p(h_C(\kappa) = \sigma) = 1/|C|$.
A typical simple hash function is to treat the key as a natural number and then to take its modulus by the number of hash codes, $|C|$. This result is then used as an index into $C$, which is ordered in some way and treated as mapping naturals to hash codes. If $C[x]$ represents the result of indexing this mapping $C$ by $x$ then this simple hash function can be written as $h_C(\kappa) = C[\kappa \mod |C|]$. One problem with such a simple hash function is that when a new hash code is added to $C$ or an existing hash code removed, the existing mapping between keys and hash codes is non-minimally altered. If $C \subset
C'$ and $|C'| = |C| + 1$ (so $|C|$ and $|C'|$ are relatively prime) then we can define the set of keys which do not get remapped as $$\{\kappa \mid \delta \in \{0 \dots (|C| - 1)\},\, \iota \in
\mathbb{N},\, \kappa = \delta + \iota \cdot |C| \cdot |C'|\}$$ Thus in any set of keys of size $|C| \cdot |C'|$, there will be on average only $|C|$ keys for which $h_C(\kappa) = h_{C'}(\kappa)$, so the probability of a key not being remapped to a different hash code is $1/|C'|$, or $p(h_C(\kappa) = h_{C'}(\kappa)) = 1/|C'|$. Clearly, as the set of hash codes grows large, the probability of a key mapping to the same hash code after the addition or removal of a new hash code approaches zero.
In many applications, this high likelihood of keys being remapped when $C$ is altered is unacceptable; so we require a different hash function. For example, in the case of a distributed cache, a hash function might be being used to determine which node contains a requested object, with the key provided to the hash function being an object identifier (e.g. a URL), and the hash codes being node identifiers (e.g. I.P. addresses). In this scenario, when the set of nodes (i.e. hash codes) is altered, we want to minimise the number of objects that must move between nodes in order to satisfy the new mapping. Consistent Hashing [@Karger1997] provides exactly this: in addition to the usual determinism and uniformity properties of hash functions, it also requires that when a hash code is added or removed, only the minimal number of keys are remapped to maintain uniformity. In the case of addition, the minimal number of keys is simply the number of keys that must be mapped to the new hash code; therefore it is not permitted to remap keys between existing hash codes.
This remapping requirement in combination with the uniformity requirement reveals further details of how the hash function should behave. For the uniformity property to be maintained after a hash code is added, each existing hash code should give up an equal proportion of their keys to become the keys of the new hash code. Once again with $C \subset C'$ and $|C'| = |C| + 1$, the uniformity property gives us $\forall \sigma \in C, \forall \kappa. \: p(h_C(\kappa) = \sigma) =
1/|C|$ and $\forall \sigma \in C', \forall \kappa. \: p(h_{C'}(\kappa)
= \sigma) = 1/|C'|$. We can now relate these by $$\forall \sigma \in
C, \forall \kappa. \: h_C(\kappa) = \sigma \implies p(h_{C'}(\kappa) =
\sigma) = \frac{|C|}{|C'|}$$ i.e. if $h_C(\kappa)$ yields hash code $\sigma$ for key $\kappa$, then the probability of $h_{C'}(\kappa)$ yielding the same $\sigma$ for the same $\kappa$ is $|C| / |C'|$. The probability of $h_{C'}(\kappa)$ yielding a hash code from $C$ is $|C|
/ |C'|$, leaving $1 / |C'|$ for the new hash code, as required. Note that there is no possibility of keys being moved between existing hash codes: each hash code loses only as many keys as are required to be donated to the new hash code; all remaining keys stay with their existing hash code thus the remapping property is satisfied. The inverse specifies the conditions of a hash code being removed: to maintain the uniformity property, the removed hash code’s keys must be equally distributed amongst the remaining hash codes, and to maintain the remapping property, no keys are remapped except those that were mapped to the now departed hash code. It is worth observing that the requirement that only $1/|C'|$ of the keys are remapped is the complement of that achieved by our unsuitable simple hash function in which only $1/|C'|$ keys are [*not*]{} remapped!
The introduction of the properties of a Consistent Hash also presented an algorithm [@Karger1997], which we refer to as the classic algorithm. Whilst this algorithm has been implemented in many different programming languages and used in many scenarios, it has several flaws which we explore in this paper. We then present a new algorithm which precisely achieves the desired properties and solves the flaws identified in the classic algorithm.
Classic Algorithm
=================
The classic algorithm places hash codes at random points around a circle. Keys to the hash function are interpreted as points on the circle, and the hash function identifies the [*next*]{} hash code point around the circle, . In this way, we can think of each hash code as owning different segments of the circle. It is usual to apply a standard hash function to keys in order to limit the keys to the range of the circle and ensure uniform distribution of keys around the circle. Whilst the hash codes are placed at random points around the circle, the points may still be found deterministically, for example generated by applying a standard hash function to the hash code names themselves.
/łin [ 30/$\delta$, 90/$\gamma$, 45/$\beta$, 135/$\epsilon$, 60/$\alpha$ ]{}
(0,0) – (:1) arc (::1); (0,0) – (:1.1);
at (:1.3) [ł]{};
(0,0) – (325:1.6); at (325:1.7) [$\kappa$]{}; (0,0) (325:1.3) arc (325:305:1.3);
Because each discrete point around the circle can only be occupied by a single hash code, addition of hash codes is not commutative. This is not an essential property, but it does have an effect on the size of the state that must be maintained for the hash function. Addition is not commutative because of the possibility that two hash codes both try to occupy the same point around the circle. When such a collision occurs, a number of solutions are possible, but one of the two hash codes must [*win*]{} the particular point otherwise the remapping property will be violated (commutativity could be achieved if neither hash code wins the contested point, but that would result in keys being transferred from the first hash code added (the initial winner), to a different existing hash code, which is illegal). If hash code points around the circle are randomly generated, then the state maintained must include all those points (when a collision occurs, a fresh random point is generated for the new hash code). If the hash code points are generated deterministically by applying a standard hash function to the hash code identifiers, then the state must maintain the order in which hash codes were added and removed so that the points can be correctly reestablished including the outcome of collisions (when a collision occurs, a number of options are available, such as hashing the concatenation of the new hash code name and an attempt-number).
This classic algorithm achieves the remapping and deterministic properties but fails to guarantee the uniformity property. Whenever a hash code is added, its determined point around the circle splits an existing segment, transferring a portion of the keys from that segment to the new hash code. No other segments around the circle are altered so all other keys remain mapped to their existing hash codes, . Similarly, when a hash code is removed, the disappearance of its point around the circle merges its segment with the [*next*]{} segment. Again, no other points around the circle are altered, so no keys are remapped between other, surviving, hash codes. As a result, the remapping property is achieved. The determinism property is immediate, provided state is maintained appropriately in light of changes to $C$ as discussed previously.
$$\begin{aligned}
\vcenter{
\hbox{
\begin{tikzpicture}[scale=1.2]
\setcounter{b}{0}
\foreach \t/\l in
{
30/$\delta$,
90/$\gamma$,
180/$\beta$,
60/$\alpha$
}
{
\setcounter{a}{\value{b}}
\addtocounter{b}{\t}
{
\draw (0,0) -- (\thea:1) arc (\thea:\theb:1);
\draw (0,0) -- (\thea:1.1);
\node at (\thea:1.3) {\l};
\pgfmathparse{0.5*\thea+0.5*\theb}
\let\midangle\pgfmathresult
\node at (\midangle:0.75) {\l};
}
}
\end{tikzpicture}
}
}
&& \overset{+\epsilon}\Longrightarrow &&
\vcenter{
\hbox{
\begin{tikzpicture}[scale=1.2]
\setcounter{b}{0}
\foreach \t/\l in
{
30/$\delta$,
90/$\gamma$,
45/$\beta$,
135/$\epsilon$,
60/$\alpha$
}
{
\setcounter{a}{\value{b}}
\addtocounter{b}{\t}
{
\draw (0,0) -- (\thea:1) arc (\thea:\theb:1);
\draw (0,0) -- (\thea:1.1);
\node at (\thea:1.3) {\l};
\pgfmathparse{0.5*\thea+0.5*\theb}
\let\midangle\pgfmathresult
\node at (\midangle:0.75) {\l};
}
}
\end{tikzpicture}
}
}\end{aligned}$$
Uniformity however is at best approximated. The location of hash code points is random, so with just two hash codes, it is very unlikely for each hash code’s points to be $180^\circ$ apart: just $1/R$ where $R$ is the range of the circle. So the likelihood of the segments being of equal size and thus uniformity achieved, is very low. If one traverses around the perimeter of the circle at constant speed, encountering a hash code point (and entering a new segment) can be modelled by a Poisson process. The interval between such encounters (i.e. the segment length) is then an exponential distribution [@Niedermayer2005; @Xiaouien2009]. The extreme left-skew of this distribution demonstrates how exceedingly unlikely it is to achieve uniformity. For example, a circle divided by 10 points, one for each of 10 hash codes, will have a mean arc length of $36^\circ$, but a median arc length of just $25^\circ$.
To address this, the classic algorithm uses several points for each hash code, . This changes the distribution of segment lengths from exponential to Erlang; an Erlang-$k$ distribution is the sum of $k$ independent exponential distributions. If $k$ points are used per hash code then the distribution of lengths forms an Erlang-$k$ distribution (the second parameter to the Erlang distribution, $\lambda$, is in this case $k
\cdot |C|$). As $k$ rises, so the skew in the distribution of lengths is reduced. However, the ratios of smallest and largest segment lengths to the mean length remains high: with $k$ as high as $|C|$, the mean length will be around $1.1$ of the smallest length, and the largest length will also be around $1.1$ of the mean length. With $k =
4 \cdot |C|$, these ratios fall to around $1.06$, and with $k = 8
\cdot |C|$, they only fall to around $1.04$.
/łin [ 10/$\beta$, 10/$\delta$, 20/$\beta$, 40/$\epsilon$, 30/$\gamma$, 60/$\epsilon$, 20/$\alpha$, 10/$\delta$, 10/$\delta$, 40/$\gamma$, 15/$\beta$, 20/$\gamma$, 10/$\alpha$, 35/$\epsilon$, 30/$\alpha$ ]{}
(0,0) – (:1) arc (::1); (0,0) – (:1.1);
at (:1.3) [ł]{};
(0,0) – (325:1.6); at (325:1.7) [$\kappa$]{}; (0,0) (325:1.3) arc (325:300:1.3);
In some scenarios it may well be acceptable to have one hash code receive $8\%$ more keys than another. Note though that this imbalance is not reduced by adding additional hash codes, indeed quite the opposite: the lower $k / |C|$ falls, the greater the spread. To reduce the imbalance, the value of $k$ needs to be determined as a multiple of the [*maximum*]{} number of hash codes. Thus if the set of hash codes is normally relatively small and only under certain conditions are many more hash codes added, then the circle will generally contain many more points than strictly necessary, due to the high $k$. In scenarios with large numbers of hash codes, the probability of points colliding rises and you may even run out of points: with 20,000 hash codes each with 200,000 points, over $93\%$ of 32-bit integers are used up, thus switching to a 64-bit circle perimeter range would become necessary.
As the classical algorithm is typically implemented by holding the points in a binary tree, the depth of the tree determines the look-up cost. Assuming any other hash functions being used to prepare the key are ${\ensuremath{\operatorname{O}\bigl(1\bigr)}}$, we should have an overall cost of ${\ensuremath{\operatorname{O}\bigl(\log_2(k
\cdot |C|)\bigr)}}$ which simplifies to ${\ensuremath{\operatorname{O}\bigl(\log_2(|C|)\bigr)}}$ as $k$ is a constant. Some implementations claim worst cost of ${\ensuremath{\operatorname{O}\bigl(1\bigr)}}$ because of the finite upper limit on the number of points around the circle, and thus a limit on the depth of the corresponding tree, but that’s arguably a consequence of limitations of the implementation, and average cost (rather than worst-case) is still ${\ensuremath{\operatorname{O}\bigl(\log_2(|C|)\bigr)}}$. Techniques do exist to reduce this for the average case, but they increase memory footprint and make adding and removing hash codes more expensive. Obviously, this does not invalidate such approaches, but we will not consider them further here.
The memory footprint of so many points is also worth considering: if an implementation holds the points in some sort of tree, 20,000 hash codes each with 200,000 points would result in a tree of depth 32, with 8.6 billion nodes (number of nodes in a tree is found by $2^{d+1}
- 1$ where $d$ is the tree depth). Assuming each node carries two 64-bit pointers to its children and a 64-bit value, we have a minimum of 192 bits per node, or 24 bytes. Such a tree then works out at a minimum memory cost of around 200 GB, or around 10 MB per hash code.
Analysing Requirements
======================
The large number of replicas in the classic algorithm is not only necessary to address the exponential distribution of segment lengths (and thus approximate uniformity given a static number of hash codes) but also to maintain approximate uniformity in light of changes to the set of hash codes, $C$. We now consider how to algorithmically place points around the circle for each hash code such that we precisely achieve and maintain uniformity, and do not rely on any random distributions.
As stated previously, when a hash code is added, it should inherit an equal number of keys from each of the existing hash codes, and when it leaves, it should equally distribute its keys to the surviving hash codes. In the classic algorithm, to achieve this with the removal of a hash code requires that that hash code must have at least as many segments as there are remaining hash codes so that each of its own segments might be followed by a segment of a each of the other hash codes.[^1]
For example, with three hash codes, $\alpha$, $\beta$, $\gamma$, we require that $\alpha$ must have at least two segments: one followed by $\beta$ and the other followed by $\gamma$. The same holds for the other two hash codes, so we must accommodate all possible pairs around the circle: $(\alpha, \beta)$, $(\beta, \alpha)$, $(\alpha, \gamma)$, $(\gamma, \alpha)$, $(\beta, \gamma)$, $(\gamma, \beta)$. One solution would be $[\alpha, \beta, \alpha, \gamma, \beta, \gamma]$, with each segment being of equal length (), though there are a number of equivalent solutions. Note how the last element of each pair forms the first element of a different pair, and thus there are six segments of the circle corresponding to the six pairs. This is a minimal solution: the required pairings cannot be achieved with fewer points around the circle. If two hash codes leave, by elimination, the one remaining hash code will inherit all the keys and therefore we do not need to worry about the distribution of a single key’s points around the circle. This explains why with three hash codes, we concern ourselves with pairs of hash codes, and not triples.
$$\begin{aligned}
\vcenter{
\hbox{
\begin{tikzpicture}[scale=1.2]
\setcounter{b}{0}
\foreach \t/\l in
{
60/$\alpha$,
60/$\beta$,
60/$\alpha$,
60/$\gamma$,
60/$\beta$,
60/$\gamma$
}
{
\setcounter{a}{\value{b}}
\addtocounter{b}{\t}
{
\draw (0,0) -- (\thea:1) arc (\thea:\theb:1);
\draw (0,0) -- (\thea:1.1);
\node at (\thea:1.3) {\l};
}
}
\draw[dashed] (0,0) -- (325:1.6);
\node at (325:1.7) {$\kappa$};
\draw[->] (0,0) (325:1.3) arc (325:305:1.3);
\end{tikzpicture}
}
}
&& \overset{-\gamma}\Longrightarrow &&
\vcenter{
\hbox{
\begin{tikzpicture}[scale=1.2]
\setcounter{b}{0}
\foreach \t/\l in
{
60/$\alpha$,
60/$\beta$,
120/$\alpha$,
120/$\beta$
}
{
\setcounter{a}{\value{b}}
\addtocounter{b}{\t}
{
\draw (0,0) -- (\thea:1) arc (\thea:\theb:1);
\draw (0,0) -- (\thea:1.1);
\node at (\thea:1.3) {\l};
}
}
\draw[dashed] (0,0) -- (325:1.6);
\node at (325:1.7) {$\kappa$};
\draw[->] (0,0) (325:1.3) arc (325:245:1.3);
\end{tikzpicture}
}
}\end{aligned}$$
With four hash codes, $\alpha$, $\beta$, $\gamma$, $\delta$, if two hash codes leave, we still require uniformity of distribution of keys to the two remaining hash codes. This is no longer about incorporating every possible [*pair*]{} of hash codes around the circle, it is now about incorporating every possible [*triple*]{} of hash codes: pairs will maintain uniformity in case of one removal, triples are required for two removals. In general, for $|C|$ hash codes, every permutation of length $|C| - 1$ must exist around the circle in such a way that all but the first element (i.e. the last $|C| - 2$ elements) of each permutation forms the first elements of the next permutation. This is known as a [*universal cycle*]{} for the $|C| - 1$ permutations of $C$, and is a well studied problem [@Jackson1993].
Permutations of length one less than the number of available symbols are known as [*shorthand*]{} permutations as the remaining symbol is implicit. For example with the symbols $\alpha$, $\beta$, $\gamma$, $\delta$, the permutation $[\gamma,\delta,\beta]$ can be considered shorthand for $[\gamma,\delta,\beta,\alpha]$. Thus we can also say that we require a universal cycle of the shorthand permutations of $C$. In general, it is always possible to find a universal cycle of shorthand permutations, and efficient algorithms exist to directly construct such permutations [@Ruskey; @Holroyd2010].
Such a cycle will work as desired in light of multiple removals of hash codes. As every hash code in the cycle is followed an equal number of times by each of the other hash codes, removal of any hash code will equally distribute its keys amongst the remaining hash codes, who’s segments will grow in size. For example, with four hash codes, $\alpha$, $\beta$, $\gamma$, $\delta$, a universal cycle of shorthand permutations is:
$$[\alpha,\beta,\gamma,\alpha,\beta,\delta,\alpha,\gamma,\beta,\alpha,\gamma,\delta,\beta,\alpha,\delta,\gamma,\beta,\delta,\gamma,\alpha,\delta,\beta,\gamma,\delta]$$
Uniformity is achieved: each hash code has six entries and thus six segments of equal length around the circle, . If we remove the hash code $\gamma$ then we are left with:
$$[\alpha,\beta,\alpha,\alpha,\beta,\delta,\alpha,\beta,\beta,\alpha,\delta,\delta,\beta,\alpha,\delta,\beta,\beta,\delta,\alpha,\alpha,\delta,\beta,\delta,\delta]$$
In our notation here, removal is represented by substitution of the removed hash code with the next surviving hash code. As each element is a segment of the circle of equal length, this makes it easier to check uniformity; each remaining hash code now has eight entries, and thus uniformity has been maintained ().
/łin [ 15/$\alpha$, 15/$\beta$, 15/$\gamma$, 15/$\alpha$, 15/$\beta$, 15/$\delta$, 15/$\alpha$, 15/$\gamma$, 15/$\beta$, 15/$\alpha$, 15/$\gamma$, 15/$\delta$, 15/$\beta$, 15/$\alpha$, 15/$\delta$, 15/$\gamma$, 15/$\beta$, 15/$\delta$, 15/$\gamma$, 15/$\alpha$, 15/$\delta$, 15/$\beta$, 15/$\gamma$, 15/$\delta$ ]{}
(0,0) – (:1) arc (::1); (0,0) – (:1.1);
at (:1.3) [ł]{};
Note how there are two $\alpha,\alpha$ pairs, two $\beta,\beta$ pairs, and two $\delta,\delta$ created by the removal of the six $\gamma$ segments. Each of these themselves are still followed by each of the remaining hash codes: one $\alpha,\alpha$ pair is followed by a $\beta$, and one by a $\delta$; similarly for the other pairs. Thus multiple removals of hash codes still result in uniformity of distribution of keys. This is not a surprising result given the nature of the permutations: the very reason why there are two instances of $\gamma$ followed by $\alpha$ (thus forming the two $\alpha,\alpha$ pairs upon removal of $\gamma$) is so that they can be followed by each of the remaining hash codes: $\beta$ and $\delta$ in this case.
If we remove another hash code, for example $\delta$, then we are now left with:
$$[\alpha,\beta,\alpha,\alpha,\beta,\alpha,\alpha,\beta,\beta,\alpha,\beta,\beta,\beta,\alpha,\beta,\beta,\beta,\alpha,\alpha,\alpha,\beta,\beta,\alpha,\alpha]$$
Again, uniformity has been maintained: we have now 12 $\alpha$s and 12 $\beta$s ().
$$\begin{aligned}
\begin{tikzpicture}[scale=1.2]
\setcounter{b}{0}
\foreach \t/\l in
{
15/$\alpha$,
30/$\beta$,
15/$\alpha$,
15/$\beta$,
15/$\delta$,
30/$\alpha$,
15/$\beta$,
30/$\alpha$,
15/$\delta$,
15/$\beta$,
15/$\alpha$,
30/$\delta$,
15/$\beta$,
30/$\delta$,
15/$\alpha$,
15/$\delta$,
30/$\beta$,
15/$\delta$
}
{
\setcounter{a}{\value{b}}
\addtocounter{b}{\t}
{
\draw (0,0) -- (\thea:1) arc (\thea:\theb:1);
\draw (0,0) -- (\thea:1.1);
\node at (\thea:1.3) {\l};
}
}
\end{tikzpicture}
&&
\begin{tikzpicture}[scale=1.2]
\setcounter{b}{0}
\foreach \t/\l in
{
15/$\alpha$,
30/$\beta$,
15/$\alpha$,
30/$\beta$,
30/$\alpha$,
15/$\beta$,
45/$\alpha$,
15/$\beta$,
45/$\alpha$,
45/$\beta$,
30/$\alpha$,
45/$\beta$
}
{
\setcounter{a}{\value{b}}
\addtocounter{b}{\t}
{
\draw (0,0) -- (\thea:1) arc (\thea:\theb:1);
\draw (0,0) -- (\thea:1.1);
\node at (\thea:1.3) {\l};
}
}
\end{tikzpicture}\end{aligned}$$
Whilst it is clear that removal will maintain uniformity when the circle is constructed from a universal cycle of shorthand permutations of $C$, it is less clear how to construct such circles additively: given the cycle for $\alpha$, $\beta$, $\gamma$, $\delta$ given in , how do you modify it to incorporate a new hash code, $\epsilon$, whilst achieving the remapping property? Equivalently, given the circle diagram of the remaining $\alpha$s and $\beta$s on the right in , it is far from clear how to construct this, and thus create the necessary spaces for later hash codes to fill. In the classic algorithm, the circle exists to achieve the remapping property both for addition and removal of hash codes. However, with the positions of each hash code point being precisely determined by the universal cycle, the circle only now serves to provide the remapping property upon removal of a hash code, not the addition: addition can no longer be achieved by splitting existing segments. Happily, our new algorithm manages to achieve the uniformity and remapping properties precisely, without needing to address this problem.
However, first let us examine the size of these cycles. Because all but one hash code from each permutation overlaps with the next hash code, each permutation contributes one hash code to the length of the cycle. The number of shorthand permutations of $C$ is the same as the number of $|C|$-length permutations of $C$, which is $|C|!$. Thus with just 12 hash codes, we have a cycle length of 479,001,600. Whilst this is less than $2^{32}$, 13 hash codes would be create a cycle length greater than $2^{32}$. This not only impacts the representation of the circle (and its memory footprint if the entire circle must be constructed and maintained), but also affects the key: in essence what this means is that a 32-bit key can only choose between up to 12 hash codes. A 512-bit key can only choose between up to 98 hash codes. This has implications for consistent hashing generally: with large numbers of hash codes and short keys, it is impossible to achieve perfect uniformity, and an approximate solution in such scenarios cannot be bettered. To achieve perfect uniformity and the remapping property in light of removals, every permutation needs an equal chance of being selected. The use of shorthand permutations is only necessary to be able to construct universal cycles out of the segments around the circle.
This factorial of $|C|$ also impacts performance. As discussed earlier, the classic algorithm is typically implemented using a binary tree to hold the points around the circle. The depth of the tree and thus the average cost of look-up is now ${\ensuremath{\operatorname{O}\bigl(\log_2(|C|!)\bigr)}}$ which is worse than ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$ (we present an intuitive proof of this later). However, with even small numbers of hash codes, the factorial results in so many nodes that it is unwise to maintain the whole tree in memory. Instead the nodes of the tree would need to be constructed by some means as the tree was traversed. This would likely result in a very different look-up cost.
The factorial also explains why the multiple of $|C|$ to define the number of replicas, $k$, in the classic algorithm must rise itself as $|C|$ rises: to approximate maintaining uniformity in light of removals, the classic algorithm must have sufficient points per hash code to approximate a universal cycle of shorthand permutations of $C$. Thus $k$ should also be a multiple of the factorial of $|C|$ to have confidence of being able to approximate such a cycle by random placement of hash code points.
New algorithm
=============
In the previous section, the universal cycle served to position the hash codes around the circle such that uniformity was achieved, and that in the event of removal of hash codes, uniformity would be maintained. It can be considered that what the hash function is actually returning is not a single hash code, but a permutation of the hash codes, with removed hash codes filtered out.
Our new algorithm explicitly returns a permutation of all the hash codes. The interpretation of a permutation as a result of the hash function is not fixed, but for our purposes, we read $[\alpha,\beta]$ as [*first try $\alpha$, then try $\beta$*]{}. Every permutation is equally probable, which achieves both the uniformity requirement and the remapping requirement in light of removal of hash codes (i.e. as before, filtering out removed hash codes from the resulting permutation will maintain uniformity). Each permutation exists as a leaf of a tree, but this is not a binary tree: whilst the root node has two children, all other nodes have one more child than does their parent, . We then subdivide the key to navigate through the tree. The remapping requirement means that if we use part of the key to decide between different orderings of particular hash codes in the resulting permutation then we must forevermore use that same part of the key to make that same decision.
$$\begin{aligned}
\vcenter{
\hbox{
\begin{tikzpicture}[scale=1.2,
level 1/.style={sibling distance=40}]
\node {$[\alpha]$}
child {node {$[\alpha,\beta]$} edge from parent node[left] {0}}
child {node {$[\beta,\alpha]$} edge from parent node[right] {1}};
\end{tikzpicture}
}
}
&& \overset{+\gamma}\Longrightarrow &&
\vcenter{
\hbox{
\begin{tikzpicture}[scale=1.2,
level 1/.style={sibling distance=100},
level 2/.style={sibling distance=30}]
\node {$[\alpha]$}
child {node {$[\alpha,\beta]$}
child {node {$[\alpha,\beta,\gamma]$} edge from parent node[left] {0}}
child {node {$[\alpha,\gamma,\beta]$} edge from parent node[left] {1}}
child {node {$[\gamma,\alpha,\beta]$} edge from parent node[right] {2}}
edge from parent node[left] {0}
}
child {node {$[\beta,\alpha]$}
child {node {$[\beta,\alpha,\gamma]$} edge from parent node[left] {0}}
child {node {$[\beta,\gamma,\alpha]$} edge from parent node[right] {1}}
child {node {$[\gamma,\beta,\alpha]$} edge from parent node[right] {2}}
edge from parent node[right] {1}
};
\end{tikzpicture}
}
}\end{aligned}$$
With one hash code, the result is trivial. With two hash codes, $\alpha$ and $\beta$, we want the answer to be the permutation $[\alpha,\beta]$ as often as the permutation $[\beta,\alpha]$. To choose between these, we use the key ($\kappa$) modulus two. With three hash codes, $\alpha$, $\beta$ and $\gamma$, we now have six permutations. As we previously used $\kappa \bmod 2$ to choose between [*$\alpha$ before $\beta$*]{} versus [*$\beta$ before $\alpha$*]{}, we must continue to do so, and must then discard that part of the key (by dividing by two). At the next layer of the tree we have two 3-way choices, each refining the previous choice by adding in the new hash code $\gamma$. Thus we use the remaining key modulus three to make this choice. The tree at the right of is shown as a table in .
$\kappa \bmod 6$ $\frac{\kappa}{2} \bmod 3$ $\kappa \bmod 2$ Permutation
------------------ ---------------------------- ------------------ -------------------------
0 0 0 $[\alpha,\beta,\gamma]$
1 0 1 $[\beta,\alpha,\gamma]$
2 1 0 $[\alpha,\gamma,\beta]$
3 1 1 $[\beta,\gamma,\alpha]$
4 2 0 $[\gamma,\alpha,\beta]$
5 2 1 $[\gamma,\beta,\alpha]$
In general, each layer of the tree adds a new hash code. A node of any particular layer can be seen to receive a permutation from its parent, and to add its new hash code in every possible position within that permutation; each node has a child for each of the possible positions at which its own hash code can be inserted. But no modifications are made to the ordering of existing hash codes within the permutation which a node receives, and it is this that achieves the remapping property. Note that in the tree of , the index of each branch indicates the distance from the [*end*]{} of the existing permutation at which the new hash code is inserted. This is an arbitrary choice: any strategy for determining the position of the new hash code within the received permutation is acceptable (and can even vary per layer), provided the strategy is both deterministic and uniform.
If we consider all keys modulus two (i.e. the values $0$ and $1$), then with just the hash codes $\alpha$ and $\beta$, we see $0$ (and thus all even keys) maps to $[\alpha,\beta]$, and $1$ (and thus all odd keys) maps to $[\beta,\alpha]$. With the hash codes $\alpha$, $\beta$ and $\gamma$, all keys modulus six (i.e. the values $0$ to $5$), and if we just consider the first element of each permutation returned, then we see $0$ and $2$ are mapped to $\alpha$ (as they were previously without the $\gamma$ hash code), $1$ and $3$ are mapped to $\beta$ (as they were previously without the $\gamma$ hash code), and $4$ and $5$ are mapped to $\gamma$. Thus we have ensured that in the transition from two to three hash codes, we only remap keys to the new value ($4$ and $5$ going to $\gamma$), and we have taken an equal (and minimal) number of keys from each of the existing hash codes (i.e. $4$ from $\alpha$ and $5$ from $\beta$), resulting in an equal distribution of keys to values. Uniformity has been maintained and the remapping property achieved.
We must also check what happens when a hash code is removed. If the hash code removed is the most recently added, then we can simply discard the lowest layer of the tree and return to the earlier configuration. Otherwise, we continue with the existing tree, but must filter out the removed hash code from the resulting permutation (depending on the implementation, this filtering could be done as the resulting permutation is constructed). With the hash codes $\alpha$, $\beta$ and $\gamma$, there are two permutations which start with $\alpha$: $[\alpha,\beta,\gamma]$ and $[\alpha,\gamma,\beta]$. If the $\alpha$ hash code is removed, we see that its keys are equally redistributed and uniformity maintained: we have one permutation where the initial $\alpha$ is followed by a $\beta$ and one where it is followed by a $\gamma$. As in the previous section, this is simply a consequence of using permutations. It should be noted that in common with the classic algorithm, our algorithm makes addition of hash codes non-commutative: each hash code is accommodated by individual layers of the tree (and thus is navigated by specific sections of the key), and so the order in which the hash codes were added matters.
If after $\alpha$ has been removed we add the hash code $\delta$, then $\delta$ can simply take the space created by the removal of $\alpha$. The remapping and uniformity properties are precisely achieved. We now can define exactly the state that our algorithm requires: a list containing current hash codes interspersed with a marker used to indicate free slots caused by hash codes being removed. The list will only grow when a hash code is added and there are no free slots, and will shrink whenever the hash code at the end of the list is removed (thus you should never have a free slot marker at the end of the list). Barring substitutions of a new hash code for a free slot marker, the list elements will be in the order in which the hash codes were added, corresponding to the layers of the tree.
The average cost of look-up is now ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$, as each hash code adds a layer to the tree. Whilst we have the same number of leaves as in the previous section (i.e. $|C|!$), we no longer use a binary tree: each node has one more child than does its parent. As a child is determined by indexing a node’s list of children by the modulus of part of the key, the selection of the child remains ${\ensuremath{\operatorname{O}\bigl(1\bigr)}}$ despite nodes having increasing numbers of children as depth increases. Consequently, we have one division, one modulus, and one indexing operation per layer of the tree. Assuming each of these are ${\ensuremath{\operatorname{O}\bigl(1\bigr)}}$, the average cost of reaching a leaf, and thus a look-up, is ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$. This then is our intuitive proof that ${\ensuremath{\operatorname{O}\bigl(\log_2(|C|!)\bigr)}}$ is worse than ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$: the binary tree from the previous section and our non-binary tree from this section both contain the same number of leaves, but the binary tree is limited to two children per node and so must use more nodes than our non-binary tree which has an additional child per node per generation. Consequently, for the same number of leaves, the binary tree must be deeper than our non-binary tree, thus the cost of navigating to a leaf must be higher. Therefore ${\ensuremath{\operatorname{O}\bigl(\log_2(|C|!)\bigr)}}$ is worse than ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$.
As we have the same number of leaf nodes as in the previous section, caused by the factorial of $|C|$, we have the same implications in the relationship between key bit length (or entropy) and the permissible number of hash codes. Thus whilst our algorithm does indeed achieve perfect uniformity and satisfies the remapping property, the cost is in the factorial relationship between the number of hash codes, and the range of the key. Equally, the number of nodes in our tree is given by $\sum\nolimits_{i=1}^{|C|}{i!}$. Whilst this is fewer nodes than the binary tree for the same number of leaves (indeed, the number of nodes of our tree tends towards half the number of nodes of the binary tree), nevertheless the number of nodes makes it impractical to maintain such a tree in memory, so once again we must construct the permutation dynamically as we descend the tree. Such an implementation is given in . The performance cost will change however: each layer of the tree will insert its hash code into the resulting permutation. The most efficient mechanism for doing this will be to build the permutation in a tree, for which insertions will on average ${\ensuremath{\operatorname{O}\bigl(\log_2(n)\bigr)}}$ where $n$ is the number of values in the tree. As we know we will need to do $|C|$ insertions and the average number of values in the tree will be $|C|/2$, we have a total cost of all the insertions of ${\ensuremath{\operatorname{O}\bigl(|C| \cdot \log_2(|C|/2)\bigr)}}$. Whilst this is a worse average cost than navigating a pre-constructed tree (which was ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$), the memory savings are significant.
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In the code listing of , as we subdivide the key we build up the permutation in a list rather than a tree. Whilst this will be less efficient than using a tree, for small values of $|C|$ the difference will be slight and the code simplified (as ever, beware large constant overheads!). Note that there is no marker provided to indicate removed hash codes; instead these can be filtered out from the resulting permutation as necessary. In this implementation, the position calculated by each layer of the tree is the distance from the [*start*]{} of the received permutation at which to insert the new hash code. Thus this will produce permutations in a different order to that of but the properties still hold, and the code is simplified.
If there is no need to return a permutation, and instead only the first element of the permutation is required as a result then further simplifications can be made to the code by avoiding construction of the permutation, and so reducing the average complexity back to ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$. We keep track of the [*current*]{} first element of the permutation, and update it at each layer if we find the position of new hash code is 0, thus replacing the old first element. This is shown in .
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However, in this simpler scenario, we need to be much more careful about removed hash codes: we cannot permit the algorithm to return the marker for a removed hash code as the result, as we have no way of knowing what would have been next in the permutation. Instead, we must cope with the removed-hash-code marker directly in the implementation itself. This is trickier as we need to consider many different combinations of each layer updating the current result. shows an example implementation. In the input list of hash codes, the Erlang atom is used to indicate the removed-hash-code marker.
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Whilst there are ten combinations of existing-result and new hash code to consider in this code, the algorithmic complexity is no worse, and so when just a single result is required, the cost of the function is ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$. Whilst this is still worse than for the classic algorithm, for the smaller sizes of $|C|$ that our algorithm is best suited for, this is unlikely to preclude use of our algorithm. Equally, in cases where there is a very high churn rate of hash codes being added and removed, the lower cost of these operations in our algorithm may favour it over the classic algorithm.
Evaluation
==========
As mentioned earlier, whilst our algorithm achieves perfect uniformity along with the remapping and determinism properties, the trade-off is higher average cost (${\ensuremath{\operatorname{O}\bigl(|C| \cdot \log_2(|C|)\bigr)}}$ (or ${\ensuremath{\operatorname{O}\bigl(|C|\bigr)}}$ if a single element is returned rather than an entire permutation) versus ${\ensuremath{\operatorname{O}\bigl(\log_2(|C|)\bigr)}}$ for the classic algorithm) and rapid consumption of the entropy of the key. If more hash codes are used than can be supported by the entropy of the key then the consequence is certain permutations will never be reached and thus certain hash codes may never appear at the front of resulting permutations. However, our algorithm can dynamically construct the result as the key is consumed, thus avoiding building the entire tree, saving memory. This is possible because the contents of each child node are determined by just the remainder of the key and the node’s hash code itself. By contrast in the classic algorithm, the contents of each child node are determined randomly by the placement of hash code points. This means at a minimum, all the points of every hash code must be held in memory for the classic algorithm.
Returning a permutation rather than a single result is in practice very useful, and has several interpretations depending on the application. For example, if the application is a distributed key-value store then the permutation would indicate an ordering of machines to try: in the case of a read operation you might choose to issue reads to the first few machines from the permutation, either to check that they all have the same value, or because due to transient load imbalances, one may reply more quickly than the others. For a write operation, the client application may well indicate that it only considers a write [*completed*]{} once it has been synchronously written to at least $N$ machines; again, the first $N$ elements from the permutation indicate exactly to which machines to issue synchronous writes.
In such key-value stores, it is very often the case that certain keys are much more frequently accessed than others. This might be due to a particularly popular URL; the effect of [*“being slash-dotted”*]{} or [*“going viral”*]{}. In these scenarios, a single key can substantially skew loading across a cluster of machines. Here again, returning a permutation from the consistent hash function can be advantageous: if each element of the permutation is a particular machine in your distributed key-value store and loading information per machine is available, the client may well be able to filter out particularly heavily loaded machines and still access the required information promptly. In an eventually consistent scenario with writes as well as reads occurring, this could result in the serving of stale data, but the trade-off would be better load balancing and improved latencies.
Performance comparisons in general of the classic algorithm and our new algorithm are of limited value as they will inevitably reflect both the suitability of each algorithm to the artificial conditions of the benchmark, and the relative amounts of effort to optimise each implementation.
Conclusion
==========
Consistent Hashing is a widely used and important technique, applicable to many applications. Hopefully this work provides a more detailed understanding as to how it can be achieved and what the trade-offs involved are.
We have shown how the classic algorithm relies on random distributions to approximately maintain the uniformity property. We then examined how, given the way in which the circle achieves the remapping property, universal cycles of shorthand permutations may be used to precisely achieve and maintain the uniformity property in light of removals of hash codes but that adding new hash codes is non-obvious. Finally, by abandoning the use of a circle, we presented our new algorithm which also relies on permutations but makes the addition of hash codes simple. We have discussed potential implementation strategies and average performance of these algorithms and shown that a cost of achieving perfect uniformity and remapping is in the factorial relationship between the number of hash codes and the key size.
[^1]: In the classic algorithm, due to the random placement of hash code points, a high number of replicas, $k$, is also necessary to have confidence the required permutations are achieved, but as we shall show, as the number of hash codes increases, the multiple of $|C|$ to define $k$ must itself rise.
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author:
- 'Jorge I. Cerdá$^{1}$'
- Jagoda Sławińska$^1$
- 'Guy Le Lay$^{2}$'
- 'Antonela C. Marele$^3$'
- 'José M. Gómez-Rodríguez$^{3,4}$'
- 'María .E. Dávila$^1$'
title: 'Unveiling the [*Penta*]{}-Silicene nature of perfectly aligned single and double strand Si-nanoribbons on Ag(110)'
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[**From the simplest cyclopentane ring and its numerous organic derivates to their common appearance in extended geometries such as edges or defects in graphene, pentagons are frequently encountered motifs in carbon related systems. Even a [*penta*]{}-graphene Cairo-type two dimensional structure has been proposed as a purely pentagonal C allotrope with outstanding properties competing with those of graphene [@cpenta]. Conversely, pentagonal Si motifs are hardly found in nature. Despite the large effort devoted to design Si-based structures analogous to those of carbon, the existence of Si pentagonal rings has only been reported in complex clathrate bulk phases [@calthrates]. Several theoretical studies have hypothesized stable Si pentagonal structures either in the form of one-dimensional (1D) nanotubes [@siwires1; @siwires2] or at the reconstructed edges of [*silicene*]{} nanoribbons [@silicene; @silicene1] or even as hydrogenated [*penta-silicene*]{} [@pentasilicene] or highly corrugated fivefold coordinated [*siliconeet*]{} [@grunberg] 2D sheets, the latter recognized as a topological insulator [@heine]. However, to date none of them have yet been synthesized. In the present work we unveil, via extended Density functional theory (DFT) [@siesta] calculations and Scanning tunneling microscopy (STM) simulations [@green; @loit], the atomic structure of 1D Si nano-ribbons grown on the Ag(110) surface. Our analysis reveals that this system constitutes the first experimental evidence of a silicon phase solely comprising pentagonal rings.**]{}
Since their discovery in 2005 [@Leandri2005] the atomic structure of Si nano-ribbons (NRs) on Ag(110) has remained elusive and strongly disputed [@chinese1; @Leandri2005; @ronci2010; @Lian2012; @Borensztein2014; @Lagarde2016; @arpes1; @colonna; @chinese]. Figure \[exp\] presents a summary of Si NRs measured with STM. The structures were obtained after Si sublimation onto a clean Ag(110) surface at RT. Panels (a) and (b) correspond to a low Si coverage image with an isolated [[*nano-dot*]{}]{} structure and a [*single strand*]{} NR (SNR) 0.8 nm wide running along the $[1\bar{1}0]$ direction with a 2$\times$ periodicity. The SNR topography consists of alternating protrusions at each side of the strand with a glide plane. At higher coverages and after a mild annealing, a dense and highly ordered phase is formed (panel (c)) consisting of [*double strand*]{} NRs (DNRs) with a 5$\times$ periodicity along the \[001\] direction again exhibiting a glide plane along the center of each DNR. The images are in perfect accord with previous works [@Leandri2005; @ronci2010; @colonna; @chinese]. Further key information on the system is provided by the high-resolution Si-$2p$ core level photoemission spectrum for the DNRs displayed in Figure \[exp\](d) –that for the SNRs is almost identical [@Davila2012]. The spectrum can be accurately fitted with only two (spin-orbit splited) components having an intensity ratio of roughly 2:1. Furthermore, previous Angular Resolved Photoemission (ARPES) experiments [@Paola2008] assigned the larger and smaller components to subsurface [Si$_{\mathrm{s}}$]{} and surface [Si$_{\mathrm{ad}}$]{} atoms, respectively, indicating that the NRs comprise two different types of Si atoms, with twice as many [Si$_{\mathrm{s}}$]{} as [Si$_{\mathrm{ad}}$]{}.
![(Color online) $5.3\times5.3$ nm$^{2}$ STM images of Si nanostructures on Ag(110). (a) a Si [[*nano-dot*]{}]{}, (b) a Si SNR and (c) an extended Si DNR phase. The insets show profiles along the solid lines passing over the maxima in the images. Tunneling parameters: (a) -1.5 V, 2.4 nA, (b) -1.8 V, 1.2 nA and (c) 1.3 V, 1.1 nA. (d) Si-$2p$ core level photoemission spectra recorded at normal emission and at 135.8 eV photon energy for the Si DNRs structure. \[exp\]](Fig1.eps){width="75mm"}
We first focus on the [[*nano-dot*]{}]{} shown in Fig. \[exp\](a), as it may be regarded as the precursor structure for the formation of the extended NRs. The [[*nano-dot*]{}]{} exhibits a local [*pmm*]{} symmetry with two bright protrusions aligned along the \[001\] direction, each of them having two adjacent dimmer features along the $[1\bar{1}0]$ direction. After considering a large variety of trial models (see ’Extended Data’ Fig \[nano\_all\]) we found that only one, shown in Figure \[nano\], correctly reproduces the experimental image both in terms of aspect and overall corrugation. It consists of a ten atom Si cluster located in a double silver vacancy generated by removing two adjacent top row silver atoms. There are four symmetry equivalent [Si$_{\mathrm{s}}$]{} atoms residing deeper in the vacancy, two [Si$^1_{\mathrm{ad}}$]{} in the middle which lean towards short silver bridge sites and four outer [Si$^2_{\mathrm{ad}}$]{} residing at long bridge sites. The formers lie 0.8 Å above the top Ag atoms and are not resolved in the STM image, while the [Si$^1_{\mathrm{ad}}$]{} and [Si$^2_{\mathrm{ad}}$]{}protrude out of the surface by 1.4 and 1.1 Å thus leading to the six bump structure in the simulated image with the [Si$^1_{\mathrm{ad}}$]{} at the center appearing brighter. Therefore, although the [[*nano-dot*]{}]{} shows marked differences with respect to the extended NRs, its structure already accounts for the presence of two distinct types of Si atoms at the surface ([Si$_{\mathrm{s}}$]{} and [Si$_{\mathrm{ad}}$]{}). Furthermore, it reveals the tendency of the Ag(110) surface upon Si adsorption to remove top row silver atoms (i.e. the initial stage in the creation of a missing row (MR)) and incorporate Si nanostructures in the troughs.
![ (a-b) Top and perspective views of the [[*nano-dot*]{}]{} structure. (c) Simulated STM topographic image and line profile along the solid line. \[nano\]](nano.eps){width="\columnwidth"}
Inspired by the [[*nano-dot*]{}]{} Ag di-vacancy structure and by recent grazing incidence X-ray diffraction (XRD) measurements [@mr] pointing towards the existence of a MR reconstruction along the $[1\bar{1}0]$ direction of the Ag surface, we considered several trial structures for the SNRs by placing Si atoms in the MR troughs ([Si$_{\mathrm{s}}$]{}) and next adding further adatoms ([Si$_{\mathrm{ad}}$]{}) on top, while maintaining a 2:1 concentration ratio between the two. Figures \[pmr\](a-b) show top and side views of the optimized geometry for the SNRs after testing several trial models (see ’Extended Data’ Fig. \[models\]). It involves a MR and six Si atoms per cell. The new paradigm is the arrangement of the Si atoms into pentagonal rings running along the MR and alternating their orientation (we denote it as the P-MR model). Despite no symmetry restriction was imposed, the relaxed P-MR SNR belongs to the $cmm$ group presenting two mirror planes plus an additional glide plane along the MR troughs (see ’Extended Data’ Fig. \[geom\] for a detailed description). Apart from a considerable buckling of 0.7 Å between the lower Si atoms residing in the MR troughs ([Si$_{\mathrm{s}}$]{}) and the higher ones ([Si$_{\mathrm{ad}}$]{}) leaning towards short bridge sites at the top silver row, the pentagonal ring may be considered as rather perfect, with a very small dispersion in the Si-Si distances ($2.35-2.37$ Å) and bond angles ranging between $92^\circ-117^\circ$; that is, all close to the $108^\circ$ in a regular pentagon. The associated STM image and line profile, panel (d), show (symmetry) equivalent protrusions 1.3 Å high at each side of the strand, in perfect agreement with the experimental image. Still, since different models may yield similar STM images, a more conclusive gauge to discriminate among them is to examine their relative formation energies. In this respect, the energetic stability of the P-MR structure is far better ($\sim 0.1$ eV/Si) than all other SNR models considered (see section ’Methods’ and ’Extended Data’ Fig. \[phasediag\]).
![Optimized geometry of the pentagonal missing row (P-MR) model (a-c) Top, side and simulated topographic STM image for the SNR phase. (d) Perspective view of a penta-silicene strand without the silver surface. (e-g) Top, side and simulated topographic STM image for the DNR array. Insets in (c) and (g) show line profiles along the blue lines indicated in the topographic maps. All STM simulations employed a sharp Si ended tip apex and set points $V=-0.2$ V and $I=1$ nA. \[pmr\]](pmr)
Within the pentagonal model the DNR structure may be naturally generated by placing two SNRs within a $c(10\times2)$ cell. However, since the P-MR SNRs are chiral, adjacent pentagonal rings may be placed with the same or with different handedness, leading to two possible arrangements among the enantiomers. Figures \[pmr\](e-g) display the optimized geometry and simulated STM topography for the most stable (by 0.03 eV/Si) P-MR DNR configuration. The pentagonal structure in each NR is essentially preserved, the main difference with respect to the SNRs being the loss of the glide plane along the MR troughs replaced by a new one along the top silver row between adjacent SNRs. There is a slight repulsion between the NRs which shifts them away from each other by around 0.2 Å. As a result, the [Si$_{\mathrm{ad}}$]{} at the outer edges of the DNR end up lying 0.07 Å higher than the inner ones making the alternating pentagons along each strand not strictly equivalent anymore. In the simulated STM image the outer maxima appear dimmer than the inner ones by 0.1 Å, which adopt a zig-zag aspect. The inversion in their relative corrugations is due to the proximity between the inner Si adatoms ($\sim4$ Å) compared to the almost 6 Ådistance between the inner and outer ones, so that the bumps of the formers overlap and lead to brighther maxima. All these features are in accordance with the experimental profiles shown in Fig. \[exp\](c). In fact, the P-MR DNR structure is the most stable among all other NR models considered for a wide range of Si chemical potentials ranging from Si-poor to -rich conditions (see ’Extended Data’ Fig. \[phasediag\]).
Figure \[electronic\] presents a summary of the electronic properties of the P-MR structure. Panel (a) shows an isosurface of the total electronic density for the SNRs. The [Si$_{\mathrm{s}}$]{} atoms in the pentagonal rings are clearly linked through an $sp^2$ type bonding (three bonds each) while the [Si$_{\mathrm{ad}}$]{}, due to the buckling, show a distorted $sp^3$ type tetrahedral arrangement making bonds with two [Si$_{\mathrm{s}}$]{} as well as with the adjacent short bridge silver atoms in the top row. Panel (b) displays ARPES spectra for the SNR and DNR phases. Both energy distribution curves reveal Si-related peaks previously attributed to [*quantum well states*]{} (QWS) originating from the lateral confinement within the NRs.
For the SNRs three states are observed at -1.0, -2.4 and -3.1 eV binding energy (BE), while for the DNRs one further peak is identified at -1.4 eV. The computed (semi-infinite) surface band structures projected on the Si pentagons (blue) and the silver MR surface (red) are superimposed in panels (c) and (d) for the SNRs and DNRs, respectively. Overall, within the expected DFT accuracy and experimental resolution, the maps satisfactorily reproduce the experimental spectra. At $\Gamma$ the SNRs present two sharp intense Si bands below the Fermi level (S1 and S3) and faint (broader) features arising from two almost degenerate bands (S4 and S5) and a dimmer state (S2). As expected, they are almost flat along $\Gamma-X$ while along $\Gamma-Y$ they present an appreciable dispersion and finally merge into two degenerate states at the high symmetry $Y$ point. The orbital character of the S2-S5 bands is mainly $p_{xy}$ and may thus be assigned to localized $sp^2$ planar bonds. Conversely, band S1 is fully dominated by the Si$_s-p_z$ states ($\pi$-band) and shows a strong downward dispersion along $\Gamma-Y$ due to hybridization with the metal $sp$ bands. Similarly, faint dispersive bands of mainly $p_z$ character hybridizing with the metal appear in the empty states region. The electronic structure for the DNRs is similar to that of the SNRs, except that the number of Si bands is doubled and most of them become splited and shifted due to the interaction between adjacent SNRs. Noteworthy is the appearance of an electron pocket (EP) at $\Gamma$ associated to a parabolic Si-$p_z$ band with onset at -0.5 eV.
![Electronic structure of the P-MR model. (a) Charge density isosurface for the SNRs with blue sticks indicating the Si pentagons. (b) Energy distribution curves around the $X$ point for the SNRs acquired at 78 eV photon energy (adapted from Ref [@arpes2]) and for the DNRs at 75 eV. (c-d) PDOS[($\vec{k}, E$) ]{}projected on the Si (blue) and Ag (red) atoms along the $Y-\Gamma-X$ $k$-path (see insets) for the SNRs and DNRs, respectively. \[electronic\]](rho-ek.eps){width="\columnwidth"}
To conclude, we have solved the long debated structure of silicon nano-ribbons on Ag(110), finding an unprecedented 1D [*penta-silicene*]{} phase which consists of adjacent inverted pentagons stabilized within the MR troughs. The model is in accordance with most of previous experimental results for this system: it involves a MR reconstruction as deduced from XRD [@mr], comprises two types of Si atoms with a ratio 2:1 between the [Si$_{\mathrm{s}}$]{} and [Si$_{\mathrm{ad}}$]{}concentrations as seen by photoemission, accurately matches the STM topographs also explaining dislocation defects between NRs (see ’Extended Data’ Fig. \[defect\]) and accounts for the QWS measured by ARPES. We have also determined the quasi-hexagonal geometry of a Si [[*nano-dot*]{}]{} inside a silver-divacancy. This precusor structure for the NRs can be considered as the limiting process for expelling surface Ag atoms in order to create a missing row along which the Si pentagons can develop. We are convinced that the discovery of this novel silicon allotrope will promote the synthesis of analogous exotic Si phases on alternative templates with promising properties [@grunberg].
This work has been funded by the Spanish MINECO under contract Nos. MAT2013-47878-C2-R, MAT2015-66888-C3-1R, CSD2010-00024, MAT2013-41636-P, AYA2012-39832-C02-01/02 and ESP2015-67842-P.
[**Author contributions:**]{} J.I.C. and J.S. performed all the theoretical calculations. A.C.M., M.E.D., and J.M.G.R. performed all the STM experiments. M.E.D. and G.L.L. performed the ARPES measurements. J.I.C. and M.E.D. conceived most of the novel model structures tested. J.I.C. and G.L.L. wrote the manuscript. All authors contributed to the manuscript and figure preparation.
Methods
=======
[**Experimental**]{}
For both types of prepared structures (isolated Si SNRs or ordered DNRs), the same procedure has been used for sample preparation: i.e. the Ag(110) substrate was cleaned in the Ultra-high vacuum (UHV) chambers (base pressure: 9$\times10^{-11}$ mbar) by repeated sputtering of Ar$^+$ ions and subsequent annealing of the substrate at 750 K, while keeping the pressure below 3$\times10^{-10}$ mbar during heating. Si was evaporated at a rate of 0.03 ML/min from a silicon source in order to form the NRs. The Ag substrate was kept at room temperature RT to form the isolated SNR 0.8 nm wide, while a mild heating of the Ag substrate at 443 K allows the formation of an ordered grating DNR 1.6 nm wide [@Davila2012].
STM measurements were done with a home-made variable temperature UHV STM [@Custance2003]. All STM data were measured and processed with the WSxM software [@Horcas2007]. High-Resolution Photoelectron Spectroscopy (HRPEs) experiments of the shallow Si-$2p$ core-levels and of the valence states, were carried out to probe, comparatively, the structure and the electronic properties of those nanostructures. The ARPES experiments were carried out at the I511 beamline of the Swedish Synchrotron Facility MAX-LAB in Sweden. The end station is equipped with a Scienta R4000 electron spectrometer rotatable around the propagation direction of the synchrotron light. It also houses low energy electron diffraction (LEED) and sputter cleaning set-ups. Further details on the beam line are given in Ref. [@Denecke1999]. In all the photoemission spectra the binding energy is referenced to the Fermi level. The total experimental resolution for core level and valence band (VB) spectra were 30 meV (h$\nu$=135.8 eV for Si-$2p$) and 20 meV (h$\nu$=75 eV for the VB), respectively. A least-square fitting procedure was used to analyze the core-levels, with two doublets, each with a spin-orbit splitting of $610\pm5$ meV and a branching ratio of 0.42. The Si-$2p$ core level collected at normal emission is dominated by the [Si$_{\mathrm{s}}$]{} component. Its full width at half maximum (FWHM) is only 68 meV while the energy difference between the two [Si$_{\mathrm{s}}$]{} and [Si$_{\mathrm{ad}}$]{} components is 0.22 eV.
[**Theory**]{}
All calculations have been carried out at the [*ab initio*]{} level within the Density Functional Theory (DFT) employing the SIESTA-GREEN package [@siesta; @green]. For the exchange-correlation (XC) interaction we considered both the Local Density [@ca] (LDA) as well as the Generalized Gradient [@pbe] (GGA) approximations. The atomic orbital (AO) basis set consisted of Double-Zeta Polarized (DZP) numerical orbitals strictly localized after setting a confinement energy of 100 meV in the basis set generation process. Real space three-center integrals were computed over 3D-grids with a resolution equivalent to a 700 Rydbergs mesh cut-off. Brillouin zone (BZ) integration was performed over $k$-supercells of around (20$\times$28) relative to the Ag-(1$\times$1) lattice while the temperature $kT$ in the Fermi-Dirac distribution was set to 100 meV.
All considered Si-NR-Ag(110) structures were relaxed employing two-dimensional periodic slabs involving nine metal layers with the NR adsorbed at the upper side of the slab. A $c(10\times2)$ supercell was employed for both the SNR and DNR structures. In all cases, the Si atoms and the first three metallic layers were allowed to relax until forces were below 0.02 eV/Å while the rest of silver layers were held fixed to their bulk positions (for which we used our LDA (GGA) optimized lattice constant of 4.07 Å (4.15 Å), slightly smaller (larger) than the 4.09 Å experimental value).
For the [[*nano-dot*]{}]{} calculations, and given that a larger unit cell is required to simulate its isolated geometry, the atomic relaxations of all the trial models (see Fig. \[nano\_all\]) were carried out for (4$\times$5) or (4$\times$6) supercells. STM topographic images were next computed for all relaxed structures after recomputing the slab Hamiltonians with highly-extended AOs for the surface atoms. Once the correct structure was identified (see Fig. \[nano\_all\]), we further optimized it increasing the unit cell to a $(6\times10)$ to remove any overlaps between image cells (see Fig. \[nano\] in the main text).
[*Band structure–*]{} In order to examine the surface band dispersion we computed $k$-resolved surface projected density of states PDOS($k,E$) maps in a semi-infinite geometry. To this end we stacked the Si-NR and first metallic layers on top of an Ag(110) bulk-like semi-infinite block via Green’s functions matching techniques following the prescription detailed elsewhere [@ysi2; @loit]. For this step we recomputed the slab’s Hamiltonian employing highly extended orbitals (confinement energy of just 10-20 meV) for the Si and Ag surface atoms in the top two layers (this way the spatial extension of the electronic density in the vacuum region is largely extended and the calculation becomes more accurate.
[*STM simulations–*]{} For the STM simulations we modeled the tip as an Ag(111) semi-infinite block with a one-atom terminated pyramid made of ten Si atoms stacked below acting as the apex (see Figure \[tip\]). Test calculations employing other tips (e.g. clean Ag or clean W) did not yield any significant changes. Highly extended orbitals were also employed to describe the apex atoms thus reproducing better the expected exponential decay of the current with the tip-sample normal distance $z_{tip}$. Tip-sample AO interactions were computed at the DFT level employing a slab including the Si NR on top of three silver layers as well as the Si tip apex. The interactions (Hamiltonian matrix elements) were stored for different relative tip-surface positions and next fitted to obtain Slater-Koster parameters that allow a fast and accurate evaluation of these interactions for any tip-sample relative position [@loit]. Our Green’s function based formalism to simulate STM images includes only the elastic contribution to the current and assumes just one single tunneling process across the STM interface; it has been extensively described in previous works [@green; @loit]. Here we employed an imaginary part of the energy of 20 meV which also corresponds to the resolution used in the energy grid when integrating the transmission coefficient over the bias window. We further assumed the so called wide band limit (WBL) at the tip [@loit] in order to alleviate the computational cost and remove undesired tip electronic features. The images were computed at different biases between -2 to +2 V scanning the entire unit cell with a lateral resolution of 0.4 Å always assuming a fixed current of 1 nA. Nevertheless, the aspect of the images hardly changed with the bias, in accordance with most experimental results.
[*Energetics–*]{} To establish the energetic hierarchy among different Si NR structures we first computed their adsorption energies (per Si atom), $E_{\mathrm{ads}}$, via the simple expression: $$\label{eq:ads}
E_{\mathrm{ads}} = \left(E_{\mathrm{tot}}(N_{Ag},N_{Si})-E_{\mathrm{surf}}(N_{Ag})-
N_{\mathrm{Si}}E^0_{\mathrm{Si}}\right)/N_{\mathrm{Si}}$$ where $N_{\mathrm{Si/Ag}}$ are the number of Si and Ag atoms in the slab containing the NR and the Ag(110) surface, $E_{\mathrm{tot}}(N_{Ag},N_{Si})$ refers to its total energy, $E_{\mathrm{surf}}(N_{Ag})$ the energy of the clean Ag surface without the NRs (but including any MRs) and $E^0_{\mathrm{Si}}$ the energy of an isolated Si atom. In the low temperature limit eq. (\[eq:ads\]) allows to discriminate between structures with the same number of silver and Si atoms.
However, a more correct approach to compare the NR’s stabilities between structures with different Si and Ag concentrations is to compute their formation energies, $\gamma$, as a function of the Si and Ag chemical potentials, $\mu_{Si/Ag}$. To this end, we employ the standard low temperature expression for the grand-canonical thermodynamic potential [@phasediag]: $$\Omega(\mu_{\mathrm{Si}}, \mu_{\mathrm{Ag}})=E_{\mathrm{tot}}(N_{\mathrm{Si}}, N_{\mathrm{Ag}})-N_{\mathrm{Si}}\mu_{\mathrm{Si}} - N_{\mathrm{Ag}}\mu_{\mathrm{Ag}}$$
The chemical potentials may be obtained via $\mu_{Ag/Si}=E^{ref}_{Ag/Si}-
E^0_{Ag/Si}$, where $E^0$ corresponds to the total energy of the isolatad atom and $E^{ref}$ to that of a reference structure acting as a reservoir of Ag or Si atoms. Here we use the bulk $fcc$ phase for silver ($\mu^{LDA}_{Ag}=-4.67$ eV and $\mu^{GGA}_{Ag}=-3.60$ eV), while that of Si is considered as a parameter (see below). The NR’s formation energy, normalized to the Ag(110)-$(1\times1)$ surface unit cell area, then takes the form: $$\label{eq:form}
\gamma=\frac{1}{N}[E_{tot}(N_{Si},N_{Ag}) -
N_{Ag} E^{ref}_{Ag} - N_{Si} \mu_{Si}] - \gamma^{sb}_{Ag}$$ with $N=10$ because the same $c(10\times2$) was used for all NR structures and $\gamma^{sb}$ accounts for the formation energy of the unrelaxed surface at the bottom of the slab, which was obtained according to: $\gamma^{sb}_{Ag}=\frac{1}{2}[E_{1\times1}(N_{Ag})- N_{Ag} E^{ref}_{Ag}]$ with $E_{1\times1}(N_{Ag})$ giving the total energy of an unrelaxed nine layers thick Ag(110)-(1$\times$1) slab.
We follow the standard procedure of treating the Si chemical potential as a parameter in eq. (\[eq:form\]) and plot the formation energies for each structure as a function of $\mu_{Si}$ in Figures \[phasediag\](a) and (b) for the LDA and GGA derived energies, respectively. However, since a reference structure for the Si reservoir is not available (and hence the absolute value of $\mu_{Si}$ is unknown) we plot the formation energies as a function of a chemical potential shift, $\Delta\mu_{Si}$, whose origin is placed at the first crossing between the formation energy of the clean Ag(110) and that of any of the NRs (in our case it corresponds to the P-MR DNR structure). Within this somewhat arbitrary choice, small or negative values of $\Delta\mu_{Si}$ would correspond to Si poor conditions, while large positive values to Si rich conditions.
![[**Si$_{10}$-Ag(111) tip:**]{} Bottom and side views of the Ag(111) tip terminated in a 10 Si atom pyramid employed for all STM simulations.[]{data-label="tip"}](tip.eps)
{width="\textwidth"}
1.0 em
atom $z$ (Å) $d_{Si-Ag}$ (Å)
-- -------------------------- --------- ------------------------ --
[Si$_{\mathrm{ad}}$]{} 1.42 2.56 ($\times2$), 2.80
[Si$_{\mathrm{s}}$]{} 0.68 2.58, 2.74
[Si$^1_{\mathrm{ad}}$]{} 1.44 2.55 ($\times2$), 2.87
[Si$^2_{\mathrm{ad}}$]{} 1.38 2.56 ($\times2$), 2.78
[Si$^1_{\mathrm{s}}$]{} 0.68 2.58, 2.73
[Si$^2_{\mathrm{s}}$]{} 0.58 2.58, 2.76
: [**Details of the P-MR/Ag(110) geometry:**]{} Relative vertical distances ($z$) with respect to the average topmost Ag layer and bond distances to the first silver nearest neighbor ($d_{Si-Ag}$) for each of the symmetry inequivalent atoms in the SNR and DNR structures –see Fig. \[geom\] for further details. \[nn\]
![[**Dislocation defects between NRs.**]{} (a) Experimental STM image showing a dislocation within the array of DNRs. The image size is $6.2\times6.2$ nm$^{2}$. The sample bias voltage is 0.9 V and the tunnel current 0.8 nA. (b) Same as (a) after superimposing the Si pentagons (blue) and the MRs (dashed solid lines). The upper and lower truncated MRs at the center are shifted from each other by one Ag lattice parameter, leading to a DNR-SNR arrangement at the top of the image and a SNR-DNR at the bottom.[]{data-label="defect"}](defect.eps)
Model
------- ------------- ----------------- ----------------- ------------- ----------------- ----------------- ------------- ----------------- -----------------
$N_{Si/Ag}$ $E^{LDA}_{ads}$ $E^{GGA}_{ads}$ $N_{Si/Ag}$ $E^{LDA}_{ads}$ $E^{GGA}_{ads}$ $N_{Si/Ag}$ $E^{LDA}_{ads}$ $E^{GGA}_{ads}$
P-MR 6/88 [**6.44**]{} [**5.70**]{} 12/86 [**6.43**]{} [**5.70**]{} 12/86 6.40 6.57
P2-MR 6/88 6.24 5.56 12/86 6.23 5.68 12/86 6.41 5.49
ZZ-MR 4/88 6.36 5.59 8/86 6.35 5.58 8/86 6.34 5.58
{width="\textwidth"}
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abstract: 'The Ott–Antonsen ansatz is a powerful tool to extract the behaviors of coupled phase oscillators, but it imposes a strong restriction on the initial condition. Herein, a systematic extension of the Ott–Antonsen ansatz is proposed to relax the restriction, enabling the systematic approximation of the behavior of a globally coupled phase oscillator system with an arbitrary initial condition. The proposed method is validated on the Kuramoto–Sakaguchi model of identical phase oscillators. The method yields cluster and chimera-like solutions that are not obtained by the conventional ansatz.'
author:
- Akihisa Ichiki
- Keiji Okumura
title: 'Diversity of dynamical behaviors due to initial conditions: exact results with extended Ott–Antonsen ansatz for identical Kuramoto–Sakaguchi phase oscillators'
---
Coupled oscillator systems are important in both the pure science of synchronization phenomena [@winfree1967biological; @dorfler2014synchronization; @RevModPhys.77.137] and engineering applications such as electric power grids [@PhysRevE.93.032222] and wireless communication networks [@Guilera]. It also plays a central role in understanding biological phenomena such as neural networks [@Novikov; @Hannaye1701047] and the synchronous rhythm of cardiomyocytes [@hayashi2017community]. The most representative models of coupled phase oscillators are the Kuramoto model [@kuramoto1984chemical] and its generalization, the Kuramoto–Sakaguchi model [@sakaguchi1986soluble]. In recent years, Ott and Antonsen [@ott2009long] proposed a powerful ansatz to analyze these models, and it has deepened the understanding of various behaviors of coupled oscillator systems. The Ott–Antonsen ansatz (OAA) has been successfully employed in systems with various natural frequency distributions [@kawamura2010phase; @martens2009exact; @hong2012mean] and systems with external driving [@childs2008stability; @schwab2012kuramoto]. The OAA essentially reduces a system consisting of numerous coupled phase oscillators to a two-dimensional nonlinear oscillator system [@ott2009long; @goldobin2018collective]. The process is conventionally understood as follows: if the initial distribution of oscillator phases is set to a two-parameter distribution family called the Poisson kernel, the phase distribution after time evolution remains in the Poisson kernel [@marvel2009identical]. Thus, the OAA strongly restricts the initial condition, although it is effective for understanding a globally coupled oscillator system. In general, the origin of diversity in the dynamical behaviors of a system can be rooted in both the native properties of oscillators and the initial condition. The restriction of the OAA hinders the full understanding of the dependence of system behaviors on initial conditions. In the present Letter, the OAA is extended to systematically relax the restrictions on the initial distribution for understanding the initial-condition dependence of complicated behaviors in phase oscillator systems.
The present Letter makes three main claims. Firstly, the Poisson kernel appearing in the OAA is claimed to be equivalent to a Cauchy–Lorentz distribution (CLD). Therefore, the OAA is interpreted as follows: if the initial phase distribution is set to a CLD, the phase distribution remains in the distribution family of CLD. Secondly, as an extension of the conventional OAA, it is claimed that if the initial phase distribution is set to a superposition of CLDs, the phase distribution remains in the superposition of CLDs. Consequently, an arbitrary initial condition can be analyzed systematically by approximating the phase distribution as a superposition of CLDs. Thirdly, compared to the conventional OAA, the extended version is more helpful to understand complicated behaviors of the system. To show the advantage, the extended OAA is employed for the Kuramoto–Sakaguchi model of identical phase oscillators, and it yields a variety of dynamical behaviors including a cluster solution [@golomb1992clustering; @gong2019repulsively] that could not be obtained by the conventional OAA.
To overview the derivation of the OAA, let us consider a system of $N$ phase oscillators globally coupled via mean-field couplings, i.e., the Kuramoto–Sakaguchi model [@sakaguchi1986soluble]: $$\begin{aligned}
\dot{\theta}_i = \omega_i - \dfrac{K}{N}\displaystyle\sum_{j=1}^N \sin\left(\theta_i - \theta_j + \alpha\right)\,,\label{original_dyn}\end{aligned}$$ where $\theta_i$ denotes the phase of the $i^{\rm th}$ oscillator, $\omega_i$ the natural frequency of the oscillator $i$, $K$ the coupling constant between the oscillators, and $\alpha$ a constant. Many interesting phenomena are known for cases with distributed natural frequencies [@kawamura2010phase; @hong2012mean; @martens2009exact; @nagai2010noise; @battogtokh2002coexistence; @abrams2004chimera]. The natural frequency distribution is assumed to be $g(\omega)$. In the thermodynamic limit, i.e., $N\to\infty$, by the self-averaging property, the empirical distribution of oscillator phases with a natural frequency $\omega$, $$\begin{aligned}
P(\theta, t|\omega) := \dfrac{1}{N_\omega}\sum_{j\in \omega}\delta\left(\theta - \theta_j(t)\right),\label{empiricalProb}\end{aligned}$$ evolves by the following nonlinear Fokker–Planck equation [@kuramoto1984chemical; @strogatz2000kuramoto; @RevModPhys.77.137]: $$\begin{aligned}
&&\dfrac{\partial P(\theta, t|\omega)}{\partial t} = -\dfrac{\partial}{\partial \theta}\left[\omega - KQ(\theta, t)\right]P(\theta, t|\omega)\,,\label{FPE}\\
&&Q(\theta, t) := \dfrac{1}{2i}\left[z^\ast(t) e^{i(\theta + \alpha)} - z(t) e^{-i(\theta+\alpha)}\right]\,.\end{aligned}$$ In Eq. (\[empiricalProb\]), $N_\omega$ denotes the number of oscillators with a natural frequency $\omega$, and the sum is taken over all oscillators of a natural frequency $\omega$. The Kuramoto order parameter $z(t)$ is defined as [@kuramoto1984chemical; @kuramoto1987statistical] $$\begin{aligned}
z(t) &:=& \displaystyle\lim_{N\to\infty}\dfrac{1}{N}\sum_{j=1}^N \exp\left[i\theta_j(t)\right]\nonumber\\
&=&\int\,d\omega g(\omega)\int\,d\theta \exp\left(i\theta\right) P(\theta, t|\omega)\,.\end{aligned}$$ It is convenient to Fourier expand the empirical distribution by $\Theta_s(\theta) = \exp\left(i s\theta\right)$: $$\begin{aligned}
P(\theta, t|\omega) =\int_{-\infty}^\infty\, ds p_s^\omega(t)\Theta_s(\theta)\,.\label{expand}\end{aligned}$$ By using the above expansion, according to the linear independence of $\Theta_s$, the Fokker-Planck equation (\[FPE\]) is expressed as $$\begin{aligned}
\dot{p}_s^\omega(t) = -is\omega p_s^\omega(t) + \dfrac{sK}{2}\left[z^\ast e^{i\alpha}p_{s-1}^\omega - z e^{-i\alpha}p_{s+1}^\omega\right]\,.\label{dyn}\nonumber\\\end{aligned}$$ Note that the expansion coefficient $p_s^\omega(t)$ is related to the characteristic function $\phi_\omega(s, t) := \int\,d\theta\exp\left(is\theta\right)P(\theta, t|\omega)$ for the empirical distribution by $p_s^\omega(t) = \phi_\omega(-s, t)/2\pi$ according to the orthogonality of the basis $\{\Theta_s\}$ ($s \ge 0$). In general, Eq. (\[dyn\]) has an infinite hierarchy and cannot be solved. In order to resolve the hierarchy, Ott and Antonsen [@ott2009long] assumed that there exists an appropriate complex variable $A_\omega(t)$ and $$\begin{aligned}
p_s^\omega(t) = A_\omega^s(t)\label{OAansatz}\end{aligned}$$ holds for all nonnegative numbers of $s \ge 0$. This ansatz for the expansion coefficients $\{p_s^\omega\}$, i.e., Eq. (\[OAansatz\]), is called the OAA. The OAA is found to be equivalent to $$\begin{aligned}
\phi_\omega(-s, t) = \exp\left[-s\gamma_\omega(t) + is\mu_\omega(t)\right]\label{OAQ}\end{aligned}$$ for all nonnegative $s \ge 0$ with appropriate real variables $\gamma_\omega(t)$ and $\mu_\omega(t)$. The simplest distribution function with such a characteristic function is a CLD. The parameters $\mu_\omega$ and $\gamma_\omega$ play the roles of the location and half-width at half-maximum of the peak in CLD, respectively.
The Ott–Antonsen manifold is a two-dimensional manifold that is the invariant of the empirical distribution family under a time evolution, and it has been discussed in relation to the Poisson kernel [@marvel2009identical]. From the above argument, the Poisson kernel is equivalent to a CLD, which is a family of distributions characterized by two parameters: peak location and half-width at half-maximum. The manifold formed by these two parameters is the Ott–Antonsen manifold. In discussions on coupled phase oscillator systems using the OAA, the natural frequency distribution $g (\omega)$ is often assumed to be a CLD [@ott2009long]. However, if the initial phase distribution in the ensemble of oscillators with the same natural frequency is taken as a CLD, the phase distribution at any time is given by a CLD, and it is not related to the frequency distribution $g(\omega)$.
By introducing a complex variable $A_\omega := \exp\left[-\gamma_\omega(t) + i\mu_\omega(t)\right]$, from the evolution equation (\[dyn\]) with the OAA (\[OAansatz\]), the evolution for $A_\omega$ is given as $$\begin{aligned}
\dot{A}_\omega = i\omega A_\omega + \dfrac{K}{2}z e^{-i\alpha} - \dfrac{K}{2}z^\ast e^{i\alpha}A_\omega^2\label{dyn_A_ori}\end{aligned}$$ with $z(t) = \int\,d\omega g(\omega)A_\omega(t)$.
When a phase distribution is assumed to be a CLD, the evolution equation (\[dyn\_A\_ori\]) is straightforwardly obtained by averaging the dynamics (\[original\_dyn\]). By introducing a complex variable $A_i = \exp\left[i \theta (t) \right]$, Eq. (\[original\_dyn\]) is rewritten in the same form as Eq. (\[dyn\_A\_ori\]): $$\begin{aligned}
\dot{A} _i = i \omega A_i + \dfrac{K}{2} z e^{-i\alpha}-\dfrac{K}{2} z ^ \ast e^{i\alpha} A_i ^ 2\,. \label{1body}\end{aligned}$$ When the phase distribution for oscillators with the natural frequency $\omega$ is given by a CLD $$\begin{aligned}
P_{\rm CL}\left(\theta| \mu_\omega(t), \gamma_\omega(t)\right) := \dfrac{1}{\pi}\dfrac{\gamma_\omega(t)}{\left[\theta - \mu_\omega(t)\right]^2 + \gamma_\omega^2(t)}\,,\end{aligned}$$ the averaged quantity $\bar{A}_\omega := \sum_{j\in\omega} A_j / N_\omega$ is given by the pole of $P_{\rm CL}\left(\theta|\mu_\omega(t), \gamma_\omega(t)\right)$ in the thermodynamic limit as $$\begin{aligned}
\bar{A}_\omega(t) = \exp\left(i\theta\right)|_{\theta = \mu_\omega(t) + i\gamma_\omega(t)} = A_\omega\,.\label{anl_con}\end{aligned}$$ Thus, the set of oscillators obeying the CLD is reduced to a single oscillator with a complex phase evolving with Eq. (\[1body\]). It is concluded that the OAA is a reduction method for degrees of freedom owing to the representative property of a pole in a CLD.
As discussed above, the conventional OAA considers only the case where the initial phase distribution is a CLD. In the analysis of systems with distributed natural frequencies [@ott2009long; @martens2009exact; @kawamura2010phase; @hong2012mean], the OAA restricts the initial distribution of a phase oscillator group with each frequency $\omega$ to a CLD. Consequently, the CLD for each frequency evolves as per Eq. (\[dyn\]).
To extend the conventional OAA, it is worth mentioning that, in Eq. (\[dyn\]), oscillators belonging to different CLDs interact only through the Kuramoto order parameter $z$. This is true even when the natural frequency distribution is significantly sharp. Then, it is possible to consider the case where the width of the natural frequency distribution around a certain frequency $\omega$ approaches zero. In this case, it is concluded that the phase distribution of oscillators with the natural frequency $\omega$ is given by a superposition of CLDs satisfying the OAA. To formulate this idea, for simplicity, let us consider a set of phase oscillators with a single natural frequency. By dividing the $N$ oscillators into $M$ groups, the empirical distribution of the $\nu^{\rm th}$ group is formally defined as $$\begin{aligned}
P_\nu(\theta, t) = \dfrac{1}{N_\nu}\displaystyle\sum_{j\in \Omega_\nu}\delta\left(\theta - \theta_j(t)\right)\,,\end{aligned}$$ where the sum is taken for all oscillators in the $\nu^{\rm th}$ group $\Omega_\nu$ and $N_\nu$ is the number of oscillators in $\Omega_\nu$ of $\mathcal{O}(N)$. Consequently, oscillators belonging to different groups interact only via the Kuramoto order parameter. If the initial distribution of $P_\nu$ is a CLD, the time evolution is exactly given by the OAA. In other words, if the initial phase distribution is given as a superposition of $M$ CLDs, the empirical distribution at any time remains to be a superposition of $M$ CLDs: $$\begin{aligned}
P(\theta, t) = \displaystyle\sum_{\nu=1}^M r_\nu P_{\rm{CL}}(\theta|\mu_\nu(t), \gamma_\nu(t))\,,\end{aligned}$$ where $r_\nu := N_\nu / N$ is the ratio of the $\nu^{\rm th}$ distribution. Because the conventional OAA holds for each group, the evolution equation is given as $$\begin{aligned}
\dot{A}_\nu &=& i\omega A_\nu + \dfrac{K}{2}z e^{-i\alpha} - \dfrac{K}{2}z^\ast e^{i\alpha}A_\nu^2\,,\label{SKOAA}\\
z(t) &=& \displaystyle\sum_{\nu=1}^M r_\nu A_\nu(t)\,,\label{def_z}\end{aligned}$$ where $A_\nu$ corresponds to the characteristic function of $P_\nu(\theta, t)$, which is assumed to be a CLD. In this framework, the Ott–Antonsen manifold is extended to $2M$-dimensions. That is, if the superposition of arbitrary CLDs is set as an initial phase distribution, the time evolution of the phase distribution is exactly given. This fact is useful for systematically approximating the behavior of a system starting from an arbitrary initial distribution. The approximated system behavior can be obtained with arbitrary accuracy if the initial phase distribution is approximated as a superposition of CLDs to the required accuracy.
To guarantee the correctness of the above extension of the OAA, it is worth mentioning its mathematical background. The conventional OAA is interpreted as follows: if the initial phase distribution is a CLD, the phase distribution remains in the CLD family at any arbitrary time. The evolution equation for the distribution function is given through the characteristic function. The characteristic function is the Fourier transform of the distribution function, and the Fourier transform is a linear transformation. Therefore, the evolution of the superposition of CLDs is given by the superposition of characteristic functions, and the OAA holds for each CLD.
Finally, in order to show the advantage of the higher-dimensional version of OAA, let us consider a variety of solutions including a cluster solution [@golomb1992clustering; @gong2019repulsively] and chimera-like solution [@abrams2004chimera] in the Kuramoto–Sakaguchi model [@sakaguchi1986soluble] of identical phase oscillators. Under the extended OAA, the variable $A_\nu$ obeys the dynamics given by Eq. (\[SKOAA\]), where the Kuramoto order parameter $z$ is evaluated using Eq. (\[def\_z\]). The conventional OAA corresponds to the case of $M=1$, and the evolution equation for $z$ is given as $$\begin{aligned}
\dot{z} = i\omega z + \dfrac{K}{2}z\left(e^{-i\alpha} - |z|^2 e^{i\alpha}\right)\,.\label{convOAz}\end{aligned}$$ Because $|z| \le 1$ and the real part of the coefficient for $z$ in the right-hand side of Eq. (\[convOAz\]) gives the growth rate of $|z|$, $|z|$ increases monotonically to $|z| \to 1$ when the coupling is attractive, i.e., $\cos \alpha > 0$. Therefore, in this case, all the oscillators become in phase. On the other hand, if the coupling is repulsive, i.e., $\cos\alpha < 0 $, $|z|$ decreases monotonically to $|z| \to 0$. In the conventional OAA, because identical oscillators are considered and $z = A_\omega = \exp\left (-\gamma + i \mu\right) $ in this case, $z \to 0$ implies a uniform distribution of oscillator phases. However, it is known that complicated behaviors appear in systems with repulsive coupling [@hansel1993clustering; @golomb1992clustering; @gong2019repulsively; @abrams2004chimera], even if the system consists of identical oscillators. A typical example is a cluster solution [@golomb1992clustering; @gong2019repulsively]. In the repulsive case, there is a possibility that multiple clusters exist and cancel each other’s phase effects to satisfy $z = 0$, but this phenomenon cannot be described by the conventional OAA, in which $z = 0$ implies a uniform phase distribution.
On the other hand, the extended OAA retains the possibility of nontrivial phase distributions satisfying $z=0$. By decomposing $A_\nu$ into $z$ and its variation around $z$ as $A_\nu = z + \Delta_\nu$, the dynamics for $z$ and $\Delta_\nu$ are obtained as $$\begin{aligned}
&&\dot{z} = i\omega z + \dfrac{K}{2}z \left(e^{-i\alpha} - |z|^2 e^{i\alpha}\right) - \dfrac{K}{2} e^{i\alpha} z^\ast\displaystyle\sum_{\sigma=1}^M r_\sigma \Delta_\sigma^2\,,\nonumber\\
\\
&&\dot{\Delta}_\nu = i\omega\Delta_\nu - K|z|^2 e^{i\alpha}\Delta_\nu + \dfrac{K}{2}e^{i\alpha} z^\ast\left(\displaystyle\sum_{\sigma=1}^M r_\sigma \Delta_\sigma^2 - \Delta_\nu^2\right)\,,\nonumber\\
\\
&&\displaystyle\sum_{\nu=1}^M r_\nu \Delta_\nu = 0\,.\end{aligned}$$ Note that, in the dynamics of $z$ for the extended OAA, a nonlinear term with respect to $\Delta_\nu$ has been added to the dynamics of $z$ for the conventional OAA Eq. (\[convOAz\]). Because of this nonlinearity, we cannot simply conclude that attractive coupling, i.e., $\cos\alpha> 0$, leads to a synchronous solution and that repulsive coupling, i.e., $\cos\alpha < 0$, leads to an asynchronous solution. By taking a complicated initial distribution, a nontrivial behavior that is not predicted by the conventional OAA may occur. In fact, in the case of $M = 3$, a nontrivial cluster solution with $z=0$ is numerically observed. The solution of Eq. (\[SKOAA\]) and the direct numerical solution of Eq. (\[original\_dyn\]) for the repulsive case are shown in Fig. \[fig\_M3cluster\]. The solutions for Eqs. (\[SKOAA\]) and (\[original\_dyn\]) agree very well. Further, the snapshots of the corresponding oscillator phases obtained using Eq. (\[original\_dyn\]) are shown in Fig. \[fig\_M3Raster\]. The oscillators in the two clusters corresponding to $A_1$ and $A_2$ are almost anti-phase, and the oscillators in the cluster corresponding to $A_3$ are almost uniformly distributed after a long time. Note that such a chimera-like [@abrams2004chimera] cluster solution cannot be obtained by the conventional OAA as mentioned above. When the initial values for $A_k$ ($k = 1, 2, 3$) are identical, the results are the same as in the conventional OAA, i.e., $A_1$, $A_2$, and $A_3$ all become to correspond to characteristic functions of uniform distributions; in other words, $A_k = 0$. As the difference between the initial values of $A_k$ increases, the solutions $A_1$, $A_2$, and $A_3$ gradually split, resulting in a nontrivial cluster solution.
![(Color online) Chimera-like cluster solution of three groups of oscillators with the same natural frequency $\omega$ in the repulsive coupling case. Top panel: time evolution of $\mu_k$. Middle panel: time evolution of $\gamma_k$. Bottom panel: time evolution of Kuramoto order parameter $z$. The parameters are set to $\omega=1.0, K=1.0, \alpha=3\pi/4,$ and $r_1 = r_2 = r_3 = 1/3$. The initial values are set to $\gamma_1 = \gamma_2 = 0.1, \gamma_3 = 5.0, \mu_1 = 0.0, \mu_2 = 3.0,$ and $\mu_3 = 1.5$. The solid, dashed, and dotted lines in the top and middle panels correspond to the solutions of Eq. (\[SKOAA\]) for $A_1$, $A_2$, and $A_3$, respectively. The circle, triangle, and square marks in the top and middle panels correspond to the numerical solutions of Eq. (\[original\_dyn\]) for $A_1$, $A_2$, and $A_3$, respectively. In the bottom panel, the solid line and circle marks correspond to the solutions of Eq. (\[SKOAA\]) and Eq. (\[original\_dyn\]), respectively. In the calculation of Eq. (\[original\_dyn\]), the numbers of oscillators were taken as $N_1 = N_2 = N_3 = 10^5$. \[fig\_M3cluster\]](Fig1_epslatex.eps){width="8.6cm"}
![Phase distributions obtained from the direct calculation of Eq. (\[original\_dyn\]) corresponding to the result shown in Fig. \[fig\_M3cluster\]. The site indices $0$ to $10^5-1$, $10^5$ to $2\times 10^5 -1$, and $2\times 10^5$ to $3\times 10^5 -1$ correspond to the oscillators belonging to the clusters of $A_1$, $A_2$, and $A_3$, respectively. The panels (a)–(d) correspond to the snapshots at $t=0, 5, 10, 15$, respectively. For the convenience of viewing, the points are plotted by thinning out at a rate of $1/250$. \[fig\_M3Raster\]](Fig2_epslatex.eps){width="8.6cm"}
Nontrivial dynamical behavior is observed even in the attractive coupling case, as shown in Fig. \[fig\_M3attractive\]. The Kuramoto order parameter $|z|$ increases with oscillation, whereas $|z|$ increases monotonically in the prediction of the conventional OAA. As shown in Fig. \[fig\_M3attractiveRaster\], it is possible to predict the chimera-like state in the transient regime by using the extended version of the OAA.
![(Color online) Nontrivial evolution of three groups of oscillators with the same natural frequency $\omega$ in the attractive coupling case. Top panel: time evolution of $\mu_k$. Middle panel: time evolution of $\gamma_k$. Bottom panel: time evolution of the Kuramoto order parameter $z$. The parameters are set to $\omega=1.0, K=1.0, \alpha=1.8\pi/4,$ and $r_1 = r_2 = r_3 = 1/3$. The initial values are set to $\gamma_1 = \gamma_2 = 1.0, \gamma_3 = 0.1, \mu_1 = 0.0, \mu_2 = 1.0,$ and $\mu_3 = 3.0$. The solid, dashed, and dotted lines in the top and middle panels correspond to the solutions of Eq. (\[SKOAA\]) for $A_1$, $A_2$, and $A_3$, respectively. The circle, triangle, and square marks in the top and middle panels correspond to the numerical solutions of Eq. (\[original\_dyn\]) for $A_1$, $A_2$, and $A_3$, respectively. In the bottom panel, the solid line and circle marks correspond to the solutions of Eq. (\[SKOAA\]) and Eq. (\[original\_dyn\]), respectively. In the numerical calculation of Eq. (\[original\_dyn\]), the numbers of oscillators were taken as $N_1 = N_2 = N_3 = 10^5$. \[fig\_M3attractive\]](Fig3_epslatex.eps){width="8.6cm"}
![Phase distributions obtained from the direct calculation of Eq. (\[original\_dyn\]) corresponding to the result shown in Fig. \[fig\_M3attractive\]. The site indices $0$ to $10^5-1$, $10^5$ to $2\times 10^5 -1$, and $2\times 10^5$ to $3\times 10^5 -1$ correspond to the oscillators belonging to the clusters of $A_1$, $A_2$, and $A_3$, respectively. The panels (a)–(h) correspond to the snapshots at $t=0, 5, 10, 15, 20, 25, 30, 35$, respectively. For the convenience of viewing, the points are plotted by thinning out at a rate of $1/250$. \[fig\_M3attractiveRaster\]](Fig4_epslatex.eps){width="8.6cm"}
Several numerical experiments have shown that the stability of $z$ is independent of $M$. Numerically, synchronous solutions of $|z| = 1$ were obtained for the attractive case, i.e., $\cos\alpha > 0$, and asynchronous solutions of $z=0$ were obtained for repulsive case, i.e., $\cos\alpha < 0$, for several initial conditions. In other words, the behavior of the Kuramoto order parameter $z$ after a long time is independent of the initial condition. However, the behavior of $z$ after a long time with respect to the general initial condition is not analytically guaranteed. The relation between the behavior of $z$ after a long time and the initial condition remains to be clarified in the future.
To conclude, the OAA, which is a method for reducing the high number of degrees of freedom of globally coupled phase oscillators to a two-dimensional manifold, has been extended for reduction to a high-dimensional manifold. The conventional two-dimensional Ott–Antonsen manifold has been clarified to be a manifold of a CLD, which is characterized by two parameters. Owing to the representative property of the poles of a CLD, the many-body problem of phase oscillators has been reduced to a single oscillator problem under the conventional OAA. By taking advantage of the linearity of the characteristic function with respect to the superposition of empirical distributions, the extension of OAA has been realized by the superposition of CLDs. Since the extended OAA is exact in the thermodynamic limit, it would be a powerful tool to investigate the behaviors of a system consisting of many phase oscillators. Moreover, this extension enables the systematic approximation of the behavior of coupled phase oscillator systems with arbitrary initial conditions. The extended OAA has been employed for the Kuramoto–Sakaguchi model of identical phase oscillators to show a variety of dynamical behaviors, such as a chimera-like cluster solution, which cannot be obtained by the conventional OAA. The conventional chimera state was found in the case where couplings depend on the distances between the oscillators [@abrams2004chimera]. On the other hand, the chimera-like state found in the present study occurs in a situation where there is no concept of distance. This fact may deepen the understanding of the origin of complicated behaviors of oscillator systems. With the extended OAA, complicated behaviors have been obtained even in a system of phase oscillators with a single natural frequency. The proposed method can be applied to a wide range of phase oscillator systems, as long as the number of oscillators is sufficiently large. In the future, it will be applied to more interesting cases with distributed natural frequencies.
This work was supported by JSPS KAKENHI grants numbered JP17H06469 and JP19K20360.
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abstract: 'We develop a new approach of the quantum phase in an Hilbert space of $\rm{fi}$nite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result, phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of the phase noise is allowed for speci$\rm{fi}$c quantum states. The concept of phase entanglement for Kloosterman pairs of phase-locked states is introduced.'
address:
- |
Laboratoire de Physique et Métrologie des Oscillateurs du CNRS,\
32 Avenue de l’observatoire, 25044 Besançon Cedex, France\
- |
Potosinian Institute of Scientific and Technological Research\
Apdo Postal 3-74, Tangamanga, San Luis Potosi, SLP, Mexico\
author:
- Michel Planat and Haret Rosu
title: |
The hyperbolic, the arithmetic\
and the quantum phase
---
Introduction
============
Time and phase are not well de$\rm{fi}$ned concepts at the quantum level. The present work belongs to a longstanding effort to model phase noise and phase-locking effects that are found in highly stable oscillators. It was unambiguously demonstrated that the observed variability, (i.e., the 1/f frequency noise of such oscillators) is related to the $\rm{fi}$nite dynamics of states during the measurement process and to the precise $\rm{fi}$ltering rules that involve continued fraction expansions, prime number decomposition, and hyperbolic geometry [@Planat1]-[@Planat3].
We add here a quantum counterpart to these effects by studying the notions of quantum phase-locking and quantum phase entanglement. The problem of de$\rm{fi}$ning quantum phase operators was initiated by Dirac in 1927 [@Dirac]. For excellent reviews, see [@Reviews]. We use the Pegg and Barnett quantum phase formalism [@Pegg] where the calculations are performed in an Hilbert space $H_q$ of $\rm{fi}$nite dimension q. The phase states are de$\rm{fi}$ned as superpositions of number states from the so-called quantum Fourier transform (or QFT) $$|\theta _p\rangle=q^{-1/2}\sum_{n=0}^{q-1}\exp(\frac{2i\pi p n
}{q})|n\rangle~. \label{eq1}$$ in which $i^2=-1$. The states $|\theta_p\rangle$ form an orthonormal set and in addition the projector over the subspace of phase states is $\sum_{p=0}^{q-1}|\theta_p\rangle
\langle\theta_p|=1_q$ where $1_q$ is the identity operator in $H_q$. The inverse quantum Fourier transform follows as $|n\rangle=q^{-1/2}\sum_{p=0}^{q-1}\exp(-\frac{2i\pi p n
}{q})|\theta _p\rangle$. As the set of number states $|n\rangle$, the set of phase states $|\theta_p\rangle$ is a complete set spanning $H_q$. In addition the QFT operator is a $q$ by $q$ unitary matrix with matrix elements $\kappa_{pn}^{(q)}=\frac{1}{\sqrt {q}}\exp(2i\pi \frac{pn}{q})$.
From now we emphasize phase states $|\theta' _p\rangle$ satisfying phase-locking properties. The quantum phase-locking operator is defined in (\[eq4\]). We $\rm{fi}$rst impose the coprimality condition $$(p,q)=1,$$ where $(p,q)$ is the greatest common divisor of $p$ and $q$. Differently from the phase states (\[eq1\]), the $|\theta'
_p\rangle$ form an orthonormal base of a Hilbert space whose dimension is lower, and equals the number of irreducible fractions $p/q$, which is given by the Euler totient function $\phi(q)$. These states were studied in our recent publication [@PLA03][^1].
Guided by the analogy with the classical situation [@Planat2], we call these irreducible states the phase-locked quantum states. They generate a cyclotomic lattice $L$ [@Craig] with generator matrix $M$ of matrix elements $\kappa_{pn}^{'(q)}$, (p,q)=1 and of size $\phi(q)$. The corresponding Gram matrix $H=M^{\dag}M$ shows matrix elements $h_{n,l}^{(q)}=c_q(n-l)$ which are Ramanujan sums $$c_q(n)=\sum_p \exp(2i\pi \frac{p}{q}
n)=\frac{\mu(q_1)\phi(q)}{\phi(q_1)},~~\rm{with}~q_1=q/(q,n).
\label{eq2}$$ where the index $p$ means summation from $0$ to $q-1$, and $(p,q)=1$. Ramanujan sums are thus de$\rm{fi}$ned as the sums over the primitive characters $\exp(2i\pi \frac{pn}{q})$, (p,q)=1, of the group $Z_q=Z/qZ$. In the equation above $\mu(q)$ is the Möbius function, which is $0$ if the prime number decomposition of $q$ contains a square, $1$ if $q=1$, and $(-1)^k$ if $q$ is the product of $k$ distinct primes [@Hardy]. Ramanujan sums are relative integers which are quasi-periodic versus $n$ with quasi-period $\phi(q)$ and aperiodic versus $q$ with a type of variability imposed by the Möbius function. Ramanujan sums were introduced by Ramanujan in the context of Goldbach conjecture [@Hardy].
They are also useful in the context of signal processing as an arithmetical alternative to the discrete Fourier transform [@Planat4]. In the discrete Fourier transform the signal processing is performed by using all roots of unity of the form $\exp(2i\pi p/q)$ with $p$ from $1$ to $q$ and taking their nth powers $e_p(n)$ as basis function. We generalized the classical Fourier analysis by using Ramanujan sums $c_q(n)$ as in (\[eq2\]) instead of $e_p(n)$. This type of signal processing is more appropriate for arithmetical functions than is the ordinary discrete Fourier transform, while still preserving the metric and orthogonal properties of the latter. Notable results relating arithmetical functions to each other can be obtained using Ramanujan sums expansion while the discrete Fourier transform would show instead the low frequency tails in the power spectrum.
In this paper we are also interested in pairs of phase-locked states which satisfy the two conditions $$(p,q)=1~~and~~p\bar{p}=-1 (mod~q).
\label{phaselocked}$$ Whenever it exists $\bar{p}$ is uniquely de$\rm{fi}$ned from minus the inverse of $p$ modulo $q$. Geometrically the two fractions $p/q$ and $\bar{p}/q$ are the ones selected from the partition of the half plane by Ford circles. Ford circles are de$\rm{fi}$ned as the set of the images of the horizontal line z=x+i, x real, under all modular transformations in the group of $2\times 2$ matrices $SL(2,Z)$ [@Planat3]. Ford circles are tangent to the real axis at a Farey fraction $p/q$, and are tangent to each other. They have been introduced by Rademacher as an optimum integration path to compute the number of partitions by means of the so-called Ramanujan’s circle method. In that method two circles of indices $p/q$ and $\bar{p}/q$, of the same radius $\frac{1}{2q^2}$ are dual to each other on the integration path \[see also Part 2 and Fig. 3\].
Hints to the hyperbolic geometry of phase noise
===============================================
The phase-locked loop, low pass filtering and $1/f$ noise
---------------------------------------------------------
A newly discovered clue for $1/f$ noise was found from the concept of a phase locked loop (or PLL) [@Planat2]. In essence two interacting oscillators, whatever their origin, attempt to cooperate by locking their frequency and their phase. They can do it by exchanging continuously tiny amounts of energy, so that both the coupling coefficient and the beat frequency should fluctuate in time around their average value. Correlations between amplitude and frequency noises were observed [@Yamoto].
One can get a good level of understanding of phase locking by considering the case of quartz crystal oscillators used in high frequency synthesizers, ultrastable clocks and communication engineering (e.g., mobile phones). The PLL used in a FM radio receiver is a genuine generator of $1/f$ noise. Close to phase locking the level of $1/f$ noise scales approximately as $\tilde{\sigma}^2$, where $\tilde{\sigma}=\sigma
K/\tilde{\omega_B}$ is the ratio between the open loop gain $K$ and the beat frequency $\tilde{\omega_B}$ times a coefficient $\sigma$ whose origin has to be explained. The relation above is explained from a simple non linear model of the PLL known as Adler’s equation $$\dot{\theta}(t)+K H(P)\sin \theta(t)=\omega_B, \label{equation1}$$ where at this stage $H(P)=1$, $\omega_B=\omega(t)-\omega_0$ is the angular frequency shift between the two quartz oscillators at the input of the non linear device (a Schottky diode double balanced mixer), and $\theta(t)$ is the phase shift of the two oscillators versus time t. Solving (\[equation1\]) and differentiating one gets the observed noise level $\tilde{\sigma}$ versus the bare one $\sigma=\delta \omega_B/\tilde{\omega_B}$. Thus the model doesn’t explain the existence of $1/f$ noise but correctly predicts its dependence on the physical parameters of the loop [@Planat2].
Besides one can get detailed knowledge of harmonic conversions in the PLL by accounting for the transfer function $H(P)$, where $P=\frac{d}{dt}$ is the Laplace operator. If $H(P)$ is a low pass filtering function with cut-off frequency $f_c$, the frequency at the output of the mixer + filter stage is such that $$\mu=f_B(t)/f_0=q_i|\nu-p_i/q_i| \le f_c/f_0,~~
\mbox{$p_i$~and~$q_i$~integers}. \label{equation2}$$ The beat frequency $f_B(t)$ results from the continued fraction expansion of the input frequency ratio $$\nu=f(t)/f_0=[a_0;a_1,a_2,\ldots a_i,a,\ldots ], \\
\label{equation3}$$ where the brackets mean expansions $a_0+1/(a_1+1/(a_2+1/\ldots+1/(a_i+1/(a+\ldots))))$. The truncation at the integer part $a=[\frac{f_0}{f_c q_i}]$ defines the edges of the basin for the resonance $p_i/q_i$; they are located at $\nu_1=[a_0;a_1,a_2,\ldots,a_i, a]$ and $\nu_2=[a_0;a_1,a_2,\ldots a_i-1,1,a]$ [@Planat1]. The two expansions in $\nu_1$ and $\nu_2$, prior to the last filtering partial quotient $a$, are the two allowed ones for a rational number. The convergents $p_i/q_i$ at level $i$ are obtained using the matrix products $$\left[\begin{array}{cc} a_0 & 1\\ 1 & 0 \end{array}\right]
\left[\begin{array}{cc} a_1 & 1\\ 1 & 0 \end{array}\right]\cdots
\left[\begin{array}{cc} a_i & 1\\ 1 & 0 \end{array}\right]
=\left[\begin{array}{cc} p_i&p_{i-1}\\ q_i&q_{i-1}
\end{array}\right]. \label{matrices}$$ Using (\[matrices\]), one can get the fractions $\nu_1$ and $\nu_2$ as $\nu_1=\frac{p_a}{q_a}$ and $\nu_2=\frac{p_i(2a+1)-p_a}{q_i(2a+1)-q_a}$, so that with the relation relating convergents $(p_i
q_{i-1}-p_{i-1}q_i)=(-1)^{i-1}$, the width of the basin of index $i$ is $|\nu_1-\nu_2|=\frac{2a+1}{q_a(q_a+(2a+1)q_i)}\simeq\frac{1}{q_a
q_i}$ whenever $a>1$.
In previous publications of one of the authors a phenomenological model for $1/f$ noise in the PLL was proposed, based on an arithmetical function which is a logarithmic coding for prime numbers [@Planat1],[@Planat2]. If one accepts a coupling coefficient evolving discontinuously versus the time $n$ as $K=K_0\Lambda(n)$, with $\Lambda(n)$ the Mangoldt function which is $\ln(p)$ if $n$ is the power of a prime number $p$ and $0$ otherwise, then the average coupling coefficient is $K_0$ and there is an arithmetical fluctuation $\epsilon(t)$ $$\begin{aligned}
&\psi(t)=\sum_{n=1}^t\Lambda(n)=t(1+\epsilon(t)),\nonumber\\
&t\epsilon(t)=-\ln(2\pi)-\frac{1}{2}\ln(1-t^{-2})-\sum_{\rho}\frac{t^{\rho}}{\rho}.
\label{Riemann}\end{aligned}$$ The three terms at the right hand side of $t\epsilon(t)$ come from the singularities of the Riemann zeta function $\zeta(s)$, that are the pole at $s=1$, the trivial zeros at $s=-2l$, $l$ integer, and the zeros on the critical line $\Re(s)=\frac{1}{2}$[@Planat1]. Moreover the power spectral density roughly shows a $1/f$ dependance versus the Fourier frequency $f$. This is the proposed relation between Riemann zeros (the still unproved Riemann hypothesis is that all zeros should lie on the critical line) and $1/f$ noise.
We improved the model by replacing the Mangoldt function by its modified form $b(n)=\Lambda(n)\phi(n)/n$, with $\phi(n)$ the Euler (totient) function [@Planat4]. This seemingly insignificant change was introduced by Hardy [@Hardy] in the context of Ramanujan sums for the Goldbach conjecture and resurrected by Gadiyar and Padma in their recent analysis of the distribution of pairs of prime numbers [@Gadiyar]. Then by defining the error term $\epsilon_B(t)$ from the cumulative modified Mangoldt function $$B(t)=\sum_{n=1}^t b(n)=t(1+\epsilon_B(t)), \label{functionB}$$ its power spectral density $S_B(f)\simeq\frac{1}{f^{2\alpha}}$ exhibits a slope close to the Golden ratio $\alpha\simeq(\sqrt5-1)/2\simeq0.618$ (see Fig. 1).
The modified Mangoldt function occurs in a natural way from the logarithmic derivative of the following quotient $$Z(s)=\frac{\zeta(s)}{\zeta(s+1)}=\sum_{n\ge
1}\frac{\phi(n)}{n^{s+1}}, \label{equation11}$$ since $-\frac{Z'(s)}{Z(s)}=\sum_{n\ge 1}\frac{b(n)}{n^s}$. This replaces the similar relation from the Riemann zeta function where $-\frac{\zeta'(s)}{\zeta(s)}=\sum_{n\ge 1}\frac{\Lambda(n)}{n^s}$.
In the studies of $1/f$ noise, the fast Fourier transform (FFT)plays a central role. But the FFT refers to the fast calculation of the discrete Fourier transform (DFT) with a finite period $q=2^l$, $l$ a positive integer. In the DFT one starts with all $q^{\rm{th}}$ roots of the unity $\exp(2i\pi
p/q)$, $p=1\ldots q $ and the signal analysis of the arithmetical sequence $x(n)$ is performed by projecting onto the $nth$ powers (or characters of *Z*/q*Z*) with well known formulas.
The signal analysis based on the DFT is not well suited to aperiodic sequences with many resonances (naturally a resonance is a primitive root of the unity: $(p,q)=1$), and the FFT may fail to discover the underlying structure in the spectrum. We recently introduced a new method based on the Ramanujan sums defined in (\[eq2\]) [@Planat4].
Mangoldt function is related to Möbius function thanks to the Ramanujan sums expansion found by Hardy [@Gadiyar] $$b(n)=\frac{\phi(n)}{n}\Lambda(n)=\sum_{q=1}^{\infty}\frac{\mu(q)}{\phi(q)}c_q(n).
\label{equab}$$ We call such a type of Fourier expansion a Ramanujan-Fourier transform (RFT). General formulas are given in our recent publication [@Planat4] and in the paper by Gadiyar [@Gadiyar]. This author also reports on a stimulating conjecture relating the autocorrelation function of $b(n)$ and the problem of pairs of prime numbers. In the special case (\[equab\]), it is clear that $\mu(q)/\phi(q)$ is the RFT of the modified Mangoldt sequence $b(n)$.
Using Ramanujan-Fourier analysis the $1/f^{2\alpha}$ power spectrum gets replaced by a new signature shown on Fig. 2, not very different of $\mu(q)/\phi(q)$ (up to a scaling factor).
The hyperbolic geometry of phase noise and $1/f$ frequency noise
----------------------------------------------------------------
The whole theory can be justified by studying the noise in the half plane $\it{H}=\{z=\nu+iy,~i^2=-1\footnote{The imaginary
symbol $i$ should not be confused with the index $i$ in integers
$p_i$, $q_i$ and in related integers.},~y>0\}$ of coordinates $\nu=\frac{f}{f_0}$ and $ y=\frac{f_B}{f_c}>0$ and by introducing the modular transformations $$z \rightarrow \gamma(z)=z'=\frac{p_i z+p'_i}{q_i z+q'_i},
~~p_iq'_i-p'_iq_i=1. \label{equation13}$$ The set of images of the filtering line $z=\nu+i$ under all modular transformations can be written as $$|z'-(\frac{p_i}{q_i} + \frac{i}{2q_i^2})|=\frac{1}{2q_i^2}.
\label{Ford}$$ Equation (\[Ford\]) defines Ford circles (see Fig. 3) centered at points $z=\frac{p_i}{q_i}+\frac{i}{2q_i^2}$ with radius $\frac{1}{2q_i^2}$ [@Rademacher]. To each $\frac{p_i}{q_i}$ a Ford circle in the upper half plane can be attached, which is tangent to the real axis at $\nu=\frac{p_i}{q_i}$. Ford circles never intersect: they are tangent to each other if and only if they belong to fractions which are adjacent in the Farey sequence $\frac{0}{1}<\cdots\frac{p_1}{q_1}<\frac{p_1+p_2}{q_1+q_2}<\frac{q_1}{q_2}\cdots<\frac{1}{1}$[@Rademacher].
The half plane $\it{H}$ is the model of Poincaré hyperbolic geometry. A basic fact about the modular transformations (\[equation13\]) is that they form a discontinuous group $\Gamma\simeq SL(2,\it{Z})/\{\pm1\}$, which is called the modular group. The action of $\Gamma$ on the half-plane $\it{H}$ looks like the one generated by two independent linear translations on the Euclidean plane, which is equivalent to a tesselation the complex plane $\it{C}$ with congruent parallelograms. One introduces the fundamental domain of $\Gamma$ (or modular surface) $\it{F}=\{z \in \it{H}:~|z|\ge 1,~|\nu|\le \frac{1}{2}\}$, and the family of domains $\{\gamma(\it{F}),\gamma \in \Gamma\}$ induces a tesselation of $\it{H}$ [@Gutzwiller].
It can be shown [@Planat3],[@Gutzwiller] that the noise amplitude is a particular type of solution of the eigenvalue problem with the non-Euclidean Laplacian $\Delta=y^2(\frac{\partial^2}{\partial
\nu^2}+\frac{\partial^2}{\partial y^2})$. The solution corresponds to the scattering of waves in the fundamental domain $F$. It can be approximated as a superposition of three contributions. The first one is an horizontal wave and is of the power law form $y^s$, the second one is also horizontal wave of the form $S(s)
y^{1-s}$ and corresponds to a reflected wave with a scattering coefficient of modulus $|S(s)|=1$, whereas the remaining part $T(y,\nu)$ is a complex superposition of waves depending of $y$ and the harmonics of $\exp(2i\pi\nu)$, but going to zero for $y
\rightarrow \infty$. Extracting the smooth part in $S(s)$ one is left with a random factor which is precisely equal to the function $Z(2s-1)$ defined in (\[equation11\]) as the quotient of two Riemann zeta functions at $2s-1$ and $2s$, respectively. An interesting case is when $s$ is on the critical line, i.e. $s=\frac{1}{2}+ik$ in which case the superposition $T(y,\nu)$ vanishes and the reflexion coefficient is $$S(k)= \exp[2i\theta(k)],~~~\mbox{with}~\theta'(k)=\frac{d\ln
S(s)}{ds}~~\mbox{at}~s=\frac{1}{2}+ik.
\label{hyperbphase}$$ The scattering of waves from the modular surface is thus similar to the phase-locking model plotted in the set of equations (\[equation1\])-(\[equation11\]). It explains the relationship between the hyperbolic phase and the $1/f$ noise found in the counting function $\theta'(k)$. The phase factor $\theta(k)$ is represented in Fig. 4.
Quantum phase-locking
=====================
The quantum phase operators
---------------------------
Going back to the quantum definition of phase states announced in the introduction one calculates the projection operator over the subset of phase-locked quantum states $|\theta' _p\rangle$ as
$$P_q^{\rm{lock}}=\sum_p|\theta'_p\rangle
\langle\theta'_p|=\frac{1}{q}\sum_{n,l} c_q(n-l)|n\rangle \langle
l|,
\label{eq3}$$
where the range of values of $n,l$ is from $0$ to $\phi(q)$. Thus the matrix elements of the projection are $q\langle n|P_q|l\rangle
= c_q(n-l)$. This sheds light on the equivalence between cyclotomic lattices of algebraic number theory and the quantum theory of phase-locked states.
The projection operator over the subset of pairs of phase-locked quantum states $|\theta' _p\rangle$ is calculated as $$P_q^{\rm{pairs}}=\sum_{p,\bar{p}}|\theta'_p\rangle
\langle\theta'_{\bar{p}}|=\frac{1}{q}\sum_{n,l} k_q(n,l)|n\rangle
\langle l|,
\label{eq3bis}$$ where the notation $p,\bar{p}$ means that the summation is applied to such pairs of states satisfying (\[phaselocked\]). The matrix elements of the projection are $q\langle
n|P_q^{\rm{pairs}}|l\rangle = k_q(n,l)$, which are in the form of so-called Kloosterman sums [@Terras99] $$k_q(n,l)=\sum_{p,\bar{p}}\exp[\frac{2i\pi}{q}(pn-\bar{p}l)].
\label{Kloos}$$ Kloosterman sums $k_q(n,l)$ as well as Ramanujan sums $c_q(n-l)$ are relative integers. They are given below for the Hilbert dimensions $q=5 (\phi(5)=4)$ and $q=6(\phi(6)=2)$. $$\begin{aligned}
&q=5:~~ c_5=\left[
\begin{array}{cccc}
~~4 & -1&-1&-1 \\
-1 & ~~4&-1&-1\\
-1&-1&~~4&-1\\
-1&-1&-1&~~4
\end{array}
\right],~~~~k_5=\left[
\begin{array}{cccc}
-1 & -1&-1&~~4 \\
-1 & ~~4&-1&-1\\
-1&-1&~~4&-1\\
~~4&-1&-1&-1
\end{array}
\right],\nonumber\\
&q=6:~~c_6=\left[
\begin{array}{cc}
2&1\\
1&2
\end{array}
\right],~~k_6=\left[
\begin{array}{cc}
-1&~~2\\
~~-2&1
\end{array}
\right].\nonumber\end{aligned}$$ One de$\rm{fi}$nes the quantum phase-locking operator as $$\Theta_q^{\rm{lock}}=\sum_p \theta_p |\theta'_p\rangle
\langle\theta'_p|=\pi
P_q^{\rm{lock}}~~\rm{with}~\theta_p=2\pi\frac{p}{q}. \label{eq4}$$ The Pegg and Barnett operator [@Pegg] is obtained by removing the coprimality condition. It is Hermitian with eigenvalues $\theta_p$. Using the number operator $N_q=\sum_{n=0}^{q-1} n|n
\rangle \langle n|$ a generalization of Dirac’s commutator $[\Theta_q,N_q]=-i$ has been obtained.
Similarly one de$\rm{fi}$nes the quantum phase operator for Kloosterman pairs as $$\Theta_q^{\rm{pairs}}=\sum_{p,\bar{p}} \theta_p |\theta'_p\rangle
\langle\theta'_{\bar{p}}|=\pi
P_q^{\rm{pairs}}~~\rm{with}~\theta_p=2\pi\frac{p}{q}.
\label{eq4bis}$$ The phase number commutator for phase-locked states calculated from (\[eq4\]) is $$C_q^{\rm{lock}}=[\Theta_q^{\rm{lock}},N_q]=\frac{\pi}{q}\sum_{n,l}(l-n)c_q(n-l)|n\rangle\langle
l|, \label{eq5}$$ with antisymmetric matrix elements $\langle
l|C_q^{\rm{lock}}|n\rangle=\frac{\pi}{q}(l-n)c_q(n-l)$.
For pairs of phase-locked states an antisymmetric commutator $C_q^{\rm{pairs}}$ similar to (\[eq5\]) is obtained with $k_q(n,l)$ in place of $c_q(n-l)$.
Phase expectation value and variance
------------------------------------
The $\rm{fi}$nite quantum mechanical rules are encoded in the expectation values of the phase operator and phase variance.
Rephrasing Pegg and Barnett, let us consider a pure phase state $|f \rangle = \sum_{n=0}^{q-1} u_n |n \rangle$ having $u_n$ of the form $$u_n=(1/\sqrt{q})\exp(i n\beta), \label{pure}$$ where $\beta$ is a real phase parameter. One de$\rm{fi}$nes the phase probability distribution $\langle\theta'_p|f\rangle^2$, the phase expectation value $\langle
\Theta_q^{\rm{lock}}\rangle=\sum_{p}\theta_p
\langle\theta'_p|f\rangle^2$, and the phase variance $(\Delta
\Theta_q ^2)^{\rm{lock}}=\sum_p (\theta_p-\langle
\Theta_q^{\rm{lock}}\rangle)^2 \langle\theta'_p|f\rangle^2$. One gets $$\begin{aligned}
&\langle\Theta_q^{\rm{lock}}\rangle=\frac{\pi}{q^2}\sum_{n,l}
c_q(l-n) \exp[i\beta(n-l)],\label{expec2}\\
&(\Delta \Theta_q^2)^{\rm{lock}}=4 \langle
\tilde{\Theta}_q^{\rm{lock}}\rangle
+\frac{\langle\Theta_q\rangle^2}{\pi}(\langle\Theta_q\rangle-2\pi),\end{aligned}$$ with the modi$\rm{fi}$ed expectation value $\langle\tilde{\Theta}_q^{\rm{lock}}\rangle=\frac{\pi}{q^2}\sum_{n,l}
\tilde{c}_q(l-n) \exp[i\beta(n-l)]$, and the modi$\rm{fi}$ed Ramanujan sums $\tilde{c}_q(n)=\sum_p (p/q)^2 \exp(2i\pi
m\frac{p}{q})$.
Fig. 5 illustrates the phase expectation value versus the dimension $q$ for two different values of the phase parameter $\beta$. For $\beta=1$ they are peaks at dimensions $q=p^r$ which are powers of a prime number $p$. The most significant peaks are fitted by the function $\pi\Lambda(q)/\ln q$, where $\Lambda(q)$ is the Mangoldt function introduced in (\[Riemann\]) of Sect.2. This observation provides the link between the arithmetical hyperbolic viewpoint and the quantum one. A deepest explanation based on the relation with quantum statistical mechanics and the work of Bost and Connes can be found in [@PlanatKMS]. For $\beta=0$ the peaks are smoothed out due to the averaging over the Ramanujan sums matrix. Fig. 6 shows the phase expectation value versus the phase parameter $\beta$. For the case of the prime number $q=13$, the mean value is high with absorption like lines at isolated values of $\beta$. For the case of the dimension $q=15$ which is not a prime power the phase expectation is much lower in value and much more random.
Fig. 7 illustrates the phase variance versus the dimension $q$. Again the case $\beta=1$ leads to peaks at prime powers. Like the expectation value in Fig. 5, it is thus reminiscent of the Mangoldt function. Mangoldt function $\Lambda(n)$ is de$\rm{fi}$ned as $\ln p$ if n is the power of a prime number $p$ and $0$ otherwise. It arises in the frame of prime number theory [@Planat1] from the logarithmic derivative of the Riemann zeta function $\zeta(s)$ as $-\frac{\zeta'(s)}{\zeta(s)}=\sum_{n=0}^{\infty}\frac{\Lambda(n)}{n^s}$. Its average value oscillates about $1$ with an error term which is explicitely related to the positions of zeros of $\zeta(s)$ on the critical line $s=\frac{1}{2}$. The error term shows a power spectral density close to that of $1/f$ noise [@Planat1]. It is stimulating to recover results reminding prime number theory in the new context of quantum phase-locking.
Finally, the phase variance is considerably smoothed out for $\beta=\pi$ and is much lower than the classical limit $\pi^2/3$. The parameter $\beta$ can thus be interpreted as a squeezing parameter since it allows to de$\rm{fi}$ne quantum phase-locked states having weak phase variance for a whole range of dimensions.
Towards discrete phase entanglement
-----------------------------------
The expectation value of quantum phase states can be rewritten using the projection operator of individual phase states $\pi_p=|\theta'_p\rangle
\langle \theta'_p|$ as follows $$\langle \Theta_q^{\rm{lock}}\rangle=\sum_p \theta_p \langle
f|\theta'_p \rangle \langle \theta'_p |f \rangle=\sum_p \theta_p
\langle f|\pi_p|f \rangle.$$ This suggests a de$\rm{fi}$nition of expectation values for pairs based on the product $\pi_p \pi_{\bar{p}}$ as follows $$\langle \Theta_q^{\rm{pairs}}\rangle=\sum_{p,\bar{p}} \theta_p
\langle f|\pi_p \pi_{\bar{p}}|f \rangle . \label{pairs}$$ It is inspired by the quantum calculation of correlations in Bell’s theorem [@Scully]. Using pure phase states as in (\[pure\]) we get $$\langle \Theta_q^{\rm{pairs}}\rangle=\frac{2
\pi}{q^2}\sum_{n,l}\tilde{k_q}(n,l) \exp[i \beta(n-l)],$$ where we introduced generalized Kloosterman sums\
$\tilde{k}_q(n,l)=\sum_{p,\bar{p}} p \exp[\frac{2 i
\pi}{q}(p-\bar{p})(l-n)]$. These sums are in general complex numbers (and are not Gaussian integers). The expectation value is real as expected. In Fig. 8 it is represented versus the dimension $q$ for two different values, $\beta=0$ and $\beta=1$, respectively. Note that the pair correlation (\[pairs\]) is very strongly dependent on $q$ and becomes quite huge at some values.
This result suggests that a detailed study of Bell’s type inequalities based on quantum phase-locked states, and their relationship to the properties of numbers, should be undertaken. Calculations involving fully entangled states $$|f\rangle= \frac{1}{q}\sum_{p,\bar{p}} |\theta_p,1\rangle \otimes
|\theta_{\bar{p}},2\rangle,$$ have to be carried out. This is left for future work.
The discrete phase: cycles in $Z/qZ$
------------------------------------
There is a scalar viewpoint for the above approach, which emphasizes well the intricate order of the group $Z/qZ$, the group of integers modulo $q$. One asks the question: what is the largest cycle in that group. For that purpose one looks at the primitive roots, which are the solutions $g$ of the equation $$g^{\alpha} \equiv 1(\textrm{mod}~q),$$ such that the equation is wrong for any $1 \le \alpha < q-1$ and true only for $\alpha=q-1$. If q=p, a prime number, and $p=7$, the largest period is thus $\phi(p)$=p-1=6, and the cycle is as given in Table I. If $q=2$, $4$, $q=p^r$, a power a prime number $>2$, or $q=2p^r$, twice the power of a prime number $>2$, then a primitive root exists, and the largest cycle in the group is $\phi(q)$. For example $g=2$ and $q=3^2$ leads to the period $\phi(9)=6<q-1=8$, as shown in Table II.
[$\alpha$]{} [1]{} [2]{} [3]{} [4]{} [5]{} [6]{} [7]{} [8]{}
------------------ ------- ------- ------- ------- ------- ------- ------- -------
[$3^{\alpha}$]{} [3]{} [2]{} [6]{} [4]{} [5]{} [1]{} [3]{} [2]{}
: $(Z/7Z)^*$ is a cyclic group of order $\phi(7)=6$.
[$\alpha$]{} [1]{} [2]{} [3]{} [4]{} [5]{} [6]{} [7]{} [8]{}
------------------ ------- ------- ------- ------- ------- ------- ------- -------
[$2^{\alpha}$]{} [2]{} [4]{} [8]{} [7]{} [5]{} [1]{} [2]{} [4]{}
: $(Z/3^2 Z)^*$, is a cyclic group of order $\phi(9)=6$.
Otherwise there is no primitive root. The period of the largest cycle in $Z/qZ$ can still be calculated and is called the Carmichael Lambda function $\lambda(q)$. It is shown in Table III for the case $g=3$ and $q=8$. It is $\lambda(8)=2<\phi(8)=4<8-1=7$.
[$\alpha$]{} [1]{} [2]{} [3]{} [4]{} [5]{} [6]{} [7]{} [8]{}
------------------ ------- ------- ------- ------- ------- ------- ------- -------
[$2^{\alpha}$]{} [3]{} [1]{} [3]{} [1]{} [3]{} [1]{} [3]{} [1]{}
: $(Z/8Z)^*$ has a largest cyclic group of order $\lambda(8)=2$.
Fig. 9 shows the properly normalized period for the cycles in Z/qZ. Its fractal character can be appreciated by looking at the corresponding power spectral density shown in Fig. 10. It has the form of a $1/f^{\alpha}$ noise, with $\alpha=0.70$. For a more refined link between primitive roots $g$, cyclotomy and Ramanujan sums see also [@Moree].
Conclusion
==========
In conclusion, we explained how useful could be the concepts of prime number theory in explaining various features of phase-locking at the classical and quantum level. In the classical realm we reminded the hyperbolic geometry of phase, which occurs when one accounts for all harmonics in the mixing and low-pass filtering process, how $1/f$ frequency noise is produced and how it is related to Mangoldt function, and thus to the critical zeros of Riemann zeta function. Then we studied several properties resulting from introducing phase-locking in Pegg-Barnett quantum phase formalism. The idea of quantum teleportation was initially formulated by Bennett et al in $\rm{fi}$nite-dimensional Hilbert space [@Bennett], but, yet independently of this, one can conjecture that cyclotomic aspects in phase-locking could play an important role in many fundamental tests of quantum mechanics related to quantum entanglement. Munro and Milburn [@Munro] already conjectured that the best way to see the quantum nature of correlations in entangled states is through the measurement of the observable canonically conjugate to photon number, i.e. the quantum phase. In their paper dealing with the Greenberger-Horne-Zeilinger quantum correlations, they presented a homodyne scheme requiring discrete phase measurement. We expect that the interplay between quantum mechanics and number theory will appear repetitively in the coming attempts to manipulate quantum information [@Wootters].
Acknowledgments {#acknowledgments .unnumbered}
===============
The third part of this paper was presented at the International Conference on Squeezed States and Uncertainty Relations in Puebla, in June 2003. The authors acknowledge Hector Moya for his invitation.
References {#references .unnumbered}
==========
[99]{}
M. Planat 2001 *Fluc. and Noise Lett.* **1** R65
M. Planat and E. Henry 2002 *Appl. Phys. Lett.* **80** 2413
Planat M and Rosu H 2003 *Phys. Lett. A* **315** 1\
(Planat M and Rosu H *Preprint* quant-ph/0304101)
Planat M 2002 Modular functions and Ramanujan sums for the analysis of $1/f$ noise in electronic circuits *Preprint* hep-th/0209243
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Hardy G H and Wright E M 1979 *An Introduction to the Theory of Numbers* (Oxford Press, Oxford) p 237
Planat M, Rosu H and Perrine S 2002 *Phys. Rev. E* **66** 56128\
(Planat M, Rosu H and Perrine S 2002 *Preprint* math-ph/0209002)
Yamoto T, Kano H and Takagi K 2001 *Noise in Physical Systems and $1/f$ fluctuations, ICNF* edited by Gijs Bosman (World Scientific: New Jersey) p 503–506
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[^1]: Some errors or misunderstandings are present in that earlier report. The summation in (3),(5),(7) and (9) should be (this is implicit) from $0$ to $\phi(q)$. The expectation value $\langle \theta_q^{\rm{lock}}\rangle$ in (8) should be squared. There are also slight changes in the plots.
|
---
abstract: 'As modern engineering systems grow in complexity, [attitudes toward]{} a modular design approach become increasingly more favorable. A key challenge to a modular design approach is the [certification]{} of robust stability under uncertainties in the rest of the network. In this paper, we consider the problem of identifying the parametric region, which guarantees stability of the connected module in the robust sense under uncertainties. We derive the conditions under which the robust stability of the connected module is guaranteed for some values of the design parameters, and present a sum-of-squares (SOS) optimization-based algorithm to identify such a parametric region for polynomial systems. Using the example of an inverter-based microgrid, we show how this parametric region [changes]{} with variations in the level of uncertainties in the network.'
author:
-
bibliography:
- 'IEEEabrv.bib'
- 'references.bib'
- 'RefMGStability.bib'
---
Introduction
============
With the growing complexity of modern engineering systems, [attitudes toward]{} reconfigurability and modular design approaches are gaining popularity. Plug-and-play design approaches have drawn attention [for use]{} in cyber-physical networks, power grids, biological networks, and process control systems [@baheti2011cyber; @farhangi2010path; @huang2011future; @litcofsky2012iterative; @bendtsen2013plug]. In the context of microgrids, and power systems in general, the plug-and-play design approach is particularly attractive because of the involvement of various stakeholders (not all resources/equipment on the network are owned by the same utility). A hierarchical design is often preferred, where a network-level assessment of the operational conditions sets certain interconnection guidelines (from the dynamic security and economic considerations) to which individual resource owners (or resource aggregators) adhere when plugging in their resource to the network [@nehrir2011review; @planas2013general; @lasseter2011smart]. As such, [a key challenge for a successful plug-and-play operation]{} is the identification of the design parameter space that certifies robust stability under various operational conditions of the network.
Unlike bulk power systems, which have adequate rotational inertia to naturally stabilize fluctuations in the network, the dynamic security of low-inertia microgrids needs to be specifically ensured via design [@Xu:2018; @mashayekh2018security]. Identification of droop-coefficients for stability certification of inverter-based microgrids have been investigated in recent works [@Schiffer:2014; @Vorobev:2018]. A centralized approach of identifying the droop-coefficients for a lossless microgrid was adopted in [@Schiffer:2014], while conditions on droop-coefficients were derived in a distributed approach for small-signal stability in [@Vorobev:2018]. However, low-to-medium voltage microgrids typically have significant line resistance-to-reactance ratios, and often operate in a nonlinear regime due to fluctuations from renewable generation, rendering the aforementioned approaches inapplicable.
Lyapunov function methods have been widely used in the context of nonlinear systems stability certification [@Lyapunov:1892; @Khalil:1996]. Extension of the theory to robust stability problems under uncertainties as well as parametric stability analysis have been proposed [@blanchini1999set; @Gahinet:1996]. More recent works have used advanced computational techniques, such as sum-of-squares (SOS) algorithms, for parametric stability analysis using Lyapunov functions [@Anderson:2015; @Wloszek:2003; @Parrilo:2000; @Tan:2006; @Anghel:2013]. Lyapunov-based methods have been applied to robust stability analysis and control problems in power grids [@vu2017framework; @wang1998robust]. Chebyshev minimax formulation has been used for identifying the parametric stability region for linear systems (with Lur’e-type nonlinearity) [@siljak1989parameter]. The construction of the design parameter space that ensures robust stability of nonlinear networks, though, still remains a challenge.
The main contributions of this article are - 1) the theoretical construction of robust stability certificates in the design parameter space for nonlinear systems, under exogenous time-varying but bounded uncertainties; and 2) an algorithmic approach to identifying the largest robust stability region in the parametric space for polynomial networks. Numerical illustrations are provided in the context of identifying droop-coefficient values for robustly stable plug-and-play design of inverter-based microgrids. The rest of this article is structured as follows: Section\[S:prelim\] provides a background of the relevant theoretical and computational methods; Section\[S:problem\] presents a description of the microgrid example and the problem formulation; Sections\[S:theory\] and \[S:algo\] describe the theoretical and algorithmic approach; while a numerical example is presented in Section\[S:example\]. We conclude this article in Section\[S:concl\]. Throughout the text, $\left|\,\cdot\,\right|$ will be used to denote both the $\mathcal{L}_2$-norm of a vector and the absolute value of a scalar; while $\nabla_x$ denotes the gradient (of a function) with respect to $x$.
Preliminaries {#S:prelim}
=============
Stability Analysis: Lyapunov Functions
--------------------------------------
Consider a nonlinear system of the form: $$\begin{aligned}
\label{E:f}
\mathcal{S}:\quad\dot{x} = f(x)\,,\quad x\in\mathcal{X}\subseteq\mathbb{R}^n\end{aligned}$$ [The equilbrium point of interest is shifted to the origin ($0\in\mathcal{X}$) and $f$ is assumed to be locally Lipschitz in $\mathcal{X}$]{}. The equilibrium at origin is said to be locally asymptotically stable if 1) for every $\nu>0$ there exists an $\epsilon>0$ such that $|x(t)|\leq\nu\,\forall t\geq 0$ for every $|x(0)|\leq\epsilon$, and 2) $\lim_{t\rightarrow\infty}\left\vert x(t)\right\vert =0\,$ for every $x(0)$ in $\mathcal{X}$. Lyapunov’s stability conditions state[@Lyapunov:1892; @Khalil:1996] :
\[T:Lyap\] Existence of a continuously differentiable radially unbounded positive definite function $\Psi\!:\!\mathcal{X}\!\rightarrow\!\mathbb{R}_{\geq 0}$ (Lyapunov function) [where $\nabla_x{\Psi}^T\!f(x)$ is negative definite in $\mathcal{X}$ guarantees asymptotic]{} stability of the origin.
Parametric Lyapunov Analysis
----------------------------
Consider a dynamical system in a parametric form:
\[E:flam\] $$\begin{aligned}
\dot{x} &= f(x,\lambda)\,,\,\quad x\in\mathcal{X}\,,\,\lambda\in\Lambda\\
\text{where}\,~\mathcal{X}&:=\lbrace x\left| \,a_i(x)\geq 0\,,\,i\in\lbrace 1,\dots,p\rbrace\right.\rbrace.\end{aligned}$$
[$\lambda\in\mathbb{R}^l$ is an $l$-dimensional vector of the design parameters and $f$ is assumed to be locally Lipschitz continuous in $\mathcal{X}$.]{}
Assume that over the range of possible values of the parameters, the equilibrium point of interest always remains at the origin, i.e., $f(0,\lambda) = 0~\forall \lambda\in\Lambda$. Stability of such systems can be analyzed in a similar treatment to that of the *absolute stability* problem [@Khalil:1996; @Anderson:2015]. We argue that the origin of the parametric system is locally asymptotically stable in $\mathcal{X}$ if there exists a continuously differentiable parametric Lyapunov function $\Psi:\mathcal{X}\times\Lambda\rightarrow\mathbb{R}_{\geq 0}$ satisfying
\[E:Vun\] $$\begin{aligned}
\Psi(x,\lambda)&\geq \phi_1(x)\quad \forall (x,\lambda)\in\mathcal{X}\times\Lambda\\
\nabla_x \Psi^Tf(x,\lambda)&\leq-\phi_2(x)\quad\forall (x,\lambda)\in\mathcal{X}\times\Lambda\end{aligned}$$
for some positive definite functions $\phi_1(\cdot)$ and $\phi_2(\cdot)$. The conditions can be extended to the situations when the equilibrium point depends on the values of the parameter. For example, if the equilibrium point of interest is an explicit function of the parameter, $x_0(\lambda)$, then the above argument holds after shifting of the state variables $\tilde{x}=x-x_0(\lambda)$.
Sum-of-Squares Optimization
---------------------------
Relatively recent studies have explored how SOS-based methods can be utilized to find Lyapunov functions by restricting the search space to SOS polynomials [@Wloszek:2003; @Parrilo:2000; @Tan:2006; @Anghel:2013]. Let us denote by $\mathbb{R}\left[x\right]$ the ring of all polynomials in $x\in\mathbb{R}^n$. A multivariate polynomial $p\in\mathbb{R}\left[x\right],~x\in\mathbb{R}^n$, is an SOS if there exist some polynomial functions $h_i(x), i = 1\ldots s$ such that $p(x) = \sum_{i=1}^s h_i^2(x)$. We denote the ring of all SOS polynomials in $x$ by $\Sigma[x]$. Whether or not a given polynomial is an SOS is a semi-definite problem which can be solved with SOSTOOLS, a MATLAB$^\text{\textregistered}$ toolbox [@sostools13], along with a semi-definite programming solver such as SeDuMi [@Sturm:1999]. An important result from algebraic geometry, called Putinar’s Positivstellensatz theorem [@Putinar:1993; @Lasserre:2009], helps in translating conditions such as in into SOS feasibility problems.
\[T:Putinar\] Let $\mathcal{K}\!\!=\! \left\lbrace x\in\mathbb{R}^n\left\vert\, k_1(x) \geq 0\,, \dots , k_m(x)\geq 0\!\right.\right\rbrace$ be a compact set, where $k_j$ are polynomials. Define $k_0=1\,.$ Suppose there exists a $\mu\!\in\! \left\lbrace {\sum}_{j=0}^m\sigma_jk_j \left\vert\, \sigma_j \!\in\! \Sigma[x]\,\forall j \right. \right\rbrace$ such that $\left\lbrace \left. x\in\mathbb{R}^n \right\vert\, \mu(x)\geq 0 \right\rbrace$ is compact. Then, $$\begin{aligned}
p(x)\!>\!0~\forall x\!\in\!\mathcal{K}
\!\implies\! p \!\in\! \left\lbrace {\sum}_{j=0}^m\sigma_jk_j \left\vert\, \sigma_j \!\in\! \Sigma[x]\,\forall j \right. \right\rbrace\!.\end{aligned}$$
Using Theorem\[T:Putinar\], we can translate the problem of checking that $p\!>\!0$ on $\mathcal{K}$ into an SOS feasibility problem where we seek the SOS polynomials $\sigma_0\,,\,\sigma_j\,\forall j$ such that $p\!-\!\sum_j\sigma_j k_j$ is [SOS. Note that any]{} equality constraint $k_i(x)\!=\!0$ can be expressed as two inequalities $k_i(x)\!\geq 0$ and $k_i(x)\!\leq\! 0$. In many cases, especially for the $k_i\,\forall i$ used throughout this work, a $\mu$ satisfying the conditions in Theorem\[T:Putinar\] is guaranteed to exist (see [@Lasserre:2009]), and need not be searched for.
Problem Description {#S:problem}
===================
Motivational Example: Microgrids
--------------------------------
Design of a networked microgrid involves solving an optimization problem that ensures operational reliability (e.g., transient stability) while achieving certain economic goals [@Barnes:2017]. Typically this translates to identifying the largest region in the space of design parameters that certify stability of the system under a set of uncertainties. Consider the case of droop-controlled inverters [@Coelho:2002; @Schiffer:2014]:
\[E:droop\] $$\begin{aligned}
\dot{\theta}_i & = \omega_i\,,\\
\tau_i\dot{\omega}_i & = -\omega_i + \lambda_i^p \left(P_i^d-P_i\right)\\
\tau_i\dot{v}_i & = v_i^d-v_i + \lambda_i^q \left(Q_i^d-Q_i\right)\end{aligned}$$
where $\lambda_i^p>0$ and $\lambda_i^q>0$ are the droop-coefficients associated with the active power vs. frequency and the reactive power vs. voltage droop curves, respectively; $\tau_i$ is the time-constant of a low-pass filter used for the active and reactive power measurements; $\theta_i\,,\,\omega_i$ and $v_i$ are, respectively, the phase angle, frequency, and voltage magnitude; $v^d_i,\,P_i^d$ and $Q^d_i$ are the nominal values of the voltage magnitude, active power, and reactive power, respectively. $P_i$ and $Q_i$ are, respectively, the active and reactive power injected into the network, related to the neighboring bus voltage phase angle and magnitude as:
\[E:PQ\] $$\begin{aligned}
P_i &= v_i{\sum}_{k\in\mathcal{N}_i} v_k\left(G_{i,k}\cos\theta_{i,k} + B_{i,k}\sin\theta_{i,k}\right)\\
Q_i &= v_i{\sum}_{k\in\mathcal{N}_i} v_k\left(G_{i,k}\sin\theta_{i,k} - B_{i,k}\cos\theta_{i,k}\right)\end{aligned}$$
where $\theta_{i,k}=\theta_i-\theta_k$, and $\mathcal{N}_i$ is the set of neighbor nodes; and $G_{i,k}$ and $B_{i,k}$ are respectively the transfer conductance and susceptance values of the line connecting the nodes $i$ and $k$. [Considering the droop-coefficients as the design parameters, the goal of this work is the algorithmic identification of the design space that ensures robust stability of the microgrid. Note that the particular choice of droop-coefficients as design parameters is for illustrative purpose only, while the proposed algorithm is generalizable to other choices of design parameters (such as line parameters and dispatched power set-points).]{}
The nominal (or desired) equilibrium is attained when $$\begin{aligned}
\forall i:\quad P_i=P_i^d,\,Q_i=Q_i^d,\,\omega_i=0,\,v_i=v_i^d\,.\end{aligned}$$ In a plug-and-play [operation]{}, it is important that the design parameters are chosen to ensure robust stability of the (possibly) *time-varying* equilibrium point of the connected inverter under bounded uncertainties in the (rest of the) network. Moreover, an additional constraint that needs to be enforced through the choice of design parameters is that the equilibrium point under uncertainties should stay *close* to the nominally desired equilibrium of $(\omega_i,v_i)=(0,v^d_i)$. This ensures that even under uncertainties, the operating conditions remain *acceptable*.
[After introducing]{} the following variables: $$\begin{aligned}
\label{E:droop_disturbance}
\delta_{1,i,k}:=v_k\cos\theta_{i,k}\,\text{ and }\,\delta_{2,i,k} := v_k\sin\theta_{i,k}\,,\end{aligned}$$ the inverter dynamics - can be reformulated in the polynomial form as follows:
\[E:droop\_poly\] $$\begin{aligned}
\!\!\!\tau_i\dot{\omega}_i & = -\omega_i \!+\! \lambda_i^p \left[P_i^d\!-\!v_i\!\!\!\sum_{k\in\mathcal{N}_i} \!(G_{i,k}\delta_{1,i,k} \!+\! B_{i,k}\delta_{2,i,k})\right]\!\!\\
\!\!\!\tau_i\dot{v}_i & = v_i^d\!-\!v_i \!+\! \lambda_i^q \left[Q_i^d\!-\!v_i\!\!\!\sum_{k\in\mathcal{N}_i} \!(G_{i,k}\delta_{2,i,k} \!-\! B_{i,k}\delta_{1,i,k})\right]\!\!\end{aligned}$$
In the reformulation, the phase angle dynamics are dropped, since the phase angle differences (represented in $\delta_{1,i,k}$ and $\delta_{2,i,k}$) are sufficient to model the power flow across networks.
Problem Formulation
-------------------
Consider an uncertain polynomial dynamical system which is represented in a parametric form as follows:
\[E:flamd\] $$\begin{aligned}
\mathcal{S}[\lambda,\delta]:~\dot{x}(t) &= f(x(t),\lambda,\delta(t))\,,\,~\left\lbrace\begin{array}{rl}
x(t)\!\!&\!\!\in\mathcal{X}\,,\\
\lambda\!\!&\!\!\in\Lambda\,,\\
\delta(t)\!\!&\!\!\in\mathcal{D}
\end{array} \right.\\
\text{where}\,~\mathcal{X}&:=\lbrace x\left| \,a_i(x)\geq 0\,,\,i\in\lbrace 1,\dots,p\rbrace\right.\rbrace\,,\\
\mathcal{D}&:=\lbrace \delta\left| \,b_i(\delta)\geq 0\,,\,i\in\lbrace 1,\dots,q\rbrace\right.\rbrace\end{aligned}$$
where $\delta(t)\in\mathbb{R}^d$ denote a $d$-dimensional vector of uncertain and (possibly) time-varying exogenous parameters, which lie in a semi-algebraic domain $\mathcal{D}$; $f,\,a_i,\,b_i$ are polynomials. For notational simplicity, we will henceforth drop the time parameter $t$ from the argument of $x$ and $\delta$, whenever obvious. Without any loss of generality, we assume that $0\in\mathcal{D}$, and that $x=0$ is an equilibrium of the system when $\delta=0$, i.e., $$\begin{aligned}
\label{E:nom_eq}
f(0,\lambda,0) = 0\quad\lambda\in\Lambda\,.\end{aligned}$$ Moreover, when $\delta\neq 0$, the equilibrium point of interest, $x_0(\lambda,\delta)$, is defined uniquely in the domain $(x,\lambda,\delta)\in\mathcal{X}\times\Lambda\times\mathcal{D}$ by the relationship: $$\begin{aligned}
\label{E:pert_eq}
x_0(\lambda,\delta):=\lbrace x\in\mathcal{X}\,|\,f(x,\lambda,\delta) = 0\rbrace~\forall (\lambda,\delta)\in\Lambda\times\mathcal{D}.\end{aligned}$$
We assume the explicit functional form $x_0(\lambda,\delta)$ to be available. Future work will address the issues when this relationship is implicit. Also note that the condition holds when droop coefficients are chosen as the design parameters values. Future efforts will consider relaxing that condition.
The problem we are interested in is identifying a set of possible values of the design parameter $\lambda$ that ensures the robust stability of the system under bounded uncertainties, i.e., find the set $\widehat{\Lambda}\subseteq{\Lambda}$ such that the following hold:
1. the equilibrium point of interest, $x_0(\lambda,\delta)$, remain within an *acceptable* region $\mathcal{X}_0\subseteq\mathcal{X}$ ($0\in\mathcal{X}_0$) for every uncertainty $\delta\in\mathcal{D}$ and for every design parameter $\lambda\in\widehat{\Lambda}$;
2. the *locally* asymptotic stability of the equilibrium point $x_0(\lambda,\delta)$ of the uncertain system $\mathcal{S}[\lambda,\delta]$ in is guaranteed for every $\delta\in\mathcal{D}$ and for every $\lambda\in\widehat{\Lambda}$.
Theoretical Construction {#S:theory}
========================
In this section, we discuss the theoretical development regarding robust stability of the connected module over some parameter range, under bounded uncertainties.
\[AS:closeness\] The system admits a unique equilibrium point $x_0(\lambda,\delta)$ inside the domain $\lbrace x|\,|x|\!\leq\!\Delta\rbrace\!\subset\!\mathcal{X}$, i.e., $$\begin{aligned}
(\lambda,\delta)\in\Lambda\times\mathcal{D}\implies\exists\, x_0(\lambda,\delta)\in\lbrace x|\,|x|\!\leq\!\Delta\rbrace\!\subset\!\mathcal{X}~\text{s.t.}~\eqref{E:pert_eq}\end{aligned}$$
Note that the value of $\Delta$ depends not only on the uncertainties, but also on the parameter values. Given some parameter value, $\Delta$ decreases as the uncertainty level goes down. On the other hand, given a range of uncertainties, we can choose the range of parameter values to lower $\Delta$.
\[AS:lyapunov\] The system admits a parametric Lyapunov function $\Psi(x,\lambda)$ satisfying the following: $$\begin{aligned}
\forall (x,\lambda,\delta)\in\mathcal{X}\!\times\!\Lambda\!\times\!\mathcal{D}:\quad \Psi(x,\lambda)\geq \phi_1(x\!-\!x_0(\lambda,\delta))\\
\nabla_x\Psi^T\!f(x,\lambda,\delta)\leq -\phi_2(x\!-\!x_0(\lambda,\delta))\end{aligned}$$ where the equilibrium of interest $x_0(\lambda,\delta)$ satisfies Assumption\[AS:closeness\]; and $\phi_{1,2}(\cdot)$ are positive definite functions.
Let us define
\[\] $$\begin{aligned}
\Gamma &:=\max\left\lbrace \gamma\,|\,|x|\leq \gamma\implies x\in\mathcal{X}\right\rbrace\\
\Gamma_\Psi(\lambda) &:=\max\left\lbrace \gamma\,|\,\Psi(x,\lambda)\leq \gamma\implies x\in\mathcal{X}\right\rbrace\end{aligned}$$
i.e., $\Gamma$ is the largest level-set of the $\mathcal{L}_2$-norm of the state $x$ contained within $\mathcal{X}$, while $\Gamma_\Psi(\lambda)$ is the maximum level-set of $\Psi(x,\lambda)$ contained within $\mathcal{X}$. Note that, $$\begin{aligned}
\Delta<\Gamma\,.\end{aligned}$$
\[P:bounded\] (Boundedness) Let us define the following:
\[E:bounds\] $$\begin{aligned}
\zeta^*(\lambda)&=\min\left\lbrace \zeta\,\left| \begin{array}{c} |x-x_0(\lambda,\delta)|\leq\Delta,\,\delta\in\mathcal{D}\\
\implies\Psi(x,\lambda)\leq\zeta\end{array}\right.\right\rbrace\\
\nu^*(\lambda)&=\min\left\lbrace \nu\,\left| \begin{array}{c} \Psi(x,\lambda)\leq\zeta^*,\,\delta\in\mathcal{D}\\
\implies |x-x_0(\lambda,\delta)|\leq\nu\end{array}\right.\right\rbrace.\end{aligned}$$
For sufficiently weak uncertainties satisfying $$\begin{aligned}
\Delta<\Gamma-\nu^*(\lambda)\,,\text{ and }\,\zeta^*(\lambda)<\Gamma_\Psi(\lambda)\,, \end{aligned}$$ there exists a $\xi>0$ for every $\nu\in[\nu^*(\lambda)+\Delta,\Gamma]$ such that $|x(0)|\leq\xi$ implies $|x(t)|\leq\nu$ for all $t\geq 0$.
From Assumption\[AS:lyapunov\], we have $$\begin{aligned}
\Psi(x(t),\lambda)-\Psi(x(0),\lambda)\leq-\int_0^t\phi_2(x(\tau)-x_0(\lambda,\delta))\,d\tau\end{aligned}$$ i.e., $\Psi(x,\lambda)$ is non-increasing in $x$ along the trajectories of the system . Note that when $x(0)=0$, $|x(0)-x_0(\lambda,\delta)|\leq\Delta$. From it follows immediately that $x(0)=0$ implies $\Psi(x(0),\lambda)\leq\zeta^*$ which implies $\Psi(x(t),\lambda)\leq\zeta^*$ for all $t\geq 0$. Applying again, we have $|x(t)-x_0(\lambda,\delta)|\leq\nu^*$, such that $$\begin{aligned}
\forall t\geq 0: ~|x(t)|\leq|x(t)-x_0(\lambda,\delta)|+|x_0(\lambda,\delta)|\leq\nu^*+\Delta\,.\end{aligned}$$ For sufficiently weak uncertainties satisfying $\Delta<\Gamma-\nu^*(\lambda)$, we have that $x(0)=0$ implies $|x(t)|<\Gamma$ for all $t\geq 0$.
Now, for every $\nu\in(\nu^*(\lambda)+\Delta,\Gamma]$, we have $$\begin{aligned}
|x(t)-x_0(\lambda,\delta)|\leq\nu-\Delta\implies |x(t)|\leq\nu\,.\end{aligned}$$ Moreover, because $\phi_1(\cdot)$ is a radially unbounded and positive definite function bounding $\Psi(x,\lambda)$ from below, for every such $\nu-\Delta>\nu^*$, we have a $\zeta>\zeta^*$ such that $$\begin{aligned}
\Psi(x(t),\lambda)\leq\zeta\implies |x(t)-x_0(\lambda,\delta)|\leq\nu-\Delta\,.\end{aligned}$$ Since $\Psi(x,\lambda)$ is non-increasing in $x$ along system trajectories, and since $\phi_1(\cdot)$ is positive definite, there exists a $\xi>0$ for every $\zeta>\zeta^*$ such that $$\begin{aligned}
|x(0)|\leq\xi&\implies\Psi(x(0),\lambda)\leq\zeta\\
&\implies\Psi(x(t),\lambda)\leq\zeta\implies|x(t)|\leq\nu\,.\end{aligned}$$ This completes the proof. [Fig.\[F:illustrate\] illustrates the different level-sets used in the derivation.]{}[ <1.5em - 1.5em plus0em minus0.5em height0.75em width0.5em depth0.25em]{}
![[An illustration of the different level-sets.]{}[]{data-label="F:illustrate"}](illustration.eps)
\[P:converge\](Convergence) Let us define the following[^1]: $$\begin{aligned}
\mu^*(\lambda):=\max\left\lbrace \mu\,\left| \begin{array}{c} |x-x_0(\lambda,\delta)|\leq\mu,\,\delta\in\mathcal{D}\\
\implies\Psi(x,\lambda)\leq\Gamma_\Psi(\lambda)\end{array}\right.\right\rbrace.\end{aligned}$$ For sufficiently weak uncertainties satisfying $\Delta<\mu^*(\lambda)/2$, there exists a finite time $T(\mu,\epsilon)$ for every $\mu\in[\Delta,\mu^*(\lambda)-\Delta]$ and $\epsilon\in(0,\Gamma-\Delta]$ such that $|x(t)|\leq\epsilon+\Delta$ for all $t\geq T(\mu,\epsilon)$ for every $|x(0)|\leq\mu$.
For every $\mu$ such that $|x(0)|\leq\mu$, we have $|x(0)-x_0(\lambda,\delta)|\leq\mu+\Delta$. Since $\phi_1(\cdot)$ is positive definite, there exists a $\rho^*$ such that $$\begin{aligned}
|x(0)-x_0(\lambda,\delta)|\leq\mu+\Delta\implies\Psi(x(0),\lambda)\leq\rho^*\,.\end{aligned}$$ For sufficiently weak uncertainties satisfying $\Delta<\mu^*(\lambda)/2$, we have $\rho^*\leq\Gamma_\Psi(\lambda)$ for every $\mu\in[\Delta,\mu^*(\lambda)-\Delta]$.
Since $\phi_1(\cdot)$ is radially unbounded and positive definite, there exists a $\rho_*\in(0,\rho^*)$ for every $\epsilon\in(0,\Gamma-\Delta]$ such that $$\begin{aligned}
\Psi(x(t),\lambda)\leq\rho_*&\implies |x(t)-x_0(\lambda,\delta)|\leq\epsilon\\
&\implies|x(t)|\leq\epsilon+\Delta\,.\end{aligned}$$ Let us define: $$\begin{aligned}
\kappa(\lambda):=\min\left\lbrace \phi_2(x-x_0(\lambda,\delta))\,|\,\Psi(x,\lambda)\in[\rho_*,\rho^*]\,,\,\delta\in\mathcal{D}\right\rbrace\!.\end{aligned}$$ Choosing $T(\mu,\epsilon)=(\rho^*-\rho_*)/\kappa$, we can show that: $$\begin{aligned}
\forall t\geq T(\mu,\epsilon):~\rho^*-\Psi(x(t),\lambda)&\geq\Psi(x(0),\lambda)-\Psi(x(t),\lambda)\\
&\geq \kappa\,t\geq \kappa\,T(\mu,\epsilon)\geq \rho^*-\rho_*\\
\implies\quad \Psi(x(t),\lambda)&\leq\rho_*\,.\end{aligned}$$ This completes the proof.[ <1.5em - 1.5em plus0em minus0.5em height0.75em width0.5em depth0.25em]{}
(Main Result) Suppose Assumptions\[AS:closeness\] & \[AS:lyapunov\] hold, and the uncertainties are sufficiently weak such that $$\begin{aligned}
\begin{array}{c}\Delta<\min\left(\Gamma-\nu^*(\lambda),\,\mu^*(\lambda)/2\right)\,,\\
\text{and }~\,\zeta^*(\lambda)<\Gamma_\Psi(\lambda)\,,\end{array}\end{aligned}$$ then the system $\mathcal{S}[\lambda,\delta]$ in satisfies the following boundedness and uniform asymptotic convergences properties: there exists a $\xi>0$ for every $\nu\in[\nu^*(\lambda)+\Delta,\Gamma]$ such that $|x(0)|\leq\xi$ implies $|x(t)|\leq\nu$ for all $t\geq 0$, and $$\begin{aligned}
\forall \mu\in[\Delta,\mu^*(\lambda)-\Delta]:~|x(0)|\leq\mu\implies\lim_{t\rightarrow\infty}|x(t)|\leq\Delta\,.\end{aligned}$$
Follows from Propositions\[P:bounded\] and \[P:converge\]. [ <1.5em - 1.5em plus0em minus0.5em height0.75em width0.5em depth0.25em]{}
Algorithmic Procedure {#S:algo}
=====================
In this section we present an algorithmic procedure to compute the largest parameter set with certified robust stability. Without any loss of generality, let us assume that $0\in\Lambda$,[^2] and that the origin is a locally asymptotically stable equilibrium point of the nominal (unperturbed) system $\mathcal{S}(0,0)$. In the rest of this article, we will restrict ourselves to the identification of the region of design parameter space in the form of $$\begin{aligned}
\label{E:lambda_hat}
\widehat{\Lambda}(\beta):=\left\lbrace \lambda\in\mathbb{R}^l\left|\,G\,\lambda\leq \beta\,h\right.\right\rbrace\,,\end{aligned}$$ where $\beta\geq 0$ is a scalar, $h=[h_i]$ is an $m$-dimensional vector of non-negative scalars, for some $m\geq 1$, i.e. $h_i\geq0\,\forall i\in\lbrace 1,2,\dots,m\rbrace$, and $G=[g_{ij}]$ is an $m\times l$ matrix. Note that $0\in\widehat{\Lambda}(0)$. Moreover, $$\begin{aligned}
\widehat{\Lambda}(\beta_1)\subseteq\widehat{\Lambda}(\beta_2)\quad\forall \beta_2\geq\beta_1\geq 0\,.\end{aligned}$$
We are interested in solving the following problem:
\[E:opt\] $$\begin{aligned}
\max_{\Psi(x,\lambda)}&\quad\beta\\
\text{subject to,}&\quad\forall (x,\lambda,\delta)\in\mathcal{X}\!\times\!\widehat{\Lambda}(\beta)\!\times\!\mathcal{D}:\nonumber\\
&\quad\Psi(x,\lambda)\geq \varepsilon_1\left|x\!-\!x_0(\lambda,\delta)\right|^2\\
&\quad\nabla_x\Psi^T\!f(x,\lambda,\delta)\leq -\varepsilon_2\left|x\!-\!x_0(\lambda,\delta)\right|^2\\
&\quad|x_0(\lambda,\delta)|^2\leq\Delta^2\end{aligned}$$
where $\mathcal{X}$ and $\mathcal{D}$ are semi-algebraic domains defined in , while $\varepsilon_{1,2}$ are small positive scalars. The first two constraints are the Lyapunov conditions, while the third constraint is to make sure that the equilibrium point under uncertainties do not move far from the nominal (desired) equilibrium point at the origin. Using Theorem\[T:Putinar\], the above problem can be recast into an SOS optimization problem as follows: $$\begin{aligned}
\label{E:opt_sos}
\underset{\Psi(x,\lambda),\lbrace s^{k1}_i\rbrace,\lbrace s^{k2}_i\rbrace,\lbrace s^{k3}_i\rbrace\,\forall k\in\lbrace 1,2,3\rbrace}{\max}\quad\beta\qquad\\
\text{subject to:}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\notag\\
\!\!\Psi(x,\lambda)- \varepsilon_1\left|x\!-\!x_0(\lambda,\delta)\right|^2-\sum_{i=1}^ps^{11}_ia_i(x)\notag\\
\!\!+\sum_{i=1}^ms^{12}_i(\sum_{j=1}^lg_{ij}\lambda_j-\beta h_i)-\sum_{i=1}^qs^{13}_ib_i(\delta)\in\Sigma[x,\lambda,\delta],\notag\\
-\nabla_x\Psi^T\!f(x,\lambda,\delta) -\varepsilon_2\left|x\!-\!x_0(\lambda,\delta)\right|^2-\sum_{i=1}^ps^{21}_ia_i(x)\notag\\
\!\!+\sum_{i=1}^ms^{22}_i(\sum_{j=1}^lg_{ij}\lambda_j-\beta h_i)-\sum_{i=1}^qs^{23}_ib_i(\delta)\in\Sigma[x,\lambda,\delta],\notag\\
\Delta^2 -\left|x_0(\lambda,\delta)\right|^2\notag\\
\!\!+\sum_{i=1}^ms^{32}_i(\sum_{j=1}^lg_{ij}\lambda_j-\beta h_i)-\sum_{i=1}^qs^{33}_ib_i(\delta)\in\Sigma[x,\lambda,\delta],\notag\end{aligned}$$ where $\lbrace s^{k1}_i\rbrace\,\forall k\in\lbrace 1,2\rbrace,\lbrace s^{k2}_i\rbrace\,\forall k\in\lbrace 1,2,3\rbrace,\lbrace s^{k3}_i\rbrace\,\forall k\in\lbrace 1,2,3\rbrace$ are multi-variate SOS polynomials from the ring $\Sigma[x,\lambda,\delta]$. There are two challenges to solving this problem: 1) the explicit functional form of $x_0(\lambda,\delta)$ may not be available in polynomial form (or at all); and 2) the decision variables are in bilinear form, such as the terms $s^{12}_i\beta,\,s^{22}_i\beta,$ and $s^{32}_i\beta$. [The first challenge can be resolved by obtaining *sufficiently close* polynomial approximation of $x_0(\lambda,\delta)$ via Taylor series expansion around $(\lambda,\delta)=(0,0)$ (or, by polynomial recasting techniques [@Antonis:2005]). The second challenge is resolved by reformulating as an iterative feasibility problem while applying a bisection-search algorithm for the maximum value of $\beta$.]{}
Example: Inverter-Based Microgrid {#S:example}
=================================
We consider a modified version of the CERTS microgrid network described in [@lasseter2011certs] as an example. Disconnecting the utility, we replace the substation by a droop-controlled inverter, with two other inverters placed alongside load banks 3 and 5 (no inverters at load banks 4 and 6). Nominal operating point (equilbrium) of the network was obtained by solving the steady-state power-flow equations . A disturbance set was created by allowing the uncertain parameters to vary within some limits around their nominal values (denoted by superscript ‘nom’) in the form of: [$$\begin{aligned}
\forall i,\,\forall k\in\mathcal{N}_i:~\left|\frac{\delta_{1,i,k}-\delta_{1,i,k}^{\text{nom}}}{\delta_{1,i,k}^{\text{nom}}}\right|\leq \alpha\,,\,\left|\frac{\delta_{2,i,k}-\delta_{2,i,k}^{\text{nom}}}{\delta_{2,i,k}^{\text{nom}}}\right|\leq \alpha\end{aligned}$$ w]{}here the value of $\alpha>0$ denotes different levels of uncertainties. The design parameter set for the droop-coefficients was chosen to be of the form with the affine constraints $$\begin{aligned}
\lambda^p_i\in(0,\beta]\,\text{ and }\,\lambda^p_i\in(0,0.2\beta]\,.\end{aligned}$$ Small positive scalars were used as the minimum values for the droop-coefficients, as per the typical norm on grid operations. Notice that when $\lambda^p_i\ll 1$ and $\lambda^q_i\ll 1$ the inverter voltage and frequency become *stiff*, not adjusting with network conditions, which is an unfavorable scenario from the network resiliency perspective. The perturbed equilibrium point is desired to remain within some domain of the form: $$\begin{aligned}
\label{E:eq_dom}
\left\lbrace (\omega_i,v_i)\,\left|\,\left(\frac{\omega_i}{\omega^{\max}}\right)^2+\left(\frac{v_i-v_i^d}{\Delta v^{\max}}\right)^2\leq c\right.\right\rbrace\end{aligned}$$ where $\omega^{\max}$ was set to $0.7\,$Hz, and $\Delta v^{\max}$ to $0.2\,$p.u.. The value of $c$ was varied to investigate different uncertainty scenarios. Note that the constraint defining the domain is equivalent to the third constraint in , albeit after scaling and shifting. The choice of $c$ influences the possible set of design parameter values (with smaller values yielding narrower design space).
Fig.\[F:c1\] shows the identified robustly stable design parameter space for the inverters under varying uncertainties in the exogenous input, for two different values of $c$, which refer to different levels of perturbations allowed on the equilibrium point ($c=1$ allows larger perturbation than $c=0.5$). The design space shrinks as the uncertainty level rises (higher value of $\alpha$) and as the allowable perturbation on the equilibrium point is reduced.
Conclusion {#S:concl}
==========
[In the context of robust plug-and-play [design]{} of nonlinear networks, we address the problem of identifying the largest region in the design parameter space that ensures asymptotic convergence of the states of the connected element under uncertainties in the network. We derive novel theoretical conditions of robust stability, as well as develop a SOS programming algorithm to identify the largest stability region in the design parameter space. Numerical illustrations are provided in the context of identifying droop-coefficient values of inverters for a plug-and-play [operation]{} of microgrids. Future work will explore the scalability and applicability of the algorithm to large-scale microrgid networks with other forms of dynamic resources (responsive loads, diesel generators).]{}
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was carried out under support from the U.S. Department of Energy as part of their Resilient Electric Distribution Grid R&D program (contract DE-AC05-76RL01830).
[^1]: Note that, by construction, $\mu^*(\lambda)<\Gamma$.
[^2]: This can be achieved by defining new parameters $\tilde{\lambda}=\lambda-\lambda^{\min}$.
|
---
abstract: 'Let a finite group $G$ act on the complex plane $(\C^2, 0)$. We consider multi-index filtrations on the spaces of germs of holomorphic functions of two variables equivariant with respect to 1-dimensional representations of the group $G$ defined by components of a modification of the complex plane $\C^2$ at the origin or by branches of a $G$-invariant plane curve singularity $(C,0)\subset(\C^2,0)$. We give formulae for the Poincaré series of these filtrations. In particular, this gives a new method to obtain the Poincaré series of analogous filtrations on the rings of germs of functions on quotient surface singularities.'
author:
- 'A. Campillo'
- 'F. Delgado'
- 'S.M. Gusein-Zade [^1]'
title: 'On Poincaré series of filtrations on equivariant functions of two variables [^2] '
---
Introduction {#sec0 .unnumbered}
============
In [@CDGa], [@DG] (see also [@CDGb] for another proof of the result of [@CDGa]) there were computed the Poincaré series (in several variables) of the so called divisorial (multi-index) filtrations on the ring ${\cal O}_{\C^2,0}$ of germs of holomorphic functions on the complex plane $\C^2$ at the origin and of the filtration defined by orders of a function on branches of a germ of a plane curve $(C,0)\subset(\C^2,0)$. In particular it was shown that the Poincaré series of the latter filtration coincides with the (multi-variable) Alexander polynomial of the (algebraic) link $(S_\eps^3, C\cap S_\eps^3)$ corresponding to the curve $(C,0)$. Here we give versions of these results for the equivariant case, i.e. when there is an action (a representation) of a finite group $G$ on the complex plane $\C^2$ and the corresponding filtrations are considered on subspaces of germs of functions equivariant with respect to 1-dimensional representations $G\to \C^*=\mbox{\bf GL}(\C, 1)$ of the group $G$. There is a problem to define an equivariant version of the monodromy zeta function. One reason to study “equivariant” Poincare series of filtrations is the hope that they may give a hint for such a definition (at least for curve singularities).
For a finite group $G$, the set $G^*$ of 1-dimentional representations $G\to\C^*=\mbox{\bf GL}(\C,1)$ of the group $G$ is a group itself. Let $R(G)$ be the ring of (complex virtual) representations of the group $G$ and let $R_1(G)$ be its subring generated by 1-dimensional representations. Elements of the ring $R_1(G)$ are formal sums $\sum n_\alpha \alpha$ over all different 1-dimensional representations $\alpha$ of the group $G$, $n_\alpha\in \Z$. The multiplication is defined by the tensor product. The rings $R(G)$ and $R_1(G)$ coincide iff the group $G$ is abelian.
Let $(V,0)$ be a germ of a complex analytic variety with an action of a finite group $G$. The group $G$ acts on the ring ${\cal O}_{V,0}$ of germs of functions on $(V,0)$ by the formula $g^*f(x)=f(g^{-1}(x))$ ($f\in {\cal O}_{V,0}$, $g\in G$, $x\in(V,0)$). Let $\alpha:G\to \C^*$ be a 1-dimensional representation of the group $G$
A germ $f\in {\cal O}_{V,0}$ is [*equivariant*]{} with respect to the representation $\alpha$ if $g^*f=\alpha(g)\cdot f$ for $g\in
G$.
The set of germs of functions on $(V,0)$ equivariant with respect to the representation $\alpha$ is a vector subspace ${\cal O}^\alpha_{V,0} \subset {\cal O}_{V,0}$. If $\alpha_1\ne\alpha_2$, one has ${\cal O}^{\alpha_1}_{V,0}\cap{\cal O}^{\alpha_2}_{V,0} =
\{0\}$. Product of functions equivariant with respect to representations $\alpha_1$ and $\alpha_2$ is a function equivariant with respect to the representation $\alpha_1\cdot\alpha_2$. If the group $G$ is abelian, ${\cal O}_{V,0} = \bigoplus\limits_{\alpha}{\cal O}^\alpha_{V,0}$.
For a vector space $A$ (finite or infinite dimensional, e.g., for ${\cal O}_{V,0}$ or ${\cal O}^\alpha_{V,0}$), a $1$-index (decreasing) filtration on $A$ is defined by a function $v: A \to \Z_{\ge 0}
\cup \{\infty\}$, such that $v(f_1+f_2)\ge \min\{v(f_1),
v(f_2)\}$, $f_i\in A$, $i=1,2$. An $s$-index filtration on $A$ is defined by $s$ such functions $v_i$, $i=1, \ldots, s$. For $\vv=(v_1, \ldots, v_s)\in \Z^s$, the corresponding subspace $J(\vv)$ is defined as $\{a\in A: v_i(a)\ge v_i, i=1,\ldots, s\}$. (To describe the filtration it is sufficient to define the subspaces $J(\vv)$ only for $\vv\in\Z^s_{\ge 0}$ (i.e., $v_i\ge 0$ for $i=1, \ldots, s$), however it is convenient to suppose $J(\vv)$ to be defined for all $\vv$ from the lattice $\Z^s$.)
Suppose that all the factor spaces $J(\vv)/J(\vv+\1)$ ($\1=(1,\ldots,1)$, $\vv\in\Z^s$) are finite dimensional. (This is equivalent to say that $\dim
A/J(\vv)$ is finite for any $\vv\in\Z^s_{\ge 0}$.) Let $d(\vv) :=
\dim J(\vv)/J(\vv+\1)$, $\vv\in \Z^r$. The [*Poincaré series*]{} of the multi-index filtration $\{J(\vv)\}$ is the series $P(t_1,\ldots, t_s)\in\Z[[t_1,\ldots, t_s]]$ defined by the formula $$\label{eq1}
P(\tt)=\frac{\left(\sum \limits_{\vv\in\Z^s}
d(\vv)\cdot\tt^{\vv}\right)\prod\limits_{i=1}^s(t_i-1)}
{\tt^{\1}-1}\,,$$ where $\tt^{\vv} = t_1^{v_1}\cdot \ldots \cdot t_s^{v_s}$ ($\tt=(t_1,\ldots,t_s)$, $\vv=(v_1,\ldots,v_s)$); see, e.g., [@CDK], [@CDGa].
Pay attention that the sum in the numerator is over all $\vv\in\Z^s$, not only over those from $\Z^s_{\ge 0}$. This sum contains monomials with negative powers of variables (for $s\ge
2$), however, being multiplied by $\prod\limits_{i=1}^s(t_i-1)$, it becomes a power series, i.e. an element of $\Z[[t_1,\ldots,
t_s]]$.
If there is a linear action (a representation) of a group $G$ on the space $A$ and the filtration $\{J(\vv)\}$ ($\vv\in\Z^s$) is invariant with respect to this representation (i.e. $g^*J(\vv)=J(\vv)$ for $g\in G$), one can define the equivariant Poincaré series of the filtration $\{J(\vv)\}$ as an element of $R(G)[[t_1,\ldots, t_s]]$. For that $d(v)$ in (\[eq1\]) should be substitute by $[J(\vv)/J(\vv+\1)]$ considered as a $G$-module (i.e. an element of the ring $R(G)$). Filtrations on the ring ${\cal O}_{V,0}$ considered here are, generally speaking, not $G$-invariant: the action of the group permutes valuations. This difficulty can be (partially) avoided by considering filtrations on the spaces of equivariant functions. This way an equivariant Poincaré series can be defined as an element of $R_1(G)[[t_1,\ldots, t_s]]$.
In the case when $A$ is the ring ${\cal O}_{V,0}$ of germs of functions on $(V,0)$ or a subspace of it, one can define a notion of integration with respect to the Euler characteristic over the projectivization $\P A$ of the space $A$ (see, e.g., [@CDGb], [@CDGc]; there the notion is defined for $A={\cal O}_{V,0}$, however there is no essential difference with the general case $A\subset {\cal O}_{V,0}$: the role of the jet spaces $J^k_{V,0}=
{\cal O}_{V,0}/{\frak m}^{k+1}$ ($\frak m$ is the maximal ideal in the ring ${\cal O}_{V,0}$) is played by the spaces $JA^k_{V,0}=
A/{\frak m}^{k+1}\cap A$). This notion generalizes the usual notion of integration with respect to the Euler characteristic ([@Viro]) and was inspired by the notion of motivic integration (see, e.g., [@DL]).
The (multi-index) filtration $\{J(\vv)\}$ on a subspace $A\subset {\cal O}_{V,0}$ is [*finitely determined*]{} if for each $\vv\in\Z^s$ there exists $N\ge 0$ such that ${\frak m}^N\cap A \subset J(\vv)$.
If the filtration $\{J(\vv)\}$ is finitely determined, one can show that $$P(\tt)=\int\limits_{\P A} \tt^{\,\vv(f)}d\chi\,;$$ see, e.g., [@CDGb], [@CDGc].
Here we compute the Poincaré series for some multi-index filtrations (divisorial ones and those defined by invariant curves) on the space of equivariant function-germs on the plane $(\C^2, 0)$ with a finite group action. For the trivial (1-dimensional) representation of the group, this gives formulae for divisorial filtrations on the quotient surface singularity $(\C^2/G, 0)$ and for the filtrations defined by curves on them somewhat different from those in $\cite{CDGc}$, $\cite{CDGd}$.
Divisorial filtrations on the spaces of equivariant functions {#sec1}
=============================================================
Let $\pi:(\widetilde V, D)\to (V,0)$ be a resolution of the germ $(V,0)$. Here $\widetilde V$ is a smooth manifold, $\pi$ is a proper analytic map which is an isomorphism outside of the exceptional divisor $D=\pi^{-1}(0)$, $D$ is a normal crossing divisor. Let $E_\sigma$ be irreducible components of the divisor $D$: $D=\bigcup_\sigma E_\sigma$. For a germ $f\in {\cal O}_{V,0}$, let $v_{\sigma}(f)$ be the multiplicity of the lifting $\widetilde f=f\circ\pi$ of the germ $f$ to the space $\widetilde V$ of the resolution along the component $E_\sigma$. The function $v_\sigma: {\cal O}_{V,0}\setminus\{0\}\to\Z_{\ge0}\cup\{+\infty\}$ defines a valuation (a [*divisorial*]{} one) on the field of quotients of the ring ${\cal O}_{V,0}$.
Suppose that the resolution $\pi:(\widetilde V, D)\to (V,0)$ is equivariant with respect to an action of a finite group $G$ on $(V,0)$, i.e., that the action of $G$ on $(V,0)$ can be lifted to an action on $(\widetilde V, D)$. In particular the group $G$ acts on the set $\{E_\sigma\}$ of irreducible components of the exceptional divisor $D$ by permutations. If the components $E_{\sigma_1}$ and $E_{\sigma_2}$ are in the same orbit of the action and a function $f\in {\cal O}_{V,0}$ is equivariant (with respect to a 1-dimensional representation $\alpha: G\to \C^*$), then $v_{\sigma_1}(f)= v_{\sigma_2}(f)$.
Let us fix $s$ components $E_1$, …, $E_s$ of the exceptional divisor $D$. The corresponding valuations $v_1$, …, $v_s$ define a multi-index filtration on the ring ${\cal O}_{V,0}$ of germs of functions on $(V,0)$ and also a filtration on the space ${\cal O}^\alpha_{V,0}$ of functions equivariant with respect to a 1-dimensional representation $\alpha$. Let $J^\alpha(\vv):=\{f\in {\cal O}^\alpha_{V,0}: v_i(f)\ge v_i, i=1,\ldots,s\}$. It is easy to see that the filtration $\{J^\alpha(\vv)\}$ is finitely determined. Let $P^\alpha(\tt)$ ($\tt=(t_1,\ldots,t_s)$) be the Poincaré series of the filtration $\{J^\alpha(\vv)\}$. One has $$P^\alpha(\tt)=\int\limits_{\P {\cal O}^\alpha_{V,0}} \tt^{\,\vv(f)}d\chi\,.$$
[**1.**]{} It is somewhat natural to suppose that for $i\ne j$ the divisors $E_i$ and $E_j$ are in different orbits of the action of the group $G$, however, from formal point of view this is not important. One can even permit that $E_i=E_j$ for some $i$ and $j$.
[**2.**]{} In fact it is sufficient to consider the case when $\{E_i\}=\{E_{\sigma}\}$, i.e. $E_1$, …, $E_s$ are all the components of the exceptional divisor $D$. The Poincaré series corresponding to a subset of components of the exceptional divisor is obtained from the one corresponding to all the components by putting additional variables equal to $1$.
The series $$P^G(\tt)=\sum\limits_{\alpha\in G^*}\alpha\cdot P^\alpha(\tt)\,\in\,R_1(G)[[t_1,\ldots,t_s]]\,,$$ where the sum is over all non-isomorphic 1-dimensional representations $\alpha$ of the group $G$, will be called the [*equivariant Poincaré series*]{} of the divisorial filtration corresponding to the components $E_1$, …, $E_s$ of the exceptional divisor $D$ of the resolution $\pi:(\widetilde V, D)\to(V, 0)$.
The discussion above implies that the equivariant Poincaré series $P^G(\tt)$ can be represented as $$\int\limits_{\coprod_\alpha \P {\cal O}^\alpha_{V,0}} \alpha(f)\cdot\tt^{\,\vv(f)}d\chi\,,$$ where $\alpha:\coprod_\alpha \P {\cal O}^\alpha_{V,0}\to R_1(G)$ is the (tautological) function which sends $\P {\cal O}^\alpha_{V,0}$ to $\alpha\in R_1(G)$.
The ring $R_1(G)$ can be described in the following way. Let the abelian group $G/(G,G)$ ($(G,G)$ is the commutator of the group $G$) be the direct sum $\bigoplus\limits_{j=1}^q \Z_{m_j}$ of the cyclic groups $\Z_{m_j}=\Z/m_j\Z$, let ${\widehat\xi}_j$ be a generator of the group $\Z_{m_j}$, let ${\xi}_j$ be an element of the group $G$ which maps to ${\widehat\xi}_j$ under the factorization by the commutator $(G,G)$, and let $u_j$ be the representation of the group $G$ defined by $u_j(\xi_j)=\exp\frac{2\pi i}{m_j}$, $u_j(\xi_k)=1$ for $k\ne j$. Then the representations $\prod\limits_{j=1}^q u_j^{\ell_j}$ with $0\le\ell_j<m_j$ form a basis of the ring $R_1(G)$ as a $\Z$-module and as a ring $R_1(G)\cong\Z[u_1,\ldots, u_q]/(u_j^{m_j}-1;\ j=1,\ldots, q)$ (see e.g. [@S]).
Divisorial filtrations on the ring ${\cal O}_{\C^2,0}$ {#sec2}
======================================================
From now on let $(V,0)=(\C^2,0)$, a finite group $G$ acts on $(\C^2,0)$. Without loss of generality we suppose this action to be defined by a representation $G\to \mbox{\bf GL}(2,\C)$. Let $\pi:(X,D)\to (\C^2,0)$ be a modification of the plane $(\C^2,0)$ by a sequence of blow-ups invariant with respect to the group action, that is the action of the group $G$ on the plane $(\C^2,0)$ lifts to its action on the space $X$ of the modification. All components $E_\sigma$ of the exceptional divisor $D$ are isomorphic to the complex projective line $\CP^1$. Let ${\stackrel{\bullet}{E}}_\sigma$ be the “smooth part" of the component $E_\sigma$, i.e. $E_\sigma$ itself minus intersection points with all other components of the exceptional divisor $D$, let ${\stackrel{\bullet}{D}}=\bigcup\limits_\sigma{\stackrel{\bullet}{E}}_\sigma$ be the smooth part of the exceptional divisor $D$, and let $\widehat{D}={\stackrel{\bullet}{D}}/G$ be the corresponding factor space, i.e. the space of orbits of the action of the group $G$ on ${\stackrel{\bullet}{D}}$.
Let $-k_\sigma$ be the self intersection number $E_\sigma\circ E_\sigma$ of the component $E_\sigma$ on the (smooth) surface $X$. Let $(E_\sigma\circ E_{\sigma^\prime})$ be the intersection matrix of the components $E_\sigma$ of the exceptional divisor $D$: $E_\sigma\circ E_\sigma=-k_\sigma$; for $\sigma\ne{\sigma^\prime}$ one has $E_\sigma\circ E_{\sigma^\prime}=1$ if the components $E_\sigma$ and $E_{\sigma^\prime}$ intersect (at a point) and $E_\sigma\circ E_{\sigma^\prime}=0$ otherwise. One has $\det(E_\sigma\circ E_{\sigma^\prime})=(-1)^{\#\sigma}$ where ${\#\sigma}$ is the number of components of the exceptional divisor $D$, i.e. the number of blow-ups which produce the modification $(X,D)$. Let $M=(m_{\sigma{\sigma^\prime}}):= -(E_\sigma\circ E_{\sigma^\prime})^{-1}$. All entries of the matrix $M$ are positive integers. The meaning of the entry $m_{\sigma{\sigma^\prime}}$ is the following. Let ${\widetilde L}_\sigma$ be a germ of a smooth curve on $X$ transversal to the component $E_\sigma$ at a smooth point of it, i.e. at a point of ${\stackrel{\bullet}{E}}_\sigma$. Let the curve $L_\sigma=\pi({\widetilde L}_\sigma)\subset(\C^2,0)$ be given by an equation $h_\sigma=0$, $h_\sigma\in{\cal O}_{\C^2,0}$. Then $m_{\sigma{\sigma^\prime}}=v_{\sigma^\prime}(h_\sigma)$ ($=v_\sigma(h_{\sigma^\prime})$). For $\sigma\ne\sigma^\prime$ the entry $m_{\sigma{\sigma^\prime}}$ is also equal to the intersection number $L_\sigma\circ L_{\sigma^\prime}$. The diagonal entry $m_{\sigma{\sigma}}$ is equal to $L_\sigma\circ L^\prime _{\sigma }$ where $L_\sigma=\pi({\widetilde L}_\sigma)$, $L^\prime_\sigma=\pi({\widetilde L}^\prime_\sigma)$ for smooth curves ${\widetilde L}_\sigma$ and ${\widetilde L}^\prime_\sigma$ transversal to the component $E_\sigma$ at different smooth points.
For a space $Z$, let $S^kZ=Z^k/S_k$ be the $k$th symmetric power of the space $Z$. If $Z$ is a quasiprojective variety, $S^kZ$ is also one. For example $S^k\C\cong\C^k$, $S^k\CP^1\cong\CP^k$.
Let $E_0$ be the first (in the order of blow-ups) component of the exceptional divisor $D$ ($E_0$ may very well coincide with one of the chosen components $E_1$, …, $E_s$). The component $E_0$ is invariant with respect to the action of the group $G$. For a point $p$ of $E_0$, i.e. for a line in $\C^2$, let $\ell_p$ be the corresponding linear function on $\C^2$: the function which vanishes at $p$; it is well defined up to multiplication by a non-zero constant. Let $G_p=\{g\in G: gp=p\}$ be the stabilizer of the point $p$ and let $Gp\cong G/G_p$ be the orbit of the point $p$. The homogeneous function $\prod\limits_{q\in Gp} \ell_q$ of degree equal to the number $\vert Gp\vert = \vert G\vert/\vert G_p\vert$ of points in the orbit of $p$ is an equivariant one (with respect to a certain 1-dimensional representation of the group $G$). Let $u_p$ be the corresponding representation.
[**1.**]{} Let the group $\Z_m$ act on $(\C^2, 0)$ by $\xi(x,y)=(\xi^k x,
\xi^{\ell} y)$ where $\xi=\exp(2\pi i/m)$ and $\gcd(k, \ell)$ is relatively prime with $m$ ($k$ and $\ell$ are the weights of the representation of the group $\Z_m$). The group $\Z_m^*$ of 1-dimentional representation of the group $\Z_m$ is the cyclic group of order $m$ generated by the representation $u$ defined by $u(\xi)=\xi$. The action of the group $\Z_m$ on the projectivization $\CP^1=(\C^2\setminus\{0\})/\C^*$ of the plane $\C^2$ has two fixed points $p_1=(1, 0)$ and $p_2=(0, 1)$ and is free outside of them. The linear function which vanishes at the point $p_1$ is the coordinate $y$. Therefore $u_{p_1}=u^{-\ell}$. In the same way $u_{p_2}=u^{-k}$. For a point $p\in\CP^1$ different from $p_1$ and $p_2$ one has $u_p=1$.
[**2.**]{} Let $G=D_2^*$ be the subgroup of $\mbox{\bf SL}(\C, 1)$ generated by the transformations $\sigma(x,y)=(ix, -iy)$ and $\tau(x,y)=(iy,ix)$, $i=\sqrt{-1}$. The group $G$ is of order $8$ and has $3$ cyclic subgroups of order $4$ generated by $\sigma$, $\tau$, and $\tau\sigma$ respectively. Intersection of any two of these subgroups is the commutator $(G,G)=\{1, \sigma^2\}$ of the group $G$ ($\sigma^2=\tau^2=(\tau\sigma)^2$). The group $G^*$ of 1-dimensional representations of the group $G$ is isomorphic to $\Z_2\times\Z_2$ and is generated by the representations $u_1$ and $u_2$ defined by $u_1(\sigma)=-1$, $u_1(\tau)=1$, $u_2(\sigma)=1$, $u_2(\tau)=-1$; $R_1(G)=
\Z[u_1,u_2]/(u_1^2-1, u_2^2-1)$.
The isotropy group of a generic point of the action of the group $G$ on the projectivization $\CP^1=(\C^2\setminus\{0\})/\C^*$ consists of 2 elements and is the commutator $(G,G)$ of the group $G$. Each cyclic subgroup of order $4$ has two fixed points which are in the same orbit of the $G$-action. They are $Q_1=(1:0)$ and $Q_2=(0:1)$ for the subgroup $\langle\sigma\rangle$, $Q_3=(1:1)$ and $Q_4=(1:-1)$ for the subgroup $\langle\tau\rangle$, $Q_5=(1:i)$ and $Q_6=(1:-i)$ for the subgroup $\langle\tau\sigma\rangle$. The pairs $(Q_1, Q_2)$, $(Q_3, Q_4)$, and $(Q_5, Q_6)$ are orbits of the action of the group $G$ on $\CP^1$.
The product of the linear functions vanishing at the points $Q_1$ and $Q_2$ is $xy$. It is invariant with respect to the transformation $\sigma$ and anti-invariant with respect to the transformation $\tau$. Therefore $u_{Q_1}=u_{Q_2}=u_2$. In the same way $u_{Q_3}=u_{Q_4}=u_1$, $u_{Q_5}=u_{Q_6}=u_2$ (the corresponding products of linear functions are $x^2-y^2$ and $x^2+y^2$ respectively). The product of linear functions vanishing at the points of an orbit of the action of the group $G$ different from those consisting of the points $Q_i$ is a homogeneous polynomial of degree $4$ and is invariant with respect to the $G$-action. Therefore for any $Q\ne Q_i$, $i=1,\ldots, 6$, one has $u_Q=1$.
Let $\{\Xi\}$ be a stratification of the space (in fact of a smooth curve) $\widehat D$ (${\widehat D}=\bigcup\Xi$) such that:
1. Each stratum $\Xi$ is connected and therefore is contained in the image of one component $E_\sigma$ under the map $j:{\stackrel{\bullet}{D}}\to{\widehat D}$ of factorization.
2. Over each strutum $\Xi$ the map $j$ is a covering.
As above, for a point $p\in {\stackrel{\bullet}{D}}$, let $G_p$ be its stabilizer $\{g\in G: gp=p\}$ and let $Gp$ be the orbit of the point $p$. For all points from a connected component of the preimage $j^{-1}(\Xi)$ of a stratum $\Xi$ the stabilizer $G_p$ is one and the same and for all points of $j^{-1}(\Xi)$ the stabilizers are conjugate to each other. For $p\in {\stackrel{\bullet}{D}}$, let $\sigma=\sigma(p)$ be the number of the component $E_\sigma$ which contains the point $p$. Let $$m_{\Xi,i}\ :=\ \sum\limits_{p\in j^{-1}({\widehat p})}
m_{\sigma(p)i}\quad \mbox{for $\widehat p \in \Xi$},$$ $\mm_{\,\Xi}:=( m_{\Xi,1}, \ldots, m_{\Xi,s})$.
For $p\in{\stackrel{\bullet}{D}}$, let $p_0$ be the corresponding point of the component $E_0$ of the exceptional divisor: if $p\not\in E_0$, $p_0$ is the point which belongs to the closure of the corresponding component of the complement $D\setminus E_0$. Otherwise $p_0=p$. Let $$u_\Xi\ :=\ u_{p_0}^{m_{\sigma(p)0}(\vert Gp\vert/\vert Gp_0\vert)}\,,$$ where $p\in j^{-1}(\Xi)$. The (1-dimensional) representation $u_{\Xi}$ is a monomial, say $\uu^{\,\ll_{\,\Xi}}$, in basic representations, $\uu=(u_1,\ldots, u_q)$, $\ll_{\,\Xi}=(\ell_1, \ldots, \ell_q)$.
\[theo1\] $$\label{eq2}
{P^G}(\tt)\ =\ \prod\limits_{\Xi}
\left(1-\uu^{\,\ll_{\,\Xi}}\tt^{\,\mm_{\,\Xi}}\right)^{- \chi(\Xi)}\,.$$
Let us fix $\VV\in \Z_{\ge 0}^s$ and suppose that we compute the Poincaré series $P^{G}(\tt)$ up to terms of degree $\VV$ in $\seq{t}1s$. Let $\P{\cal O}^\alpha_{\C^2,0}(\VV)$ be the set $\{f\in \P{\cal O}^\alpha_{\C^2,0} : \vv(f)\le \VV \}$. We can make additional blow-ups of intersection points of components of the exceptional divisor so that, for any $f\in\OO_{\C^2,0}$ with $\vv(f)\le \VV$, the strict transform of the curve $\{f=0\}$ intersects the exceptional divisor only at smooth points. We shall keep the notation $(X,D)$ for this modification as well explaining later why new-born components do not affect the answer. The space of $G$-invariant effective divisors on ${\stackrel{\bullet}{D}}$, i.e. of $G$-invariant unordered collection of points of ${\stackrel{\bullet}{D}}$ with some multiplicities, can be in the natural way identified with the space $$Y^G\ =\ \prod\limits_{\Xi}\left(\bigcup\limits_{k=0}^\infty S^k \Xi \right) =
\bigcup\limits_{\{k_{\Xi}\}} \left(\prod\limits_{\Xi} S^{k_{\Xi}} \Xi \right)\,.$$ The space $Y^G$ can be considered as a semigroup with the union of collections as the semigroup operation. Let $$I_G:\,\coprod\limits_\alpha\,\P{\cal O}^\alpha_{\C^2,0}(\VV)\to Y^G$$ be the map which sends a (non-zero) function $f\in{\cal O}^\alpha_{\C^2,0}(\VV)$ to the intersection of the strict transform of the zero-level curve ${\{f=0\}}$ with the exceptional divisor $D$ (a $G$-invariant collection of the intersection points counted with the corresponding multiplicities).
Let $\vv: Y^G \to \Z_{\ge 0}^s$ be the map (a semigroup homomorphism) which sends a point of the component $\prod\limits_{\Xi} S^{k_{\Xi}} \Xi$ of the space $Y^G$ to $\sum\limits_{\Xi} k_{\Xi} \mm_{\,\Xi}$. Let $\uu: Y^G\to G^{*}$ be the map to the group $G^*$ of one-dimensional representations of the group $G$ which sends a point of the component $\prod\limits_{\Xi} S^{k_{\Xi}} \Xi$ of the space $Y^G$ to $\prod\limits_{\Xi} u_{\Xi}^{k_{\Xi}}$. One can easily see that $\vv\circ I_G = \vv$ and $\uu\circ I_G = \uu$ where $\vv$ and $\uu$ in the right hand sides of the equations are the corresponding maps from $\coprod\limits_\alpha\,\P{\cal O}^\alpha_{\C^2,0}(\VV)$. The image of the map $I_G$ is the union of all components of the space $Y^G$ with $\vv\le\VV$. For a space (a quasi-algebraic variety) $Z$ one has $$\sum\limits_{k=0}^\infty \chi(S^kZ)t^k = (1-t)^{-\chi(Z)}\,.$$ This implies that $$\int\limits_{Y^G} \uu\,\tt^{\,\vv} d\chi = \prod\limits_\Xi \left(1-\uu^{\ll_{\,\Xi}}\tt^{\mm_{\,\Xi}}\right)^{-\chi(\Xi)}\,.$$ Moreover, all additional strata corresponding to components of the exceptional divisor born under additional blow-ups have the Euler characteristics equal to zero and therefore do not participate in the right hand side of the equation.
For any point $y\in Y^G$ with $\vv(y)\le\VV$, its preimage $I_G^{-1}(y)$ in $\coprod\limits_\alpha\,\P{\cal O}^\alpha_{\C^2,0}(\VV)$ is an affine space: [@CDGb Proposition 2]. With the Fubini formula this implies that the integrals of $\uu\,\tt^{\,\vv}$ over the spaces $Y^G$ and $\coprod\limits_\alpha\,\P{\cal O}^\alpha_{\C^2,0}(\VV)$ coincide up to terms of degree $\ge\VV$. Therefore Equation \[eq2\] is correct up to terms of degree $\ge\VV$ for any $\VV\in\Z_{\ge 0}^s$. This implies the statement.
Examples
========
[**1.**]{} Let $G=\Z_3$ act on the plane $(\C^2, 0)$ by $\xi(x,y)=(\xi
x, \xi^{-1} y)$, where $\xi=\exp(2\pi i/3)$ is the generator of the group $\Z_3$. The factor space $(\C^2, 0)/\Z_3$ is a surface singularity of type $A_2$. Blowing up the origin we glue-in the exceptional divisor $\CP^1$ with the self-intersection number $(-1)$. The action of the group $\Z_3$ on the space of the modification has two fixed points corresponding to the coordinate axes in $(\C^2,0)$. At one of them (corresponding to the $x$-axis) the representation of the group $\Z_3$ has weights $(1,1)$, at the other it has weights $(-1, -1)$. After blowing up these two points one gets a modification $X$ of the plane $\C^2$ whose exceptional divisor $D$ consists of three components with the self-intersection numbers $-1$, $-3$, and $-1$. The information about the space $X$ and the $\Z_3$-action on it is encoded in Figure \[fig1\].
$$\unitlength=1.00mm
\begin{picture}(80.00,20.00)(0,0)
\thicklines
\put(20,10){\line(1,0){40}}
\put(20,10){\circle*{2}}
\put(40,10){\circle*{2}}
\put(60,10){\circle*{2}}
\put(20,10){\vector(-1,0){22}}
\put(60,10){\vector(1,0){22}}
\put(16,5){$-1$}
\put(36,5){$-3$}
\put(56,5){$-1$}
\put(4,12){$(-1,0)$}
\put(26,12){$(0,2)$}
\put(44,12){$(-2,0)$}
\put(66,12){$(0,1)$}
\end{picture}$$
The vertices correspond to the components of the exceptional divisor. The arrows correspond to the strict transforms of the coordinate axes in $(\C^2, 0)$. The numbers under the vertices are the self-intersection numbers of the corresponding components of the exceptional divisor. The numbers over each edge ($\mbox{mod } 3$) are the weights of the representation of the group $\Z_3$ at the corresponding intersection (fixed) point; each of these numbers corresponds to the action on the component which it is close to.
The group $\Z_3$ acts trivially on the first and on the third components of the exceptional divisor (counted from the left). The action on the second one is non-trivial and has two fixed points. The fact that at each fixed point of the $\Z_3$-action on the space $X$ one of the weights of the representation is equal to zero implies that the factor space $X/\Z_3$ is smooth and is a resolution of the $A_2$-singularity $(\C^2, 0)/\Z_3$.
In general, for an action of a group (say, of a cyclic one) on the plane $(\C^2, 0)$, it is not possible to get an equivariant modification $X$ of the plane such that the factor space $X/G$ is non-singular and therefore is a resolution of the singularity $(\C^2, 0)/G$. The factor space $X/G$ may have singular points: see e.g. Example 2 below.
One has $$-\left(E_i\circ E_j\right)= \left(
\begin{array}{ccc}
2 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 2
\end{array}
\right)\,.$$ As the stratification $\{\Xi\}$ of the factor space $\widehat{D}={\stackrel{\bullet}{D}}/\Z_3$ one can take the one whose three strata are isomorphic to the projective line $\CP^1$ minus two points (and therefore have the Euler characteristics equal to zero) and two strata consists of one point each: the images of the intersection points of the strict transforms of the coordinate axes in $\C^2$ with the exceptional divisor $D$. For these two strata the representations $u_\Xi$ are equal to $u$ and $u^{-1}$ respectively. Therefore $$P^{\Z_3}(t_1, t_2, t_3) =
(1-u\,\tt^{(2,1,1)})^{-1}(1-u^{-1}\tt^{(1,1,2)})^{-1}\,.$$
The exceptional divisor of the resolution $(X,0)/\Z_3\to(\C^2,0)/\Z_3$ is the image of the exceptional divisor $D$ of the modification $(X,0)\to(\C^2,0)$ and also consists of three components. The self-intersection numbers of these components can be determined in the following way. On the first and on the third components of the exceptional divisor $D$ the $\Z_3$-action is trivial. Therefore they map isomorphically on their images. The group $Z_3$ acts only on the normal bundles to these components. The normal bundles to their images are the third powers of those over the preimages. Therefore the self-intersection numbers of the corresponding components of $D/\Z_3$ are equal to $(-3)$. On the second component of $D$ the $\Z_3$-action is not trivial. This implies that the self-intersection number of its image (the factor under the $\Z_3$-action) is $3$ times less than the self-intersection number of the preimage and is equal to $(-1)$. If, in $(X,0)/\Z_3$, we blow-down the second component of the exceptional divisor (with the self-intersection number $(-1)$), we get the standard (minimal) resolution of the $A_2$-singularity. To get the Poincaré series of the divisorial filtration corresponding to it one should take the part of the series $P^{\Z_3}(T_1^{1/3}, 1, T_2^{1/3})$ corresponding to the trivial representation. The reason for the change of variables $T_1=t_1^3$, $T_2=t_2^3$ is the difference between normal bundles to the corresponding components of $D$ and to their images described above. This way one gets $$P_{A_2}(T_1,
T_2)=\frac{(1-\TT^{(3,3)})}{(1-\TT^{(2,1)})(1-\TT^{(1,2)})(1-\TT^{(1,1)})}$$ (cf. [@CDGc]).
Now let $G=\Z_5$ act on $(\C^2, 0)$ by $\xi(x,y)=(\xi x,
\xi^{-1} y)$ where $\xi=\exp(2\pi i/5)$. The factor space $(\C^2,
0)/\Z_5$ is a surface singularity of type $A_4$. Blowing up the origin, after that all fixed points of the modification and then all fixed points of the second one we arrive to a modification the exceptional divisor of which has 7 irreducible components. The weights of the $\Z_5$-action at the fixed points which are not intersection points of the components of the exceptional divisor (they are intersection points of the exceptional divisor with the strict transforms of the coordinate axes in $\C^2$) are $(1,1)$ and $(-1,-1)$ respectively. Blowing up these points one gets the modification described in Figure \[fig2\].
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\put(120,10){\line(1,0){15}}
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\put(135,10){\vector(1,0){15}}
\put(12,3){\small $-1$}
\put(27,3){\small $-2$}
\put(42,3){\small $-3$}
\put(57,3){\small $-1$}
\put(72,3){\small $-5$}
\put(87,3){\small $-1$}
\put(102,3){\small $-3$}
\put(117,3){\small $-2$}
\put(132,3){\small $-1$}
{\footnotesize
\put(0,14){ $(-1,0)$}
\put(15,14){ $(0,-1)$}
\put(32,14){ $(1,3)$}
\put(45,14){ $(-3,0)$}
\put(62,14){ $(0,2)$}
\put(75,14){ $(-2,0)$}
\put(92,14){ $(0,3)$}
\put(103.5,14){ $(-3,-1)$}
\put(122,14){ $(1,0)$}
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}
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The factor space of this modification is not smooth: it has two quotient cyclic singularities of type $(1,3)$ in the terminology of [@P], corresponding to the intersection points (edges) depicted by dashed lines. Minimal resolution of such a point has the exceptional divisor consisting of two components with the self intersection numbers $-2$ and $-3$ (see [@P]). One can easily see that additional blow-ups at these points do not lead to a modification the factor space of which is smooth. Just as in Example 1 one has $$P^{\Z_5}(t_1, \ldots , t_9) =
(1-u\,\tt^{(4,3,2,3,1,2,1,1,1)})^{-1}(1-u^{-1}\tt^{(1,1,1,2,1,3,2,3,4)})^{-1}\,.$$ The components of the exceptional divisor $D$ which correspond to the components of the standard (minimal) resolution of the $A_4$-singularity are those with numbers 1,4,6, and 9. One can verify this resolving the two singular points of $X/\Z_5$ and watching the sequence of blow-downs which leads to the minimal resolution of the $A_4$-singularity. On all of them the $\Z_5$-action is trivial and therefore they map on the four components of the resolution of the $A_4$-sigularity isomorphically. The Poincaré series of the divisorial filtration of (the minimal resolution of) the $A_4$-singularity is the part of the series $P^{\Z_5}(T_1^{1/5}, 1, 1, T_2^{1/5},1, T_3^{1/5}, 1, 1, T_4^{1/5})$ corresponding to the trivial representation. This gives $$P_{A_4}(T_1,T_2,T_3,T_4) =
\frac{(1-\TT^{(5,5,5,5)})}{
(1-\TT^{(1,1,1,1)})\,
(1-\TT^{(1,2,3,4)})\,
(1-\TT^{(4,3,2,1)})}\;$$ (cf [@CDGc]).
[**3.**]{} Let $G={\Bbb D}_2^*$ be the subgroup of $\mbox{\bf SL}(\C, 1)$ generated by the transformations $\sigma(x,y)=(ix, -iy)$ and $\tau(x,y)=(iy,ix)$, $i=\sqrt{-1}$ (see Example 2 of Section \[sec2\]). The factor of the plane $\C^2$ by the $G$-action has a singular point of type $D_4$ (see, e.g., [@P]).
Let us blow-up the origin in $\C^2$. On the exceptional divisor (the projective line $(\C^2\setminus\{0\})/\C^*$) there are 6 points $Q_1$, …, $Q_6$ whose stabilizers are cyclic subgroups of $G$ of order $4$ (see Section \[sec2\], Example 2). The stabilizer of any other point of the exceptional divisor consists of 2 elements and is the commutator $(G,G)$ of the group $G$. Blowing-up the points $Q_i$, $i=1, \ldots, 6$, one gets $6$ new components of the exceptional divisor. On a component corresponding to a fixed point $Q_i$ of a cyclic subgroup $H$ of order $4$ the action of the group $H$ is not trivial. It has two fixed points one of which is the intersection point with the exceptional divisor of the initial blow-up. The other one is the intersection point with the strict transform of the line in $\C^2$ corresponding to the point $Q_i\in(\C^2\setminus\{0\})/\C^*$. Blowing-up these $6$ points which do not belong to the exceptional divisor of the initial blow-up, one gets $6$ new components of the exceptional divisor. On the one of them corresponding to a fixed point of a cyclic subgroup of order $4$ the action of this subgroup is trivial. We arrive to the modification described in Figure \[fig3\]. The arrows correspond to the strict transforms of the lines in $\C^2$ corresponding to the points $Q_i$, $i=1, \ldots, 6$. The numbers in parenthesis are the weights of the action of the corresponding cyclic group of order 4 at the corresponding point. Let us choose one component of the exceptional divisor from each $G$-orbit. (The multiplicities of any equivariant function on components from the same $G$-orbit are equal.) These 7 components are indicated on the left hand side of Figure \[fig3\].
$$ =1.00mm
(140.00,40)(3,0) (15,18)[(1,0)[15]{}]{} (30,18)[(2,3)[8.3]{}]{} (30,18)[(2,-3)[8.3]{}]{} (30,18)[(1,0)[15]{}]{} (30,18)[(-2,3)[8.3]{}]{} (30,18)[(-2,-3)[8.3]{}]{}
(15,18)[(-1,0)[7.5]{}]{} (45,18)[(1,0)[7.5]{}]{} (38.3,30.5)[(2,3)[4.15]{}]{} (21.7,30.5)[(-2,3)[4.15]{}]{} (38.3,5.5)[(2,-3)[4.15]{}]{} (21.7,5.5)[(-2,-3)[4.15]{}]{}
(25,14)[(31,8)]{}
(15,18) (22.5,18) (30,18) (38.3,30.5) (34.15,24.25) (38.3,5.5) (34.15,11.75)
(45,18) (37.5,18) (21.7,30.5) (25.85,24.25) (21.7,5.5) (25.85,11.75)
[(29.5,14.5)[$1$]{} (37,14.5)[$2$]{} (44.5,14.5)[$3$]{} (26.75,24.25)[$4$]{} (22.6,30.25)[$5$]{} (26.75,10.5)[$6$]{} (22.6,4.5)[$7$]{} ]{}
(30,8.5)[(1,0)[4]{}]{} (30,8.5)[(-1,0)[4]{}]{} (20.2,21.4)[(2,3)[3]{}]{} (22.2,24.4)[(-2,-3)[3]{}]{} (39.8,21.4)[(-2,3)[3]{}]{} (37.8,24.4)[(2,-3)[3]{}]{}
(58,16.5)[$\Rightarrow$]{}
(70,18)[(20,0)]{} (90,18)[(1,0)[40]{}]{} (70,18) (90,18) (110,18) (110,18)[(1,0)[20]{}]{}
(67,13)[$-7$]{} (87,13)[$-2$]{} (107,13)[$-1$]{}
(74,20)[$(2,-1)$]{} (95,20)[$(1,0)$]{} (115,20)[$(0,1)$]{}
(65,8)[(70,20)]{}
$$
A stratification $\{\Xi\}$ of the factor space $\widehat{D}=
{\stackrel{\bullet}{D}}/G$ can be chosen in the following way.
1. The factor space $\Xi_1$ of the smooth part of the initial (“central") component of the exceptional divisor. It has the Euler characteristic equal to $-1$. The corresponding representation $u_{\Xi_1}$ is trivial.
2. Three strata consisting of one point each: the images of the intersection points of the exceptional divisor $D$ with the strict transforms of the lines corresponding to the points $Q_i$, $i=1,\dots, 6$. The corresponding representations $u_{\Xi_i}$ are: $u_2$ for the stratum corresponding to the fixed points of the subgroup $\langle\sigma\rangle$, $u_1$ for that corresponding to the subgroup $\langle\tau\rangle$, $u_1u_2$ for that corresponding to the subgroup $\langle\tau\sigma\rangle$.
3. Six strata with the Euler characteristics equal to zero. They are images of the smooth parts of 12 components of the exceptional divisor $D$ different from the central one minus intersection points with the strict transforms of the lines corresponding to the points $Q_i$.
Theorem \[theo1\] gives $$P^G(\seq t17) = \frac{1- \tt^{(4,4,4,4,4,4,4)}}{
(1-u_2\,\tt^{(2,3,4,2,2,2,2)})
(1-u_1\,\tt^{(2,2,2,3,4,2,2)})
(1-u_1 u_2\,\tt^{(2,2,2,2,2,3,4)})} \, .$$
The components of the exceptional divisor $D$ of the modification which corresponds to the components of the standard (minimal) resolution of the $D_4$ singularity $(\C^2,0)/G$ are those with numbers 1, 3, 5, and 7. Therefore $P_{D_4}(T_1,T_2,T_3,T_4)$ is the part of the series $P^G(T_1^{1/2},1,T_2^{1/4},1,T_3^{1/4},1,T_4^{1/4})$ corresponding to the trivial representation. One gets $$P_{D_4}(\TT) = \frac{(1-\TT^{(2,1,1,1)})(1+\TT^{(3,2,2,2)})}{
(1-\TT^{(2,2,1,1)})
(1-\TT^{(2,1,2,1)})
(1-\TT^{(2,1,1,2)})
}$$ (cf [@CDGc]).
Filtrations defined by $G$-invariant curves
===========================================
Let $(C, 0)\subset(\C^2,0)$ be a plane curve singularity invariant with respect to the action of the group $G$ on $(\C^2,0)$. Let $C=\bigcup_{i=1}^r C_i$ be the representation of the curve $C$ as the union of its irreducible components. Let $\varphi_i:(\C,0)\to(\C^2, 0)$ be the parametrization (uniformization) of the component $C_i$ of the curve $C$, i.e. $\mbox{Im}\, \varphi_i =C_i$, $\varphi_i$ is an isomorphism between $\C$ and $C_i$ outside of the origin. For a germ $f\in \OO_{\C^2,0}$ let $v_i=v_i(f)$ be the power of the leading term in the power series decomposition of the germ $f\circ \varphi_i: (\C,0)\to \C$: $f\circ \varphi_i (\tau) = a \tau^{v_i} + \mbox{terms of higher degree}$, $a\neq 0$. If $f\circ \varphi_i (\tau) \equiv 0$, $v_i(f):=\infty$. The functions (valuations) $v_i$ define a multi-index filtration on the ring $\OO_{\C^2,0}$ of germs of functions of two variables and also on the subspace $\OO^\alpha_{\C^2,0}$ of functions equivariant with respect to a 1-dimensional representation $\alpha$ of the group $G$. Let $P_C^\alpha(t_1, \ldots, t_r)$ be the Poincaré series of this filtration on the space ${\OO}^\alpha_{\C^2,0}$, let $$P_C^G(\tt) :=\sum_\alpha \alpha\cdot P_C^\alpha(\tt)\in
R_1(G)[[t_1, \ldots, t_r]]$$ (the sum is over all non-isomorphic 1-dimensional representations $\alpha$ of the group $G$) be the [*equivariant Poincaré series*]{} of the filtration on the ring $\OO_{\C^2,0}$.
One can easily see that if the components $C_{i_1}$ and $C_{i_2}$ of the curve $C$ are in one and the same orbit of the $G$-action on the set of components, $v_{i_1}(f)=v_{i_2}(f)$ for any $f\in \OO^\alpha_{\C^2,0}$. This means that in all the monomials of the equivariant Poincaré series $ P_C^G(\tt)$ the exponents at the variables $t_{i_1}$ and $t_{i_2}$ coincide. Therefore without any loss of generality one can choose one component from each orbit of the $G$-action.
Let $\pi:(X,D)\to(\C^2,0)$ be a $G$-invariant embedded resolution of the curve $C$ (i.e. the total transform $\pi^{-1}(C)$ of the curve $C$ is a normal crossing divisor on the space $X$ of the resolution and the group $G$ acts on the space (a surface) $X$ as well). Let $D=\bigcup_\sigma E_\sigma$ be the representation of the exceptional divisor $D=\pi^{-1}(0)$ as the union of its irreducible components. Let $\{\Xi\}$ be a stratification of the factor space $\widehat{D}={\stackrel{\bullet}{D}}/G$ described in section \[sec2\]. Let ${\stackrel{\circ}{E}}_\sigma$ be $E_\sigma$ itself minus intersection points with all other components of the total transform $\pi^{-1}(C)$ of the curve $C$ (${\stackrel{\circ}{E}}_\sigma\subset{\stackrel{\bullet}{E}}_\sigma$), let ${\stackrel{\circ}{D}}=\bigcup\limits_\sigma{\stackrel{\circ}{E}}_\sigma$, and let $\widehat{D}^\prime:={\stackrel{\circ}{D}}/G$ be the space of orbits of the action of the group $G$ on $\widehat{D}^\prime$. Let $\{\Xi^\prime\}$ be a stratification of the space $\widehat D^\prime$ with the same properties as the stratification $\{\Xi\}$ of the space $\widehat D$ in Section \[sec2\].
One has ${\stackrel{\circ}{D}}\subset {\stackrel{\bullet}{D}}$, $\widehat{D}' \subset \widehat{D}$ and one can suppose that each stratum of the stratification $\{\Xi'\}$ is a part of a stratum of the stratification $\{\Xi\}$. For a stratum $\Xi'$, let $\ll_{\,\Xi'}$ be equal to $\ll_{\,\Xi}$ for the corresponding stratum $\Xi$. For $i=1, \ldots, r$, let $E_{\sigma(i)}$ be the component of the exceptional divisor $D$ of the resolution which intersects the component $C_i$ of the curve $C$. Let $m_{\Xi',i} := m_{\Xi,\sigma(i)}$, $\mm_{\,\Xi'} := (m_{\Xi',1}, \ldots, m_{\Xi',r})$.
\[theo2\] $$P^G_C(\tt)\ =\ \prod\limits_{\Xi^\prime}
\left(1-\uu^{\ll_{\,\Xi^\prime}}\tt^{\mm_{\,\Xi^\prime}}\right)^{-
\chi(\Xi^\prime)}\,.$$
The [**proof**]{} is essentially the same as the one of Theorem \[theo1\].
[12]{} Campillo A., Delgado F., Gusein-Zade S.M. The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. J., v.117, no.1, 125–156 (2003). Campillo A., Delgado F., Gusein-Zade S.M. The Alexander polynomial of a plane curve singularity and integrals with respect to the Euler characteristic. Internat. J. Math., v.14, no.1, 47–54 (2003). Campillo A., Delgado F., Gusein-Zade S.M. Poincaré series of a rational surface singularity. Inventiones Math., v.155, no.1, 41–53 (2004). Campillo A., Delgado F., Gusein-Zade S.M. Poincaré series of curves on rational surface singularities. Commentarii Math. Helvetici, v.80, no.1, 95–102 (2005). Campillo A., Delgado F., Kiyek K. Gorenstein property and symmetry for one-dimensional local Cohen–Macaulay rings. Manuscripta Mathematica, v.83, no.3–4, 405–423 (1994). Delgado F., Gusein-Zade S.M. Poincaré series for several plane divisorial valuations. Proc. Edinb. Math. Soc. (2), v.46, no.2, 501–509 (2003). Denef J., Loeser F. Germs of arcs on singular algebraic varieties and motivic integration. Inventiones Math., v.135, no.1, 201–232 (1999). Pinkham, H. Singularités de Klein – I, II. In: Séminaire sur les singularités des surfaces. Lecture Notes in Math. 777, pp.1–20. Springer, Berlin–Heidelberg–New York, 1980. Serre, J.P. Représentations linéaires des groupes finis. Hermann, Paris, 1978. Viro O.Y. Some integral calculus based on Euler characteristic. Topology and Geometry – Rohlin seminar. Lecture Notes in Math. v.1346, Springer, Berlin–Heidelberg–New York, 1988, pp.127–138.
Addresses:
University of Valladolid Dept. of Algebra, Geometry and Topology 47005 Valladolid, Spain E-mail: campillocpd.uva.es, fdelgadoagt.uva.es
Moscow State University Faculty of Mathematics and Mechanics Moscow, 119992, Russia E-mail: sabirmccme.ru
[^1]: Partially supported by the grant MCYT: MTM2004-00958. The third author partially supported by the grants RFBR–04–01–00762, NSh–4719.2006.1.
[^2]: Math. Subject Class. 14B05, 16W70, 16W22.
|
---
abstract: 'We develop a dynamical approach to study the Casimir–Polder force between a initially bare molecule and a magnetodielectric body at finite temperature. Switching on the interaction between the molecule and the field at a particular time, we study the resulting temporal evolution of the Casimir–Polder interaction. The dynamical self-dressing of the molecule and its population-induced dynamics are accounted for and discussed. In particular, we find that the Casimir–Polder force between a chiral molecule and a perfect mirror oscillates in time with a frequency related to the molecular transition frequency, and converges to the static result for large times.'
author:
- Pablo Barcellona
- Roberto Passante
- Lucia Rizzuto
- Stefan Yoshi Buhmann
title: 'Dynamical Casimir–Polder interaction between a chiral molecule and a surface'
---
\[Introduction\]Introduction
============================
Casimir and Casimir–Polder forces (CP) are electromagnetic interactions between neutral macroscopic bodies and/or molecules due to the quantum fluctuations of the electromagnetic field [@Casimir48; @CasimirPolder48; @Milonni94]. The presence of perfect boundaries (perfect conductors) modifies the possible wavelength of vacuum fluctuations and leads to observable effects like the lifetime and frequency shift of an atom in an excited state [@bartonqed], the interatomic potential between two atoms [@power], the anomalous gyromagnetic ratio [@Kreuzer; @bennet], which are all different from the vacuum case. The interaction between a ground-state molecule and a perfect electric mirror is always attractive while the interaction between an excited molecule and an perfect electric mirror shows an oscillating distance-dependence. This has been confirmed in measurements of the force between an excited ion and a metallic mirror [@wilson; @bushev].
Recently, the attention in the literature has been directed towards chiral molecules which posses distinctive optical properties including optical rotation as well as circular dichroism. Chiral molecules are molecules without point or plane symmetry; the two distinct mirror images of a chiral molecule are called enantiomers. Many of the processes crucial to life involve chiral molecules whose chiral identity plays a central role in their chemical reactions, the wrong enantiomer reacts in a different way and does not produce the required result. Spectroscopically, enantiomers have identical properties and distinguishing between them is a non-trivial task when using normal spectroscopy. A frequently used method to separate enantiomers in an industrial setting is chiral chromotography [@gil]. Recently, several laser schemes have been proposed to separate mixtures of enantiomers, and the effect of molecular rotation on enantioseparation has been studied [@jacob]. Furthermore Casimir–Polder forces between chiral molecules in absorptive and dispersive chiral medium have shown discriminatory effects, which might be used to separate enantiomers [@jenkins; @craig; @salam; @butcher; @salam2; @salam3].
In this article, we consider the dynamical Casimir–Polder interaction between a chiral molecule and a metal or dielectric body at finite temperature using a dynamical approach, with the molecule exhibiting electric, magnetic and chiral polarizabilities. The dynamical CP force between an enantiomer and a perfect chiral mirror is a possible system for distinguishing and separating enantiomers, because the dispersion energy between these systems depends on the relative handedness of the molecule with respect to that of the molecules constituting the chiral mirror. Therefore, enantiomers that pass at low speeds near the chiral mirror will be attracted or repelled in opposite directions and will be separated based on their chirality.
We assume to switch on the interaction between the molecule and the field at a particular time and study the resulting time evolution of the Casimir–Polder interaction. Even if the interaction with the free field is always present, our assumption to switch on suddenly the interaction with the body-assisted field at $t_0=0$ can be a good approximation of the more realistic cases of a rapid change of some parameter characterizing the strength of the atom-field interaction or of putting the atom at some distance from the macroscopic body, obtaining a partially dressed atom or molecule [@GPPP95]. The dynamics of the force could be observed in principle on time-scales of femto-seconds for typical molecules and nano-seconds for Rydberg atoms.
The dynamical self-dressing has been considered for an electric ground-state atom near an electric perfect conductor [@vasile], or for a partially dressed atomic state [@messina]. Also, the dynamical CP interaction between a neutral atom and a real surface has been recently investigated [@Haakh14]. The CP force of a chiral molecule near a body has so far been considered only in the static case [@butcher]. In this paper, we consider the dynamical self-dressing for a chiral molecule near a body; our approach includes finite temperature, arbitrary geometries of the body and arbitrary internal molecular states. As a simple application, we consider the CP interaction between an initially bare ground-state chiral molecule and a perfect chiral plate at zero temperature. We will show that the Casimir–Polder interaction at large times can be attractive or repulsive depending on the chiralities of the molecule and the medium. This differs from the stationary CP interaction between an electric molecule and an electric perfect plate, which is always attractive [@CasimirPolder48; @barton]. Furthermore, the time-dependent approach allows us to follow the temporal evolution of the CP force due to the initial conditions.
The article is organised as follows. In Sect. \[Sec2\], we consider the Heisenberg dynamics of the molecule and the body-assisted field, mutually coupled. Then, in Sect. \[Sec3\], we consider the dynamical Casimir–Polder interaction between a molecule with electric, magnetic and chiral responses and a body of arbitrary shape at finite temperature. In Sect. \[Sec4\], we apply these results to a particular case: the dynamical CP interaction between an initially bare chiral ground-state molecule and a perfectly reflecting chiral plate at zero temperature. We close with some conclusion in Sect. \[Sec5\].
\[Sec2\]Dynamics of the molecule and the body-assisted field
============================================================
We consider the mutually coupled temporal evolution of a single molecule and the body-assisted field. The body-field system is prepared at uniform temperature $T$, and the molecule in an arbitrary incoherent superposition of internal energy eigenstates. The dynamics of the molecule can be described with time-dependent flip operators, defined by $A_{mn} = \left| m \right\rangle \left\langle n \right|$, where $\left| n \right\rangle $ is an energy eigenstate.
In order to evaluate the dynamical CP force between the molecule and the body, we must first solve the molecule-field dynamics to obtain the flip operators and the field operators in the Heisenberg picture. The total Hamiltonian is the sum of three terms, the molecule and the field Hamiltonians and the interaction term in the dipole approximation: $H = H_A + H_F + H_{AF}$, where $$\begin{aligned}
\nonumber
H_A=& \sum\limits_n E_nA_{nn} \\ \nonumber
H_F =& \sum\limits_{\lambda = e,m} \int \mathrm{d}^3r
\int\limits_0^\infty
\mathrm{d}\omega \hbar \omega \mathbf{f}_\lambda ^\dag \left(
\mathbf{r},\omega \right)
\cdot \mathbf{f}_\lambda \left( \mathbf{r},\omega \right) \\
H_{AF} =& - \mathbf{d}\cdot \mathbf{E}\left( \mathbf{r}_A \right)
- \mathbf{m}\cdot \mathbf{B}\left( \mathbf{r}_A \right)\end{aligned}$$ where $\textbf{f}_\lambda \left( \mathbf{r},\omega \right)$ is the annihilation operator for the elementary electric and magnetic excitations of the system [@butcher], $\mathbf{d}$ and $\mathbf{m}$ are respectively the molecule’s electric and magnetic dipole moments, and $\textbf{r}_A$ the position of the molecule.
We introduce the Fourier component of the electric field $\mathbf{E}\left( \mathbf{r},\omega
\right)$, $\mathbf{E}\left( \mathbf{r}\right)=\int_0^\infty \text{d}\omega \mathbf{E}\left( \mathbf{r},\omega
\right) +\textup{h.c.} $\
The commutators between electromagnetic fields read [@butcher]: $$\begin{aligned}
\nonumber
&\left[ \mathbf{E}\left( \mathbf{r},\omega \right),\mathbf{E}^\dag
\left( \mathbf{r}',\omega ' \right) \right] = \frac{\hbar \mu
_0}{\pi } \,\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}',\omega
\right)\omega ^2\delta \left( \omega - \omega ' \right) \\ \nonumber
&\left[ \mathbf{E}\left( \mathbf{r},\omega \right),\mathbf{B}^\dag
\left( \mathbf{r}',\omega ' \right) \right] = \\ \nonumber
& \qquad \qquad -\frac{\mathrm{i}\hbar \mu _0}{\pi }\,\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r},\mathbf{r}',\omega \right)
\times \overleftarrow \nabla '\omega \delta \left(\omega - \omega '
\right) \\ \nonumber
&\left[ \mathbf{B}\left( \mathbf{r},\omega \right),\mathbf{E}^\dag
\left( \mathbf{r}',\omega ' \right) \right] =\\ \nonumber
& \qquad \qquad -\frac{\mathrm{i}\hbar \mu _0}{\pi } \, \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}',\omega \right)
\omega \delta \left( \omega - \omega ' \right) \\ \nonumber
& \left[ \mathbf{B}\left( \mathbf{r},\omega \right),\mathbf{B}^\dag
\left( \mathbf{r}',\omega ' \right) \right] =\\
& \qquad \qquad - \frac{\hbar \mu
_0}{\pi } \,\nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r},\mathbf{r}',\omega \right)
\times \overleftarrow \nabla '\delta \left( \omega - \omega '
\right)
\label{comm}\end{aligned}$$ where ${\mbox{\textbf{\textit{\textsf{G}}}}}$ is the classical Green tensor of the electromagnetic field and $\left[ {\mbox{\textbf{\textit{\textsf{G}}}}} \times \overleftarrow \nabla '
\right]_{ij} = \text{G}_{ik}\varepsilon _{jkl}\overleftarrow \partial
/\partial x'_l$, the Heisenberg equations for the coupled molecule–field dynamics read: $$\begin{aligned}
\nonumber
&\dot A_{mn} = \mathrm{i}\omega _{mn}A_{mn} + \frac{\mathrm{i}}{\hbar
} \,\mathbf{K}_{mn} \cdot \mathbf{E}\left( \mathbf{r}_A \right) +
\frac{\mathrm{i}}{\hbar } \,\mathbf{Q}_{mn} \cdot \mathbf{B}\left(
\mathbf{r}_A \right) \\ \nonumber
&\dot {\mathbf{E}}\left( \mathbf{r},\omega \right) = -\mathrm{i}\omega
\mathbf{E}\left( \mathbf{r},\omega \right) + \frac{\mathrm{i}\mu
_0}{\pi } \, \omega ^2\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r},\mathbf{r}_A,\omega \right) \cdot \mathbf{d} \\ \nonumber
& \qquad \qquad+ \frac{\mu _0}{\pi
} \,\omega \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A,\omega
\right) \times \overleftarrow \nabla ' \cdot \mathbf{m} \\ \nonumber
&\dot {\mathbf{B}}\left( \mathbf{r},\omega \right) = -
\mathrm{i}\omega \mathbf{B}\left( \mathbf{r},\omega \right) +
\frac{\mu _0}{\pi } \,\omega \nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r},\mathbf{r}_A,\omega \right) \cdot \mathbf{d} \\
& \qquad \qquad -\frac{\mathrm{i}\mu _0}{\pi } \,\nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r},\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla
' \cdot \mathbf{m} \end{aligned}$$ where $\mathbf{K}_{mn} = \left[ A_{mn},\mathbf{d} \right] $, $\mathbf{Q}_{mn} = \left[ A_{mn},\mathbf{m} \right] $. $\nabla$ and $\overleftarrow \nabla '$ operators act only on the first and second arguments of the Green’s tensor; for example ${\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla ' = \left. {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}',\omega \right) \times \overleftarrow \nabla ' \right|_{\mathbf{r}' = \mathbf{r}_A}$ Note that an electric dipole moment can produce a magnetic field and a magnetic dipole moment can create an electric field; these cross contributions are the relevant ones for the chiral part of the Casimir force.
In order to include the Lamb shifts and the dissipation of the molecular system we require the master equations for the populations $p_n\left( t \right) = \left\langle A_{nn}\left( t
\right) \right\rangle $ and the coherences $\sigma_{nm}\left( t
\right) = \left\langle A_{nm}\left( t \right) \right\rangle $, where the expectation value is taken over the field thermal state and the molecular internal state. The evolution of the populations are governed by the decay rates and the oscillations of the coherences are governed by the molecular transition frequencies. The electric field at the position of the molecule consists of two terms: the radiation reaction and the free field. As shown in the literature for a purely electric atom, the radiation reaction field gives rise to frequency shifts and spontaneous decay for molecule [@Buhmann2; @ackerhalt]. We thus renormalise the field by splitting off the radiation reaction: $$\begin{aligned}
\nonumber
&\left\langle \dot A_{mn} \right\rangle = \left[ \text{i}\tilde
\omega _{mn} - \left( \Gamma _n +\Gamma _m
\right)/2
\right]\left\langle A_{mn} \right\rangle \\
& \qquad \qquad+ \frac{\text{i}}{\hbar
} \left\langle \mathbf{K}_{mn} \cdot \mathbf{E}^{\left( 0
\right)}\left( \mathbf{r}_A \right) \right\rangle +
\frac{\text{i}}{\hbar }\left\langle \mathbf{Q}_{mn} \cdot
\mathbf{B}^{\left( 0 \right)}\left( \mathbf{r}_A \right) \right\rangle\end{aligned}$$ where $m \ne n$ and the expectation value is taken over the field thermal state and the molecular internal state. $\tilde \omega _{mn}$ are the Lamb-shifted molecular frequencies and $\Gamma_k$ the decay rates, which have electric, magnetic and chiral contributions. Our model hence takes into account the dissipation of the molecular system; in this case there is only one channel of decay due to the interaction of the molecule with the bath of electromagnetic modes.
We integrate these equations of motion with respect to the time, starting from the initial time $t_0=0$ at which the molecule and the field are uncoupled, to obtain the free and induced flip operator and electromagnetic fields: $$\begin{aligned}
\nonumber
& \left\langle A_{mn}\left( t \right)\right\rangle = \left\langle
A_{mn}^{\left( 0 \right)}\left( t \right)\right\rangle + \frac{\text{i}}{\hbar }\int\limits_0^t \text{d} t_1f_{mn}\left( t - t_1 \right)\\ \nonumber
& \quad \times \left\langle \mathbf{K}_{mn}\left( t_1 \right)
\cdot \mathbf{E}^{(0)}\left( \mathbf{r}_A,t_1 \right)
+ \mathbf{Q}_{mn}\left( t_1 \right) \cdot \mathbf{B}^{(0)}\left( \mathbf{r}_A,t_1 \right) \right\rangle \\ \nonumber
& \mathbf{E}\left( \mathbf{r},\omega \right) = \mathbf{E}^{\left( 0
\right)}\left( \mathbf{r},\omega \right)
+ \sum\limits_{m,n} \int\limits_{0}^t \mathrm{d}t_1 \mathrm{e}^{ -
\mathrm{i}\omega \left( t - t_1 \right)}A_{mn}\left( t_1 \right) \\
\nonumber
& \qquad \qquad \times \left[ \frac{\text{i}\mu _0}{\pi }\,\omega ^2
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A,\omega \right) \cdot \mathbf{d}_{mn} \right.\\
\nonumber
&\qquad \qquad \qquad \quad \left. + \frac{\mu _0}{\pi }\,\omega
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A,\omega \right)
\times \overleftarrow \nabla ' \cdot \mathbf{m}_{mn} \right] \\ \nonumber
&\mathbf{B}\left( \mathbf{r},\omega \right) = \mathbf{B}^{\left( 0
\right)}\left( \mathbf{r},\omega \right)
+ \sum\limits_{m,n} \int\limits_{0}^t \mathrm{d}t_1 \mathrm{e}^{ -
\mathrm{i}\omega \left( t - t_1 \right)}A_{mn}\left( t_1 \right) \\ \nonumber
& \qquad \qquad \times \left[ \frac{\mu _0}{\pi }\,
\omega \nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A,\omega \right) \cdot \mathbf{d}_{mn} \right. \\
&\qquad \qquad \qquad \quad \left. - \frac{\text{i}\mu _0}{\pi }\,
\nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A,\omega \right)
\times \overleftarrow \nabla ' \cdot \mathbf{m}_{mn} \right]
\label{eqn1}\end{aligned}$$ where $\textbf{d}_{mn}$, $\textbf{m}_{mn}$ are the matrix elements of the electric and magnetic dipole operators between the states $\left| m \right\rangle $ and $\left| n \right\rangle $. We will consider time-reversal symmetric systems, where $\textbf{d}_{mn}$ is real and $\textbf{m}_{mn}$ purely imaginary [@lloyd] ($\textbf{d}_{mn}=\textbf{d}_{nm}$, $\textbf{m}_{mn}=-\textbf{m}_{nm}$). Furthermore, we have defined the function: $$\displaystyle
f_{mn}\left( t \right) = \text{e}^{\left[ \text{i}\tilde \omega_{mn} -
\left( \Gamma _n +\Gamma _m
\right)/2 \right]t}$$ The flip operator and the fields are the sum of the free terms, as they would be in absence of coupling, and induced terms. The molecule and field systems depend on their history because of their coupling.
\[Sec3\]Dynamical Casimir–Polder force
======================================
We consider the electromagnetic force due to the interaction of a molecule exhibiting electric, magnetic and chiral polarisabilities with the body-assisted field. The field is in a thermal state with temperature $T$, while the molecule is in a generic internal state.
The CP force between the molecule and the bodi(es) is due to the exchange of a single photon: it is emitted, reflected by the bodi(es) and reabsorbed by the molecule (Fig. \[fig\]). The respective Feynman diagram must contain two interaction vertices which represent the emission and reabsorption of one photon. The electric contribution involves two electric-dipole interactions and the magnetic contribution involves two magnetic-dipole interactions. The chiral interaction involve one electric-dipole interaction and one magnetic-dipole interaction, in other words: the interaction must depend on cross terms with one electric dipole moment and one magnetic dipole moment.
![Casimir–Polder force: exchange of a single photon between the chiral molecule and the body.[]{data-label="fig"}](fig1.pdf)
The dynamical CP force for a fixed molecule is: $$\begin{gathered}
\mathbf{F} = \left. \nabla \left\langle \mathbf{d}(t) \cdot \mathbf{E}
\left( \mathbf{r},t \right) \right\rangle \right|_{\mathbf{r} = \mathbf{r}_A}
+ \left. \nabla \left\langle \mathbf{m} (t) \cdot
\mathbf{B}\left( \mathbf{r},t \right) \right\rangle \right|_{\mathbf{r} = \mathbf{r}_A}\end{gathered}$$ where all operators are obtained by solving the Heisenberg equations and the expectation value is taken over the thermal field state and the internal molecular state.
We can express the electric field in terms of its free part and the source field due to the molecule (see Eq. (\[eqn1\])): $$\begin{gathered}
\mathbf{F}(t) = \int\limits_0^\infty \text{d} \omega \sum\limits_{m,n} \left. \nabla \left\langle
A_{mn}\left( t \right)\mathbf{d}_{mn} \cdot \mathbf{E}^{\left( 0 \right)}
\left( \mathbf{r},\omega ,t \right) \right\rangle \right|_{\mathbf{r} = \mathbf{r}_A}\\
+ \int\limits_0^\infty \text{d} \omega \sum\limits_{m,n} \nabla
\left. \left\langle A_{mn}\left( t \right)\mathbf{m}_{mn} \cdot
\mathbf{B}^{\left( 0 \right)} \left( \mathbf{r},\omega ,t \right) \right\rangle
\right|_{\mathbf{r} = \mathbf{r}_A} \\
+ \frac{\mathrm{i}\mu _0}{\pi }\sum\limits_{m,n}
\sum\limits_{p,q} \int\limits_0^\infty \mathrm{d}\omega
\int\limits_{0}^t \mathrm{d}t_1 \mathrm{e}^{ - \mathrm{i}\omega
\left( t - t_1 \right)}\left\langle A_{mn}\left( t \right)A_{pq}\left(
t_1 \right) \right\rangle \\
\times \nabla \left\{ \omega
^2 \mathbf{d}_{mn} \cdot \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right)
\cdot \mathbf{d}_{pq} \right. \\
- \mathbf{m}_{mn} \cdot \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right) \times \overleftarrow \nabla ' \cdot \mathbf{m}_{pq} \\
- \mathrm{i}\omega
\mathbf{d}_{mn} \cdot \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla
' \cdot \mathbf{m}_{pq} \\
\left.- \mathrm{i}\omega \mathbf{m}_{mn} \cdot \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right) \cdot \mathbf{d}_{pq}\right\} + \mathrm{c.c.}
\label{eq8}\end{gathered}$$ As already mentioned, $\nabla$ and $\overleftarrow \nabla '$ operators act only on the first and second arguments of the Green’s tensor, respectively: $$\begin{aligned}
\nonumber
&\nabla {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A \right) = \left. \nabla {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A\right) \right|_{\mathbf{r} = \mathbf{r}_A} \\
&\nabla '{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A \right) = \left. \nabla' {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}' \right) \right|_{\mathbf{r}' = \mathbf{r}_A}\end{aligned}$$ In the first two terms in (\[eq8\]), we use the dynamical equations (\[eqn1\]) for the flip operator: $$\begin{gathered}
\mathbf{F} (t)= \frac{\text{i}}{\hbar }\int\limits_0^\infty
\text{d}\omega \int\limits_0^\infty \text{d}\omega ^\prime
\sum\limits_{m,n} \int\limits_{0}^t \text{d}t_1 f_{mn}\left( t - t_1
\right) \\
\times \nabla \left\langle \left[ \mathbf{E}^{\left( 0 \right)\dag }
\left( \mathbf{r}_A,\omega ',t_1 \right)
\cdot \mathbf{K}_{mn}\left( t_1 \right) \right. \right.\\
\left. + \mathbf{B}^{\left( 0 \right)\dag }
\left( \mathbf{r}_A,\omega ',t_1 \right)
\cdot \mathbf{Q}_{mn}\left( t_1 \right) \right]\\
\left. \left. \times \left[ \mathbf{d}_{mn}
\cdot \mathbf{E}^{\left( 0 \right)}\left( \mathbf{r},\omega ,t \right)
+ \mathbf{m}_{mn} \cdot \mathbf{B}^{\left( 0 \right)}\left( \mathbf{r},\omega ,t \right) \right]
\right\rangle \right|_{\mathbf{r} = \mathbf{r}_A} \\
+ \frac{\mathrm{i}\mu _0}{\pi }\sum\limits_{m,n} \sum\limits_{p,q}
\int\limits_0^\infty \mathrm{d}\omega
\int\limits_{0}^t \mathrm{d}t_1 \mathrm{e}^{ - \mathrm{i}\omega
\left( t - t_1 \right)}
\left\langle A_{mn}\left( t \right)A_{pq}\left( t_1 \right)
\right\rangle \\
\times \nabla \left\{ \omega
^2 \mathbf{d}_{mn} \cdot \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right)
\cdot \mathbf{d}_{pq} \right. \\
- \mathbf{m}_{mn} \cdot \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right) \times \overleftarrow \nabla ' \cdot \mathbf{m}_{pq} \\
- \mathrm{i}\omega
\mathbf{d}_{mn} \cdot \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla
' \cdot \mathbf{m}_{pq} \\
\left.- \mathrm{i}\omega \mathbf{m}_{mn} \cdot \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right) \cdot \mathbf{d}_{pq}\right\} + \mathrm{c.c.}\end{gathered}$$ The thermal expectation value of two positive-frequency electromagnetic fields is zero.
Next, we use the known formula for the field fluctuations [@butcher]: $$\begin{aligned}
\nonumber
&\left\langle \mathbf{E}^{(0)\dag }\left( \mathbf{r},\omega
\right)\mathbf{E}^{(0)}\left( \mathbf{r}',\omega ' \right)
\right\rangle = \\ \nonumber
& \qquad \qquad \frac{\hbar \mu _0}{\pi } \,\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r},\mathbf{r}',\omega \right)\omega ^2\delta \left( \omega -
\omega ' \right)n\left( \omega \right) \\ \nonumber
&\left\langle \mathbf{E}^{(0)\dag }\left( \mathbf{r},\omega
\right)\mathbf{B}^{(0)}\left( \mathbf{r}',\omega ' \right)
\right\rangle = \\ \nonumber
& \qquad \qquad \frac{\mathrm{i}\hbar \mu _0}{\pi
} \,\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}',\omega \right)
\times \overleftarrow \nabla '\omega \delta \left(\omega - \omega '
\right) n\left( \omega \right)\\ \nonumber
&\left\langle \mathbf{B}^{(0)\dag }\left( \mathbf{r},\omega
\right)\mathbf{E}^{(0)}\left( \mathbf{r}',\omega ' \right)
\right\rangle= \\ \nonumber
& \qquad \qquad \frac{\mathrm{i}\hbar \mu _0}{\pi } \, \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}',\omega \right)
\omega \delta \left( \omega - \omega ' \right)n\left( \omega \right)
\\ \nonumber
&\left\langle \mathbf{B}^{(0)\dag }\left( \mathbf{r},\omega
\right)\mathbf{B}^{(0)}\left( \mathbf{r}',\omega ' \right)
\right\rangle = \\
& \qquad \qquad - \frac{\hbar \mu _0}{\pi } \,\nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}',\omega \right)
\times \overleftarrow \nabla '\delta \left( \omega - \omega '
\right) n\left( \omega \right)\end{aligned}$$ where $n\left( \omega \right)$ is the Bose-Einstein distribution: $$n\left( \omega \right) = \frac{1}{\mathrm{e}^{\hbar \omega /k_BT} -
1}\,.$$ We also perform the expectation value on the internal molecular state, which is an incoherent superposition of energy eigenstates $\left| n \right\rangle $, with probabilities $p_n$. Furthermore, the two-time correlation function can be simplified with the Lax regression theorem [@onsager; @lax] ($t_1
\leq t$): $$\begin{gathered}
\left\langle A_{mn}\left( t \right)A_{pq}\left( t_1 \right)
\right\rangle = f_{mn}\left( t - t_1 \right)\left\langle A_{mn}\left(
t_1 \right)A_{pq}\left( t_1 \right) \right\rangle =\\ f_{mn}\left( t -
t_1 \right)\delta _{np}\left\langle A_{mq}\left( t_1 \right)
\right\rangle \end{gathered}$$ Approximating $\left\langle A_{mn}\left(t_1 \right) \right\rangle \simeq \text{e}^{\text{i}\tilde \omega_{mn}\left( t_1 - t \right)}\left\langle A_{mn}\left( t \right) \right\rangle $, the CP force is a weighted sum of the CP forces associated with each eigenstate $
\mathbf{F} (t)= \sum\limits_n p_n(t) \mathbf{F}_n(t) $: $$\begin{gathered}
\mathbf{F}_n(t) = \frac{\text{i}\mu _0}{\pi }\int\limits_0^\infty
\text{d}\omega \sum\limits_k \int\limits_{0}^t \text{d}t_1
\text{e}^{ - \text{i}\omega \left( t - t_1 \right)}\\
\times \left\{ f_{nk}\left(
t - t_1 \right)\left[ 1 + n\left( \omega \right) \right] -
f_{kn}\left( t - t_1 \right)n\left( \omega \right) \right\} \\
\nonumber
\times \nabla \left\{ \omega
^2 \mathbf{d}_{nk} \cdot \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right)
\cdot \mathbf{d}_{kn} \right. \\
- \mathbf{m}_{nk} \cdot \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right) \times \overleftarrow \nabla ' \cdot \mathbf{m}_{kn} \\
\left.- 2 \mathrm{i}\omega \mathbf{m}_{nk} \cdot \nabla \times
\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right) \cdot \mathbf{d}_{kn}\right\} + \mathrm{c.c.}\end{gathered}$$ where $p_n\left( t \right)=\left\langle A_{nn}\left( t \right) \right\rangle $ is the population of energy-state $\left| n \right\rangle $. To obtain this expression we have used the Green tensor reciprocity theorem ${\mbox{\textbf{\textit{\textsf{G}}}}}^T\left( \mathbf{r},\mathbf{r}' \right) = {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}',\mathbf{r} \right)$, and for the chiral part the property $\left[ {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla ' \right]^T$ $= - \nabla \times {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right)$. The term proportional to the Bose-Einstein distribution describes the interaction between the molecule and the thermal field, while the term independent of $n$ describes the interaction with the vacuum field.
Now we introduce $\nabla_A$ which acts on both arguments of the Green tensor. Exploiting Onsager reciprocity, the relation $$\begin{gathered}
\nabla _A {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A \right) \\
= \left. \nabla {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r},\mathbf{r}_A \right) \right|_{\mathbf{r} = \mathbf{r}_A}
+ \left. \nabla ' {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}' \right) \right|_{\mathbf{r}' = \mathbf{r}_A} \\
= \nabla {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A \right)
+ \nabla '{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A \right)\\
= \nabla {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A \right)
+ \nabla {\mbox{\textbf{\textit{\textsf{G}}}}}^T\left( \mathbf{r}_A,\mathbf{r}_A \right)\end{gathered}$$ holds. For a time-reversal symmetric molecule, we can hence make the replacement: $$\nabla {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \to
\frac{1}{2}\nabla_A {\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right)$$ After performing the time-integrals we obtain: $$\begin{aligned}
\nonumber
\mathbf{F}(t) &= \sum\limits_n p_n (t)\left[ \mathbf{F}_n^e(t) + \mathbf{F}_n^m(t) + \mathbf{F}_n^c(t) \right], \\ \nonumber
\mathbf{F}^e_n (t)&= \frac{\mu _0}{2\pi }\int\limits_0^\infty
\text{d}\omega \omega^2 \sum\limits_k \Psi_{kn}(\omega, t)\\ \nonumber
& \qquad \times \nabla_A \mathbf{d}_{nk}\textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right) \mathbf{d}_{kn}, \\ \nonumber
\mathbf{F}^m_n (t)&=- \frac{\mu _0}{2\pi }\int\limits_0^\infty
\text{d}\omega \sum\limits_k \Psi_{kn}(\omega, t) \\ \nonumber
& \qquad \times \nabla_A \mathbf{m}_{nk}\nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla
'\mathbf{m}_{kn}, \\ \nonumber
\mathbf{F}^c_n (t)&= -\frac{\text{i}\mu _0}{\pi }\int\limits_0^\infty
\text{d}\omega \omega \sum\limits_k \Psi_{kn}(\omega, t) \\
& \qquad \times \nabla_A \mathbf{m}_{nk}\nabla
\times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega
\right)\mathbf{d}_{kn}\end{aligned}$$ where $$\begin{gathered}
\Psi_{kn}(\omega, t)= \frac{1 - \text{e}^{ -
\text{i}t\left( \omega + \omega _{kn}^{\left( - \right)}
\right)}}{\omega + \omega _{kn}^{\left( - \right)}}\left[ 1 +
n\left( \omega \right) \right] \\
- \frac{1 - \text{e}^{ -
\text{i}t\left( \omega - \omega _{kn}^{\left( + \right)}
\right)}}{\omega - \omega _{kn}^{\left( + \right)}}n\left( \omega
\right) +\mathrm{c.c.} \end{gathered}$$ and $\omega _{kn}^{\left( \pm \right)} = \tilde \omega _{kn} \pm
\text{i}\left( \Gamma _n + \Gamma _k \right)/2$. As explained before, for time-reversal symmetric systems, the electric dipole elements are real and the magnetic dipole elements purely imaginary. We have separated the electric, magnetic and chiral contributions: electric contribution contains two electric dipole moments, the magnetic contribution two magnetic dipole moments and chiral contribution contains cross terms with one electric dipole moment and one magnetic dipole moment.
We observe that the force depends on time in two ways: firstly, the populations of the internal molecular states may depend on time. For example, a molecule initially prepared in some excited state will unavoidably decay to the ground state, so the population of the excited state is one for short times but zero for large times. The time scale of this population-induced dynamics of the force is set by the life times of the initially populated states. For a ground-state molecule the populations of the energy levels are constant in time and there is no population-induced dynamics.
The time-dependent exponentials on the other hand describe the dynamical self-dressing of the molecule which is the focus of this work. The self-dressing dynamics operates on the much shorter time scales of the order of the inverse molecular transition frequencies $1/\omega_{nk}$. The dynamical self-dressing has been considered for a single non-absorbing electric molecule in front a plate [@vasile]; our approach is generalized for finite temperature, arbitrary geometry of the body and it accounts for molecular absorption.
For times much larger than $1/\omega_{nk}$, the exponential function is rapidly oscillating and averages to zero. The electric part of the CP force then converges to the value obtained in the literature with a dynamical approach where the time-dependence is solely due to population-induced dynamics [@buh; @Buhmann2]. Our new results hence generalise the previous dynamical approach to include the self-dressing of the molecule as well as the magnetic and the chiral parts of the interactions. The static limit of our result for the chiral contribution extends previous results from time-independent perturbation theory to finite temperature and absorbing molecules [@butcher].
As a simple example, we consider an isotropic non-absorbing molecule. The electric, magnetic and chiral parts of the dynamical Casimir–Polder interaction are: $$\begin{aligned}
\nonumber
\mathbf{F}_n (t) &=\mathbf{F}_n^e (t) +\mathbf{F}_n^m (t) +\mathbf{F}_n^c(t) \\ \nonumber
\quad \mathbf{F}_n^e (t) &= \frac{\mu _0}{3\pi }\int\limits_0^\infty \mathrm{d}\omega \omega ^2\sum\limits_k \mathbf{d}_{nk} \cdot \mathbf{d}_{kn} \Psi' _{kn}\left( \omega ,t \right) \\ \nonumber
&\quad \times \nabla_A \textup{Tr}\left\{ \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left(
\mathbf{r}_A,\mathbf{r}_A,\omega \right) \right\}, \\ \nonumber
\quad \mathbf{F}_n^m(t) &= - \frac{\mu _0}{3\pi }\int\limits_0^\infty \text{d} \omega \sum\limits_k \mathbf{m}_{nk} \cdot \mathbf{m}_{kn} \Psi' _{kn}\left( \omega ,t \right)\\ \nonumber
&\quad \times \nabla_A \textup{Tr}\left\{\nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \times \overleftarrow \nabla ' \right\}, \\ \nonumber
\quad \mathbf{F}_n^c (t) &=- \frac{2\mu _0}{3\pi }\int\limits_0^\infty \mathrm{d}\omega \omega \sum\limits_k R_{nk}\Psi' _{kn}\left( \omega ,t \right) \\
&\quad \times \nabla_A \textup{Tr} \left\{ \nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \right\}
\label{form1}\end{aligned}$$ where $$\begin{gathered}
\Psi ' _{kn}\left( \omega ,t \right) = \frac{1 + n\left( \omega \right)}{\omega _{kn} + \omega }\left( 1 - \cos \left[ \left( \omega _{kn} + \omega \right)t \right] \right)\\ + \frac{n\left( \omega \right)}{\omega _{kn} - \omega} \left( 1 - \cos \left[ \left( \omega _{kn} - \omega \right)t \right] \right) ,\end{gathered}$$ $\textup{Tr}$ is the trace, ${R_{nk}} = \operatorname{Im} \left( {{{\mathbf{d}}_{nk}} \cdot {{\mathbf{m}}_{kn}}} \right)$ is the rotatory strength and $\omega_{kn}$ the transition frequency between the state $\left| k \right\rangle $ and $\left| n \right\rangle $.\
If either the medium ($\textup{Tr} \left\{ \nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \right\} = 0$), or the particle ($R_{nk} = 0$) is achiral there will be no chiral component to the Casimir–Polder potential. This can be thought of as an application of to the Curie dissymmetry principle (originally formulated for crystal symmetries): the CP potential cannot distinguish between molecules of different handedness if the medium does not possess chiral properties itself.
Under reflection, the electric dipole moment changes sign, while the magnetic dipole moment does not. The electric and magnetic parts of the dynamical interaction hence do not change if the molecule is substituted with its enantiomer (mirror image), but the chiral part of the CP force changes sign. This shows the discriminatory effect for the chiral part of the dynamical interaction.
\[Sec4\]Chiral molecule in front a perfect mirror
=================================================
The interaction between a ground-state electric molecule and a perfectly conducting electric plate at zero-temperature has been investigated in the literature [@vasile]; the results can be recovered with our model but we will not focus on this point here. We consider instead the interaction between a ground-state chiral molecule and and a perfectly reflecting chiral plate at zero temperature, $n(\omega)=0$. As the population of the ground state is constant in time, the only dynamics of the Casimir–Polder force arises due to self-dressing.
The Green’s tensor of the perfectly reflecting chiral plate is known, and it depends only in the distance $d$ between the molecule and the mirror [@butcher; @Buhmann]: $$\begin{gathered}
\frac{\partial }{\partial d}\left\{ \omega \nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \right\}= \\
\pm \frac{3c}{8\pi d^4}\left[ \cos x + x\sin x -
\frac{1}{3} \,x^2\cos x \right]_{x = 2d\omega /c}
\label{eq13}\end{gathered}$$ where the sign $+$ or $-$ refers to plates of positive and negative chirality, respectively. Note that the trace of the Green tensor scales differently for small and larges distances leading to different dependences of the force on the distance.
After inserting this expression into Eq. (\[form1\]), we next need to perform the frequency integrals for the three terms in the above Eq. (\[eq13\]). This task can be simplified considerably by expressing the Green’s tensor in terms of a differential operator: $$\begin{gathered}
\frac{\partial }{\partial d}\left\{ \omega \nabla \times \textup{Im}{\mbox{\textbf{\textit{\textsf{G}}}}}\left( \mathbf{r}_A,\mathbf{r}_A,\omega \right) \right\} = \\
\pm \frac{3c}{8\pi d^4} \,\mathop {\lim }\limits_{m \to 1} \left[ 1 -
\frac{\partial }{\partial m} + \frac{1}{3} \frac{\partial ^2}{\partial
m^2} \right]\left. \cos \left( mx \right) \right|_{x = 2d\omega /c}\end{gathered}$$ Inserting the Green’s tensor in this form, the chiral interaction reads: $$\begin{gathered}
\mathbf{F}^c = \mp \frac{1}{4\pi ^2\varepsilon _0cd^4} \,
\mathop {\lim }\limits_{m \to 1} \left[ 1 - \frac{\partial }{\partial
m} + \frac{1}{3}\frac{\partial ^2}{\partial m^2} \right] \\
\times \sum\limits_k R_{0k} \int\limits_0^\infty \text{d} x \frac{\cos \left( mx \right)}{x + x_k}
\left( 1 - \cos \left[ \left( x + x_k \right)a \right] \right) \hat d \end{gathered}$$ where $x = 2d\omega /c$, $x_k = 2d\omega_k /c$ and $a=ct/(2d)$ and $\hat d= \textbf{d}/d$.
Stationary case: large times
----------------------------
For times much larger than $1/\omega_k$, the cosine function oscillates rapidly and its contribution vanishes; this situation corresponds to a totally dressed molecule.
We introduce the auxiliary functions, for $m,y>0$: $$\begin{aligned}
\nonumber
&\text{F}\left( m,y \right) = \int\limits_0^\infty \mathrm{d}x
\frac{\sin \left( mx \right)}{x + y} \\ \nonumber
&\qquad =\sin \left( my \right)\operatorname{Ci}\left(
my \right) - \cos \left( my \right)\left[ \operatorname{Si} \left( my \right) -
\frac{\pi }{2} \right] \\ \nonumber
&\text{G}\left( m,y \right) = \int\limits_0^\infty \mathrm{d}x
\frac{\cos \left( mx \right)}{x + y} \\
&\qquad = - \cos \left( my
\right)\operatorname{Ci}\left( my \right) - \sin \left( my \right)\left[ \operatorname{Si} \left( my
\right) - \frac{\pi }{2} \right]\end{aligned}$$ where $\operatorname{Si}$ and $\operatorname{Ci}$ are the sine and cosine integral functions. For large times the CP force converges to the following static force: $$\begin{gathered}
\nonumber
\mathbf{F}^c_{t \to \infty } =\mp \frac{1}{4\pi ^2\varepsilon
_0cd^4} \, \sum\limits_k R_{0k}\\
\times \mathop {\lim }\limits_{m \to 1} \left[ 1 - \frac{\partial
}{\partial m} + \frac{1}{3}\frac{\partial ^2}{\partial m^2}
\right]\text{G}\left( m,x_k \right) \hat d \\
= \mp \frac{1}{3\pi ^2\varepsilon _0cd^4}\sum\limits_k R_{0k}\\
\times \left[ 1 - 2\operatorname{Ci}\left( 2k_kd \right)f\left( k_kd \right) + \left(
2\operatorname{Si}\left( 2k_kd \right) - \pi \right)g\left( k_kd \right) \right]
\hat d \end{gathered}$$ where $k_k = \frac{\omega _k}{c}$ is the molecular wave number and we have introduced the auxiliary functions: $$\begin{aligned}
\nonumber
&f\left( x \right) = \frac{3x}{4} \,\sin \left( 2x \right) + \left(
\frac{3}{8} - \frac{x^2}{2} \right)\cos \left( 2x \right) \\
& g\left( x \right) = \frac{3x}{4} \, \cos \left( 2x \right) - \left(
\frac{3}{8} - \frac{x^2}{2} \right)\sin \left( 2x \right)
\label{fg}\end{aligned}$$ This is an alternative, slightly more explicit form for the result known in the literature [@Buhmann; @butcher].
As an example of a chiral molecule, consider dimethyl disulphide $(\mathrm{CH}_3)_2\mathrm{S}_2$. The dipole and rotatory strengths for each transition have been numerically calculated for various orientations [@rauk]. As an example, we have chosen the first transition when the orientation between the two $\mathrm{CH}_3-\mathrm{S}-S$ planes is $90^\circ$. The transition frequency between the excited state and the ground state is $\omega_{10} = 9.17 \cdot 10^{15} \mathrm{Hz}$, the square of the dipole moment $\left| \mathbf{d}_{01} \right|^2 = 8.264 \cdot 10^{ -
60}\left( \mathrm{Cm} \right)^2$ and the rotatory strength is $R_{10} = 3.328 \cdot 10^{-64}
\mathrm{C}^2\mathrm{m}^3\mathrm{s}^{-1}$. Fig. \[fig1\] shows the chiral Casimir–Polder force for a ground-state dimethyl disulphide molecule above a perfect mirror of negative chirality. The stationary chiral CP force between the ground-state molecule and the medium is repulsive due to the chosen opposite chiralities of molecule and mirror. This differs from the CP interaction between an electric molecule and a perfectly conducting electric plate, which is attractive [@CasimirPolder48; @barton; @vasile].
![Stationary chiral Casimir–Polder interaction between a ground-state dimethyl disulphide and a perfect mirror of negative chirality.[]{data-label="fig1"}](fig2.pdf)
The distance-dependence of the chiral CP force can be reduced to simple power laws in the retarded and non-retarded limits. In the non-retarded limit $d
\ll \lambda_k$ or equivalently ${x_k} \ll 1$, we may approximate $\mathrm{G}\left( m,x_k \right) \to - \gamma - \log \left( mx_k
\right)$ where $\gamma$ is the Euler–Mascheroni constant. The force is then [@butcher; @Buhmann]: $$\mathbf{F}_{non - ret} = \pm \frac{1}{4\pi
^2\varepsilon _0cd^4}\sum\limits_k R_{0k}\log \left( \frac{\omega
_kd}{c} \right) \hat d$$
In the retarded limit $d \gg \lambda_k$ or equivalently $x_k \gg 1$, we have $\mathrm{G}\left( m,x_k \right) \to \frac{1}{\left(
mx_k
\right)^2}$ and the force is [@butcher; @Buhmann]: $$\mathbf{F}_{ret} = \mp \frac{5c}{16\pi ^2\varepsilon
_0d^6}\sum\limits_k \frac{R_{0k}}{\omega _k^2} \, \hat d$$ As expected, the retarded interaction decreases more rapidly due to the finite velocity of the light: during the time in which the virtual photon has been exchanged, the molecule will evolve. This associated loss of correlation leads to a more rapidly decreasing force.
Due to the unusual $\log \left( \frac{\omega _kd}{c} \right)/d^4$ dependence of the force in the non-retarded limit, the chiral potential grows more rapidly than the electric and magnetic potentials when approaching the surface.
The chiral force changes sign if the molecule is substitute with its entaniomer, or when the plate of negative chirality is substituted with one of negative chirality. This discriminatory effect is also observed in the dynamic case, which we will consider in the next section.
Dynamical case
--------------
We now consider the dynamical situation in which the interaction with the perfect chiral plate starts at the initial time $t_0=0$ and we ask for the dynamical Casimir–Polder force between the chiral molecule and the mirror. In this case, the bare molecular state is not an eigenstate of the total Hamiltonian, and thus it evolves in time (dynamical self-dressing), yielding a time-dependent force between the mirror and the molecule.
To evaluate the force, we use the trigonometric relation $$\begin{gathered}
\nonumber
\cos \left( mx \right)\left( 1 - \cos \left[ \left( x + x_k \right)a
\right] \right) = \cos \left( mx \right)\\
- \frac{\cos \left( ax_k
\right)}{2} \,\left\{ \cos \left[ \left( m + a \right)x \right] + \cos
\left[ \left( m - a \right)x \right] \right\} \\
+ \frac{\sin \left( ax_k \right)}{2} \,\left\{ \sin \left[ \left( m + a
\right)x \right] - \sin \left[ \left( m - a \right)x \right] \right\}\end{gathered}$$ The force has two different expressions before and after the back-reaction time ($t=2d/c$), which is the time needed for light emitted by the atom to be reflected by the mirror and return to the molecule. For $t<2d/c$ and $t>2d/c$ the chiral dynamical CP force is:
$$\begin{aligned}
\nonumber
\mathbf{F}^c_{t < 2d/c} &=\mp \frac{1}{4\pi ^2\varepsilon
_0cd^4}\sum\limits_k R_{0k} \mathop {\lim }\limits_{m \to 1} \left[ 1
- \frac{\partial }{\partial m} + \frac{1}{3}\frac{\partial
^2}{\partial m^2} \right] \left\{ \mathrm{G}\left( m,x_k \right)
+ \right. \\ \nonumber
& \qquad \left. - \frac{\cos \left( ax_k \right)}{2} \, \left[ \mathrm{G}\left( m
+ a,x_k \right) + \mathrm{G}\left( m - a,x_k \right) \right] +
\frac{\sin \left( ax_k \right)}{2} \, \left[ \mathrm{F}\left( m + a,x_k
\right) - \mathrm{F}\left( m - a,x_k \right) \right] \right\} \hat d
\\ \nonumber
&= \mp \frac{1}{3\pi ^2\varepsilon _0cd^4}\sum\limits_k R_{0k} \left[
1 - \frac{8d^4\cos \left( \omega _kt \right)}{\left( c^2t^2 - 4d^2
\right)^2} \right. + \frac{2d^2\cos \left( \omega _kt \right) +
d^2\omega _kt\sin \left( \omega _kt \right)}{c^2t^2 - 4d^2} \\
\nonumber
& \qquad - \left[ 2\operatorname{Ci}\left( 2k_kd \right) - \operatorname{Ci}\left( 2k_kd - \omega
_kt \right) - \operatorname{Ci}\left( 2k_kd + \omega _kt \right) \right]f\left( k_kd
\right) \\
& \qquad \left. + \left[ 2\operatorname{Si}\left( 2k_kd \right) - \operatorname{Si}\left( 2k_kd -
\omega _kt \right) - \operatorname{Si}\left( 2k_kd + \omega _kt \right)
\right]g\left( k_kd \right) \right]\hat d\end{aligned}$$
$$\begin{aligned}
\nonumber
\mathbf{F}^c_{t > 2d/c} &=\mp \frac{1}{4\pi ^2\varepsilon
_0cd^4}\sum\limits_k R_{0k} \mathop {\lim }\limits_{m \to 1} \left[ 1
- \frac{\partial }{\partial m} + \frac{1}{3}\frac{\partial
^2}{\partial m^2} \right] \left\{ \mathrm{G}\left( m,x_k \right)
+ \right. \\ \nonumber
&\qquad \left. - \frac{\cos \left( ax_k \right)}{2} \, \left[ \mathrm{G}\left(
a+m,x_k \right) + \mathrm{G}\left( a-m,x_k \right) \right] +
\frac{\sin \left( ax_k \right)}{2} \, \left[ \mathrm{F}\left( a+m,x_k
\right) + \mathrm{F}\left( a-m,x_k \right) \right] \right\} \hat d \\
\nonumber
&= \mp \frac{1}{3\pi ^2\varepsilon _0cd^4}\sum\limits_k R_{0k} \left[
1 - \frac{8d^4\cos \left( \omega _kt \right)}{\left( c^2t^2 - 4d^2
\right)^2} \right. + \frac{2d^2\cos \left( \omega _kt \right) +
d^2\omega _kt\sin \left( \omega _kt \right)}{c^2t^2 - 4d^2} \\
\nonumber
& \qquad - \left[ 2\operatorname{Ci}\left( 2k_kd \right) - \operatorname{Ci}\left( \omega _kt-2k_kd
\right) - \operatorname{Ci}\left( \omega _kt+2k_kd \right) \right]f\left( k_kd
\right) \\
& \qquad \left. + \left[ 2\operatorname{Si}\left( 2k_kd \right) + \operatorname{Si}\left( \omega
_kt-2k_kd \right) - \operatorname{Si}\left( \omega _kt+2k_kd \right)-\pi
\right]g\left( k_kd \right) \right]\hat d\end{aligned}$$
where the functions $f,g$ are defined by Eq. (\[fg\]). It is easy to show that $\mathbf{F} \to 0$ for $t \to 0$; this is due to the fact that we switch on the interaction at the initial time $t_0=0$.
For subsequent times, the force increases exhibiting an oscillatory behaviour in time. Depending on the time, the force can be attractive and repulsive for a given distance, contrary to the static case where it has a definite sign. This is illustrated in Fig. \[fig2\], where we display the chiral CP force at fixed distance from the mirror before the back-reaction time.
![Chiral dynamical Casimir–Polder interaction between an ground-state dimethyl disulphide and a negative perfect chiral medium for $d=0.1 \mu\mathrm{m}$ and $t<2d/c=0.67 \mathrm{fs}$.[]{data-label="fig2"}](fig3.pdf)
\
For large times, the force converges to the static force; this corresponds to a totally dressed molecule. Fig. \[fig3\] shows the dynamics of the chiral force after the backreaction.
![Chiral dynamical Casimir–Polder interaction between an ground-state dimethyl disulphide and a negative perfect chiral medium for $d=0.1\,\mu\mathrm{m}$ and $t>2d/c=0.67 \mathrm{fs}$.[]{data-label="fig3"}](fig4.pdf)
To interpret our results, recall that the CP force is due to the exchange of one virtual photon between the molecule and the mirror. Different expressions for the force are needed before and after the backreaction time because the photon needs a finite time in order to be reflected and absorbed by the molecule. Note that the force is non-vanishing even before the backreaction time, because the molecule interacts with the field modes which incorporate the presence of the conducting wall and hence instantaneously feels the presence of the mirror. This is because we are evaluating the force on the atom, which responds to the local field at its position and thus it is immediately influenced by a change of the atom’s physical parameters. We expect that, if the force on the conducting wall were evaluated, it would be influenced by a change of the atomic parameters (or by a sudden switching on of the atom-field interaction) only after the causality time $t=d/c$.
The force is divergent on the light cone $t=2d/c$, because in the frequency-integral we include arbitrarily large frequencies: this divergence is not surprising, being due to the assumption of a point-like molecule (dipole approximation) and to the idealised nature of the material. A real material is transparent for large frequencies, providing a natural cut-off to regularize the integral; moreover, also the inclusion of a finite size of the atom/molecule would provide a natural ultraviolet cutoff given by the appropriate atomic form factor.
\[Sec5\]Conclusions
===================
Using a dynamical approach, we have obtained the electric, magnetic and the chiral parts of the time-dependent Casimir–Polder interaction between an initially bare chiral molecule and a body at finite temperature, the molecule initially being prepared in a generic internal state. The force depends on time because the populations of the excited states of the molecule depend on time (population-induced dynamics), but also because of the initial boundary condition (self-dressing induced dynamics).
As an example we have considered the particular case of the interaction between an initially bare ground-state chiral molecule and perfectly reflecting chiral plate at zero temperature. Here, the force is time-dependent only because of self-dressing. The dynamical CP can be attractive or repulsive depending on time, contrary to the static case where it has a definite sign for a given distance.
The dynamical interaction is due to the exchange of one virtual photon between the chiral molecule and the mirror. A characteristic time scale of the dynamical CP force is the time taken by the virtual photon to be emitted by the molecule, reflected by the mirror, and reabsorbed by the molecule (back-reaction time). The dynamical interaction oscillates in time and the scale of these oscillations is related to the molecular transition frequency. The dynamical effect we have considered could in principle be measured by switching on the interaction between the molecule and the field at the initial time $t_0=0$. Even if this is an idealized situation, it could be approximated by the more realistic case of a rapid change of some parameter characterizing the atom–field interaction (strength and/or orientation of the atomic dipole moments, atomic transition frequency by Stark shift, for example) or putting the atom at some distance from the macroscopic body [@MPRSV14]. Another possibility to obtain a dynamical effect could be introducing the mirror at $t_0=0$; however, because in our formalism the presence of the mirror is included in the boundary conditions and not in the system’s dynamics, this would require a different approach based on a transformation of field operators and modes relating old and new ones (i.e. before and after switching on the mirror), similarly to the dynamical Casimir effect. The use of chiral Rydberg atoms, which have low transition frequencies and large polarizabilities, could be a simpler system to measure the dynamical Casimir force, because in this case the dynamical force evolves on longer timescales [@gallagher] ($\tau=10^{-9}s$).
We have shown that the chiral Casimir–Polder interaction shows a discriminatory effect because it changes sign if the molecule is substituted by its enantiomer and the dynamical force can hence allow us to distinguish different enantiomers.
We finally remark that in our approach the medium is considered macroscopically with the electromagnetic Green tensor. Our model can be generalized to different molecular internal states and different geometries in which the Green’s tensor is known.
Acknowledgement
===============
RP and LR gratefully acknowledge financial support by the Julian Schwinger Foundation and by MIUR. SYB is grateful for support by the DFG (grant BU 1803/3-1) and the Freiburg Institute for Advanced Studies.
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|
---
abstract: |
This paper considers a network formation model when links are measured with error. We focus on a game-theoretical model of strategic network formation with incomplete information, in which the linking decisions depend on agents’ exogenous attributes and endogenous positions in the network. In the presence of link misclassification, we derive moment conditions that characterize the identified set of the preference parameters associated with homophily and network externalities. Based on the moment equality conditions, we provide an inference method that is asymptotically valid with a single network of many agents.
Keywords: Misclassification, Network formation models, Strategic interactions, Incomplete information
author:
- 'Luis E. Candelaria[^1]'
- 'Takuya Ura[^2]'
bibliography:
- '../bib\_missperp.bib'
title: Identification and Inference of Network Formation Games with Misclassified Links
---
Introduction {#sec1}
============
Researchers across different disciplines have recorded that measurement error of links is a pervasive problem in network data (e.g., @holland/leinhardt:1973, @moffitt2001policy, @kossinets:2006, @ammermueller/pischke:2009, @wangetal:2012, @angrist:2014, @depaula:2017, @advani/malde:2018). Although strategic network formation models are essential for learning about the creation of linking connections and peer effects with an endogenous network structure, to the best of our knowledge, there has been no work addressing the misclassification problem in strategic network formation models. In this paper, we consider identification and inference in a game-theoretical model of strategic network formation with potentially misclassified links.
We focus on a simultaneous game with imperfect information in which agents decide to form connections to maximize their expected utility (cf. @leung2015two, and @ridder/sheng:2015). The agents’ decisions are interdependent since the utility attached to establishing a specific link depends on the agents’ observed attributes and positions in the network through link externalities (such as reciprocity, in-degree, and out-degree statistics). The misclassification problem will affect the decisions to forming network links in two different ways. First, the binary outcome variable of link, which represents an individual’s optimal decision, is misclassified. Second, the link misclassification problem prevents us from directly identifying the belief system of an agent about others’ linking decisions, which the agent’s decision relies on. In this sense, the measurement error problem occurs on the left- and right-hand side of the equation describing the linking decisions as in Lemma \[lemma1\].
We propose a novel approach for analyzing network formation models, which is robust to misclassification. Specifically, we characterize the identified set for the structural parameters, including the preference parameters concerning homophily and network externalities. A notable innovation from our approach is that we derive the relationship between the choice probabilities of observed network connections and the belief system (Lemma \[belief\_lemma\]). This result is crucial to control for the endogeneity of the equilibrium beliefs and reduce the model to a single agent decision model in the presence of misspecification.
We also introduce an inference method that is asymptotically valid as long as we observe one network with a large number of agents. Given a finite support of the exogenous attribute, the identified set is characterized by a finite number of unconditional moment equalities. Based on these moment equalities, we construct a test statistic whose asymptotic null distribution is the $\chi^2$ distribution with known degrees of freedom.
Our methodology contributes to the growing econometric literature that studies strategic formation of networks. See @graham:2015, @chandrasekhar:2016, and @depaula:2017 for a recent survey. The network formation model considered in this paper builds on the framework of strategic interactions with incomplete information introduced by @leung2015two and extended by @ridder/sheng:2015. The analysis in our paper addresses the problems arising due to link misclassification in their models.
This paper is also related to the literature of mismeasured discrete variables, e.g., misclassified binary outcome variable [@hausman/abrevaya/schott-morton:1998], and misclassified discrete treatment variable [@mahajan:2006; @lewbel:2007; @chen/hu/lewbel:2008; @hu:2008]. Specifically, our approach to misclassified links is based on [@molinari:2008], which offers a general bounding strategy with misclassified discrete variables.
There are a few papers in the literature of social interactions that have examined the presence of measurement error in network data ([@Chandrasekhar_Lewis_2014; @kline2015identification; @LewbelNorrisPendakurQu2018]). However, the results in these papers cannot be applied directly to our framework since they have a different object of interest. In particular, they have primarily focused on studying peer effects given a network of interactions, and do not investigate directly the underlying process that drives the formation of the network connections. In contrast, our paper studies the effects of link misclassification in a model of strategic network formation.
The remainder of this paper is organized as follows. Section 2 describes the network formation model as a game of incomplete information. Section 3 characterizes the identified set of the structural parameters. Section 4 introduces an inference method based on the representation of the identified set.
Network formation game with misclassification {#sec2}
=============================================
We extend [@leung2015two] and [@ridder/sheng:2015] to model the formation of a directed network with misclassified links. Particularly, our approach follows [@leung2015two] for simplicity.
The network consists of a set of $n$ agents, which we denote by $\mathcal{N}_n=\{1,\ldots,n\}$. We assume that each pair of agents $(i,j)$ with $i,j \in \mathcal{N}_n$ is endowed with a vector of exogenous attributes $X_{ij} \in \mathbb{R}^d$ and an idiosyncratic shock $\varepsilon_{ij} \in \mathbb{R}$. Let $X=\{X_{ij}: i \in \mathcal{N}_{n}\}\in \mathcal{X}^n$ be a profile of attributes that is common knowledge to all the agents in the network, and $\varepsilon_i=\{\varepsilon_{ij}:j\in\mathcal{N}_n\}$ is a profile of idiosyncratic shocks that is agent $i$’s private information. Let $\varepsilon=\{\varepsilon_i:i\in\mathcal{N}_n\}$.
The network is represented by a $n\times n$ adjacency matrix $G_n^{\ast}$, where the $ij$th element $G^{\ast}_{ij,n} = 1$ if agent $i$ forms a direct link to agent $j$ and $G^{\ast}_{ij,n}=0$ otherwise. We assume that the network is directed, i.e., $G^{\ast}_{ij,n}$ and $G^{\ast}_{ji,n}$ may be different. The diagonal elements are normalized to be equal to zero, i.e., $G^\ast_{ii,n}=0$. The network $G_{n}^\ast$ is potentially misclassified and the researcher observes $G_{n}$, which is a proxy for $G_{n}^\ast$.
Given the network $G_{n}^\ast$ and information $(X,\varepsilon_i)$, agent $i$ has utility $$\begin{aligned}
U_{i}(G_{i,n}^\ast, G_{-i,n}^\ast, X, \varepsilon_{i})
= \frac{1}{n} \sum_{j=1}^{n} G_{ij,n}^\ast
\left[
\left(
G_{ji,n}^\ast,\frac{1}{n}\sum_{k\ne i}G_{kj,n}^\ast,\frac{1}{n}\sum_{k\ne i}G_{ki,n}^\ast G_{kj,n}^\ast, X_{ij}'
\right)\beta_0
+ \varepsilon_{ij}
\right],\end{aligned}$$ where $G^\ast_{i,n}=\{G^{\ast}_{ij,n}: j\in\mathcal{N}_n\}$, $G^\ast_{-i,n} = \{G^\ast_{j,n}: j\neq i\}$, and $\beta_0$ is an unknown finite dimensional vector in a parameter space $\mathcal{B}$.
Agent $i$’s marginal utility of forming the link $G_{ij,n}^\ast$ depends on a vector of network statistics, the profile of exogenous attributes, and the link-specific idiosyncratic component. The first component in the vector of network statistics capture the utility obtained from a reciprocate link with agent $j$, $G^{\ast}_{ij,n}$. The second network statistic is the weighted in-degree of agent $j$, $\frac{1}{n}\sum_{k\ne i}G_{kj,n}^\ast$, captures the utility of connecting with agents of high centrality in the network. The last network statistic capture the utility of being connected to the same agents, $\frac{1}{n}\sum_{k\ne i}G_{ki,n}^\ast G_{kj,n}^\ast$. The profile of exogenous attributes captures the preferences for homophily on observed characteristics. Finally, $\varepsilon_{ij}$ is an unobserved link-specific component affecting agent $i$’s decision of linking with agent $j$.
Let $\delta_{i,n}(X, \varepsilon_{i})$ denote a generic agent $i$’s pure strategy, which maps the information $(X,\varepsilon_{i})$ to an action in $\mathcal{G}^{n} = \{0,1\}^{n}$. Let $\sigma_{i,n}(g_{i,n}^\ast\mid X) = Pr(\delta_{i,n}(X,\varepsilon_{i})=g_{i,n}^\ast\mid X)$ be the probability that agent $i$ chooses $g_{i,n}^\ast \in \mathcal{G}^{n}$ given $X$, and let $\sigma_{n}(X) = \{ \sigma_{i,n}(g_{i,n}^\ast\mid X), i \in \mathcal{N}_{n}, g_{i,n}^\ast\in \mathcal{G}^{n} \}$. We call $\sigma_{n}(X)$ a belief profile. Given a belief profile $\sigma_n$ and the information $(X,\varepsilon_i)$, the agent $i$ chooses $g_{i,n}^\ast$ from $\mathcal{G}^{n}$ to maximize the expected utility of $U_{i}(g_{i,n}^\ast, \delta_{-i,n}(X,\varepsilon_{-i}), X, \varepsilon_{i})$ given $(X,\varepsilon_{i},\sigma_{n})$.
In an $n$-player game, a Bayesian Nash Equilibrium $\sigma_{n}(X)$ is a belief profile that satisfies $$\begin{aligned}
\sigma_{i,n}(g_{i,n}^\ast\mid X) = Pr(\delta_{i,n}(X,\varepsilon_{i})=g_{i,n}^\ast\mid X, \sigma_{n})\end{aligned}$$ for all attribute profiles $X \in \mathcal{X}^n$, actions $g_{i,n}^\ast \in \mathcal{G}^n$, and agents $i \in \mathcal{N}_{n}$, where $$\delta_{i,n}(X,\varepsilon_{i})= \operatorname*{arg\,max}_{g_{i,n}^\ast \in \mathcal{G}^n} E\left[U_{i}(g_{i,n}^\ast, \delta_{-i,n}(X,\varepsilon_{-i}), X, \varepsilon_{i}) \mid X, \varepsilon_{i} ,\sigma_n\right].$$
We impose the following assumptions, which are also used by [@leung2015two] and [@ridder/sheng:2015].
\[Ass1\] The following hold for any $n$,
1. For any $A_{1}, A_{2} \subset \mathcal{N}_{n}$ disjoint, $\{X_{ij}: i,j \in A_{1}\}$ and $\{X_{kl}: k,l \in A_{2}\}$ are independent.
2. $\{\varepsilon_{ij}: i,j\in\mathcal{N}_{n}\}$ are marginally distributed with the standard normal distribution with the cdf $\Phi$ and the pdf $\phi$. Further, $\{\varepsilon_{i}: i \in \mathcal{N}_{n} \}$ are mutually independent.
3. $\varepsilon$ and $X$ are independent.
4. Attributes $\{X_{ij}:i,j\in\mathcal{N}_{n}\}$ are identically distributed with a probability mass function bounded away from zero.
We focus on a symmetric equilibrium, where an equilibrium profile $\sigma_{n}$ is symmetric if $\sigma_{i,n}(g_{i,n}^\ast \mid X)=\sigma_{\pi(i),n}(\pi(g_{\pi(i),n}^\ast)\mid \pi(X))$ for any $i \in \mathcal{N}_{n}$, $g_{i,n}^\ast \in \mathcal{G}^n$, and permutation $\pi \in \Pi$.[^3] Given Assumption \[Ass1\], @leung2015two [Theorem 1] and @ridder/sheng:2015 [Proposition 1] show the existence of a symmetric equilibrium.
\[Ass2\] For any $n$, the agents play a symmetric equilibrium $\sigma_n$, i.e., there exists $\{\delta_{i,n}:i\in \mathcal{N}_{n}\}$ such that for any $i \in \mathcal{N}_{n}$ the following holds: (i) $G_{i,n}^\ast = \delta_{i,n}(X,\varepsilon_{i})$, (ii) $\sigma_{i,n}(g_{i,n}^\ast \mid X)=Pr(\delta_{i,n}(X,\varepsilon_{i})=g_{i,n}^\ast\mid X, \sigma_{n})$, (iii) $ \delta_{i,n}(X,\varepsilon_{i})= \operatorname*{arg\,max}_{g_{i,n}^\ast \in \mathcal{G}^n} E\left[U_{i}(g_{i,n}^\ast, \delta_{-i,n}(X,\varepsilon_{-i}), X, \varepsilon_{i}) \mid X, \varepsilon_{i} ,\sigma_n\right]$, and (iv) $\sigma_n$ is symmetric.
The next lemma characterizes the optimal decision rule for the formation of each link in the network.
\[lemma1\] Under Assumption \[Ass1\] and \[Ass2\] , $G_{ij,n}^\ast=\mathbf{1} \left\{(Z_{ij,n}^\ast)'\beta_0+ \varepsilon_{ij}\geq 0\right\}$, where $$\gamma_{ij,n}^\ast = E \left[ \left(G_{ji,n}^\ast,\frac{1}{n}\sum_{k\ne i}G_{kj,n}^\ast,\frac{1}{n}\sum_{k\ne i}G_{ki,n}^\ast G_{kj,n}^\ast\right)' \mid X, \sigma_n \right]$$ and $$Z_{ij,n}^\ast=\left(\begin{array}{c}\gamma_{ij,n}^\ast\\X_{ij}\end{array}\right).$$
Notice that given the misclassification problem, both the optimal action $G_{ij,n}^\ast$ and the equilibrium beliefs about the network statistics $\gamma_{ij,n}^\ast$ in the optimal decision rule will be misclassified.
We assume that the conditional distribution of the observed network $G_n$ is related to that of the true state of network, $G_n^\ast$, as follows.
\[Ass3\] There are two non-negative real numbers $\rho_{0}$ and $\rho_{1}$ with $\rho_{0}+\rho_{1}<1$ such that the following two statements hold for every $n$ and every $i,j,k\in\mathcal{N}_n$. (i) $G_{ki,n}$ and $G_{kj,n}$ are independent given $(G_{ki,n}^\ast, G_{kj,n}^\ast,X,\sigma_n)$. (ii) $Pr(G_{ij,n}\ne G_{ij,n}^\ast\mid G_{ij,n}^\ast,X,\sigma_n)=\rho_01\{G_{ij,n}^\ast=0\}+\rho_11\{G_{ij,n}^\ast=1\}$.
Condition $(i)$ in Assumption \[Ass3\] requires that the observed linking decision $G_{ki,n}$ and $G_{kj,n}$ are conditional independent given the true state of the links $G_{ki,n}^\ast$ and $G_{kj,n}^\ast$, and information $X,\sigma_n$. Condition $(i)$ in Assumption \[Ass3\] characterizes the misclassification probabilities.
The following statement is a key observation in our analysis, which relates the observed network statistics, $\gamma_{ij,n}$, to the payoff relevant network statistics, $\gamma_{ij,n}^\ast$.
\[belief\_lemma\] If Assumptions \[Ass1\]-\[Ass3\] hold, then $\gamma_{ij,n}^\ast=c(\rho_0,\rho_1)+C(\rho_0,\rho_1)\gamma_{ij,n}$ for every $i,j$, where $$\gamma_{ij,n}=E\left[\left(G_{ji,n},\frac{1}{n}\sum_{k\ne i}G_{kj,n},\frac{1}{n}\sum_{k\ne i}G_{ki,n} G_{kj,n},\frac{1}{n}\sum_{k\ne i}(G_{ki,n}+G_{kj,n})\right)'\mid X, \sigma_n\right]$$ $$c(r_0,r_1)=
-
\left(\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
\end{array}\right)
\left(\begin{array}{cccc}
1-r_0-r_1&0&0&0\\
0&1-r_0-r_1&0&0\\
0&0&(1-r_0-r_1)^2&r_0(1-r_0-r_1)\\
0&0&0&1-r_0-r_1\\
\end{array}\right)^{-1}
\left(\begin{array}{c}
r_0\\
r_0\\
r_0^2\\
r_0
\end{array}\right)$$ $$C(r_0,r_1)=
\left(\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
\end{array}\right)
\left(\begin{array}{cccc}
1-r_0-r_1&0&0&0\\
0&1-r_0-r_1&0&0\\
0&0&(1-r_0-r_1)^2&r_0(1-r_0-r_1)\\
0&0&0&1-r_0-r_1\\
\end{array}\right)^{-1}.$$
Notice that the first three components in $\gamma_{ij,n}$ are the observed analog to the statistics in $\gamma_{ij,n}^\ast$ since they determined by the observed network $G_{n}$. The last component in $\gamma_{ij,n}$ is the sum of the in-degrees of agents $i$ and $j$, and it is the result of controlling for the unobserved network statistics $\frac{1}{n}\sum_{k\ne i}G_{ki,n}^\ast G_{kj,n}^\ast$. Precisely, the last two statistics in $\gamma_{ij,n}$ control for the beliefs about the unobserved network statistic $\frac{1}{n}\sum_{k\ne i}G_{ki,n}^\ast G_{kj,n}^\ast$. The intuition behind this result is similar to the one found in polynomial regression models with mismeasured continuous covariates [@hausman/ichimura/newey/powell:1991].
Assumptions \[Ass1\]-\[Ass3\] imply the following relationship between the distributions of $G_{ij,n}$ and $G_{ij,n}^\ast$, which will be used in our identification analysis. Since we observe $G_{ij,n}$ in the dataset but the outcome of interest is $G_{ij,n}^\ast$, it is crucial to connect these two objects.
\[Ass3\_lemma\] Under Assumptions \[Ass1\]-\[Ass3\], $Pr(G_{ij,n}=1\mid X_{ij},\gamma_{ij,n}, \gamma_{ij,n}^\ast)=(1-\rho_0)Pr(G_{ij,n}^\ast=0\mid X_{ij},\gamma_{ij,n}^\ast)+\rho_1Pr(G_{ij,n}^\ast=1\mid X_{ij},\gamma_{ij,n}^\ast)$.
Identification Analysis {#sec3}
=======================
We characterize the identified set based on the joint distribution $P_{0,n}$ of $(G_{ij,n},X_{ij},\gamma_{ij,n})$.[^4] In this section, we treat $\gamma_{ij,n}$ as observed because we can estimate it as follows. For a generic value $x$ in the support of $X_{ij}$, we can define $$\hat{p}(x)=\frac{1}{n^2}\sum_{i,j}1\{X_{ij}=x\}$$ $$\hat\gamma(x)=\frac{\frac{1}{n^2}\sum_{i,j}\left(G_{ji,n},\frac{1}{n}\sum_{k}G_{kj,n},\frac{1}{n}\sum_{k}G_{ki,n} G_{kj,n},\frac{1}{n}\sum_{k}(G_{ki,n}+G_{kj,n})\right)'1\{X_{ij}=x\}}{\hat{p}(x)},$$ $\hat{p}(x)$ is an estimator for $Pr(X_{ij}=x)$ and $\hat\gamma_{ij}=\hat\gamma(X_{ij})$ is an estimator for $\gamma_{ij,n}$. Then we can estimate the distribution of $(G_{ij,n},X_{ij},\gamma_{ij,n})$ using the empirical distribution of $(G_{ij,n},X_{ij},\hat\gamma_{ij})$.
To formalize our identification analysis, we introduce several notations. Denote by $\mathcal{P}^\ast$ the set of joint distributions of $(G_{ij,n},G_{ij,n}^\ast,X_{ij},\gamma_{ij,n},\gamma_{ij,n}^\ast,\varepsilon_{ij})$. Define the parameter space $\Theta=\mathcal{B}\times\{(r_0,r_1): r_0,r_1\geq 0,r_0+r_1<1\}$, where $\mathcal{B}$ is the parameter space for $\beta_0$. Denote by $\mathcal{P}$ the set of joint distributions of $(G_{ij,n},X_{ij},\gamma_{ij,n})$.
Based on Assumptions \[Ass1\]-\[Ass3\] and Lemmas \[lemma1\]-\[Ass3\_lemma\], we impose the following three conditions on the true joint distribution $P_{0,n}^\ast$ of the variables $(G_{ij,n},G_{ij,n}^\ast,X_{ij},\gamma_{ij,n},\gamma_{ij,n}^\ast,\varepsilon_{ij})$ and the true parameter value $\theta_0=(\beta,\rho_0,\rho_1)$.
\[independence\_assn\] Under $P^\ast$ the following holds: (i) $\varepsilon_{ij}$ is normally distributed with mean zero and variance one. (ii) $\varepsilon_{ij}$ and $(X_{ij},\gamma_{ij,n}^\ast)$ are independent.
\[linear\_index\] $G_{ij,n}^\ast=\mathbf{1} \left\{(Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\right\}$ a.s. $P^\ast$, where $$\begin{aligned}
Z_{ij,n}^\ast&=&\left(\begin{array}{c}\gamma_{ij,n}^\ast\\X_{ij}\end{array}\right)\\\end{aligned}$$
\[misclas\_prop\] (i) $P^\ast(G_{ij,n}=1\mid X_{ij},\gamma_{ij,n}, \gamma_{ij,n}^\ast)=(1-r_0)P^\ast(G_{ij,n}^\ast=0\mid X_{ij},\gamma_{ij,n}^\ast)+r_1P^\ast(G_{ij,n}^\ast=1\mid X_{ij},\gamma_{ij,n}^\ast)$. (ii) $\gamma_{ij,n}^\ast=c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n} \; a.s. \; P^\ast$.
For each element $P$ of $\mathcal{P}$, we are going to define the identified set based on the above three conditions.
For each distribution $P\in\mathcal{P}$, the identified set $\Theta_I(P)$ is defined as the set of all $\theta=(b,r_0,r_1)$ in $\Theta$ for which there is some joint distribution $P^\ast\in\mathcal{P}^\ast$ such that Condition \[independence\_assn\], \[linear\_index\], and \[misclas\_prop\] hold, and that the distribution of $(G_{ij,n},X_{ij},\gamma_{ij,n})$ induced from $P^\ast$ is equal to $P$.
Note that $\Theta_I(P)$ does not depend on the sample size $n$, but the identified set $\Theta_I(P_{0,n})$ based on the data distribution $P_{0,n}$ does.
The identified set is characterized as follows.
\[theorem\_identification\] Given a joint distribution $P\in\mathcal{P}$, $\Theta_I(P)$ is equal to the set of $\theta\in\Theta$ satisfying $$\label{moment_TheoremID}
E_P[G_{ij,n}-r_0-(1-r_0-r_1)\Phi((c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n})'b_1+X_{ij}'b_2)\mid X_{ij},\gamma_{ij,n}]=0.$$
If the links were measured without error, the moment equation in Eq. (\[moment\_TheoremID\]) would become $E_P[G_{ij,n}-\Phi([\gamma_{ij,n}]_{123}'b_1+X_{ij}'b_2)\mid X_{ij},\gamma_{ij,n}]=0$, where $[\gamma_{ij,n}]_{123}$ is a vector composed by the first three components of $\gamma_{ij,n}$. The specification without measurement error is used as the basis for the maximum likelihood estimation in [@leung2015two].
The identified set in Theorem \[theorem\_identification\] relies on the assumption that we know the distribution of $\varepsilon_{ij}$. In Appendix B, we characterize the identified set in a semiparametric framework.
Inference {#sec4}
=========
In this section, we propose a confidence interval for $\theta$ based on the identification analysis in Theorem \[theorem\_identification\] and derive its asymptotic coverage when we observe one single network with many agents. As in [@leung2015two] and [@ridder/sheng:2015], we use the asymptotic arguments based on a symmetric equilibrium.
Consider the unconditional sample analog of the moment condition in Eq. is $$\hat{m}_n(\theta)=\frac{1}{n^2}\sum_{i,j}\left(G_{ij,n}-r_0-(1-r_0-r_1)\Phi((c(r_0,r_1)+C(r_0,r_1)\hat\gamma_{ij})'b_1+X_{ij}'b_2)\right)\mathbf{1}_{ij},$$ where $x_1,\ldots,x_J$ are all the support points for $X_{ij}$ and $\mathbf{1}_{ij}=(1\{X_{ij}=x_1\},\ldots,1\{X_{ij}=x_J\})'$. We estimate the variance of $\hat{m}_n(\theta)$ by $$\hat{S}(\theta)=\left(\frac{1}{n}\sum_{i=1}^n\hat\psi_i(\theta)\hat\psi_i(\theta)'-\left(\frac{1}{n}\sum_{i=1}^n\hat\psi_i(\theta)\right)\left(\frac{1}{n}\sum_{i=1}^n\hat\psi_i(\theta)\right)'\right).$$ where $$\begin{aligned}
\hat\psi_i(\theta)
&=&
\frac{1}{n}\sum_{j\ne i}G_{ij,n}\mathbf{1}_{ij}\\&&-(1-r_0-r_1)\frac{1}{n^2}\sum_{l,j}\left(\phi((c(r_0,r_1)+C(r_0,r_1)\hat\gamma_{lj,n})'b_1+X_{lj}'b_2)
b_1'C(r_0,r_1)
\hat\psi_{\gamma,i, n}(X_{lj})
\right)\mathbf{1}_{lj}\end{aligned}$$ and $$\begin{aligned}
\hat\psi_{\gamma,k, n}(x)
&=&
\frac{1}{n^2} \sum_{i_{1},j_1}
\frac{1\{X_{i_{1},j_1}=x\}}{\hat{p}(x)}
\left(
\begin{array}{c}
0\\
G_{kj_1}\\
G_{ki_{1}}G_{kj_1}\\
G_{ki_{1}}+G_{kj_1}
\end{array}
\right)
+
\frac{1}{n} \sum_{i_{1}}
\frac{1\{X_{i_{1},k}=x\}}{\hat{p}(x)}
\left(
\begin{array}{c}
G_{ki_{1}}\\
0\\
0\\
0
\end{array}
\right).\end{aligned}$$ For a size $\alpha\in(0,1)$, a confidence interval for $\theta$ is constructed as $$CI_n(\alpha)=\{\theta\in\Theta: n\hat{m}_n(\theta)'\hat{S}(\theta)^{-1}\hat{m}_n(\theta)\leq c_{\alpha}\},$$ where $c_\alpha$ is the $1-\alpha$ quantile of $\chi^2_J$. The following theorem demonstrates the asymptotic coverage for the confidence interval $CI_n(\alpha)$.
\[theorem\_inference\] Suppose that the minimum eigenvalue of $Var(\psi_i(\theta_0)\mid X,\sigma_n)$ is bounded away from zero, and that $\liminf\min_{x}\hat{p}(x)>0$. Under Assumptions \[Ass1\]-\[Ass3\], $$\liminf_{n\rightarrow\infty}Pr(\theta_0\in CI_n(\alpha)\mid X,\sigma_n)\geq 1-\alpha.$$
Conclusion {#sec6}
==========
We study a network formation models with potentially misclassified links. Specifically, we focus on a strategic game of network formation with incomplete information. In the presence of network misclassification, we derive the moment equality conditions which characterize the identified set of the preference parameters associated with homophily and network spillovers. Based on the moment equality conditions, we provide an inference method which is asymptotically valid when a single large network is available.
Proofs {#sec_a1}
======
Proof of Lemmas in Section \[sec2\]
-----------------------------------
By Assumption \[Ass2\], $$\begin{aligned}
G_{i,n}^\ast
&=
\operatorname*{arg\,max}_{g_{i,n}^\ast \in \mathcal{G}^n}E\left[U_{i}(g_{i,n}^\ast, G_{-i,n}^\ast, X, \varepsilon_{i}) \mid X, \varepsilon_{i}, \sigma_n \right] \\
&=
\operatorname*{arg\,max}_{g_{i,n}^\ast \in \mathcal{G}^n} \frac{1}{n} \sum_{j=1}^{n} g_{ij,n}^\ast
\left[(Z_{ij,n}^\ast)'\beta_0 + \varepsilon_{ij} \right]. \end{aligned}$$ Therefore, $G_{ij,n}^\ast = \mathbf{1} \left\{(Z_{ij,n}^\ast)'\beta_0+ \varepsilon_{ij}\geq 0\right\}$.
Define $$D(r_0,r_1)=
\left(\begin{array}{cccc}
1-r_0-r_1&0&0&0\\
0&1-r_0-r_1&0&0\\
0&0&(1-r_0-r_1)^2&r_0(1-r_0-r_1)\\
0&0&0&1-r_0-r_1\\
\end{array}\right).$$ By Assumption \[Ass3\], we can derive $$\begin{aligned}
E\left[G_{ki,n} G_{kj,n}\mid X, \sigma_n\right]
= \rho_0^2+(1-\rho_0-\rho_1)^2 E\left[G_{ki,n}^\ast G_{kj,n}^\ast\mid X, \sigma_n\right]+\rho_0(1-\rho_0-\rho_1) E\left[G_{ki,n}^\ast+G_{kj,n}^\ast\mid X, \sigma_n\right].
\end{aligned}$$ Therefore, $$\begin{aligned}
\gamma_{ij,n}
&=&
\left(\begin{array}{c}
E\left[G_{ji,n}\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{kj,n}\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{ki,n} G_{kj,n}\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{ki,n}+G_{kj,n}\mid X, \sigma_n\right]
\end{array}\right)
\\
&=&
\left(\begin{array}{c}
\rho_0\\
\rho_0\\
\rho_0^2\\
\rho_0
\end{array}\right)
+
D(\rho_0,\rho_1)
\left(\begin{array}{c}
E\left[G_{ji,n}^\ast\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{kj,n}^\ast\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{ki,n}^\ast G_{kj,n}^\ast\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{ki,n}^\ast+G_{kj,n}^\ast\mid X, \sigma_n\right]
\end{array}\right).\end{aligned}$$ Since $D(\rho_0,\rho_1)$ is invertible given $1-\rho_0-\rho_1\ne 0$, it follows that $$\left(\begin{array}{c}
E\left[G_{ji,n}^\ast\mid X\right]\\
\frac{1}{n}\sum_{k}E\left[G_{kj,n}^\ast\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{ki,n}^\ast G_{kj,n}^\ast\mid X, \sigma_n\right]\\
\frac{1}{n}\sum_{k}E\left[G_{ki,n}^\ast+G_{kj,n}^\ast\mid X, \sigma_n \right]
\end{array}\right)
=
D(\rho_0,\rho_1)^{-1}
\left(
\gamma_{ij,n}-
\left(\begin{array}{c}
\rho_0\\
\rho_0\\
\rho_0^2\\
\rho_0
\end{array}\right)
\right).$$ The first three component of the right hand side on the above equation is $\gamma_{ij,n}^\ast$, so $$\begin{aligned}
\gamma_{ij,n}^\ast
&=&
\left(\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
\end{array}\right)
D(\rho_0,\rho_1)^{-1}
\left(
\gamma_{ij,n}-
\left(\begin{array}{c}
\rho_0\\
\rho_0\\
\rho_0^2\\
\rho_0
\end{array}\right)
\right)\\
&=&
c(\rho_0,\rho_1)+C(\rho_0,\rho_1)\gamma_{ij,n}.\end{aligned}$$
It suffices to show that $Pr(G_{ij,n}=1\mid X_{ij},\gamma_{ij,n}, \gamma_{ij,n}^\ast,X,\sigma_n)=(1-\rho_0)Pr(G_{ij,n}^\ast=0\mid X_{ij},\gamma_{ij,n}^\ast)+\rho_1Pr(G_{ij,n}^\ast=1\mid X_{ij},\gamma_{ij,n}^\ast)$. Since $(X_{ij},\gamma_{ij,n}, \gamma_{ij,n}^\ast)$ are a function of $X,\sigma_n$, it follows that $$Pr(G_{ij,n}=1\mid X_{ij},\gamma_{ij,n}, \gamma_{ij,n}^\ast,X,\sigma_n)
=
Pr(G_{ij,n}=1\mid X,\sigma_n).$$ Using Assumptions \[Ass1\]-\[Ass3\], $$\begin{aligned}
Pr(G_{ij,n}=1\mid X,\sigma_n)
&=&
\rho_0Pr(G_{ij,n}^\ast=0\mid X,\sigma_n)+(1-\rho_1)Pr(G_{ij,n}^\ast=1\mid X,\sigma_n)\\
&=&
\rho_0Pr((Z_{ij,n}^\ast)'b + \varepsilon_{ij}<0\mid X,\sigma_n)+(1-\rho_1)Pr((Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\mid X,\sigma_n)\\
&=&
\rho_0Pr((Z_{ij,n}^\ast)'b + \varepsilon_{ij}<0\mid Z_{ij,n}^\ast)+(1-\rho_1)Pr((Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\mid Z_{ij,n}^\ast),\end{aligned}$$ where the first equality follows from Assumption \[Ass3\], the second follows from Lemma \[lemma1\], and the last follows from the independence between $\varepsilon$ and $X$.
Proof of Theorem \[theorem\_identification\]
--------------------------------------------
To show that every element $\theta$ of $\Theta_I(P)$ satisfies Eq. , we can see the following equalities $$\begin{aligned}
P(G_{ij,n}=1\mid X_{ij},\gamma_{ij,n})
&=&
P^\ast(G_{ij,n}=1\mid X_{ij},\gamma_{ij,n}) \\
&=&
P^\ast(G_{ij,n}=1\mid X_{ij}, \gamma_{ij,n}, \gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)P^\ast(G_{ij,n}^\ast=1\mid X_{ij}, \gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)P^\ast((Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\mid X_{ij}, \gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)\Phi((\gamma_{ij,n}^\ast)'b_1+X_{ij}'b_2)\\
&=&
r_0+(1-r_0-r_1)\Phi((c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n})'b_1+X_{ij}'b_2),\end{aligned}$$ where the first equality follows from $P=P^\ast$ for the observables $(G_{ij,n},X_{ij},\gamma_{ij,n})$, the second equality follows because $\gamma_{ij,n}^\ast$ is a function of $ \gamma_{ij,n}$ in Condition \[misclas\_prop\](ii), the third equality follows from Condition \[misclas\_prop\](i), the fourth equality follows from Condition \[linear\_index\], the fifth equality follows from Condition \[independence\_assn\], and the last equality follows from Condition \[misclas\_prop\](ii). The rest of the proof is going to show that every element $\theta$ of $\Theta$ satisfying Eq. belongs to $\Theta_I(P)$.
Define the joint distribution $P^\ast$ in the following way. The marginal distribution of $\varepsilon_{ij}$ is standard normal. The conditional distribution of $(\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij})$ given $\varepsilon_{ij}$ is $$\label{A_equ}
P^\ast((\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij})\in B \mid \varepsilon_{ij})=P((\gamma_{ij,n},c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n},X_{ij})\in B)$$ for all the measurable sets $B$. The conditional distribution of $G_{ij,n}^\ast$ given $(\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})$ is $$\label{B_equ}
P^\ast(G_{ij,n}^\ast=1\mid \gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})=1\{(Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\}.$$ The conditional distribution of $G_{ij,n}$ given $(G_{ij,n}^\ast,\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})$ is $$\label{C_equ}
P^\ast(G_{ij,n}=1\mid G_{ij,n}^\ast,\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})
=
\begin{cases}
1-r_0&\mbox{ if }G_{ij,n}^\ast=0\\
r_1&\mbox{ if }G_{ij,n}^\ast=1.
\end{cases}$$
Note that $(P^\ast,\theta)$ satisfies Conditions 1-3, because Condition 1(i) follows because $\varepsilon_{ij}$ is normally distributed under $P^\ast$, Condition 1(ii) follows from Eq. (\[A\_equ\]). Condition 2 follows from Eq. (\[B\_equ\]). Condition 3(i) follows from Eq. (\[B\_equ\]) and (\[C\_equ\]), and Condition 3(ii) follows from Eq. (\[A\_equ\]).
The distribution of $(G_{ij,n},X_{ij},\gamma_{ij,n})$ induced from $P^\ast$ is equal to $P$. The distribution of $(X_{ij},\gamma_{ij,n})$ induced from $P^\ast$ is equal to that from $P$, by the construction of $P^\ast((\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij})\in B \mid \varepsilon_{ij})$. The equality of $P^\ast(G_{ij,n}=1\mid Z_{ij,n})=P(G_{ij,n}=1\mid Z_{ij,n})$ a.s. under $P^\ast$ is shown as follows. Note that $$\label{D_equ}
\gamma_{ij,n}^\ast=c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n}\mbox{ a.s. under }P^\ast.$$ Then $$\begin{aligned}
P^\ast(G_{ij,n}=1\mid Z_{ij,n})
&=&
P^\ast(G_{ij,n}=1\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
(1-r_0)P^\ast(G_{ij,n}^\ast=0\mid Z_{ij,n},\gamma_{ij,n}^\ast)+r_1P^\ast(G_{ij,n}^\ast=1\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)P^\ast(G_{ij,n}^\ast=1\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)E_{P^\ast}[P^\ast(G_{ij,n}^\ast=1\mid Z_{ij,n},\gamma_{ij,n}^\ast,\varepsilon_{ij})\mid Z_{ij,n},\gamma_{ij,n}^\ast]\\
&=&
r_0+(1-r_0-r_1)P^\ast((Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)\Phi((Z_{ij,n}^\ast)'b)\\
&=&
r_0+(1-r_0-r_1)\Phi((c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n})'b_1+X_{ij}'b_2)\\
&=&
P(G_{ij,n}=1\mid Z_{ij,n}),\end{aligned}$$ where the first and seventh equalities follow from Eq. (\[D\_equ\]), the second follows from Eq. (\[C\_equ\]), the fifth follows from Eq. (\[B\_equ\]), and the last follows from Eq. (\[moment\_TheoremID\]).
Proof of Theorem \[theorem\_inference\]
---------------------------------------
In the proof of this theorem, all the statements are conditional on $X$ and $\sigma_n$. We use the norm for matrices and vectors. For any vector, the norm is understood as the Euclidean norm, and for any matrix the norm is induced by the Euclidean norm. Theorem \[theorem\_inference\] follows from Lemma \[chi2dist\_conv\].
Define $$\begin{aligned}
u_{ij}(\theta_0)
&=&
(c(\rho_0,\rho_1)+C(\rho_0,\rho_1)\gamma_{ij,n})'\beta_1+X_{ij}'\beta_2\\
\hat{u}_{ij}(\theta_0)
&=&
(c(\rho_0,\rho_1)+C(\rho_0,\rho_1)\hat\gamma_{ij})'\beta_1+X_{ij}'\beta_2.\end{aligned}$$ For a generic random variable RV, define $$\begin{aligned}
RV^\dagger = RV - E[RV\mid X, \sigma_n], \end{aligned}$$ and note that $E[RV^\dagger\mid X, \sigma_n]=0$. Define $$\begin{aligned}
\psi_{\gamma,k, n}(x)
&=
\frac{1}{n^2} \sum_{i,j}
\left(\frac{1\{X_{i,j}=x\}}{\hat{p}(x)} \right)
\left(
\begin{array}{c}
0\\
G_{kj,n}^{\dagger}\\
(G_{ki,n}G_{kj,n})^{\dagger}\\
(G_{ki,n}+G_{kj,n})^{\dagger}
\end{array}
\right)
+
\frac{1}{n} \sum_{i}
\left(\frac{1\{X_{i,k}=x\}}{\hat{p}(x)} \right)
\left(
\begin{array}{c}
G_{ki,n}^{\dagger}\\
0\\
0\\
0
\end{array}
\right)\end{aligned}$$ $$\begin{aligned}
\psi_k(\theta_0)
&=&
\frac{1}{n}\sum_{j\ne k}\left(G_{kj,n}-\rho_0-(1-\rho_0-\rho_1)\Phi(u_{kj}(\theta_0))\right)\mathbf{1}_{kj}\\&&-(1-\rho_0-\rho_1)\frac{1}{n^2}\sum_{i,j}\left(\phi(u_{kj}(\theta_0))\beta_1'C(\rho_0,\rho_1) \psi_{\gamma,k, n}(X_{ij})\right)\mathbf{1}_{ij}\end{aligned}$$ $$\begin{aligned}
\tilde\psi_k(\theta_0)
&=&
\frac{1}{n}\sum_{j\ne k}G_{kj,n}\mathbf{1}_{kj}-(1-\rho_0-\rho_1)\frac{1}{n^2}\sum_{i,j}\left(\phi(u_{ij}(\theta_0))\beta_1'C(\rho_0,\rho_1)\hat\psi_{\gamma,k,n}(X_{ij})\right)\mathbf{1}_{ij}.\end{aligned}$$
\[gammastar\_symmetric\] $$\label{gammastar_symmetry}
1\{X_{i_{1},j_1}=X_{ij}\}
\left(
\begin{array}{c}
E[G_{j_1i_{1},n}^\ast\mid X, \sigma_{n}]-E[G_{ji,n}^\ast\mid X, \sigma_{n}]\\
\frac{1}{n}\sum_{k} (E[G_{kj_1,n}^\ast\mid X, \sigma_{n}] - E[G_{kj,n}^\ast\mid X, \sigma_{n}])\\
\frac{1}{n}\sum_{k} (E[G_{ki_{1},n}^\ast G_{kj_1,n}^\ast\mid X, \sigma_{n}] - E[G_{ki,n}^\ast G_{kj,n}^\ast\mid X, \sigma_{n}])
\\
\frac{1}{n}\sum_{k} \left(E\left[G_{ki_1,n}^\ast+G_{kj_1,n}^\ast\mid X, \sigma_n\right] - E\left[G_{ki,n}^\ast+G_{kj,n}^\ast\mid X, \sigma_n\right]\right)
\end{array}
\right)
=0.$$
This result follows from symmetry of the equilibrium and it is shown in a similar way to Lemma 1 in @leung2015two.
\[bounds\_many\] $$\max\{\|\hat\psi_{\gamma,k, n}(X_{ij})\|,\|\psi_{\gamma,k, n}(X_{ij})\|\}\leq \frac{\sqrt{7}}{\min_{x}\hat{p}(x)}$$ $$\max_{i}\{\|\tilde\psi_i(\theta_0)\|,\|\hat\psi_i(\theta_0)\|,\|\psi_i(\theta_0)\|\}\leq 1+(1-\rho_0-\rho_1)\phi(0) \|\beta_1'C(\rho_0,\rho_1)\|\frac{\sqrt{7}}{\min_{x}\hat{p}(x)}.$$
The bound for $\|\hat\psi_{\gamma,k, n}(X_{ij})\|$ is derived as follows. (The proof for $\|\psi_{\gamma,k, n}(X_{ij})\|$ is similar.) $$\begin{aligned}
\|\hat\psi_{\gamma,k, n}(X_{ij})\|
&\leq
\frac{\sqrt{7}}{\min_x\hat{p}(x)}.\end{aligned}$$
The bound for $\|\tilde\psi_i(\theta)\|$ is derived as follows. (The proof for $\|\hat\psi_i(\theta)\|$ is similar.) $$\begin{aligned}
\|\tilde\psi_i(\theta)\|
&\leq&
\max_{j\ne i}\left|G_{ij,n}\right|
+\max_{l,j}\left|\phi(u_{lj}(\theta_0))
\beta_1'C(\rho_0,\rho_1)
\hat\psi_{\gamma,i, n}(X_{lj})
\right|\\
&\leq&
1+(1-\rho_0-\rho_1)\phi(0) \|\beta_1'C(\rho_0,\rho_1)\|\frac{\sqrt{7}}{\min_{x}\hat{p}(x)}.\end{aligned}$$ The bound for $\|\psi_i(\theta)\|$ is derived as follows. $$\begin{aligned}
\|\psi_i(\theta_0)\|
&\leq&
\max_{j\ne i}\left|G_{ij,n}-\rho_0-(1-\rho_0-\rho_1)\Phi(u_{ij}(\theta_0))\right|\\&&+(1-\rho_0-\rho_1)\max_{l,j}\|\phi(u_{ij}(\theta_0))\beta_1'C(\rho_0,\rho_1)\| \|\psi_{\gamma,i, n}(X_{lj})\|\\
&\leq&
1+(1-\rho_0-\rho_1)\phi(0)\|\beta_1'C(\rho_0,\rho_1)\|\frac{\sqrt{7}}{\min_{x}\hat{p}(x)}.\end{aligned}$$
\[gamma\_influe\] $$\hat\gamma_{ij}-\gamma_{ij,n}=\frac{1}{n}\sum_{k} \psi_{\gamma,k, n}(X_{ij})$$ and $$\hat\gamma_{ij}-\gamma_{ij,n}=O_p(n^{-1/2})\mbox{ given $X$ and $\sigma_n$}.$$
First, we are going to show $$\label{gamma_symmetry}
1\{X_{i_{1},j_1}=X_{ij}\}
\left(
\begin{array}{c}
E[G_{j_1i_{1},n}\mid X, \sigma_{n}]-E[G_{ji,n}\mid X, \sigma_{n}]\\
\frac{1}{n}\sum_{k} (E[G_{kj_1,n}\mid X, \sigma_{n}] - E[G_{kj,n}\mid X, \sigma_{n}])\\
\frac{1}{n}\sum_{k} (E[G_{ki_{1},n} G_{kj_1,n}\mid X, \sigma_{n}] - E[G_{ki,n} G_{kj,n}\mid X, \sigma_{n}])\\
\frac{1}{n}\sum_{k} (E[(G_{ki_{1},n}+G_{kj_1,n})\mid X, \sigma_{n}] - E[(G_{ki,n}+G_{kj,n})\mid X, \sigma_{n}])
\end{array}
\right)
=0.$$ It follows from Lemma \[gammastar\_symmetric\] and Assumption \[Ass3\].
Using Eq. , we have $$\begin{aligned}
\hat\gamma_{ij}-\gamma_{ij,n}
&=
\frac{1}{n^2}\sum_{i_{1},j_1}
\frac{
1\{X_{i_{1},j_1}=X_{ij}\}
}{
\frac{1}{n^2}\sum_{i_{1},j_1}1\{X_{i_{1},j_1}=X_{ij}\}
}
\left(
\begin{array}{c}
G_{j_1i_{1},n}-E[G_{ji,n}\mid X, \sigma_{n}]\\
\frac{1}{n}\sum_{k} (G_{kj_1,n} - E[G_{kj,n}\mid X, \sigma_{n}])\\
\frac{1}{n}\sum_{k} (G_{ki_{1},n} G_{kj_1,n} - E[G_{ki,n} G_{kj,n}\mid X, \sigma_{n}])\\
\frac{1}{n}\sum_{k} ((G_{ki_{1},n}+G_{kj_1,n}) - E[(G_{ki,n}+G_{kj,n})\mid X, \sigma_{n}])
\end{array}
\right)
\\
&=
\frac{1}{n^2}\sum_{i_{1},j_1}
\frac{
1\{X_{i_{1},j_1}=X_{ij}\}
}{
\frac{1}{n^2}\sum_{i_{1},j_1}1\{X_{i_{1},j_1}=X_{ij}\}
}
\left(
\begin{array}{c}
G_{j_1i_{1},n}^\dagger\\
\frac{1}{n}\sum_{k} G_{kj_1,n}^\dagger \\
\frac{1}{n}\sum_{k} (G_{ki_{1},n} G_{kj_1,n})^\dagger\\
\frac{1}{n}\sum_{k} ((G_{ki_{1},n}+G_{kj_1,n}))^\dagger
\end{array}
\right)
\\
&
=
\frac{1}{n}\sum_{k} \psi_{\gamma,k, n}(X_{ij}).\end{aligned}$$
By Lyapunov’s central limit theorem, it suffices to show that $E[\psi_{\gamma,k, n}(X_{ij})\mid X,\sigma_n]=0$ and that $\psi_{\gamma,k, n}(X_{ij})$ is independent across $k$ given $X$ and $\sigma_n$. The equality $E[\psi_{\gamma,k, n}(X_{ij})\mid X,\sigma_n]=0$ follows from $$\begin{aligned}
E[\psi_{\gamma,k, n}(X_{ij})\mid X,\sigma_n]
&=
\frac{1}{n^2} \sum_{i_{1},j_1}
\left(\frac{1\{X_{i_{1},j_1}=X_{ij}\}}{\hat{p}(X_{ij})} \right)
\left(
\begin{array}{c}
0\\
E\left[ G_{kj_1,n}^{\dagger} \mid X,\sigma_n \right]\\
E\left[(G_{ki_{1},n}G_{kj_1,n})^{\dagger} \mid X,\sigma_n \right]\\
E\left[(G_{ki_{1},n}+G_{kj_1,n})^{\dagger} \mid X,\sigma_n \right]
\end{array}
\right)
\\
&+
\frac{1}{n} \sum_{i_{1}}
\left(\frac{1\{X_{i_{1},k}=X_{ij}\}}{\hat{p}(X_{ij})} \right)
\left(
\begin{array}{c}
E\left[ G_{ki_{1},n}^{\dagger} \mid X,\sigma_n \right]\\
0\\
0\\
0
\end{array}
\right)
\\
& =0\end{aligned}$$ since $E\left[ RV^{\dagger} \mid X,\sigma_n \right] =0$ by definition of $RV^\dagger$. The conditional independence of $\psi_{\gamma,k, n}(X_{ij})$ across $k$ is shown as follows. Note that $\psi_{\gamma,k, n}(X_{ij})$ does not depend on $G_{-k,n}$, so it is a function of $\varepsilon_k$, $X$ and $\sigma_n$. Therefore, it follows from Assumptions \[Ass1\] that $\psi_{\gamma,k, n}(X_{ij})$ is independent across $k$ given $X$ and $\sigma_n$.
\[diff\_psi\] $\max_{i}\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|=o_p(1)$.
Note that $$\begin{aligned}
\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)
&=&
-(1-\rho_0-\rho_1)\frac{1}{n^2}\sum_{l,j}\left(
\phi(\hat{u}_{lj}(\theta_0))
-\phi(u_{lj}(\theta_0))
\right)
\beta_1'C(\rho_0,\rho_1)
\hat\psi_{\gamma,i, n}(X_{lj})
\mathbf{1}_{lj}.\end{aligned}$$ Then $$\begin{aligned}
\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|
&\leq&
\|\beta_1'C(\rho_0,\rho_1)\|
\max_{l,j}\left|
\phi(\hat{u}_{lj}(\theta_0))
-\phi(u_{lj}(\theta_0))
\right|
\|\hat\psi_{\gamma,i, n}(X_{lj})\|\\
&\leq&
\phi(0)\|\beta_1'C(\rho_0,\rho_1)\|\max_{l,j}
\max\{|\hat{u}_{lj}(\theta_0)|,|u_{lj}(\theta_0)|\}|\hat{u}_{lj}(\theta_0)-u_{lj}(\theta_0)|
\|\hat\psi_{\gamma,i, n}(X_{lj})\|,\end{aligned}$$ where the last inequality follows from the mean value expansion of the normal pdf $\phi$: $|\phi(u_1)-\phi(u_2)|\leq\max_{u_1\leq u\leq u_2}|\phi'(u)||u_1-u_2|\leq\phi(0)\max\{|u_1|,|u_2|\}|u_1-u_2|$. Since $$\begin{aligned}
|u_{lj}(\theta_0)|
&\leq&
(\|c(\rho_0,\rho_1)\|+\|C(\rho_0,\rho_1)\|\|\gamma_{lj,n})\|\|\beta_1\|+\|X_{lj}\|\|\beta_2\|\\
&\leq&
(\|c(\rho_0,\rho_1)\|+\sqrt{7}\|C(\rho_0,\rho_1)\|)\|\beta_1\|+\max_{x}\|x\|\|\beta_2\|\\
|\hat{u}_{lj}(\theta_0)|
&\leq&
(\|c(\rho_0,\rho_1)\|+\sqrt{7}\|C(\rho_0,\rho_1)\|)\|\beta_1\|+\max_{x}\|x\|\|\beta_2\|\\
|\hat{u}_{lj}(\theta_0)-u_{lj}(\theta_0)|
&=&
|C(\rho_0,\rho_1)(\hat\gamma_{lj}-\gamma_{lj,n})'\beta_1|\\
&\leq&
\|C(\rho_0,\rho_1)\|\|\beta_1\|\max_{lj}\|\hat\gamma_{lj}-\gamma_{lj,n}\|, \end{aligned}$$ it follows that $$\max_{i}\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|=O_p(\max_{lj}\|\hat\gamma_{lj}-\gamma_{lj,n})\|)=o_p(1).$$
\[psi\_ind\] $\psi_i(\theta_0)$ is independent across $i$ given $X$ and $\sigma_n$.
$\psi_i(\theta_0)$ does not depend on $G_{-i,n}$, so it is a function of $\varepsilon_i$, $X$ and $\sigma_n$. It implies the statement of this lemma.
Conditional on $X$ and $\sigma_n$, $$\hat{m}_n(\theta_0)
=
\frac{1}{n}\sum_{i=1}^n\psi_i(\theta_0)+o_p(n^{-1/2}).$$
Note that $$\hat{m}_n(\theta_0)-\frac{1}{n}\sum_{i=1}^n\psi_i(\theta_0)
=
(1-\rho_0-\rho_1)\frac{1}{n^2}\sum_{i,j}\left(
\Phi(\hat{u}_{ij}(\theta_0))
-
\Phi(u_{ij}(\theta_0))
-
\phi(u_{ij}(\theta_0))\beta_1'C(\rho_0,\rho_1)(\hat\gamma_{ij}-\gamma_{ij,n})
\right)\mathbf{1}_{ij}$$ By the second-order Taylor expansion of the normal cdf $\Phi$, $$\Phi(u_1)=\Phi(u_2)+\phi(u_2)(u_2-u_1)+R(u_1,u_2)$$ where $$|R_{ij}|\leq\frac{1}{2}\max_{u_1\leq u\leq u_2}\phi'(u)|u_1-u_2|^2\leq\frac{1}{2}\phi(0)\max\{|u_1|,|u_2|\}|u_1-u_2|^2.$$ Since $$\begin{aligned}
\max\{|u_{lj}(\theta_0)|,|\hat{u}_{lj}(\theta_0)|\}
&\leq&
(\|c(\rho_0,\rho_1)\|+\sqrt{7}\|C(\rho_0,\rho_1)\|)\|\beta_1\|+\max_{x}\|x\|\|\beta_2\|\\
|\hat{u}_{lj}(\theta_0)-u_{lj}(\theta_0)|
&\leq&
\|C(\rho_0,\rho_1)\|\|\beta_1\|\max_{lj}\|\hat\gamma_{lj}-\gamma_{lj,n}\|, \end{aligned}$$ it follows that $$\|\hat{m}_n(\theta_0)-\frac{1}{n}\sum_{i=1}^n\psi_i(\theta_0)\|
=O_p(\max_{lj}\|\hat\gamma_{lj}-\gamma_{lj,n})\|^2)
=O_p(n^{-1}).$$
\[clt\_m\] Conditional on $X$ and $\sigma_n$, $$\hat{m}_n(\theta_0)=o_P(1)$$ and $$Var(\psi_i(\theta_0)\mid X,\sigma_n)^{-1/2}\sqrt{n}\hat{m}_n(\theta_0)\rightarrow_dN(0,I).$$
By Lemmas \[bounds\_many\] and \[psi\_ind\] and Lyapunov’s central limit theorem, it suffices to show $E[\psi_i(\theta_0)\mid X,\sigma_n]=0$. It follows from $$\begin{aligned}
E[\psi_i(\theta_0)\mid X,\sigma_n]
&=&
\frac{1}{n}\sum_{j\ne i}\left(E[G_{ij,n}\mid X,\sigma_n]-\rho_0-(1-\rho_0-\rho_1)\Phi(u_{ij}(\theta_0))\right)\mathbf{1}_{ij}
\\&&
-(1-\rho_0-\rho_1)\frac{1}{n^2}\sum_{l,j}\left(
\phi(u_{lj}(\theta_0))\beta_1'C(\rho_0,\rho_1)
E[\psi_{\gamma,i, n}(X_{lj})\mid X,\sigma_n]
\right)\mathbf{1}_{lj}\\
&=&
0, \end{aligned}$$ because $$\begin{aligned}
E[G_{ij,n}\mid X,\sigma_n]
&=&
\rho_0+(1-\rho_0-\rho_1)\Phi(u_{ij}(\theta_0))\\
E[\psi_{\gamma,i, n}(X_{lj})\mid X,\sigma_n]
&=&
\frac{1}{n^2} \sum_{i_{1},j_1}
\left(\frac{1\{X_{i_{1},j_1}=X_{ij}\}}{\hat{p}(X_{ij})} \right)
\left(
\begin{array}{c}
0\\
E[G_{kj_1,n}^{\dagger}\mid X,\sigma_n]\\
E[(G_{ki_{1},n}G_{kj_1,n})^{\dagger}\mid X,\sigma_n]\\
E[(G_{ki_{1},n}+G_{kj_1,n})^{\dagger}\mid X,\sigma_n]
\end{array}
\right)
\\
&&
+
\frac{1}{n} \sum_{i_{1}}
\left(\frac{1\{X_{i_{1},k}=X_{ij}\}}{\hat{p}(X_{ij})} \right)
\left(
\begin{array}{c}
E[G_{ki_{1},n}^{\dagger}\mid X,\sigma_n]\\
0\\
0\\
0
\end{array}
\right)\\
&=&
0.\end{aligned}$$ Note that $E[RV^{\dagger}\mid X,\sigma_n]=0$ by the definition of $RV^\dagger$.
\[var\_conv\] Conditional on $X$ and $\sigma_n$, $$\hat{S}(\theta_0)=Var(\psi_i(\theta_0)\mid X,\sigma_n)+o_p(1).$$
First, we are going to show $\hat{S}(\theta_0)=\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\tilde\psi_i(\theta_0)'-
\left(\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\right)
\left(\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\right)'
+o_p(1)$. Since $$\begin{aligned}
&&
\hat{S}(\theta_0)-\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\tilde\psi_i(\theta_0)'+\left(\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\right)\left(\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\right)'
\\
&=&
\frac{1}{n}\sum_{i=1}^n(\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0))(\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0))'
\\&&+\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)(\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0))'
\\&&+\frac{1}{n}\sum_{i=1}^n(\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0))\tilde\psi_i(\theta_0)'
\\&&-\left(\frac{1}{n}\sum_{i=1}^n(\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0))\right)\left(\frac{1}{n}\sum_{i=1}^n(\hat\psi_i(\theta_0))\right)'
\\&&-\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0))\left(\frac{1}{n}\sum_{i=1}^n(\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0))\right)'\end{aligned}$$ it follows that $$\begin{aligned}
&&
\left\|\hat{S}(\theta_0)-\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\tilde\psi_i(\theta_0)'+\left(\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\right)\left(\frac{1}{n}\sum_{i=1}^n\tilde\psi_i(\theta_0)\right)'
\right\|\\
&\leq&
\max_{i}\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|^2
\\&&+3\max_{i}\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|\max_{i}\|\tilde\psi_i(\theta_0)\|
\\&&+\max_{i}\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|\max_{i}\|\hat\psi_i(\theta_0)\|.\end{aligned}$$ Thus it suffices to show $\max_{i}\|\hat\psi_i(\theta_0)-\tilde\psi_i(\theta_0)\|=o_p(1)$ and $\max_{i}\{\|\tilde\psi_i(\theta_0)\|,\|\hat\psi_i(\theta_0)\|\}=O_p(1)$. They are shown in Lemmas \[bounds\_many\] and \[diff\_psi\].
Second, we are going to show $\hat{S}(\theta_0)=Var(\tilde\psi_i(\theta_0)\mid X,\sigma_n)+o_p(1)$. It suffices to show $E[\|\tilde\psi_i(\theta_0)\|^4\mid X,\sigma_n]<\infty$. By the triangle inequality, $$\begin{aligned}
E[\|\tilde\psi_i(\theta_0)\|^4\mid X,\sigma_n]^{1/4}
&\leq&
\frac{1}{n}\sum_{j\ne i}E[\left\|G_{ij,n}\right\|^4\mid X,\sigma_n]^{1/4}\\&&+\frac{1}{n^2}\sum_{l,j}E[\left\|\phi(u_{ij}(\theta_0))\beta_1'C(\rho_0,\rho_1)\hat\psi_{\gamma,i, n}(X_{lj})\right\|^4\mid X,\sigma_n]^{1/4}\\
&\leq&
\frac{1}{n}\sum_{j\ne i}\left(E[\left\|G_{ij,n}\right\|^4\mid X,\sigma_n]^{1/4}\right)\\&&+\frac{1}{n^2}\sum_{l,j}\phi(u_{ij}(\theta_0))\beta_1'C(\rho_0,\rho_1)E[\left\|\hat\psi_{\gamma,i, n}(X_{lj})\right\|^4\mid X,\sigma_n]^{1/4}\\
&\leq&
1+\frac{1}{n^2}\sum_{l,j}\phi(u_{ij}(\theta_0))\beta_1'C(\rho_0,\rho_1)E[\left\|\hat\psi_{\gamma,i, n}(X_{lj})\right\|^4\mid X,\sigma_n]^{1/4}\\
&<&
\infty,\end{aligned}$$ where the last inequality follows from Lemma \[bounds\_many\].
Third, we are going to show that $Var(\tilde\psi_i(\theta_0)\mid X,\sigma_n)=Var(\psi_i(\theta_0)\mid X,\sigma_n)$. Note that $\tilde\psi_i(\theta_0)-\psi_i(\theta_0)$ is a function of $X$ and $\sigma_n$, so the conditional variances are the same.
\[chi2dist\_conv\] Conditional on $X$ and $\sigma_n$, $$n\hat{m}_n(\theta)'\hat{S}(\theta)^{-1}\hat{m}_n(\theta)\rightarrow_d\chi^2_{J}.$$
It follows from Lemma \[clt\_m\] and \[var\_conv\].
Semiparametric Identification Analysis {#sec_a2}
======================================
Given $P\in\mathcal{P}$, we are going to characterize the identified set in the semiparametric model.
For each distribution $P\in\mathcal{P}$, the identified set $\Theta_{I,SP}(P)$ is defined as the set of all $\theta=(b,r_0,r_1)$ in $\Theta$ for which there is some joint distribution $P^\ast\in\mathcal{P}^\ast$ such that Condition \[independence\_assn\], \[linear\_index\](ii), and \[misclas\_prop\] holds, and that the distribution of $(G_{ij,n},X_{ij},\gamma_{ij,n})$ induced from $P^\ast$ is equal to $P$.
Given $P\in\mathcal{P}$, $\Theta_{I,SP}(P)$ is equal to the set of $\theta\in\Theta$ satisfying the following statements a.s.: $$\begin{aligned}
&&r_0\leq E_P\left[G_{ij,n}\mid Z_{ij,n}\right]\label{a_0_ineq}\\
&&r_1\leq E_P\left[1-G_{ij,n}\mid Z_{ij,n}\right]\label{a_1_ineq}\\
&&E_P\left[G_{ij,n}\mid Z_{ij,n}\right]=\Lambda\left((c(r_0,r_1)+\gamma_{ij,n}'C(r_0,r_1))'b_1+X_{ij}'b_2\right)\label{SSSinequality}\end{aligned}$$ for some weakly increasing and right-continuous function $\Lambda$.
First, we are going to show that every element $\theta$ of $\Theta_{I,SP}(P)$ satisfies the conditions in (\[a\_0\_ineq\])-(\[SSSinequality\]). Denote by $\Lambda^\ast$ the cdf of $-\varepsilon_{ij}$. Based on the assumptions, $$\begin{aligned}
E_{P^\ast}\left[G_{ij,n}\mid Z_{ij,n}\right]
&=&
r_0+(1-r_0-r_1)E_{P^\ast}\left[G_{ij,n}^\ast\mid Z_{ij,n}\right]\\
&=&
r_0+(1-r_0-r_1)\Lambda^\ast((c(r_0,r_1)+\gamma_{ij,n}'C(r_0,r_1))'b_1+X_{ij}'b_2).\end{aligned}$$ Define $\Lambda(v)=r_0+(1-r_0-r_1)\Lambda^\ast(c(r_0,r_1)'b_1+v)$ and we have $$E_{P^\ast}\left[G_{ij,n}\mid Z_{ij,n}\right]=\Lambda(\gamma_{ij,n}'C(r_0,r_1)'b_1+X_{ij}'b_2).$$ Since $\Lambda^\ast$ is strictly increasing, $\Lambda$ is also strictly increasing. Therefore, $$E_{P^\ast}\left[G_{i_1j_1}\mid Z_{i_1j_1}\right]
\geq
E_{P^\ast}\left[G_{i_2j_2}\mid Z_{i_2j_2}\right]
\iff
\gamma_{i_1j_1}'C(r_0,r_1)'b_1+X_{i_1j_1}'b_2
\geq
\gamma_{i_2j_2}'C(r_0,r_1)'b_1+X_{i_2j_2}'b_2,$$ which implies the condition (\[SSSinequality\]). The two inequalities in (\[a\_0\_ineq\]) and (\[a\_1\_ineq\]) are shown as follows: $$\begin{aligned}
E_{P^\ast}\left[G_{ij,n}\mid Z_{ij,n}\right]
&=&
r_0+(1-r_0-r_1)E_{P^\ast}\left[G_{ij,n}^\ast\mid Z_{ij,n}\right]\geq r_0\\
E_{P^\ast}\left[1-G_{ij,n}\mid Z_{ij,n}\right]
&=&
r_1+(1-r_0-r_1)E_{P^\ast}\left[1-G_{ij,n}^\ast\mid Z_{ij,n}\right]\geq r_1.\end{aligned}$$ where the inequalities follow from $1-r_0-r_1\geq 0$.
Next, we are going to show that every element $\theta\in\Theta$ satisfying (\[a\_0\_ineq\])-(\[SSSinequality\]), belongs to $\Theta_{I,SP}(P)$. By the condition (\[SSSinequality\]) as well as Conditions (\[a\_0\_ineq\]) and (\[a\_1\_ineq\]), there is a weakly increasing and right-continuous function $\Lambda:\mathbb{R}\rightarrow [r_0,1-r_1]$ such that $$\label{equE}
E_P\left[G_{ij,n}\mid (c(r_0,r_1)+\gamma_{ij,n}'C(r_0,r_1))'b_1+X_{ij}'b_2\right]=\Lambda\left((c(r_0,r_1)+\gamma_{ij,n}'C(r_0,r_1))'b_1+X_{ij}'b_2\right).$$ Denote by $\Lambda^\ast$ the cdf satisfying $\Lambda(v)=r_0+(1-r_0-r_1)\Lambda^\ast(c(r_0,r_1)'b_1+v)$.
Define the joint distribution $P^\ast$ in the following way. Define the cdf of $\varepsilon_{ij}$ such that $\Lambda^\ast$ is the cdf of $-\varepsilon_{ij}$. The conditional distribution of $(\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij})$ given $\varepsilon_{ij}$ is $$\label{equA}
P^\ast((\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij})\in B \mid \varepsilon_{ij})=P((\gamma_{ij,n},c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n},X_{ij})\in B)$$ for all the measurable sets $B$. The conditional distribution of $G_{ij,n}^\ast$ given $(\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})$ is $$\label{equB}
P^\ast(G_{ij,n}^\ast=1\mid \gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})=1\{(Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\}.$$ The conditional distribution of $G_{ij,n}$ given $(G_{ij,n}^\ast,\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})$ is $$\label{equC}
P^\ast(G_{ij,n}=1\mid G_{ij,n}^\ast,\gamma_{ij,n},\gamma_{ij,n}^\ast,X_{ij},\varepsilon_{ij})
=
\begin{cases}
1-r_0&\mbox{ if }G_{ij,n}^\ast=0\\
r_1&\mbox{ if }G_{ij,n}^\ast=1.
\end{cases}$$
Note that $(P^\ast,\theta)$ satisfies Conditions 1(ii), 2 and 3, because Condition 1(ii) follows from Eq. (\[equA\]), Condition 2 follows from Eq. (\[equB\]), Condition 3(i) follows from Eq. (\[equB\]) and (\[equC\]), and Condition 3(ii) follows from Eq. (\[equA\]).
The distribution of $(G_{ij,n},X_{ij},\gamma_{ij,n})$ induced from $P^\ast$ is equal to $P$. The distribution of $(X_{ij},\gamma_{ij,n})$ induced from $P^\ast$ is equal to that from $P$, by Eq. (\[equA\]). The equality of $P^\ast(G_{ij,n}=1\mid Z_{ij,n})=P(G_{ij,n}=1\mid Z_{ij,n})$ a.s. under $P^\ast$ is shown as follows. Note that $$\label{equD}
\gamma_{ij,n}^\ast=c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n} \mbox{ a.s. under }P^\ast$$ Then $$\begin{aligned}
P^\ast(G_{ij,n}=1\mid Z_{ij,n})
&=&
P^\ast(G_{ij,n}=1\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
(1-r_0)P^\ast(G_{ij,n}^\ast=0\mid Z_{ij,n},\gamma_{ij,n}^\ast)+r_1P^\ast(G_{ij,n}^\ast=1\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)P^\ast(G_{ij,n}^\ast=1\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)E_{P^\ast}[P^\ast(G_{ij,n}^\ast=1\mid Z_{ij,n},\gamma_{ij,n}^\ast,\varepsilon_{ij})\mid Z_{ij,n},\gamma_{ij,n}^\ast]\\
&=&
r_0+(1-r_0-r_1)P^\ast((Z_{ij,n}^\ast)'b + \varepsilon_{ij}\geq 0\mid Z_{ij,n},\gamma_{ij,n}^\ast)\\
&=&
r_0+(1-r_0-r_1)\Lambda^\ast((Z_{ij,n}^\ast)'b)\\
&=&
r_0+(1-r_0-r_1)\Lambda^\ast((c(r_0,r_1)+C(r_0,r_1)\gamma_{ij,n})'b_1+X_{ij}'b_2)\\
&=&
P(G_{ij,n}=1\mid Z_{ij,n}), \end{aligned}$$ where the first and seventh equalities follow from Eq. (\[equD\]), the second follows from Eq. (\[equC\]), the fifth follows from Eq. (\[equB\]), and the last follows from Eq. (\[equE\]).
[^1]: Department of Economics, University of Warwick, Coventry, CV4 7AL, U.K.; Email: L.Candelaria@warwick.ac.uk
[^2]: Department of Economics, University of California, Davis, One Shields Avenue, Davis, CA 95616-5270; Email: takura@ucdavis.edu
[^3]: Define permutation functions as follows. Fix any $k,l \in \mathcal{N}_{n}$, and let $g_{i,n}^\ast \in \mathcal{G}^n$. Define $\pi_{kl}: \mathcal{N}_{n} \mapsto \mathcal{N}_{n}$ as a permutations of the indices $k$ and $l$. Specifically, it maps the index $k$ to the index $l$, $l$ to $k$, and $i$ to itself for any $i \neq k,l$. Define $\pi_{kl}^{X}$ as a function that maps each component $X_{ij} \in \mathbb{R}^d$ to $X_{\pi_{kl}(i)\pi_{kl}(j)}$; $\pi_{kl}^{a}$ as a function that permutes the $k$th and $l$th elements of any $g_{i,n}^\ast\in \mathcal{G}^n$. Hence, $\pi_{kl}^{X}$ swaps the attributes of agents $k$ and $l$; and $\pi_{kl}^{a}$ swaps the links $G_{ik,n}^\ast$ and $G_{il,n}^\ast$ for any $i$. $\pi(\cdot)$ denote a generic element of $\Pi = \left\{ (\pi_{kl}, \pi_{kl}^{X}, \pi_{kl}^{a}); k,l \in \mathcal{N}_{n} \right\}$. In this paper, we abuse the notation $\pi(\cdot)$ so that it denotes any of the three components of an element in $\Pi$.
[^4]: We could aim to characterize the identified set based on the joint distribution of $\{(G_{ij,n},X_{ij},\gamma_{ij,n}):i,j\in\mathcal{N}_n\}$. Since we cannot estimate the joint distribution about $n$ agents from a sample of $n$ agents, however, the identified set based on $\{(G_{ij,n},X_{ij},\gamma_{ij,n}):i,j\in\mathcal{N}_n\}$ is not immediately useful for an inference. In contrast, $P_{0,n}$ can be estimated.
|
---
abstract: 'This paper shows a novel method to precisely measure the laser power using an optomechanical system. By measuring a mirror displacement caused by the reflection of an amplitude modulated laser beam, the number of photons in the incident continuous-wave laser can be precisely measured. We have demonstrated this principle by means of a prototype experiment uses a suspended 25mg mirror as an mechanical oscillator coupled with the radiation pressure and a Michelson interferometer as the displacement sensor. A measurement of the laser power with an uncertainty of less than one percent (1$\sigma $) is achievable.'
address: |
$^1$ National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan\
$^2$ Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan\
$^3$ Department of Physics, Syracuse University, NY 13244, USA\
$^4$ University of California, Berkeley, CA 94720, USA\
$^5$ Ochanomizu University Graduate School of Humanities and Sciences, Bunkyo, Tokyo 112-8610, Japan\
author:
- |
Kazuhiro Agatsuma,$^{1, 2, *}$ Daniel Friedrich,$^2$ Stefan Ballmer,$^3$\
Giulia DeSalvo,$^4$ Shihori Sakata,$^2$ Erina Nishida,$^5$\
and Seiji Kawamura$^2$
title: Precise measurement of laser power using an optomechanical system
---
[99]{}
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Introduction
============
Optical power meters used to measure the laser power are among the essential tools for optics experiments nowadays. The absolute value of the laser power was originally measured using phototubes that are based on the photoelectric effect [@Maiman1961]. Subsequently, calorimeters were used to determine the laser power (even for the masers) by measuring a temperature increase that corresponds to absorbed laser energy [@Li1962]. The semiconductors and thermopile sensors, which are based on conversions of radiation into the electrical current and of thermal gradient into the electrical current, respectively, are applied for optical power measurements, but they require a calibration referring to a primary standard. Calorimeters continue to be developed and have defined the primary standards of the laser power and energy with a traceability to SI units for nearly half a century. The current primary standard is based on a family of isoperibol calorimeters [@West1970]. These standards have a 0.25% uncertainty with 1$\sigma $ (it is a probability of 68% that the true value is included within its uncertainty), on broadband wavelengths (193nm - 10.6$\mu $m) [@NIST]. The world best measurements, as long as we know, were performed using cryogenic radiometers under some limited wavelengths and powers with an associated uncertainty of 0.01% (1$\sigma $) [@Livigni1998; @POWR]. Although typical commercial radiometers (power meters) refer to these primary standards, they still have an uncertainty of few percent.
Recently, the application range of power meters is spreading, so that a power measurement has become relevant to the quality of implemented experiments. Free electron lasers, for example, have a broad range of wavelength (around 0.1nm - 1000$\mu $m) [@FEL; @FEL2011]. Therefore, it is interesting to develop primary standards for such wavelengths (see e.g. [@Kato2012]). In addition, photodiodes (PDs) with high quantum efficiency (QE) (close to 99%) are required for optical experiments that can improve their sensitivity by means of squeezed light, such as gravitational wave (GW) detectors [@Kimble2002; @Vahlbruch2008; @GEO2010]. The QE of a PD is determined by measuring the incident laser power. Hence, the uncertainty of QE measurement is limited by the power measurement, i.e. its precise measurement (below one percent) is essential. Also, photon pressure calibrator, which is used for a displacement calibration of GW detectors, requires a precise value of the laser power as a reference of the calibration [@Clubley2001; @Mossavi2006].
We propose a new technique to measure the laser power using an optomechanical-coupled oscillator, which allows to determine the laser power from a displacement measurement. In addition, we have demonstrated this technique by constructing a prototype to investigate its feasibility. The derived uncertainty is below one percent (1$\sigma $), which is better than that of current commercial power meters. Our apparatus can be applied for arbitrary laser wavelengths by adapting material and coating of the oscillator. This technique has the potential to be an alternative method to realize a primary standard. First, theory and concept are shown. Subsequently, experimental results are discussed, which is followed by a detailed analysis of uncertainties that contribute to the overall measurement uncertainty.
Theory
======
Concept
-------
For an optomechanical system as sketched in Fig.\[fig:Concept\] with an input laser and a suspended mirror coupled through the radiation pressure, the laser power can be related with displacement of the mirror shaken by input photons. The displacement of a mirror pushed by radiation pressure in the frequency domain can be expressed as [@Clubley2001] $$d \tilde{X} = \frac{ 2 P_\mathrm{m} }{ c \, m \, \omega ^2 }. \label{eq:Opmech}$$ Here, $ d \tilde{X} $ is the Fourier transformation of the displacement of the mirror, $ m $ the mass of the mirror, $ c $ the speed of light, $ P_\mathrm{m} $ the intensity-modulated laser power, and $ \omega $ the angular frequency of the modulation. In Eq.(\[eq:Opmech\]), the mirror response is regarded as the free-oscillation mass (‘free mass’). The modulated laser power incident on the mirror can be derived by measuring the displacement of the moving mirror at the modulation frequency.
![Sketch of the optomechanical system investigated in this work. By measuring the displacement of a suspended mirror the incident modulated laser power can be determined. []{data-label="fig:Concept"}](Concept.eps){width="8cm"}
The procedure to measure the DC laser power requires a characterization of the modulation strength at the measurement frequency. For this purpose, a reference receiver like a PD is used to determine the ratio of the DC power and the modulated power. If the PD is put in the measurement region (see Fig.\[fig:Concept\]) after obtaining a value of the modulated laser power, the output voltage of the PD is related to the modulated power. The ratio of them provides $ C_\mathrm{m} \equiv P_\mathrm{m}/V_\mathrm{m} $; $ C_\mathrm{m} $ is the conversion factor between the voltage and laser power, $ V_\mathrm{m} $ is the spectrum value of the output voltage of the PD at the modulation frequency. If the PD response is independent of the frequency, the conversion factor is used for the signal not only at the modulation frequency but also DC. A well calibrated PD can be used as a precise power meter, for even other purposes.
For a known mechanical response of the mirror as well as a known power modulation one can actuate the mirror by a well defined amount. This is known as a photon pressure calibrator and is, for example, used in GW detectors [@Clubley2001; @Mossavi2006]. On the other hand, for a measured displacement of the mirror one can instead derive the modulated laser power, which is the concept of the novel power meter investigated in this work. In contrast to GW detectors, we use a lightweight mirror in the order of 25mg in order to increase the optomechanical response in the experiment. As a displacement sensor we have used a Michelson Interferometer as explained in the following section.
Theoretical model
-----------------
In the following, it is outlined how the incident laser power can be derived from the displacement of a suspended mirror. This includes a brief discussion of the optomechanical model for a quasi free mass as well as of the readout via a Michelson interferometer, which is the basis of the investigated concept. Further information can be found in section\[Sec:Uncertainty\], where the uncertainty contribution of each parameter is discussed in detail.
Generally, the incident laser can be reflected at the mirror under non-normal incidence and off center. These two effects can be accounted for by modifying Eq.(\[eq:Opmech\]) yielding off centering of the beam spot on the mirror. Also, imperfectness of the reflectivity of the mirror should be taken into account. Thus, the optomechanical response becomes $$d\tilde{X} = \frac{ 2 P_\mathrm{m} \alpha _{\mathrm{r}} \cos{\phi } }{ c \, m \, \omega ^2 } (1 + R_\mathrm{c}), \label{eq:Disp_photon}$$ where, $ \alpha _{\mathrm{r}} $ is the transfer efficiency of the momentum from photons to the mirror motion, $ \phi $ the incident angle of the input laser to the mirror and $ R_\mathrm{c} $ the rotational effect from off centering of the beam spot on the mirror. Details about the latter effect can be found in section\[Sec:Rc\]. If the mirror is suspended by wires, the response of the mirror is regarded as that of a free mass for frequencies much higher than the pendulum resonance frequency. However, the response of the mirror has a small deviation from the perfect free mass even in such frequency region. A more general expression uses $ H_{\mathrm{m}} (\omega ) $ as the mechanical response below instead of the free mass expression $ 1/(m \omega ^2) $ in Eq.(\[eq:Disp\_photon\]).
We selected the Michelson interferometer (MI) as a displacement sensor because of its simplicity and a suitable sensitivity. When the so-called ‘mid-fringe’ state is assumed, the readout signal is related with the displacement via $$d\tilde{X} = ( \lambda \big{/} 2 \pi V_{\mathrm{pp}} ) dV_{\mathrm{PD}} , \label{eq:PD_MI}$$ where $ dV_{\mathrm{PD}} $ is the output voltage from the PD and $ V_{\mathrm{pp}} $ is the peak-to-peak voltage when the arm length is changed over half a wavelength $ \lambda $. Combining Eq.(\[eq:Disp\_photon\]) and Eq.(\[eq:PD\_MI\]) yields $$P_\mathrm{m} = \frac{ c \lambda }{ 4 \pi V_{\mathrm{pp}} H_{\mathrm{m}} \alpha _{\mathrm{r}} (1 + R_\mathrm{c}) \cos{\phi } } \, dV_{\mathrm{PD}} . \label{eq:Power}$$ Note that if the mid-fringe state is kept by position control, $ dV_{\mathrm{PD}} $ with activated control loop is different from the one without control loop.
![Block diagram for the MI control. The symbols $ S, A $ and $ H_{\mathrm{PD}} $ show each transfer function of the servo filter, actuator response and PD response, respectively. []{data-label="fig:Blockdiagram"}](BlockDiagram.eps){width="6.5cm"}
To include the effect of the control loop, the term $ dV_{\mathrm{PD}} $ in Eq.(\[eq:Power\]) has to be replaced by $ V_{\mathrm{f}} G_{\mathrm{CL}} T_{\mathrm{AH}} $ because $ d\tilde{X} = V_{\mathrm{f}} G_{\mathrm{CL}} T_{\mathrm{AH}} / H_{\mathrm{PD}} $ (see Fig.\[fig:Blockdiagram\]). Here, $ V_{\mathrm{f}} $ is the feedback signal of the MI, $ G_{\mathrm{CL}} = (1+G)/G $ the closed-loop gain, $ G $ the loop gain and $ T_{\mathrm{AH}} $ the transfer function from actuator to PD. Finally, we obtain $$P_{\mathrm{m}} = \frac{ c \lambda \, V_{\mathrm{f}} G_{\mathrm{CL}} T_{\mathrm{AH}} }
{ 4 \pi V_{\mathrm{pp}} H_{\mathrm{m}} \alpha _{\mathrm{r}} (1 + R_\mathrm{c}) \cos{\phi } } . \label{eq:Power_2}$$ The absolute values of each transfer function are used in Eq.(\[eq:Power\_2\]). We focus on the uncertainty of the power measurement in this paper. The uncertainty in $ \lambda $ is included in the uncertainty evaluation of $ V_{\mathrm{f}} $. The displacement readout part ($ V_{\mathrm{f}} $, $ G_{\mathrm{CL}} $, $ T_{\mathrm{AH}} $ and $ V_{\mathrm{pp}} $) is evaluated by means of a prototype experiment to estimate actual effects, which are difficult to evaluate by just calculation, from electrical noise, laser intensity noise, thermal drift and so on. The optomechanical response part ($ H_{\mathrm{m}} $, $ \alpha _{\mathrm{r}} $, $ R_\mathrm{c} $ and $ \phi $) is analyzed using both modeling and measurements. A detailed discussion is given in section\[Sec:Uncertainty\].
Experimental setup
==================
When a light weight mirror is used, the displacement due to the radiation pressure of the incident light is increased (see Eq.(\[eq:Disp\_photon\])). We use a tiny flat mirror of about 25mg as the optomechanical-coupled oscillator. The displacement sensor is a MI that consists of the tiny mirror and a one-inch flat mirror (‘large mirror’) as end mirrors in the two arms, respectively. A schematic view of the experimental setup is shown in Fig.\[fig:Setup\]. The laser (Innolight Inc. Mephisto) has a nominal power of 500mW and a wavelength of 1064nm. The laser beam is divided into two paths that are used for the Michelson interferometer ($\sim$50mW) and to actuate the tiny mirror ($\sim$450mW). The latter one is modulated in power by means of an acousto-optic modulator (AOM), which provided a power modulation index of about 17%. In the MI light path, the laser power that remains at the front of the tiny mirror is about 10mW. The MI is housed in a sealed chamber, which is put on vibration isolation stacks, to be soundproof in air. A preliminary experiment can be found in reference [@Agatsuma2012].
![Schematic view of the experimental setup. Acronym explanation: HWP, half wave plate; QWP, quarter wave plate; BS, beam splitter; PBS, polarized beam splitter; FI, faraday isolator; and AOM, acousto-optic modulator. $ L1 $ and $ L2 $ show length measurements for the incident angle. There are three photo detectors; Symmetric port, Asymmetric port and AOM port on an optical table. The solid red line indicates a beam trace for the MI and the dotted red line is that for an intensity modulation by AOM. []{data-label="fig:Setup"}](QE_SchematicVIEW_3.eps){width="\linewidth"}
![The tiny mirror suspension system. (a) photographs of the 25mg mirror and the double pendulum. (b) model of the double pendulum.[]{data-label="fig:DoublePend"}](SuspensionModel_3.eps){width="\linewidth"}
Figure\[fig:DoublePend\](a) shows photographs of the small suspension system. The tiny mirror has a cylindrical shape with a diameter of 3mm, a thickness of 1.5mm and is suspended as a double pendulum developed in [@Sakata2010]. The lower suspension is a silica fiber with a diameter of 10$ \mu $m, glued on the mirror using UV cured resin. The middle mass of 25mg, made of aluminum, is suspended by a 10$ \mu $m tungsten wire and is eddy current damped by the surrounding small magnets. The second mirror of the MI is a conventional one inch mirror with a flat surface and a weight of 48g (including optical bracket). This suspension system is also a double pendulum with eddy current damping. The large mirror and its optical bracket are suspended by 50$ \mu $m tungsten wires. The intermediate mass of 57g is suspended by 70$ \mu $m tungsten wires.
The large mirror is actuated to keep the MI on its operation point by controlling the differential length of the two interferometer arms. This feedback control is called ‘lock’. For this purpose, four sets of coil-magnet actuators are used for position control of the large mirror. The error signal for the mid-fringe lock is derived by applying an electrical offset to the PD output at the symmetric port. The offset is adjusted to provide a zero crossing at the center of the fringe. A linear response for the displacement of the mirror is guaranteed as long as the mid-fringe state is maintained. A contrast of about 95% is kept in this experiment.
Experimental results
====================
Sensitivity and radiation pressure response
-------------------------------------------
![The sensitivity of the MI and the displacement shaken by radiation pressure at 66, 72, 82 and 233Hz. The orange dashed line shows $f ^{-2}$ response of the free mass from Eq.(\[eq:Disp\_photon\]). []{data-label="fig:Sens"}](Sensitivity_3.eps){width="\linewidth"}
In Fig.\[fig:Sens\], apparent peak signals due to the power modulation are measured together with the displacement of the MI without power modulation, which is called the sensitivity (background). Four frequencies (66, 72, 82 and 233Hz) have been chosen for the amplitude modulation because a sufficient signal to noise ratio is provided. Furthermore, they show a $f ^{-2}$ response (orange dashed line) as it is predicted for a free mass (a quasi free mass, in a rigorous expression as discussed in section\[Sec:Hm\]). In our measurement, the spectra and standard deviations are obtained from the time series data of 1kHz sampling with an anti-aliasing filter of 500Hz. The flat-top window is used when the Fourier transformation is applied to the data to evaluate precise values of the peak structure.
Laser power measurement
-----------------------
The modulated laser was partially extracted for the PD on the AOM port ($ \mathrm{PD_{AOM}} $) instead of on the measurement region. Here, $ C_\mathrm{m}^* = P_\mathrm{m} / V_{\mathrm{PDAOM}} $ becomes the corresponding conversion factor. Once $ C_\mathrm{m}^* $ is determined, $ \mathrm{PD_{AOM}} $ can be a real time monitor of the laser power at the front of the small mirror. A high power beyond a PD receivable can also be sent to the measurement region by utilizing this $ \mathrm{PD_{AOM}} $. It means we can see the response signal with reinforcing the optomechanical coupling. The conversion factor $ C_\mathrm{m}^* $ is a constant as long as the condition of the path of the modulated beam is kept. Hence, the optical components in the beam path are not related with any uncertainty contribution. If power loss is caused by mirror contamination, $ C_\mathrm{m}^* $ should be measured again. The results presented in the following are based on measurements with a modulation frequency of 72Hz, as it showed the lowest uncertainty in our setup.
The modulated power $ P_\mathrm{m} $ is calculated using Eq.(\[eq:Power\_2\]) and the peak value at 72Hz in Fig.\[fig:Sens\]. The result was $ P_\mathrm{m} = 39$mW, which corresponds to $ 2.1 \times 10^{17} $ photons. The conversion factor of 0.371W/V was obtained by dividing the value of modulated power by the output voltage of $ \mathrm{PD_{AOM}} $ at 72Hz. The resultant DC voltage of the $ \mathrm{PD_{AOM}} $ was 0.802V (environmental offset was subtracted) when the modulation of AOM was turned off, which corresponds to the laser power of 298mW at a close position from the tiny mirror. Note that the above estimation did not include the rotational effect as shown in Eq.(\[eq:Power\_2\]) by $ R_\mathrm{c} $.
Quantum efficiency measurement {#Sec:QE}
------------------------------
As an application of this power measurement, the QE can be estimated as explained in the following. The QE of a PD is defined as the conversion efficiency from the light power to the current; $ QE = N_\mathrm{e} / N_\mathrm{P} $. Here, $ N_\mathrm{e} $ is the number of electrons in the output current of the PD and $ N_\mathrm{P} $ the number of photons in the input light. $ N_\mathrm{P} $ is related to a continuous-wave laser with power $ P $ by $ N_\mathrm{P} = P / ( h \nu ) $ using Planck’s constant $ h $ and the frequency of the photon $ \nu $. This QE corresponds to the inverse of the conversion factor ($ C_\mathrm{m}^{-1} $) by multiplying the inverse of an inner resistance $ R $ of the PD. This implies that a precise QE of the PD can also be obtained using our power meter. Note that the relation of $ QE = (C_\mathrm{m} R)^{-1} $ can be used when the PD is calibrated in the measurement region. To derive the QE of the $ \mathrm{PD_{AOM}} $, we need the laser power ratio between the AOM port and the measurement region.
After calibrating the $ \mathrm{PD_{AOM}} $ as a power meter, another PD (PD tank) to measure the QE was put in the measurement region. Usually, PDs cannot receive the incident power over 10mW. A neutral density filter is inserted close to the AOM output in order to reduce the laser power. Since the $ \mathrm{PD_{AOM}} $ can monitor such a change of the power including a drift, as long as the power linearity of the PD is guaranteed, the incident laser power at the measurement region can be obtained with the same uncertainty as the above calibration. The measured value of the QE of the PD tank is 0.301, corresponding to the photo sensitivity of 0.258A/W. In order to also measure the QE of $ \mathrm{PD_{AOM}} $, the positions of both PDs were exchanged. The beam spot sizes on the PD surfaces are adjusted to be almost same size (0.2mm) by measuring beam profiles and adjusting the PD positions, in case the response of the PDs change by the beam spot size. Comparing with the $ \mathrm{PD_{AOM}} $, the PD tank has an efficiency ratio of 0.939 and a laser power ratio of 102. The $ \mathrm{PD_{AOM}} $ has a QE of 0.320, corresponding to 0.275A/W. Both PDs are Si PIN photodiode (S3759 produced by HAMAMATSU).
According to the specification sheet, the expected photo sensitivity is the range from 0.3 to 0.38A/W, which comes from individual differences. The results are lower than the specification sheet by 20%, at least. The reason seems the unmeasured rotational effect, corresponding to an off centering of about 0.4mm, and low temperature in the laboratory (about 10 degree lower than that of the specification sheet). Also, note that the beam position on the PD surface should be considered to achieve more careful measurements. In order to obtain an accurate value of the QE, above items should be treated quantitatively. Using this method, the QE can be measured precisely in principle. Now, our focus is to evaluate the inevitable uncertainty of the power measurement, which is analyzed in the following section.
Uncertainty evaluation {#Sec:Uncertainty}
======================
Definition and method
---------------------
We use the standard uncertainty to evaluate each measurement. According to a document of ISO [@GUM2008], the standard uncertainty is defined as uncertainty of the result of a measurement expressed as a standard deviation. There are two ways of defining the standard uncertainty, the so-called Type A and Type B evaluation. Type A evaluation is used for the statistical uncertainty, which has a Gaussian distribution giving the standard deviation. Type B evaluation is used for the results that are not obtained from repeated observations, e.g. an upper and lower limit. Since the uniform probability density function is used under Type B evaluation, the uncertainty is divided by $ \sqrt{3} $ to correspond to 1$\sigma $ of the standard deviation.
The propagation law of uncertainty (combined standard uncertainty $ \sigma _Y $) is expressed as $$\sigma _Y = \sqrt{
\sum_{i=1}^{s} \left( \frac{\partial Y}{\partial \xi_i} \sigma_i \right) ^2 +
2 \sum_{i=1}^{s-1} \sum_{j=i+1}^{s} \frac{\partial Y}{\partial \xi_i} \frac{\partial Y}{\partial \xi_j} \sigma_i \sigma_j r( \xi_i, \xi_j ) ,
}
\label{eq:Sigma}$$ where $ Y $ is a certain complex function that consists of $ \xi_i $ ($ i = 1, 2, .., s $), $ \sigma_i $ the standard uncertainty of each component and $ r( \xi_i, \xi_j ) $ the estimated correlation coefficient associated with $ \xi_i $ and $ \xi_j $. Using both Eq.(\[eq:Power\]) and Eq.(\[eq:Sigma\]), the total uncertainty of the laser power is written as $$\frac{\sigma _P}{P} = \sqrt{
\left( \frac{\sigma _{dV}}{dV_{\mathrm{PD}}} \right) ^2 +
\left( \frac{\sigma _{V} }{ V_{\mathrm{pp}} } \right) ^2 +
\left( \frac{\sigma _H}{H_{\mathrm{m}}} \right) ^2 +
\left( \frac{\sigma _\alpha}{\alpha_{\mathrm{r}} } \right) ^2 +
\left( \sigma _\phi \tan \phi \right) ^2 +
\left( \frac{\sigma _R}{ 1 + R_\mathrm{c} } \right) ^2 .
} \label{eq:Sigma_P}$$ It is assumed that ingredients of uncertainty have no correlation to each other so that the second term of Eq.(\[eq:Sigma\]) vanishes. In order to keep the uncertainty of the laser power within one percent, the relative uncertainty of below 0.3% is required because the number of effective components is about ten when taking into account the different contributions included in the first term of Eq.(\[eq:Sigma\_P\]). Here, the relative uncertainty is a ratio between each component and its standard deviation ($\sigma _i / \xi_i $). Hence, the relative uncertainty of 0.3% is a naive requirement (criterion) and of course a lower value is desirable.
Deviation from mid-fringe lock $ dV_{\rm{PD}} $
------------------------------------------------
The voltage at the error point obtained from the PD at the symmetric port of the MI depends on the arm length difference $x$ and can be written as $$V_{\rm{PD}} = \frac{1}{2} V_\mathrm{pp} \sin{ \left[ 2 \pi \Big( \frac{x}{\lambda /2} \Big) \right] } . \label{eq:PD_MI_sin}$$ The PD response $ H_{\rm{PD}} $ is produced by its derivative. If a perfect condition of the mid-fringe ($ x = \lambda /8 + \lambda n/2 $) ($ n $ is integer) is kept, Eq.(\[eq:PD\_MI\]) is yielded using relations of $ dV_{\rm{PD}} = H_{\rm{PD}} d\tilde{X} $ and $ H_{\rm{PD}} = 2 \pi V_\mathrm{pp} / \lambda $. We used an electrical offset to adjust the operation point of the interference to the mid-fringe. The perfect mid-fringe is desirable because a linear response of the MI is obtained around that point. However, the electrical circuit may cause a drift of the offset, changing the gradient of linear response due to the term of $ \cos(p) $ in the derivative of Eq.(\[eq:PD\_MI\_sin\]) where $ p $ is phase of offset (see Fig.\[fig:TiltCorr\](a)). In addition, a large residual error signal cause a nonlinear effect because of actual sinusoidal response in Eq.(\[eq:PD\_MI\_sin\]). Namely, the PD response is rewritten as $$H_{\rm{PD}} = \frac{2\pi V_\mathrm{pp}}{\lambda } k r_\mathrm{s} , \label{eq:H_PD}$$ where $ k = \cos{(p)} $ and $ r_\mathrm{s} $ is difference between the sinusoid and linear. The uncertainty term from the PD response is expanded with the relation of $ dV_{\mathrm{PD}} = V_{\mathrm{f}} G_{\mathrm{CL}} T_{\mathrm{AH}} $ including deviation terms $ k $ and $ r_\mathrm{s} $ as $$\left( \frac{\sigma _{dV} }{ dV_{\mathrm{PD}} } \right) ^2
=
\left( \frac{\sigma _{V_\mathrm{f} } }{V_\mathrm{f} } \right) ^2
+ \left( \frac{ \sigma _{G_{CL}} }{G_\mathrm{CL}} \right) ^2
+ \left( \frac{\sigma _T}{T_\mathrm{AH}} \right) ^2
+ \left( \frac{\sigma_k}{k} \right) ^2
+ \left( \frac{\sigma_{r_\mathrm{s}}}{r_\mathrm{s}} \right) ^2 . \label{eq:PD_res_err}$$ Note that the uncertainty of $ k $ and $ r_\mathrm{s} $ are assumed to be a constant value over the measurements of the calibration ($ V_\mathrm{f} $, $ G_\mathrm{CL} $ and $ T_\mathrm{AH} $). Since $ k $ and $ r_\mathrm{s} $ could change from measurement to measurement, they should be determined at anytime the setup is calibrated. The largest value should be used as a modest estimation.
![Deviation from the mid-fringe lock. (a) effect of $ k $ and $ r_\mathrm{s} $. Here, $ p $ is an offset phase of an operation point, $ q $ an average phase of a residual error signal and $ r_\mathrm{s} $ the difference between an ideal linear response and an actual sinusoidal response; (b) required precision for the measurement of p from Eq.(\[eq:Ep\]). []{data-label="fig:TiltCorr"}](TiltCorr_Sin.eps){width="\linewidth"}
The drift in the offset from the servo circuit changes the gradient of linear response due to the term of $ k $ in Eq.(\[eq:H\_PD\]). At the ideal operation point (mid-fringe), $ k = 1 $. The offset phase shift corresponding to the 0.3% change in the slope is calculated by $ p = \arccos{(0.997)} = 0.078 $. This gives a normalized output voltage of 0.078 in the first order linear approximation. This means that the 0.3% criterion is satisfied if a drift of the offset can be kept around the ideal operation point within 7.8% of $ V_\mathrm{pp} /2 $. The uncertainty is calculated by $ \sigma_k = 1 - \cos{(p)} $. When the offset is larger than 7.8% of $ V_\mathrm{pp} /2 $, one should use the actual offset-phase, which can be derived from Eq.(\[eq:PD\_MI\_sin\]). The phase shift $ p $ is obtained from a output voltage of $ dV_\mathrm{PD} $ with a relation of $ \sin(p) = 2V_\mathrm{p}/V_\mathrm{pp} $ ($ V_\mathrm{p} \geq 0 $), where $ V_\mathrm{p} $ is the offset voltage. The deviation of $ k $ is expressed as $ k - \sigma_k = \cos{(p + E_\mathrm{p})} $ and $ k + \sigma_k = \cos{(p - E_\mathrm{p})} $, when $ 0 < p < \pi/2 $. Here, $ E_\mathrm{p} $ is upper and lower limit of the uncertainty of the measurement of $ p $. In order to satisfy the 0.3% criterion, the measurement of $ p $ is according to the following two inequalities: $$\begin{array}{c}
( k - \sigma_k )/k = \cos{(p + E_\mathrm{p} )}/\cos{(p)} \geq 0.997 \\
( k + \sigma_k )/k = \cos{(p - E_\mathrm{p} )}/\cos{(p)} \leq 1.003 , \label{eq:Ep}
\end{array}$$ when $ 0.078 < p < \pi/2 $. The equal signs of the above inequalities are illustrated in Fig.\[fig:TiltCorr\](b). This graph indicates the required accuracy for the measurement of $p$ to satisfy the 0.3% criterion (for example, $p$ should be measured within an uncertainty of 0.01rad, when $p = 0.3$). The reason for the two curves in Fig.\[fig:TiltCorr\](b) can be understood as follows. In the case of a small offset, a negative deviation goes to the ideal operation region (the slope changes a little), and a positive deviation goes to a deviate region (the slope changes a lot). To stay within the 0.3% criterion different precisions $ E_\mathrm{p} $ are required that can be derived from Eq.(\[eq:Ep\]). In the case of a large offset, the requirement for the precision becomes more severe. By applying Type B evaluation, such requirement is mitigated by $ \sqrt{3} $. Actually, our measurements of the calibration factors, $ T_\mathrm{AH} $, $ G_\mathrm{CL} $ and $ V_\mathrm{f} $ have about 3% of drift around the zero offset: $ ( \sigma_k / k ) ^2 = [ (1 - \cos{0.03}) / (\sqrt{3} \cos{0.03}) ] ^2 = ( 0.03\,\% ) ^2 $.
Let us consider a nonlinear effect by a residual error signal. When the transfer functions are measured, a large signal is injected to the control loop to increase the signal to noise ratio. In this situation a residual error signal may exceed the linear range of the mid-fringe, turning to be sinusoidal shape. In order to satisfy the criterion of 0.3%, a condition of $ r_\mathrm{s} = \sin{(q)} / q \geq 0.997 $ ($ 0 \leq q \leq \pi/2 $) is imposed, which implies $ q \leq 0.13 $. Here, $ q $ is a phase shift from the operation point. The deviation of $ r_\mathrm{s} $ becomes $ \sigma _{r_\mathrm{s}} = 1 - \sin{(q)} / q $. For general offset $ p $, a phase shift $ q $ is obtained from the relation $ q \cos{(p)} = 2V_\mathrm{q}/V_\mathrm{pp} $, where $ V_\mathrm{q} $ is the residual error signal, and then, the inequality $$r_\mathrm{s} = \frac{\sin{(p + q)} - \sin{(p - q)}}{2 q \cos{(p)}} \geq 0.997$$ is used as the 0.3% criterion. This also sets a limit on $ q $ at 0.13 regardless of the parameter $ p $. This means that the residual error signal $ V_\mathrm{q} $ should be suppressed within $ k \times (13\,\%) $ of the sinusoidal peak $ V_\mathrm{pp} /2 $. In our measurement, the peak to peak voltage is $ V_\mathrm{pp} = 3.44 $V and the standard deviation of the residual voltage is 0.094V. This is the largest value in all measurements of the calibration factors. It corresponds to 5.5% of the sinusoidal peak and $ q = 0.055 $ by the small angle linear approximation. Using Type A evaluation, the contribution factor becomes $ ( \sigma _{r_\mathrm{s}} / r_\mathrm{s} ) ^2 = [ (0.055 - \sin{0.055}) / \sin{0.055} ] ^2 = ( 0.05\,\% ) ^2 $.
Calibration factors $ V_\mathrm{f}, G_\mathrm{CL}, T_\mathrm{AH}, V_\mathrm{pp} $
---------------------------------------------------------------------------------
Instead of the error signal from a PD, the feedback signal is used for the calibration factor of the displacement, so that the noise from the servo circuit can be suppressed. The data was averaged by using 31 FFTs based on data segments of 16s length. Figure\[fig:FB\] shows the feedback signal in the case of the power modulation at 72Hz. There is an apparent peak at 72Hz, which is 40 times larger than the noise floor. The noise floor includes the readout noise and the residual motion of both the tiny mirror and large mirror due to disturbances like the seismic noise, acoustic vibration, detector noise, circuit noise and the frequency noise of the laser. These noises contribute to the peak value by $ (1/40)^2 \approx 0.06\,\% $. The measured standard deviation of the peak signal is 0.30% of the peak value. This fluctuation includes the effects from the laser power like the intensity noise and thermal drift of the laser power. In total, the contribution term becomes $ ( \sigma _{V_\mathrm{f}} / V_\mathrm{f} ) ^2 = ( 0.30\,\% )^2 + ( 0.06\,\% )^2 $. The feedback signal was recorded at the same time as the output signal of the $ \mathrm{PD_{AOM}} $ to connect power fluctuation with the signal of $ \mathrm{PD_{AOM}} $ directly.
![Feedback signal with modulated laser at 72Hz. Left: covering a wide frequency. Right: zoomed-in around 70Hz. The black arrow marks the peak at 72Hz. The black line indicates the noise level without the laser modulation. []{data-label="fig:FB"}](FB_72Hz.eps){width="\linewidth"}
Figure\[fig:CLG\_StD\](a) shows a measurement of the closed loop gain $ G_\mathrm{CL} $. A random noise was injected to measure this transfer function with a wide band of frequencies. As a result, the standard deviation is 0.48% at 72Hz as shown in Fig.\[fig:CLG\_StD\](b). The contribution term becomes $ ( \sigma _G / G_\mathrm{CL} ) ^2 = ( 0.44\,\% )^2 $. Also, the loop gain $ G $ was measured. The unity gain frequency is about 500Hz with a phase margin of 50 degrees.
![Analysis of the closed loop transfer function. (a) absolute value of the closed loop gain; (b) standard deviation of the closed loop gain. Black arrows indicate the measurement points. []{data-label="fig:CLG_StD"}](CLG_StD.eps){width="\linewidth"}
Since the transfer function $ T_\mathrm{AH} $ (from $ V_\mathrm{f} $ to $ dV_\mathrm{PD} $ in Fig.\[fig:Blockdiagram\]) includes the response of the PD $ V_\mathrm{pp} $, both values should be measured without substantial delay to avoid any change of the conditions. If the uncertainty of $ V_\mathrm{pp} $ is increased, the uncertainty of $ T_\mathrm{AH} $ is also increased. This means that there exists a positive correlation between $ T_\mathrm{AH} $ and $ V_\mathrm{pp} $. By applying Eq.(\[eq:Sigma\]) to Eq.(\[eq:Power\_2\]), the correlation term between $ T_\mathrm{AH} $ and $ V_\mathrm{pp} $ could yield a negative sign, which reduces total uncertainty, by the second term of Eq.(\[eq:Sigma\]) because $ r( T_\mathrm{AH}, V_\mathrm{pp} ) $ is positive and the partial derivative of Eq.(\[eq:Power\_2\]) with regard to $ V_\mathrm{pp} $ is negative. To obtain the correlation coefficient, the simultaneous measurement of both parameters are needed. However, they cannot be measured at the same time because $ T_\mathrm{AH} $ is measured in lock whereas $ V_\mathrm{pp} $ is measured out of lock.
![Analysis of the actuator response. (a) comparison of $ A $ between two measurement methods (monolithic and broadband), which are absolute values and divided by each PD response. Four monochromatic measurements are superimposed as the red solid line. Black allows are the measurement points; (b) output signal from the MI without lock. []{data-label="fig:ACT_Vpp"}](ACT_Vpp.eps){width="\linewidth"}
This is why, they were treated as independent measurements to assume the worst case and Eq.(\[eq:Sigma\_P\]) is kept. Figure\[fig:ACT\_Vpp\](a) shows the actuator response $ A = \lambda T_\mathrm{AH} / ( 2 \pi V_\mathrm{pp} ) $. The efficiency can be changed in various ways, for example, by changing the gap between coils and magnets, and by altering the balance of the four coils. These appear as the uncertainty of the measurement when the large mirror is shaken for the measurement of the transfer function. In order to reduce the uncertainty, the transfer function is measured at each monochromatic frequency that we select for uncertainty evaluation. A broadband measurement was also performed using random noise as the input signal, which has a somewhat large uncertainty (1.3% - 1.9%). It was consistent with the monochromatic measurements within their uncertainty. The resonance structure around 120Hz comes from a coupling with the angular motion of the large mirror. The broadband response is used for only making the displacement curve (Fig.\[fig:Sens\]) but not used for the precise uncertainty evaluation. The monochromatic measurement indicates $ ( \sigma _T / T_\mathrm{AH} )^2 = ( 0.49\,\% )^2 $ at 72Hz. This is the largest uncertainty of all measurement. Figure\[fig:ACT\_Vpp\](b) is a part of measured output signal from the MI without lock. A fluctuation of the contrast of the MI appears in the change of the peak to peak value of the output $ V_\mathrm{pp} $. The peak to peak value is measured for 50s before and after measuring the transfer function $ T_\mathrm{AH} $. The values for maximum and minimum voltage of a fringe have been measured using sufficiently long segments of about 0.5s as covered in Fig.\[fig:ACT\_Vpp\](b), then the average and standard deviation are obtained using those data. The result is $ 3.437 \pm 0.011 $V and, subsequently, $ ( \sigma _V / V_\mathrm{pp} ) ^2 = ( 0.33\,\% )^2 $.
There is a correlation between $ G_\mathrm{CL} $ and $ T_\mathrm{AH} $ because $ G_\mathrm{CL} $ includes $ T_\mathrm{AH} $ according to Fig.\[fig:Blockdiagram\]; $ G_\mathrm{CL} = (1+G)/G = (1 + S T_\mathrm{AH})/ (S T_\mathrm{AH}) $. Since $(\partial{G_\mathrm{CL}} / \partial{G}) / G_\mathrm{CL} = -1/{[G(1+G)]}$, the relative uncertainty of $G$ is attenuated by $(1+G)$ in $ G_\mathrm{CL} $. If the origin of the relative uncertainty of $G$ comes from only $ T_\mathrm{AH} $, $ \sigma_G / G = \sigma_T / T_\mathrm{AH} = 0.49\,\% $ is attenuated by $(1+G)$. It means the correlation function is $r(G_\mathrm{CL}, T_\mathrm{AH}) = 1/(1+G)$. In this measurement, the relative correlation contribution becomes $ \{ 2 \times 0.44 \times 0.49 / (1+10) \}^2 = (0.04\,\%)^2 $ from Eq.(\[eq:Sigma\]) due to $G = 10$ at 72Hz. According to this estimation, it is apparent that the relative contribution term of $ G_\mathrm{CL} $ (0.44% at 72Hz) mainly comes from the uncertainty of the servo gain $S$. The fluctuation of $S$ does not affect the measurement of $ T_\mathrm{AH} $ as far as its coherence is kept. Thus the maximum correlation is $1/(1+G)$.
Mechanical response $H_{\mathrm{m}}$ {#Sec:Hm}
-------------------------------------
In order to realize a quasi free-mass response of the 25mg mirror, it is suspended as a double pendulum with eddy current damping at the intermediate mass. Actually, the suspended mass is regarded as almost a free mass in the high frequency region ($ H_{\mathrm{m}} \approx 1/(m \omega ^2) $). Although parameter deviations of the the suspension system have the strongest impact on $ H_{\mathrm{m}} $ around the suspension resonance, they also influence the mechanical transfer function even at higher frequencies. Based on a mechanical model these deviations are investigated in the following. Figure\[fig:DoublePend\](b) shows a model of the double pendulum. The equation of motion is written as $$m_1 \ddot{x} _1 = - \frac{ (m_1 + m_2) g }{ l_1 } x_1 - \frac{ m_2 g }{ l_2 } (x_1 - x_2) - \Gamma _1 \dot{x} _1 ,$$ $$m_2 \ddot{x} _2 = - \frac{ m_2 \, g }{ l_2 } (x_2 - x_1) - \Gamma _2 \dot{x} _2 + F ,$$ where $ m $ is the mass, $ x $ the displacement, $ l $ the length of the suspension wire, $ \Gamma $ the damping coefficient of the eddy current or air, $ g $ the acceleration due to gravity at the Earth’s surface and $ F $ the force from the radiation pressure. The subscripts of 1 and 2 denote the upper pendulum and lower pendulum, respectively. After the Fourier transformation, these equations are rewritten as $$A
\left(
\begin{array}{c}
\tilde{x} _1 \\
\tilde{x} _2
\end{array}
\right)
=
\left(
\begin{array}{c}
0 \\
\tilde{F}
\end{array}
\right) , \hspace{1cm}
A ^{-1}
\equiv
\left(
\begin{array}{cc}
a_{\mathrm{m}} & b_{\mathrm{m}} \\
c_{\mathrm{m}} & d_{\mathrm{m}}
\end{array}
\right).$$ \[eq:Matrix\] Here, the matrix $A$ includes complex forms of above parameters. The response of the test mass becomes $$H_{\mathrm{m}} (\omega ) \equiv \frac{ \tilde{x} _2 }{ \tilde{F} } = d_{\mathrm{m}} .$$
![Effects from parameter deviations of the double pendulum on its mechanical response. (a) comparison between the free mass response and mechanical response with the default setting; (b-f) deviations from the default setting in case the possible differences exist in the parameters of the damping factor ($ Q $) at the intermediate mass, the test mass, the intermediate mass, the length of the upper wire, and of the lower wire, respectively. []{data-label="fig:MechRes"}](MechRes_3.eps){width="\linewidth"}
The default-setting parameters of our small pendulum are $ m_1 = 25 $mg, $ m_2 = 24.63 $mg, $ l_1 = 10 $mm, $ l_2 = 10 $mm and $ Q = 300 $. The quality factor $ Q $ is related with the damping coefficient by $ \Gamma _1 = m_1 \omega _1 / Q $ using $ \omega _1 = \sqrt{ g / l_1 } $. The quality factor of 300 is the measured value of another single pendulum with 25mg mirror and 10$\mu $m suspension wire in air. This is the highest value in the case as the middle mass is damped by only air. The quality factor of the lower pendulum is fixed at 300 with regard to $ \Gamma _2 $. Figure\[fig:MechRes\](a) shows the default setting of the mechanical response $ H_{\mathrm{m}} $, comparing with that of the free mass $ 1/(m_2 \, \omega ^2) $. The differences found are 0.58%, 0.48%, 0.37% and 0.05%, at 66, 72, 82 and 233Hz, respectively. These are small but quite large compared with the requirement. To avoid such deviation effect due to the suspension system, we use the default-setting response (red solid line of Fig.\[fig:MechRes\]) as the mechanical response $ H_{\mathrm{m}} $ instead of the free mass response. Also, the quasi free-mass response is ensured by the $f ^{-2}$ response of the peaks in Fig.\[fig:Sens\]. It means that there is no strange and large deviation from the expected response at the observation frequencies.
The response $ H_{\mathrm{m}} $ has still some components of uncertainty due to an imperfection in making the double pendulum. Figure\[fig:MechRes\](b-f) show the impact of deviations from the mechanical parameter on the mechanical response $ H_{\mathrm{m}} $. The estimation ranges are set for the wire lengths $l_1$ and $l_2$ to be $ \pm 2 $mm, for the intermediate mass $m_1$ to be $ \pm 5 $mg and for the $ Q $ from 1 to 300. Type B evaluation is used for these upper and lower limits. As expected, marked differences can be seen at a region near the resonance. The contributions from each uncertainty at 72Hz are 0.001% from $m_1$, 0.00001% from $l_1$, 0.12% from $l_2$ and 0.00001% from $Q$. The imperfection effect from the length of the lower fiber has relatively large effect. It is apparent that an uncertainty of the test mass, $ m_2 $, causes a large effect over whole frequency region directly as shown in Fig.\[fig:MechRes\](c). The weight of the 25mg mirror have been estimated by measuring its dimensions (the diameter and thickness) using slide calipers. The actual value is $ 24.63 \pm 0.16 $mg, corresponding to 0.65% uncertainty. From the above discussion, the total contribution of the uncertainty becomes $(\sigma _H / H_{\mathrm{m}} ) ^2 \approx (0.12/\sqrt{3})^2 + (0.65/\sqrt{3})^2 = (0.38\,\%)^2 $.
Transfer efficiency of momentum $ \alpha _\mathrm{r} $
-------------------------------------------------------
The reflection, scattering and absorption have different transfer efficiencies of the momentum from photons to the mirror. The transfer efficiency of the reflection is $ 2 \Delta P $, as $ \Delta P$ is the momentum of one photon. The scattering has a range from $ \Delta P$ to $ 2 \Delta P $ because of the uncertainty in the scattered direction. On the other hand, the absorption is $ \Delta P$. The total transfer efficiency of momentum $ \alpha _\mathrm{r} $ is measured including these uncertainty. The incident, reflected and transmitted laser powers were measured by repeating ten times using a commercial power meter (Coherent PowerMax, PS19Q) that has 2.5% uncertainty according to the specification sheet. Thus the uncertainty of the reflectivity becomes $ 2.5\,\% \times (\sqrt{2} / \sqrt{10}) = 1.12\,\% $ where the $ \sqrt{2} $ comes from the two power measurements of the incident and reflected. The reflectivity, transmittance and loss are $ R = 0.9941 ^{+0.0053} _{-0.0111} $, $ T = (657 \pm 5) \times 10^{-6} $, and $ L = 0.0053^{+0.0111} _{-0.0053}$, respectively. Here the relation of $ R + T + L = 1 $ and the law of the conservation of energy are assumed. Since we cannot distinguish the origin of the loss from the scattering and absorption, the upper and lower limits are estimated. The upper limit of the transfer efficiency $ \alpha _\mathrm{r} $ becomes 0.9993 by assuming the no loss material. The lower limit of it is derived from $ R + L/2 = (0.9941 - 0.0111) + (0.0053 + 0.0111)/2 = 0.9912 $ by assuming the origin of the loss is absorption only. Hence the transfer efficiency is $ \alpha _\mathrm{r} = 0.9952 \pm 0.0041 $ as the center of this range. This results in $ ( \sigma _{\alpha } / \alpha _\mathrm{r} ) ^2 = ( 0.24\,\% )^2 $ using Type B evaluation.
The beam spot of the modulated laser has a diameter of 0.4mm on the tiny mirror. The reflection area is sufficient because the reflection coating covers 80% of the surface (2.4mm) on the tiny mirror. Even if there is an off center of the beam spot, the reflection area can be kept to 1mm at least. In this case, the loss of the power becomes 0.0004% and is negligible.
Measurement of incident angle $\phi $
--------------------------------------
The incident angle $\phi $ is projected on the horizontal direction $ \phi_\mathrm{H} $ and vertical direction $ \phi_\mathrm{V} $ to measure the angles, i.e. $ \cos{\phi } \equiv \cos{\phi_\mathrm{H} }\cos{\phi_\mathrm{V} } $. The lengths of $L1$ and $L2$ in Fig.\[fig:Setup\] were measured using a ruler to decide the incident angle of the horizontal direction. The results are $L1 = 388 \pm 1$mm and $L2 = 202 \pm 1$mm, whose uncertainty stem from the minimum unit of the ruler. It corresponds to the angle $ \phi_\mathrm{H} = 0.481 \pm 0.004 $rad. This kind of measurement is classified as Type B evaluation. The contribution term for the standard uncertainty is $ \sigma _{\phi_\mathrm{H}} \tan \phi_\mathrm{H} = \left( 0.004/\sqrt{3} \right) \tan{0.481} = 0.11\,\% $. In addition to the horizontal direction, the vertical angle was also measured to be $ \phi_\mathrm{V} = 0.046 \pm 0.002 $rad. The contribution term is $ \sigma _{\phi_\mathrm{V}} \tan \phi_\mathrm{V} = 0.004\,\% $. The horizontal term is dominant because of its large angle of $ \phi_\mathrm{H} $.
Rotational effect $ R_\mathrm{c} $ {#Sec:Rc}
-----------------------------------
The actual mirror is not a point mass but a rigid body. This gives additional rotational degrees of freedom. If a laser beam hits the mirror surface off center, the mirror rotate around its center of mass because of the induced torque. This rotational motion is added to the translational direction by a product of the rotational angle and the off-centering distance. The observation spot on the mirror surface is also important if a laser interferometer is used as a displacement sensor. This coupling effect from the mirror rotation to the longitudinal direction of the MI was named rotational effect above. The induced angular tilt by the incident laser power can change the contrast of the MI. However, the position of equilibrium was not changed further since the DC laser power was kept constant during all measurements (with and without modulation). Thus, the rotational effect arises solely from the power modulation.
![Illustration of the rotational effect. The red arrows show the displacement of the mirror. []{data-label="fig:Rotation"}](Rotation.eps){width="5.5cm"}
The rotational effect is defined as a displacement ratio between the translation and rotation caused by the radiation pressure [@Hild_phD]. It is written as $ R_\mathrm{c} = X_\mathrm{R} / X_\mathrm{L} = \pm (m d_\mathrm{c} d_\mathrm{pp})/I $ as can be derived based on Fig.\[fig:Rotation\]. Here, $ X_\mathrm{L} $ is the displacement for the longitudinal direction with respect to MI due to the radiation pressure, $ X_\mathrm{R} $ is the longitudinal displacement caused by the rotation, $ m $ the mass of the mirror, $ I $ the moment of inertia given by $ I = m (r^2/4 + l^2/12) $, $ r $ the mirror radius, $ l $ the thickness of the mirror, $ d_\mathrm{c} $ the off-centering distance of the beam spot of the MI on the tiny mirror and $ d_\mathrm{pp} $ the distance of off-centering spot of the intensity modulated laser. The sign shows the relation of the beam spot positions between the MI and intensity modulation. In order to suppress the rotational effect, precise adjustment of the beam centering is needed. If both off-centering distances are comparable ($ d_\mathrm{c} = d_\mathrm{pp} = d_\mathrm{0} $), the inequality becomes simply $ (m d_\mathrm{0} ^2)/I \leq 0.003 $ for the 0.3% criterion, which has solution $ d_\mathrm{0} \leq 47$$\mu $m. The rotational effect can be suppressed below 0.3% uncertainty when the beam centering is adjusted within about 50$\mu $m including its uncertainty. By implementing Type B evaluation, the requirement is extended up to 80$\mu $m. In section\[Sec:QE\], there was a discrepancy between the measurement result and the value from the specification sheet of the photodiode. This can be explained by the rotational effect with a beam off-centering of about $d_\mathrm{0} = 400$$\mu $m when other effects (temperature and uniformity of PD surface) are not taken into account. If the off-centering distances are quite large, a finite value of $ R_\mathrm{c} $ is used and its uncertainty should be evaluated. The uncertainty of the rotational effect is expressed as $$\sigma _R =
\sqrt{
2 \left[ \frac{d_\mathrm{0}}{(r^2/4 + l^2/12)} \sigma _d \right] ^2 +
\left[ \frac{d_\mathrm{0} ^2}{(r^2/4 + l^2/12)^2} \frac{r}{2} \sigma _r \right] ^2 +
\left[ \frac{d_\mathrm{0} ^2}{(r^2/4 + l^2/12)^2} \frac{l}{6} \sigma _l \right] ^2
} ,
\label{eq:Rc_err}$$ where $ \sigma _d $, $ \sigma _r $ and $ \sigma _l $ are the measurement uncertainties of $ d_\mathrm{0} $, $r$ and $l$, respectively. If the slide calipers are used for measuring the dimension of the tiny mirror, the second and third term become negligible compared to the first term. Also, the beam position of the intensity modulation can be adjusted precisely by comparing it with the MI. The first term should be unfolded for each beam position. Then the contribution term becomes $$\left[ \frac{\sigma _R}{ (1 + R_\mathrm{c}) } \right] ^2
\approx
\frac{1}{ (1 + R_\mathrm{c})^2 }
\left\{
\left[ \frac{d_\mathrm{c}}{(r^2/4 + l^2/12)} \frac{ \sigma _{d_\mathrm{c}} }{\sqrt{3} } \right] ^2 +
\left[ \frac{d_\mathrm{pp}}{(r^2/4 + l^2/12)} \frac{ \sigma _{d_\mathrm{pp}} }{\sqrt{3} } \right] ^2
\right\} ,$$ where the factor of $ \sqrt{3} $ is the coefficient from Type B evaluation. The position uncertainty of 10$ \mu $m is obtained from one pixel when the area of $1 \times 1$$\mathrm{cm}^2$ is photographed using a CCD camera with 1M pixel. If we adjust the beam centering of the intensity modulation within $d_\mathrm{pp} = (100 \pm 10)$$ \mu $m, the uncertainty of this term becomes 0.08%. The dominant part is the beam position of the MI because it is more difficult to adjust beam centering with keeping a good contrast. In order to achieve the 0.3% criterion, the centering of the beam from the interferometer $ d_\mathrm{c} $ needs to be adjusted within $ (390 \pm 10)$$ \mu $m. This is achievable. Incidentally, in the case of a modest requirement (measurement uncertainty of 20$ \mu $m), the contribution part of $ d_\mathrm{pp} $ becomes 0.15% by assuming the beam centering of the power modulation with $(100 \pm 20)$$ \mu $m and the uncertainty of $ d_\mathrm{c} $ is kept on 0.30% by assuming the centering of the MI with $(200 \pm 20)$$ \mu $m. These are also feasible values.
A rotation of the tiny mirror can change the reflected-beam angle for the MI. This effect is included in the uncertainty evaluation of $ V_\mathrm{f} $ as a fluctuation of the peak signal. Also, if the two beam spots on the tiny mirror are adjusted to the same position, a local deformation of the mirror could appear by the radiation pressure [@Hild2007]. It is omitted because that effect appears in a high frequency region compared with our measurement region.
Conclusion and discussion
=========================
We have proposed an optomechanical power meter, which can count the number of photons by measuring the displacement of a mirror pushed by a modulated laser beam. As a prototype test, a suspended 25mg mirror is used for an optomechanical-coupled oscillator and its displacement is measured using the Michelson interferometer. Table\[tbl:Budget\] shows the summary of the uncertainty contribution. The total standard uncertainty is 0.92% plus the quadrant sum of a term from the rotational effect, which is not measured but discussed. Hence, we do not claim the 1% accuracy in this prototype experiment because of the lack of the beam centering information. If the uncertainty due to the rotational effect is assumed to be 0.30% for $ d_\mathrm{c} $ and 0.15% for $ d_\mathrm{pp} $, the total uncertainty becomes 0.98%. When $ P_\mathrm{m} $ is converted to the DC power, the uncertainty of $ \mathrm{PD_{AOM}} $ is added. The measured contribution term is $ (\sigma_{V_\mathrm{m}} / V_\mathrm{m})^2 = (0.006\,\%)^2 $ at 72Hz, which is negligible. According to our demonstration and calculation analysis, we can conclude that it is achievable for our new kind of power meter to measure the laser power within one percent uncertainty (1$\sigma $). Furthermore, average of four comparable measurements (at 66, 72, 82 and 233Hz in this case) improve the uncertainty by $ \sqrt{4} $, i.e. 0.5% uncertainty is obtained. Each measurement can be regarded as independent measurement since main contributions of the uncertainty come from the frequency specific calibration factors.
-- ---------------------------------------------------------------------------------------------------------- -- --
**[Displacement sensor]{} & &\
$dV_{\mathrm{PD}}$ & Deviation from mid-fringe & &\
$ k $ & Offset and drift & B & $ 0.03\,\% $\
$ r_\mathrm{s} $ & Residual error signal & A & $ 0.05\,\% $\
$dV_{\mathrm{PD}}$ & Calibration factors & &\
$ V_\mathrm{f} $ & Feedback signal (peak value) & A & $ 0.30\,\% $\
& Feedback signal (noise floor) & A & $ 0.06\,\% $\
$ G_\mathrm{CL} $ & Closed loop gain & A & $ 0.44\,\% $\
$ T_\mathrm{AH} $ & Transfer function from actuator to PD & A & $ 0.49\,\% $\
$ V_\mathrm{pp} $ & Peak to peak of MI & A & $ 0.33\,\% $\
$ (G_\mathrm{CL}, T_\mathrm{AH}) $ & Correlation term & A & $ 0.04\,\% $\
& **[Optomechanical response]{} & &\
$H_{\mathrm{m}}$ & Mechanical response & &\
$ l_2 $ & Lower-fiber length & B & $ 0.07\,\% $\
$ m_2 $ & Mass of tiny mirror & B & $ 0.38\,\% $\
$ \alpha _\mathrm{r} $ & Transfer efficiency of momentum & B & $ 0.24\,\% $\
$ \phi_\mathrm{H} $ & Incident angle (horizontal) & B & $ 0.11\,\% $\
$ R_\mathrm{c} $ & Rotational effect & &\
$ d_\mathrm{c} $ & Beam centering of MI & B & $ \frac{1}{ 1 + R_\mathrm{c} }
\left[ \frac{d_\mathrm{c}}{(r^2/4 + l^2/12)} \frac{ \sigma _{d_\mathrm{c}} }{\sqrt{3} } \right] $\
$ d_\mathrm{pp} $ & Beam centering of intensity modulation & B & $ \frac{1}{ 1 + R_\mathrm{c} }
\left[ \frac{d_\mathrm{pp}}{(r^2/4 + l^2/12)} \frac{ \sigma _{d_\mathrm{pp}} }{\sqrt{3} } \right] $\
\
\
****
-- ---------------------------------------------------------------------------------------------------------- -- --
: Uncertainty budget of the prototype experiment at 72Hz.[]{data-label="tbl:Budget"}
There are some means to improve this measurement. Commercial precision weighing balances can measure the weight of mass with 0.01% uncertainty [@Balance]. The uncertainty of the closed loop gain is expected to be reduced by the monochromatic measurement. By moving the position of the AOM to the MI light way, the intensity noise is canceled using the subtraction of the symmetric port from the antisymmetric port. It means that the uncertainty of the feedback signal can be reduced and the incident angle vanishes. The uncertainty of the actuator response can be reduced by adjusting the gap between coils and magnets to the insensitive region of efficiency, and also making a good suspension of the actuation mirror. Increasing the number of reflectivity measurements reduces the uncertainty in the transfer efficiency of momentum. A short length MI, including the path up to a PD, is favorable to stabilize the contrast. Averaging the measurements at other frequencies can improve the precision, thus a refined setup can be expected to reach the comparable uncertainty of the current primary standard.
The available power of this apparatus is limited by specific conditions. The maximum power is limited by the damage threshold of the coating of the oscillator (tiny mirror). If a higher power is needed to measure properly, the mirror mass should also be increased to keep the response of the displacement sensor linear. The minimum power is limited by the signal to noise ratio of the radiation pressure response. The available wavelength depends on the reflecting or absorbing condition of the oscillator. In this paper, our mirror coating is optimized to reflect a laser beam with a wavelength of 1064nm. This reflection coating is changeable and should be optimized for the laser wavelength that is to be measured. To measure the X-ray region, the absorbing material is better to use though the transfer efficiency of the momentum is reduced by a factor of two in such case. For the radio wave region, a metal material can be used for reflection. Of course, it is important to investigate the transfer efficiency of the momentum precisely. To determine the DC power of the incident laser, we need to determine the modulation index of the power modulator or to prepare the reference receiver like $ \mathrm{PD_{AOM}} $. In case a reference receiver is used, the available power and wavelength depend on the properties of the receiver. Optical choppers can be substituted for the AOM to generate the power modulation.
We have displayed a new method using an optomechanical response to measure the laser power via a displacement of an oscillator. In our technique, each parameter can be measured/estimated precisely although there are many parameters contributing to the uncertainty as shown in table\[tbl:Budget\]. Even with this prototype experiment, no vacuum or cryogenic instruments are needed to obtain the level of one-percent uncertainty. By adjusting the mirror material and coating, it can be applied to an arbitrary laser wavelength. These characteristics can expand the application range of power meters. This method could be used not only to determine a primary standard, but also to make a new kind of commercial power meter in the future.
Acknowledgments
===============
We are grateful to Keita Kawabe, Stefan Hild and Koji Arai for many comments and helpful discussions. Also, we thank Joris van Heijningen for a careful reading. S. Kawamura was supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science, G. DeSalvo was supported by the U.S. National Science Foundation under the University of Florida REU program. S. Sakata was supported by Young Scientists B.
|
---
abstract: 'We prove Stanley’s conjecture that, if $\delta_n$ is the staircase shape, then the skew Schur functions $s_{\delta_n / \mu}$ are non-negative sums of Schur $P$-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function $s_{\delta_n / \delta_{n-2}}$, we discuss connections with Eulerian numbers and alternating permutations.'
address:
- 'Department of Mathematics, San Francisco State University.'
- 'LaCIM, Université du Québec à Montréal.'
author:
- Federico Ardila
- 'Luis G. Serrano'
title: |
Staircase skew Schur functions\
are Schur $P$-positive
---
introduction
============
The Schur functions $s_{\lambda}$, indexed by partitions $\lambda$, form a basis for the ring $\Lambda$ of symmetric functions. These are very important objects in algebraic combinatorics. They play a fundamental role in the study of the representations of the symmetric group and the general linear group, and the cohomology ring of the Grassmannian [@Ful97]. The Schur $P$-functions $P_{\lambda}$, indexed by strict partitions, form a basis for an important subring $\Gamma$ of $\Lambda$. They are crucial in the study of the projective representations of the symmetric group, and the cohomology ring of the isotropic Grassmannian [@HH92] [@Joz91].
The goal of this paper is to prove the following conjecture of Richard Stanley [@Stanconj]: If $\delta_n$ is the staircase shape and $\mu \subset \delta_n$, then the *staircase skew Schur function* $s_{\delta_n / \mu}$, which belongs to the ring $\Gamma$, is a nonnegative sum of Schur $P$-functions. We find a combinatorial interpretation for the coefficients in this expansion in terms of Shimozono’s compatible fillings [@Shi99]. Furthermore, we discuss connections between the special case of the skew Schur function $s_{\delta_n / \delta_{n-2}}$ and alternating permutations, and show an expansion of these in terms of the elementary symmetric functions. The paper is organized as follows. In Section \[sec:prelim\] we recall some basic definitions, including Schur and Schur $P$-functions. In Section \[sec:staircase\] we discuss the staircase Schur functions and prove that they are indeed in the subring $\Gamma$ of $\Lambda$ generated by the Schur $P$-functions. In Section \[sec:thm\] we state our main result, Theorem \[th:main\], which states that the (non-negative integer) coefficients of $P_{\lambda}$ is the number of “$\delta_n/\mu$-compatible" fillings of the shifted shape $\lambda$. In Section \[sec:jdt\] we prove the key proposition that, in the particular case of staircase skew shapes $\delta_n/\mu$, jeu de taquin respects $\delta_n/\mu$–compatibility. Finally in Section \[sec:proof\] we prove Theorem \[th:main\].
The Schur P-positivity of staircase Schur functions has also been proved independently by Elizabeth Dewitt and will appear in her forthcoming thesis [@Dew12]
**Acknowledgments.** We would like to thank Richard Stanley for telling us about his conjecture and about Proposition \[prop:e\]. [@Stanconj] We also thank Ira Gessel, Peter Hoffman, Tadeusz Józefiak, Bruce Sagan, and John Stembridge for valuable conversations.
Preliminaries {#sec:prelim}
=============
A *partition* is a sequence $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l) \in \ZZ^l$ with $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_l > 0$. The *Ferrers diagram*, or *shape* of $\lambda$ is an array of square cells in which the $i$-th row has $\lambda_i$ cells, and is left justified with respect to the top row. The *size* of $\lambda$ is $| \lambda | := \lambda_1 + \lambda_2 + \cdots + \lambda_l$. We denote the number of rows of $\lambda$ by $\ell(\lambda) := l$.
A *strict partition* is a sequence $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l) \in \ZZ^l$ such that $\lambda_1 > \lambda_2 > \cdots > \lambda_l > 0$. The *shifted diagram*, or *shifted shape* of $\lambda$ is an array of square cells in which the $i$-th row has $\lambda_i$ cells, and is shifted $i-1$ units to the right with respect to the top row.
For example, the shape $(5,3,2)$ and the shifted shape $(5,3,2)$, of size $10$ and length $3$, are shown below. $$\young(~~~~~,~~~,~~) \quad \quad \young(~~~~~,:~~~,::~~)$$ A *skew (shifted) diagram* (or shape) $\lambda / \mu$ is obtained by removing a (shifted) shape $\mu$ from a larger shape $\lambda$ containing $\mu$.
A *semistandard Young tableau* or *SSYT* $T$ of shape $\lambda$ is a filling of a Ferrers shape $\lambda$ with letters from the alphabet $X = \{1 < 2 < \cdots \}$ which is weakly increasing along the rows and strictly increasing down the columns.
A *shifted semistandard Young tableau* or *shifted SSYT* $T$ of shape $\lambda$ is a filling of a shifted shape $\lambda$ with letters from the alphabet $X' = \{1' < 1 < 2' < 2 < \cdots \}$ such that:
- rows and columns of $T$ are weakly increasing;
- each $k$ appears at most once in every column;
- each $k'$ appears at most once in every row;
- there are no primed entries on the main diagonal.
If $T$ is a filling of a shape $\lambda$, we write $\sh(T) := \lambda$. The *content* of a (shifted) SSYT $T$ is the vector $(a_1, a_2, \ldots)$, where $a_i$ is the number of times the letters $i$ and $i'$ appear in $T$.
A (shifted) SSYT is standard, if it contains the letters $1,2, \ldots, |\lambda|$, each exactly once. In the shifted case, these letters are all unprimed. If that is the case, we call it a (shifted) SYT. A *skew (shifted) Young tableau* is defined analogously.
The following are examples of a SSYT and a shifted SSYT, both having shape $\lambda = (5,3,2)$ and content $(2,1,1,2,2,1,0,0,1)$. $${\Yvcentermath1}\young(11235,445,69) \quad \quad \young(112{{\mbox{$3'$}}}5,:445,::6{{\mbox{$9'$}}})$$
In a SYT or a shifted SYT $T$, the pair of entries $(i,j)$, where $i < j$, forms an *ascent* if $j$ is located weakly north and strictly east of $i$. We abbreviate and say that $j$ is *northEast* of $i$. The pair $(i,j)$ forms a *descent* if $j$ is located strictly south and weakly west, or *Southwest*, of $i$. Note that $(i,j)$ could be neither an ascent nor a descent.
When $j=i+1$, the pair $(i,i+1)$ must be either an ascent or a descent, and we abbreviate and call $i$ an ascent or a descent as appropriate. An entry $i$ forms a *peak* if $i-1$ is an ascent and $i$ is a descent.
\[ex:ascentdescent\] The figure below shows a SYT of shape $\delta_4:=(4,3,2,1)$ and a shifted SYT of shape $(5,3,2)$, both with descent set $(2,4,5,7,9)$, ascent set $(1,3,6,8)$, and peak set $(2,4,7,9)$. $${\Yvcentermath1}\young(1247,359,6{{$10$}},8) \quad \quad \young(12479,:358,::6{{$10$}})$$
For a (shifted) Young tableau $T$ with content $(a_1, a_2, \ldots)$, we let $x^T = x_1^{a_1} x_2^{a_2} \cdots.$ For each partition $\lambda$, the *Schur function* $s_{\lambda}$ is defined as the generating function for semistandard Young tableaux of shape $\lambda$, namely $$s_{\lambda} = s_{\lambda} (x_1, x_2, \ldots) := \sum_{\sh (T) = \lambda} x^T.$$
It is well known (see e.g., [@Sta99]) that the *power sum symmetric functions* $p_i = p_i(x_1,x_2,\ldots) := x_1^i + x_2^i + \cdots$ are a generating set, and the Schur functions $s_{\lambda}$ are a linear basis, for the ring $\Lambda$ of symmetric functions.
For each strict partition $\lambda$, the *Schur $P$-function* $P_{\lambda}$ is defined as the generating function for shifted Young tableaux of shape $\lambda$, namely $$P_{\lambda} = P_{\lambda} (x_1, x_2, \ldots) := \sum_{\sh (T) = \lambda} x^T.$$ The Schur $P$-functions form a basis for the subring $\Gamma$ of $\Lambda$ generated by the odd power sums, $\Gamma := \QQ[p_1, p_3, \ldots]$. This ring also has the presentation $$\Gamma = \{f \in \Lambda \, : \, f(t,-t,x_1,x_2,\ldots) = f(x_1,x_2,\ldots)\}.$$ See, e.g., [@HH92]. The *skew Schur functions* $s_{\lambda / \mu}$ and the *skew Schur $P$-functions* $P_{\lambda / \mu}$ are defined similarly for a skew (shifted) shape $\lambda / \mu$.
The skew Schur functions $s_{{\delta_n}/{\delta_{n-2}}}$ and $s_{{\delta_n}/{\mu}}$. {#sec:staircase}
====================================================================================
The staircase $\delta_n$ is the shape $(n, n - 1,\ldots,2,1)$. Denote $s_{{\delta_n}/{\delta_{n-2}}}=: F_{2n-1}$, and let $\F = \F(x_1, x_2, \ldots) := \sum_{n \geq 1} F_{2n-1}$.
The symmetric function $F_{2n-1}$ is one of the main subjects of study of this paper. It has nice expansions in terms of the power and elementary symmetric functions.
A permutation $a_1a_2\ldots a_n$ of $\{1,\ldots, n\}$ is said to be *alternating* if $a_1<a_2>a_3<a_4>\cdots$.
\[prop:p\] ([@Fou76]) Let $E_k$ be the number of alternating permutations of $\{1, \ldots, k\}$, and let $z_{\lambda} := \prod_{i \ge 1} \frac{i^{m_i}}{m_i!}$ for the partition $\lambda = 1^{m_1} 2^{m_2} \cdots$. We have $$F_{2n-1} = \sum_{\lambda \in OP(2n-1)} \frac{E_{l(\lambda)}}{z_{\lambda}} p_\lambda$$ where $OP(2n-1)$ is the set of partitions of $2n-1$ into odd parts.
The following proposition expresses the $\F$ in terms of the elementary symmetric functions. Equivalent formulas appear in [@Car73], [@Ges77], [@Ges90 p. 9] and [@GJ83 Corollary 4.2.20].
\[prop:e\] We have $$\F = \frac{e_1-e_3+e_5- \cdots}{1-e_2+e_4-\cdots},$$ where $e_k = \sum_{i_1 < \cdots < i_k} x_{i_1} \cdots x_{i_k}$ is the $k$-th elementary symmetric function.
Consider a SSYT $T$ of shape ${\delta_n}/{\delta_{n-2}}$ with $n \geq 2$ which contains a $1$. Let the leftmost $1$ occur in the (top entry on the) $k$th column. When we remove that $1$, we are left with a SSYT of shape ${\delta_k}/{\delta_{k-2}}$ containing no $1$s and a SSYT of shape ${\delta_{n-k}}/{\delta_{n-k-2}}$. It follows that $$\F(x_1, x_2, \ldots) - \F(x_2, x_3, \ldots) = x_1 + \F(x_2, x_3, \ldots) x_1\F(x_1, x_2, \ldots).$$ Denoting $\F_i:= \F(x_i, x_{i+1}, \ldots)$, we rewrite this as $\F_1 = \frac{x_1+\F_2}{1-x_1\F_2}$, which gives that $\arctan \F_1 = \arctan x_1 + \arctan \F_2$ as formal power series. Iterating, we obtain $$\arctan \F_1 = \arctan x_1 + \arctan x_2 + \cdots,$$ from which the desired formula follows by taking the tangent of both sides.
More importantly for us, we observe that $\F$ is in the subring $\Gamma$ of $\Lambda$.
[@Stanconj] The skew Schur functions $F_{2n-1}$ and, more generally, the *staircase skew Schur functions* $s_{{\delta_n}/{\mu}}$, are in the subring $\Gamma$ of $\Lambda$.
From the equation $e_k(t, -t, x_1, x_2, \ldots) = e_k - t^2e_{k-2}$ and Proposition \[prop:e\] it follows that $\F(t, -t, x_1, x_2, \ldots)= \F(x_1, x_2, \ldots)$, which proves that $F_{2n-1} \in \Gamma$.
For the general case, one can mimic the proof of [@Sta99 Prop. 7.17.7] for the particular case of $\mu = \emptyset$. Namely, by the Murnaghan–Nakayama rule [@Sta99 Theorem 7.17.3], the coefficient of $p_{\alpha}$ in $s_{\delta_n / \mu}$ is equal to $\sum_{T} (-1)^{ht(T)}$ where $T$ runs over all border strip tableaux of shape $\delta_n / \mu$ and type $\alpha$, and $ht(T)$ is the height of $T$. If $\alpha$ has any even part, reorder the parts such that this even part is the last nonzero entry. But then, one can see that there is no border strip tableau of shape $\delta_n / \mu$ and content $\alpha$, since any border strip of shape $\delta_n / \mu$ must have odd size. Thus, the coefficient of $p_{\alpha}$ is zero. We conclude that $s_{\delta_n / \mu} \in \Gamma$.
From the previous lemma, it follows that $F_{2n-1}$ and, more generally, $s_{{\delta_n}/{\mu}}$ have expansions in terms of the Schur $P$-functions. The purpose of this paper is to clarify this expansion.
Main result {#sec:thm}
===========
A (shifted) standard Young tableau is *alternating* if every odd number is an ascent and every even number is a descent.
\[ex:alternating\] The following are the only two shifted standard Young tableaux of size $7$ which are alternating: $${\Yvcentermath1}\young(1246,:357) \quad , \quad \young(1246,:35,::7).$$
The following is a special case of our main result, Theorem \[th:main\].
\[th:main1\] The skew Schur function $s_{{\delta_n}/{\delta_{n-2}}}$ can be expressed as a nonnegative sum of Schur $P$-functions. We have $$s_{\delta_n / \mu} = \sum_{U \in \textsf{AltShSYT}(2n-1)} P_{\sh(U)},$$ where $(2n-1)$ is the set of shifted SYT of size $2n-1$ which are alternating.
From Example \[ex:alternating\] it follows that $$s_{\delta_4 / \delta_2} = P_{43} + P_{421}.$$ Similarly, $$s_{\delta_5 / \delta_3} = P_{54} + 2P_{531}+P_{432}$$ because the shifted SYT of size $9$ which are alternating are: $${\Yvcentermath1}\young(12468,:3579)\, , \quad
\young(12468,:357,::9)\, , \quad
\young(12468,:359,::7) \,, \quad
\young(1246,:358,::79) \,.$$
For a skew shape $\lambda / \mu$ of size $n$, the *standard filling* $T_{\lambda / \mu}$ is given by filling the shape with the entries $1,2,\ldots,n$, starting from the bottom row and moving up, subsequently filling each row from left to right. To distinguish it from the SYTs, we color it [blue]{}.[^1]
For the shape $54321 / 32$, we have $${\Yvcentermath1}T_{54321 / 32} = {\textcolor{blue}}{\young(:::9{{$10$}},::78,456,23,1)}.$$
[@Shi99] A (shifted or unshifted) SYT $U$ of size $|\lambda| - |\mu|$ is said to be *$\lambda / \mu$–compatible* if
- whenever $T_{\delta_n/\mu}$ contains [${\Yvcentermath1}\young(i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})$]{}, $i$ is a descent in $T$.
- whenever $T_{\delta_n/\mu}$ contains [${\Yvcentermath1}\young(j,i)$]{}, $(i,j)$ is an ascent in $T$.
\[re:alternating\] Note that a (shifted) standard Young tableau is alternating if and only if it is $\delta_n / \delta_{n-2}$–compatible.
\[ex:compatible\] The following are the only two $54321 / 32$–compatible shifted standard Young tableaux: $${\Yvcentermath1}\young(12479,:358{{$10$}},::6) \quad \text{and} \quad \young(12479,:358,::6{{$10$}})$$
The following is our main result.
\[th:main\] For any shape $\mu \subset \delta_n$, the skew Schur function $s_{\delta_n / \mu}$ can be expressed as a nonnegative linear combination of Schur $P$-functions. We have $$s_{\delta_n / \mu} = \sum_{U \in \textsf{CompShSYT}(\delta_n / \mu)} P_{\sh(U)},$$ where $(\delta_n / \mu)$ is the set of shifted SYT tableau which are $\delta_n / \mu$–compatible.
In light of Example \[ex:compatible\], Theorem \[th:main\] says that $$s_{54321 / 32} = P_{541} + P_{532}.$$
Note that Theorem \[th:main1\] is a special case of Theorem \[th:main\], by Remark \[re:alternating\].
Jeu de taquin and $\delta_n/\mu$–compatibility {#sec:jdt}
==============================================
Let $T$ be a SYT. Consider $T$ as a skew shifted SYT in the shifted plane. Denote by $\jdt(T)$ its *shifted jeu de taquin rectification*, inspired by the notation and terminology in [@Sta99 A1.2].
\[ex:jdt\] $${\Yvcentermath1}\jdt \left( \young(1247,359,6{{$10$}},8) \right) = \young(12479,:358,::6{{$10$}}).$$ Recall that in each step or *slide* or jeu de taquin, we choose an empty internal corner, move the smaller of its (one or two) neighbors into this empty cell, then fill the resulting cell in the same way, and continue until we reach an external corner, and obtain a skew SYT. We do this subsequently until we obtain a shifted SYT, which turns out to be independent of the choices made [@Sta99 A1.2]. For instance, we can compute the jeu de taquin rectification above as follows: $${\Yvcentermath1}\young(\,\,\,1247,:\,\,359,::\,6{{$10$}},:::8) \mapsto
\young(\,\,\,1247,:\,\,359,::68{{$10$}}) \mapsto
\young(\,\,\,1247,:\,359,::68{{$10$}}) \mapsto
\young(\,\,1247,:\,359,::68{{$10$}})$$ $${\Yvcentermath1}\mapsto \young(\,\,1247,:3589,::6{{$10$}})
\mapsto \young(\,1247,:3589,::6{{$10$}})
\mapsto \young(12479,:358,::6{{$10$}})\,.$$
Our crucial technical lemma says that $\delta_n / \mu$–compatibility is well behaved under jeu de taquin. This is not true for $\lambda / \mu$–compatibility in general: for $${\Yvcentermath1}\jdt \left(\young(1,2)\right) = \young(12)\, ,$$ jeu de taquin makes the tableau lose its $(2)/\emptyset$–compatibility and gain $(1,1)/\emptyset$–compatibility.
We first give a short argument for the special case of $\delta_n/\delta_{n-2}$, and then a different (and necessarily more intricate) argument for the general case.
\[prop:compatible\] A standard Young tableau $T$ is alternating if and only if $\jdt(T)$ is alternating.
The *reading word* ${\operatorname{read}}(T)$ of a tableau $T$ is the word formed by subsequently reading each row from left to right, starting from the bottom row and moving up. Notice that $i$ is an ascent (descent) in $T$ if and only if it is an *ascent* (*descent*) in ${\operatorname{read}}(T)$, in the sense that $i$ appears before (after) $i+1$ in the word.[^2]
Now consider a skew shifted SYT $T$ and its shifted jeu de taquin rectification $U=\jdt(T)$. By and [@Hai89 Theorem 6.10], ${\operatorname{read}}(T)$ and ${\operatorname{read}}(U)$ are equivalent modulo the Sagan–Worley relations [@Sag87]: $$\begin{aligned}
ab \cdots & \approx & ba \cdots \qquad \quad \quad \textrm{ for } a<b, \\
\cdots bac \cdots & \approx & \cdots bca \cdots \qquad \, \textrm{ for } a<b<c, \\
\cdots cab \cdots & \approx & \cdots acb \cdots \qquad \, \textrm{ for } a<b<c ,\end{aligned}$$ where the letters represented by $\cdots$ remain the same.
We now prove that jeu de taquin preserves peaks, by proving that ${\operatorname{read}}(T)$ and ${\operatorname{read}}(U)$ have the same peaks. Since the Sagan-Worley moves are reversible, we only need to check that a move cannot turn a peak $i$ into a non-peak. This follows from the following observation: $i$ is a peak in a permutation if and only if it is preceded by both $i-1$ and $i+1$. This property cannot be changed by any of the Sagan-Worley relations: The first relation cannot involve $i$, and the second and third can never change the relative order of two consecutive numbers.
Finally notice that a tableau of size $2n-1$ is alternating if and only if its set of peaks is $\{2, 4, \ldots, 2n-2\}$. This property is preserved by jeu de taquin rectification.
The previous proof relies heavily on the description of alternating tableau in terms of peaks; notice that the Sagan-Worley relations do not respect the ascents and descents. We do not know how to extend this argument to the setting of $\delta_n/\mu$–compatibility. To settle this general case, we will carry out a careful analysis of the jeu de taquin algorithm from the point of view of $\delta_n/\mu$–compatibility. We will keep referring back to the following:
[|p[10cm]{}|]{}
For a $\delta_n/\mu$–compatible tableau $T$:
$\bullet$ If $T_{\delta_n/\mu}$ contains [${\Yvcentermath1}\young(j,i)$]{}, then $(i,j)$ is an **ascent** in $T$:
$\begin{array}{c}
\begin{picture}(40,40)(0,0)
\put(10,10){\makebox{$i$}}
\put(35,25){\makebox{$j$}}
\put(1,32){\line(1,0){30}}
\put(31,2){\line(0,1){30}}
\end{picture} \\
\mbox{$i$ is southWest of $j$,}
\end{array}$ $\begin{array}{c}
\begin{picture}(40,40)(0,0)
\put(0,0){\makebox{$i$}}
\put(20,20){\makebox{$j$}}
\put(7,0){\line(1,0){30}}
\put(7,0){\line(0,1){30}}
\end{picture} \\
\mbox{$j$ is northEast of $i$.}
\end{array}$
$\bullet$ If $T_{\delta_n/\mu}$ contains [${\Yvcentermath1}\young(i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})$]{}, then $(i,i+1)$ is a **descent** in $T$:
$\begin{array}{c}
\begin{picture}(40,40)(0,0)
\put(0,0){\makebox{$i+1$}}
\put(20,20){\makebox{$i$}}
\put(0,10){\line(1,0){35}}
\put(0,10){\line(0,1){30}}
\end{picture} \\
\mbox{$i$ is Northeast of $i+1$,}
\end{array}$ $\begin{array}{c}
\begin{picture}(40,40)(5,0)
\put(10,10){\makebox{$i+1$}}
\put(35,25){\makebox{$i$}}
\put(0,22){\line(1,0){40}}
\put(40,0){\line(0,1){22}}
\end{picture} \\
\mbox{$i+1$ is Southwest of $i$.}
\end{array}$
\
\[prop:compatible\] A standard Young tableau $T$ is $\delta_n / \mu$–compatible if and only if $\jdt(T)$ is $\delta_n / \mu$–compatible.
During the procedure of jeu de taquin rectification, we call the move (up or left) of a single number a *move*, and a series of (upward and leftward) moves transforming an inner corner into an outer corner a *slide*. For instance $${\Yvcentermath1}\young(\,\,1247,:\,359,::68{{$10$}}) \mapsto \young(\,\,1247,:3589,::6{{$10$}})$$ is a slide consisting of four moves. We will prove that a slide cannot affect the $\delta_n/\mu$–compatibility of a skew shifted SYT, which will show the desired result.
For the sake of contradiction, assume that a slide of jeu de taquin, which transformed a tableau $T_1$ into a tableau $T_2$, affected $\delta_n / \mu$-compatibility. There are two (not mutually exclusive) cases, namely:
- the tableau gained/lost an ascent $(i,j)$ prescribed by [${\Yvcentermath1}\young(j,i)$]{} in $T_{\delta_n / \mu}$, or
- the tableau gained/lost a descent $(i, i+1)$ prescribed by [${\Yvcentermath1}\young(i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})$]{} in $T_{\delta_n / \mu}$.
We will study these two cases separately.
**Case 1:** The tableau gained or lost an ascent $(i,j)$ prescribed by $T_{\delta_n / \mu}$, with $i<j$.
Assume that $i$ is minimal among all such ascents. We consider four subcases, namely when $i$ remains still and $j$ moves left or up, and when $i$ moves left or up.
**Case 1.1:** During the slide, $i$ did not move and $j$ moved left.
The area southWest of $j$ before the move of $j$ contains the area southWest of $j$ after the move, so the tableau must have lost the descent: Before the move $i$ was southWest of $j$, and after the move it is not, making the tableau $T_1$ lose its $\delta_n/\mu$-compatibility when it turned into $T_2$. Since $i$ did not move, it must have been on the column directly left of $j$’s column, and below $j$. But then the move put $j$ above $i$ in $T_2$, a contradiction.
**Case 1.2:** During the slide, $i$ did not move and $j$ moved up.
Before the move $i$ was not southWest of $j$, and after the move it is. The tableau gained a prescribed ascent, making $T_2$ a $\delta_n/\mu$–compatible tableau.
Since $i$ did not move, it must have been on the row directly above $j$’s row, and strictly left of $j$. In $T_2$, $i$ and $j$ are on the same row. If there was a number $x$ between them, it would satisfy $i<x<j$. In $T_{\delta_n/\mu}$, because $i$ is directly below $j$, $x$ would have to be either directly east of $i$ (and therefore Southwest of $i$ in $T_2$) or directly west of $j$ (and therefore Northeast of $j$ in $T_2$) – a contradiction in either case. It follows that the slide looked like this: $${\Yvcentermath1}\young(:\,,iy,:j) \quad \mapsto \quad
\young(:y,i\,,:j)
\mapsto \quad
\young(:y,ij,:\,)$$ where the number $y$ must move up since $i$ does not move. We have $i<y<j$ which, by the argument in the previous paragraph, means that $${\Yvcentermath1}T_{\delta_n/\mu} \quad \textrm{ contains } {\textcolor{blue}}{\quad \young(y{\cdots}j,\,\,i)}$$ In $T_2$, $j$ is Southwest of $j-1$, which is Southwest of $j-2$, , $\cdots$, which is Southwest of $y$. But $y$ and $j$ are adjacent, so $y=j-1$. Now $${\Yvcentermath1}T_{\delta_n/\mu} \quad \textrm{ contains } {\textcolor{blue}}{\quad \young({{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}j,{{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}i)}$$ which forces $i-1$ to be Northeast of $i$ and southWest of $j-1$ in $T_2$; *i.e.*, directly above $i$. Therefore the slide looked like: $${\Yvcentermath1}\young({{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}z,i{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},:j) \quad \mapsto \quad
\young({{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},i\,,:j) \quad \mapsto \quad
\young({{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},ij,:\,)$$ where the number $z$ must have moved up, or else it would be between $i-1$ and $i$.
We conclude that this slide also made the tableau gain the (smaller) ascent $(i-1,j-1)$, while leaving $i-1$ still and moving $j-1$ up. This contradicts the minimality of $i$.
**Case 1.3:** During the slide, $i$ moved left.
Here we gained the prescribed ascent $(i,j)$, making $T_2$ $\delta_n/\mu$–compatible. In $T_2$, $j$ must be on the column to the right of $i$’s column. It cannot be higher than $i$, or else it would have been on the same column and above $i$ in $T_1$. Therefore it must be directly to the right of $i$, having slid into $i$’s old position. Since $j$ was not northEast of $i$, it must have slid up from below $i$, so the slide looked like this: $${\Yvcentermath1}\young(\,i,:j) \quad \mapsto \quad
\young(ij,:\,)$$ We need to consider two subcases.
**Case 1.3.1:** There is no cell to the left of $j$ in $T_1$. From the shape of $\delta_n$, $j+1$ is to the right of $j$ in $T_{\delta_n/\mu}$, so it must be Southwest of $j$ in $T_2$. The only possibility is that it is directly below $j$, and was to the right of $j$ in $T_1$. The slide must have looked like: $${\Yvcentermath1}\young(\,ix,:j{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(ijx,:{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})$$ where $j<x<j+1$, a contradiction.
**Case 1.3.2:** There is a cell to the left of $j$ in $T_1$. The number in it must be between $i$ and $j$, and by the same argument of Case 1.2, it must actually equal $i+1$, and $${\Yvcentermath1}T_{\delta_n/\mu} \quad \textrm{ contains } \quad {\textcolor{blue}}{\young(j{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}},i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})}$$ By $\delta_n/\mu$-compatibility, $j+1$ must be directly below $j$ and to the right of $i+1$ in $T_2$, making the slide look like: $${\Yvcentermath1}\young(\,i,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}j) \quad \mapsto \quad
\young(ij,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})$$ Note that $j+1$ could not be to the right of $j$ in $T_1$, or else the number directly above it would have to be between $j$ and $j+1$. So $j+1$ must have been below $j$ and slid up: $${\Yvcentermath1}\young(\,i,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}j,:{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(ij,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})\,.$$ Therefore the slide introduced the ascent $(i+1,j+1)$ stipulated by $T_{\delta_n/\mu}$, leaving $i+1$ still and moving $j+1$. As we saw in Case 1.2, this is impossible.
**Case 1.4:** During the slide, $i$ moved up.
Here we lost the ascent $(i,j)$ when we go from the $\delta_n/\mu$–compatible tableau $T_1$ to $T_2$. Then $j$ must be on the same row as $i$ in $T_1$; arguing as above, it must actually be directly to the right of $j$. The number $x$ directly above $j$ must have stayed still, so the slide looks like: $${\Yvcentermath1}\young(\,x,ij) \quad \mapsto \quad
\young(ix,??)\,.$$ where $j$ may or may not have moved left, so we do not specify the bottom row in $T_2$. As in the previous cases, $i<x<j$ implies that $x=j-1$ and that $${\Yvcentermath1}T_{\delta_n/\mu} \quad \textrm{ contains } \quad {\textcolor{blue}}{\young({{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}j,{{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}i)}\, ,$$ and $\delta_n/\mu$-compatibility then gives that the slide was $${\Yvcentermath1}\young({{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},ij) \quad \mapsto \quad
\young(i{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},??)\,.$$ If $i-1$ slid up, then the prescribed ascent $(i-1,j-1)$ would also be lost by moving $i-1$ up, contradicting the minimality of $i$. Therefore the slide was $${\Yvcentermath1}\young(\,{{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},:ij) \quad \mapsto \quad
\young({{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}i{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},:??)\,.$$ If there was a cell to the left of $i$ in $T_1$, the number in it would need to be between $i-1$ and $i$; so this move went along the bottom left diagonal of the board. Also, $${\Yvcentermath1}T_{\delta_n/\mu} \quad \textrm{ contains } \quad {\textcolor{blue}}{\young({{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}j,{{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}i,h)}$$ for some $h$. By $\delta_n/\mu$-compatibility, $h$ must have been to the left of $i-1$ in $T_1$. Because we are at the bottom of the board, the slide must have looked like this: $${\Yvcentermath1}\young(h{{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},:ij) \quad \mapsto \quad
\young(h,{{\small\mbox{{$i$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}}i{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{-}{}$} {\small\!\!\!$1$}}}},:??)\,.$$ But then the prescribed ascent $(h,i-1)$ was lost by moving $h$ up, contradicting the minimality of $i$.
**Case 2:** The tableau gained or lost a descent $(i,i+1)$ prescribed by $T_{\delta_n / \mu}$.
Again, we consider the same four subcases as above:
**Case 2.1:** During the slide, $i$ did not move and $i+1$ moved up.
Before the move $i$ was Northeast of $i+1$, and after the move it is not. Since $i$ did not move, it must have been on the row directly above $i+1$ and east of it. Therefore the move placed $i+1$ on the same row and to the left of $i$, a contradiction.
**Case 2.2:** During the slide, $i$ did not move and $i+1$ moved left.
In this case the tableau must have gained the descent: Before the move $i$ was not Northeast of $i+1$, and after the move it is. Since $i$ did not move, it is on the same column and (necessarily directly) above $i+1$ after the move. The slide must have looked like this: $${\Yvcentermath1}\young(:i,\,x{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(:i,x\,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(:i,x{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}\,)$$ which gives $i<x<i+1$, a contradiction.
**Case 2.3:** During the slide, $i$ moved up.
Before the move, $i+1$ was not Southwest of $i$, and after the move it is. Since $i+1$ could not have been on the same row and to the left of $i$ before, it must have been directly to the right of $i$, and must have slid into $i$’s old position: $${\Yvcentermath1}\young(\,,i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(i,\,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(i,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}\,)\,.$$ But then the cell northeast of these is part of the tableau: $${\Yvcentermath1}\young(\,x,i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(ix,\,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(ix,{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}\,)\,,$$ and the number $x$ in it satisfies $i<x<i+1$, a contradiction.
**Case 2.4:** During the slide, $i$ moved left.
Before the move, $i+1$ was Southwest of $i$, and after the move it is not. Therefore $i+1$ must have been on the same column as $i$ and (necessarily directly) below it. The slide must have looked like this: $${\Yvcentermath1}\young(\,i,*{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young(i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}},*)$$ The tableaux cannot contain the cell with the asterisk, because the number in it would need to be between $i$ and $i+1$. Therefore this part of the slide is happening along the lower diagonal of the tableaux.
Because $i+1$ is not a descent in $T_1$, it must be the rightmost number in its row in $T_{\delta_n/\mu}$. Given the shape of $\delta_n$, $${\Yvcentermath1}T_{\delta_n/\mu} \quad \textrm{ contains } \quad {\textcolor{blue}}{\young(i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}},{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}})}\, .$$ for some $j+1<i$. Then $(j+1,i)$ must be an ascent in $T_1$, which implies that the slide looked like this
$${\Yvcentermath1}\young(\,,{{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}i,:{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}}) \quad \mapsto \quad
\young({{\small\mbox{{$j$\!\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}},i{{\small\mbox{{$i$\!\!} {\tiny $\stackrel{+}{}$} {\small\!\!\!$1$}}}},:\,)\,.$$ This means that this slide made the tableau lose the prescribed ascent $(j+1,i)$ which, as we saw in Case 1, leads to a contradiction.
Proof of Theorem \[th:main\] {#sec:proof}
============================
We will need the following two theorems on the Schur expansions of skew Schur functions and Schur $P$-functions.
\[th:shimozono\] We have $$s_{\lambda / \mu} = \sum_{\nu} c^{\lambda}_{\mu,\nu} s_{\nu},$$ where the Littlewood–Richardson coefficient $c^{\lambda}_{\mu,\nu}$ is equal to the number of SYT $T$ of shape $\nu$ which are $\lambda / \mu$–compatible.
The proof is in [@Shi99], but note that Shimozono’s definition of $\lambda / \mu$-compatibility differs by ours in the sense that it reverses ascents and descents. The change is easily made by considering the reverse alphabet $\cdots > 3 > 2 > 1$.
\[th:stembridge\] Fix a shifted SYT $U$ of shape $\lambda$. We have $$P_{\lambda} = \sum_{\mu} g^{\lambda}_{\mu} s_{\mu},$$ where $g^{\lambda}_{\mu}$ is the number of SYT $T$ of shape $\mu$ such that $\jdt(T) = U$.
We have now assembled all the ingredients to prove the main theorem.
Denote the set of shifted standard Young tableaux of shape $\delta_n / \mu$ by , and the set of (shifted) standard $\delta_n / \mu$–compatible tableaux by (). By Theorem \[th:shimozono\] we have $$\begin{aligned}
s_{\delta_n / \mu} & = & \sum_{T \in \textsf{CompSYT}} s_{\sh(T)} \\
& = & \sum_{U \in \textsf{ShSYT}}\,\, \sum_{T \in \textsf{CompSYT}\, : \, U = \jdt(T)} s_{\sh(T)}.\end{aligned}$$ By Proposition \[prop:compatible\] and Theorem \[th:stembridge\] respectively, this equals $$\begin{aligned}
s_{\delta_n / \mu}
&=& \sum_{U \in \textsf{CompShSYT}}\,\, \sum_{T \in \textsf{SYT} \, : \, U = \jdt(T)} s_{\sh(T)} \\
& = & \sum_{U \in \textsf{CompShSYT}} P_{\sh(U)}\end{aligned}$$ as we wished to prove.
Further Work
============
- As mentioned earlier, the Schur and the Schur $P$-functions are related to the representations and the projective representations of the symmetric group, and to the cohomology of the Grassmannian and the isotropic Grassmannian. The representation theoretic and geometric significance of Theorem \[th:main\] should be explored.
- Theorem \[th:main\] implies that if $\lambda / \mu$ is a disjoint union of staircase skew shapes and their $180$ degree rotations, then $s_{\lambda / \mu}$ is Schur $P$-positive. It is natural to wonder whether these are the only skew Schur functions which are Schur $P$-positive. In fact, Dewitt [@Dew12] has proved the stronger statement that these are the only skew Schur functions which are *linear* combinations of Schur P-functions.
[xxx]{}
L. Carlitz, Enumeration of up-down sequences, *Discrete Math* **4** (1973), 273–286.
E. Dewitt, Identities Relating Schur $s$-Functions and $Q$-Functions, Ph.D. Thesis, University of Michigan, 2012.
H. O. Foulkes, Enumeration of permutations with prescribed up-down and inversion sequences, *Discrete Math* **15** (1976), 235–252.
W. Fulton. Young tableaux. London Mathematical Society Student Texts 35. Cambridge University Press, 1997.
I. M. Gessel, Generating Functions and Enumeration of Sequences, Ph.D. Thesis, Massachusetts Institute of Technology, 1977.
I. M. Gessel, Symmetric functions and P-recursiveness, *J. Combin. Theory Ser. A* **53** (1990), 257–285.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, *John Wiley & Sons*, New York, 1983 (Dover Reprint, 2004).
M. D. Haiman, Dual equivalence with applications, including a conjecture of Proctor, *Discrete Math* **99** (1992), 79–113.
M. D. Haiman, On mixed insertion, symmetry, and shifted Young tableaux, *J. Combin. Theory Ser. A* **50** (1989), 196–225.
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T. Józefiak, Schur $Q$-functions and cohomology of isotropic Grassmannians. *Math. Proc. Camb. Phil. Soc.* **109** (1991), 471–478.
B. Sagan, Shifted tableaux, Schur $Q$-functions, and a conjecture of R. Stanley, *J. Combin. Theory Ser. A* **45** (1987), 62–103.
M. Shimozono, Multiplying Schur Q-functions, *J. Combin. Theory Ser. A* **87** (1999), no. 1, 198–232.
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[^1]: If you printed this paper in black and white, you are not missing much.
[^2]: This is sometimes called a *right ascent* (*right descent*) of the word.
|
---
abstract: 'The present status of theoretical and experimental investigations of the decay rate of a positronium is considered. The increasing interest to this problem has been caused by the disagreement of the calculated value of $\Gamma_3 (o-Ps)$ and the recent series of precise experiments. The necessity of new calculations on the basis of the quantum field methods in bound state theory is pointed out with taking into account the dependence of the interaction kernel on relative energies.'
---
Preprint IFUNAM FT-93-017\
May 1993\
(Russian version: December 1992)\
\
$^{*,\,\dagger}$\
\
and\
[**RUDOLF N. FAUSTOV**]{}\
\
and\
[**YURI N. TYUKHTYAEV**]{}$^{\ddagger}$\
& [*Nuclear Physics\
Saratov State University, Astrakhanskaya str. , 83\
Saratov 410071 RUSSIA*]{}\
[**Submitted to “Modern Physics Letters A”**]{}
KEYWORDS: quantum electrodynamics, positronium, decay rate, bound states\
PACS: 11.10.St, 12.20.Ds, 12.20.Fv, 13.40.Hq\
—————————————————————-\
$^{*}$ On leave from: [*Dept.Theor.*]{} & [*Nucl. Phys., Saratov State University\
and Sci.*]{} & [*Tech. Center for Control and Use of Physical Fields and Radiations\
Astrakhanskaya str. , 83, Saratov 410071 RUSSIA*]{}
$^{\dagger}$ Email: valeri@ifunam.ifisicacu.unam.mx\
dvoeglazov@main1.jinr.dubna.su
$^{\ddagger}$ Email: postmaster@ccssu.saratov.su
The quantum-electrodynamic systems, consisting of particle and anti-particle, have specific features. Apart from a scattering channel, the annihilation channel appears in this case. A positronium atom, which is a specimen of these systems, has no stability. The life time of a positronium (or the decay rate) is the subject of precise experimental and theoretical investigations. The charge parity of a positronium, $C = (-1)^{L+S}$ ($L $ is the eigenvalue of an angular momentum operator, $S$ is the eigenvalue of a total spin operator for a system under consideration), is a motion constant. Consequently, all its states are separated into the charge - even states ($S=1$) and the charge - odd ones ($S=-1$). The positronium total spin is also conserved and the energy levels are classified as the singlet levels ($S=0$, parapositronium) and as the triplet ones ($S=1$, orthopositronium). The S – state ($L=0$) parapositronium has a positive parity and the S – state orthopositronium has a negative parity. As a consequence of conservation of a charge parity in electromagnetic interaction a parapositronium is disintegrated into the even number of photons and an orthopositronium is decayed into the odd one.
At the present time, the essential disagreement is turned out between the theoretical and experimental values for the decay rate of an orthopositronium. The theoretical predictions are [@pg1]-[@pg4]:\
$$\begin{aligned}
\lefteqn{\Gamma^{theor}_3(o-Ps)=\frac{\alpha^6
mc^2}{\hbar}\frac{2(\pi^2-9)}{9\pi} \left
[1+A_3\frac{\alpha}{\pi}-\frac{1}{3}\alpha^2
ln\alpha^{-1}+B_3(\frac{\alpha}{\pi})^2+\ldots \right ] = }
\nonumber\\
&=&\Gamma_0+\frac{m\alpha^7}{\pi^2}\left \{ {-1.984(2) \choose
-1.9869(6)}\right\}+\frac{m\alpha^8}{\pi}
ln\alpha^{-1}\left [-\frac{4}{9}\zeta(2)+\frac{2}{3}\right
]+\frac{m\alpha^8}{\pi^3}{\cal X}+\ldots \nonumber\\
&=&\left \{ {7.0386(2) \choose 7.03830(7)} \right \}\,\mu s^{-1},\end{aligned}$$ where $$\begin{aligned}
A_3^{\cite{pg3}}&=&-10.266\pm 0.011,\\
A_3^{\cite{pg4}}&=&-10.282\pm 0.003.\end{aligned}$$ Taking into account the modern value of $\alpha$, the fine structure constant, the result can be recalculated [@pr11]: $$\Gamma^{theor}_{3}=7.038\,31(5)\,\mu s^{-1}.$$
The last experimental measurings are [@pg21; @pg22][^1]: $$\begin{aligned}
\Gamma^{exp}_{\cite{pg21}}(o-Ps)&=&7.0514(14)\,\mu s^{-1}\\
\Gamma^{exp}_{\cite{pg22}}(o-Ps)&=&7.0482(16)\,\mu s^{-1}.\end{aligned}$$ The result of Ref. [@pg21] has 9.4 standard deviation from the predicted theoretical decay rate and the result of [@pg22] has 6.2 standard deviation. The coefficient $B_3=1$ in the $O(\alpha^8)$ term can contribute $3.5\cdot10^{-5}\mu s^{-1}$ (or 5 ppm of $\Gamma_3$) only. To take off the above disagreement the coefficient $B_3$ has to be equal to about $\simeq 250\pm 40$, what is very unlikely, indeed. However, this opportunity is pointed out in [@pg22] just not to be rejected [*a priori*]{}. Therefore, the calculation of $B_3$ coefficient is desirable enough now.
For the first time the main contribution in the orthopositronium decay rate has been calculated in [@pg1] : $$\Gamma_0(o-Ps)=-2Im(\Delta
E_{3\gamma})=\frac{2}{9\pi}(\pi^2-9)m\alpha^6=7,211\,17\,\mu s^{-1}.$$ The corrections of the $O(\alpha)$ order of the magnitude to this quantity had been calculating by numerical method [@pg2; @pg4; @pg3a; @pg3b] at first, but later some of them has been figure out analytically in [@pg3; @a35] and [@pr7]-[@pr9] in the Feynman gauge. The corrections, which come from the diagrams with the self-energy and vertex insertions, have been calculated by Adkins [@pr8; @pr9]: $$\begin{aligned}
\Gamma_{OV}&=&\Gamma_0\frac{\alpha}{\pi} \left \{
D+\frac{3}{4(\pi^2-9)}\left [ -26-\frac{115}{3}ln2
+\frac{91}{18}\zeta(2)+\frac{443}{54}\zeta(3)+
\frac{3419}{108}\zeta(2)
ln2-\right .\right .\nonumber\\
&-&\left.\left. R \right ] \right
\}=\Gamma_0\frac{\alpha}{\pi}\left
[ D+2.971\,138\,5(4)\right ],\\
\Gamma_{SE}&=&\Gamma_{0}\frac{\alpha}{\pi} \left \{ -D-4+\frac{3}{ 4
(\pi^2-9)
} \left [ -7+\frac{67}{3}ln
2+\frac{805}{36}\zeta(2)-
\frac{1049}{54}\zeta(3)-\right.\right.\nonumber\\
&-&\left.\left.\frac{775}{54}\zeta(2)ln 2 \right ] \right
\}=\Gamma_0\frac{\alpha}{\pi}\left [ -D+0.784\,98\right ],\\
\Gamma_{IV}&=&\Gamma_{0}\frac{\alpha}{\pi}\left \{\frac{1}{2}
D+\frac{3}{4
(\pi^{2}-9) } \left [
-4-\frac{34}{2}ln2-\frac{841}{36}\zeta(2)+\frac{1253}{36}\zeta(2)ln
2+\frac{1589}{54}\zeta(3)+\right.\right.\nonumber\\
&+&\left.\left.\frac{17}{40}\zeta^{2}(2)-\frac{7}{8}\zeta(3)ln
2+\frac{5}{2}\zeta(2)ln^{2} 2-\frac{1}{24} ln^{4} 2-a_{4} \right ]
\right
\}=\nonumber\\
&=&\Gamma_{0}\frac{\alpha}{\pi}\left
[\frac{1}{2}D+0.160\,677\right
],\end{aligned}$$ where $$R=\int \limits^{1}_{0} dx\frac{ln(1-x)}{2-x}\left
[\zeta(2)-Li_2(1-2x)\right
]=-1.743\,033\,833\,7(3),\\$$ $$a_4=Li_4(\frac{1}{2})=\sum_{n=1}^{\infty}\frac{1}{n^4
2^n}=0.517\,479\,061\,674,$$ $$\zeta(2)=\frac{\pi^2}{6},\quad \zeta(3)=1.202\,056\,903\,2,$$ and $$D=\frac{1}{2-w}-\gamma_E+ln(4\pi)$$ is standard expression of a dimensional regularization ($2\omega$ is a space dimension). The indices “$IV$” and “$OV$” designate the insertions in the internal photon-electron vertex and in the outer ones, correspondingly. The above results are co-ordinated with the Stroscio’s result [@pr7] when $$\Gamma_0\frac{\alpha}{\pi}\left [-D-4-2 ln (\lambda^2/m^2)\right ]$$ being added to the last one. This is necessary because of the different regularization procedures which have been used in [@pr7] and in [@pr8; @pr9], correspondingly.
Recently, the calculations of these corrections have been finished [@pr11] in the Fried – Yennie gauge: $$\begin{aligned}
\Gamma_{SE}&=&\frac{m\alpha^7}{\pi^2}\left
[-\frac{13}{54}\zeta(3)+\frac{461}{108}\zeta(2)ln2
-\frac{251}{72}\zeta(2)-\frac{29}{6}ln2+\frac{9}{2}\right
]=\nonumber\\
&=&\frac{m\alpha^7}{\pi^2}(-0.007\,132\,904)=\Gamma_0\frac{\alpha}{\pi
}(-0.036\,911\,113),\\
\Gamma_{OV}&=&\frac{m\alpha^7}{\pi^2}\left
[-\frac{88}{54}\zeta(3)-\frac{299}{216}\zeta(2)ln2
+\frac{49}{18}\zeta(2)+\frac{13}{6}ln2-2-\frac{1}{6}R\right
]=\nonumber\\
&=&\frac{m\alpha^7}{\pi^2}(0.732\,986\,380)=\Gamma_0\frac{\alpha}{\pi}
(3.793\,033\,599).\end{aligned}$$ The contributions from the remained diagrams ( with a radiative insertion in a vertex of an internal photon; with two radiative photons spanned; the diagram taking into account boundary effects and the annihilation diagram, see Fig. I in \[5b\] ), have been calculated numerically. Totally, the $O(\alpha)$ corrections are joined to give $$\frac{m\alpha^7}{\pi^2}\left [-1.987\,84(11)\right ]=
\Gamma_0\frac{\alpha}{\pi}\left [-10.286\,6(6)\right ].$$ Then we have[^2] : $$\Gamma_{3,~\cite{pr11}}^{theor}(o-Ps)=7.038\,236(10)\, \mu s^{-1}.$$ The above result is the most precise theoretical result available at the present moment.
To solve the existing disagreement between theory and experiment, the 5 – photon mode of $o-Ps$ decay and the 4 – photon mode of $p-Ps$ decay have been under consideration in [@pru1; @pru2][^3]. The following theoretical evaluations have been obtained: $$\begin{aligned}
\frac{\Gamma_5^{\cite{pru1}}(o-Ps)}{\Gamma_3(o-Ps)}&=&0.177
(\frac{\alpha}{\pi})^2\simeq 0.96\cdot10^{-6},\\
\frac{\Gamma_4^{\cite{pru1}}(p-Ps)}{\Gamma_2(p-Ps)}&=&0.274
(\frac{\alpha}{\pi})^2\simeq 1.48\cdot10^{-6},\end{aligned}$$ and $$\begin{aligned}
\Gamma_5^{\cite{pru2}}(o-Ps)&=&0.018\,9(11)\alpha^2\Gamma_0,\\
\Gamma_4^{\cite{pru2}}(p-Ps)&=&0.013\,89(6)m\alpha^7.\end{aligned}$$ They are in agreement with each other and with the results of the previous papers [@pru3][^4] : $$\Gamma_4^{~\cite{pru3}}(p-Ps) = 0.013\,52\,m\alpha^7 = 11.57\cdot
10^{-3}
s^{-1}.$$
In the connection with the present situation with respect to the decay rate of $o-Ps$ investigations of alternative decay modes for this system (e.g. $o-Ps\rightarrow\gamma+a$, $a$ is an axion, a pseudo-scalar particle with mass $m_a<2m_e$) are of present interest [@pru6]-[@pru6d]. In the article [@pru6c] the following experimental limits of the branching of decay width have been obtained: $$Br=\frac{\Gamma(o-Ps\rightarrow \gamma+a)}{\Gamma(o-Ps\rightarrow
3\gamma)}
< 5\cdot 10^{-6} - 1\cdot 10^{-6}\quad (30\, ppm),$$ provided that $m_a$ is in the range 100 – 900 keV. In the case of the axion mass less than 100 keV (which is implied by the Samuel’s hypothesis [@pru6ca]; according to the cited paper [^5] $m_a< 5.7 \,keV$, $g_{ae^{+} e^{-}} \sim 2\cdot 10^{-8}$) the limits of $Br$ are the following ones [@pru6d] : $$\begin{aligned}
Br&=&7.6\cdot 10^{-6},\quad \mbox{if}\,\,\,m_a \sim 100\, keV,\\
Br&=&6.4\cdot 10^{-5}, \quad \mbox{if}\,\,\, m_a < 30\, keV.\end{aligned}$$ These limits are about 2 orders less than the value which is necessary to remove the disagreement.
Finally, a decay $o-Ps \rightarrow nothing$ (that is into weak-interacting non-detected particles) [^6] has been investigated in [@pru6e]. The obtained result $$\frac{\Gamma(o-Ps\rightarrow nothing)}
{\Gamma(o-Ps\rightarrow 3\gamma)}< 5.8\cdot 10^{-4} \quad (350\, ppm)$$ expels the opportunity that this decay mode is an origin of disajustment between theory and experiment.
The decay of $o-Ps$ into two photons, which breaks the CP – invariance, as else mentioned in [@pru7a; @pru7b], was experimentally rejected in [@pru7] [^7].
Let us mention, the contribution of weak interaction has been studied in [@pw1]. However, because of the factor $m_e^2/M^2_W \sim G_F \cdot m_e
\simeq 3 \cdot
10^{-12}$ it cannot influence final results. In the cited article the weak decay modes are estimated as $$\frac{\Gamma(p-Ps\rightarrow 3\gamma)}{\Gamma(p-Ps\rightarrow
2\gamma)}
\simeq\frac{\Gamma(o-Ps\rightarrow 4\gamma)}{\Gamma(o-Ps\rightarrow
3\gamma)}
\simeq\alpha(G_F m_e^2 g_V)^2\simeq 10^{-27},$$ where $G_F$ is the Fermi constant for weak interaction, $$g_V=1-4sin^2\Theta_W\simeq 0.08,$$ $\Theta_W$ is the Weinberg angle. The current experimental limits are [@pw2; @pw3] : $$\begin{aligned}
\frac{\Gamma(p-Ps\rightarrow 3\gamma)}{\Gamma(p-Ps\rightarrow
2\gamma)}&\leq&2.8\cdot 10^{-6},\\
\frac{\Gamma(o-Ps\rightarrow 4\gamma)}{\Gamma(o-Ps\rightarrow
3\gamma)}&\leq&8\cdot 10^{-6}.\end{aligned}$$
In the Table I all experimental results for the $o-Ps$ decay rate , known to us, are presented [^8].\
Table I. The experimental results for the $o-Ps$ decay rate.\
Year Reference $\Gamma_3(o-Ps),\,\mu s^{-1}$ Error,ppm Technique
------ ----------- ------------------------------- ----------- ------------------
1968 [@pg11b] 7.262(15) 2070 gas
1973 [@pg11] 7.262(15) 2070 gas
1973 [@pg12] 7.275(15) 2060 gas
1976 [@pg13] 7.104(6) 840 powder $SiO_{2}$
1976 [@pg14] 7.09(2) 2820 vacuum
1978 [@pg15] 7.056(7) 990 gas
1978 [@pg16] 7.045(6) 850 gas
1978 [@pg17] 7.050(13) 1840 vacuum
1978 [@pg17a] 7.122(12) 1680 vacuum
1982 [@pg18] 7.051(5) 710 gas
1987 [@pg19] 7.031(7) 1000 vacuum
1987 [@pg20] 7.0516(13) 180 gas
1989 [@pg21] 7.0514(14) 200 gas
1990 [@pg22] 7.0482(16) 230 vacuum
Regarding the results for the decay rate of a parapositronium, the situation was highly favorable until the last years. The theoretical value, which was found out else in the fifties [@pg52r; @pg52], is equal to $$\Gamma_2^{theor}(p-Ps)=-2Im(\Delta
E_{2\gamma})=\frac{1}{2}\frac{\alpha^5
mc^2}{h}\left [ 1-\frac{\alpha}{\pi}(5-\frac{\pi^2}{4})\right ]=
7.9852\,
ns^{-1}.$$ The above value, confirmed in [@pg53; @pg54], coincides with the direct experimental result up to the good accuracy: $$\Gamma^{exp}_{\cite{pg18}}(p-Ps)=7.994 \pm 0.011\, ns^{-1}.$$
The experimental values of the parapositronium decay rate are showed in the Table II [^9].\
Table II. The experimental results for the $p-Ps$ decay rate.\
Year Reference $\Gamma_2(p-Ps),\,ns^{-1}$ Error, $\%$ Technique
------ ----------- ---------------------------- ------------- -----------
1952 [@p2wh] 7.63(1.02) 13 gas
1954 [@p5a] 9.45(1.41) 15 gas
1970 [@pg55] 7.99(11) 1.38 gas
1982 [@pg18] 7.994(11) 0.14 gas
In the articles [@pg3; @pg54] it was pointed out that it is necessary to add the logarithmic corrections on $\alpha$ to the Harris and Brown’s result. In the article \[13a\] these corrections to the $\Gamma_3(o-Ps)$ and $\Gamma_2(p-Ps)$ have been calculated again, with the result of the decay rate of a parapositronium differing with the one found out before [@pg3; @pg54]: $$\begin{aligned}
\Gamma_{2}^{\cite{a35}}(p-Ps, \alpha^2
ln\alpha)&=&\frac{m\alpha^5}{2}\cdot
2\alpha^2 ln\alpha^{-1},\\
\Gamma_{2}^{\cite{pg3,pg54}}(p-Ps, \alpha^2
ln\alpha)&=&\frac{m\alpha^5}{2}\cdot\frac{2}{3}\alpha^2
ln\alpha^{-1}.\end{aligned}$$
Finally, the quite unexpected (and undesirable) result, presented in the Remiddi’s (and collaborators) talk [@pg55a] should be mentioned. The calculations, carried out by the authors of cited paper, lead to the additional contribution: $$\Gamma_{2}^{\cite{pg55a}}(p-Ps, \alpha ln\alpha)=
\frac{m\alpha^5}{2}(\frac{\alpha}{\pi})2 ln \alpha,$$ which is explained by authors to appear as a result of taking into account the dependence of an interaction kernel on the relative momenta.[^10]
The above-mentioned shows us at the necessity of a continuation of calculations of the decay rates of an orthopositronium as well as a parapositronium employing more accurate relativistic methods, one of which is the quasipotential approach in quantum field theory [@Log; @Kad] giving the opportunity taking into account of binding effects.
[**Acknowledgements**]{}
The authors express their sincere gratitude to B. A. Arbuzov, E. E. Boos, D. Broadhurst, R. Fell, V. G. Kadyshevsky, A. Sultanayev for the fruitful discussions; to the Saratov’s Scientific and Technological Center and the CONACYT ( Mexico ) for financial support. We strongly appreciate also the technical assistance of S. V. Khudyakov, G. Loyola and A. S. Rodin.
One of us (V. D.) is very grateful to his colleagues in the Laboratory of Theoretical Physics at the JINR (Dubna) and in the Departamento de Física Teórica at the IFUNAM for the creation of the excellent working conditions.
[999]{}
[^1]: See Table I for the preceding experimental results.
[^2]: The uncalculated yet $O(\alpha^8)$ corrections are not accounted here.
[^3]: As a consequence of a conservation of an angular momentum and an isotropic properties of coordinate space an orthopositronium has to decay into the odd number of photons and a parapositronium has to decay into the even one, as outlined before.
[^4]: The result [@pru4] is not correct, four times less than the above cited results. The explanation of this was given in [@pru2].
[^5]: The proposed values do not influence the agreement of theoretical and experimental results of an anomalous magnetic moment (AMM) of an electron.
[^6]: Analogously to Glashow’s hypothesis of the decay into invisible “mirror” particle [@pru6cb].
[^7]: The physics ground of these speculations is a possible existence of an unisotropic vector field with non-zero vacuum expectation [@pru7c], with whom an electron and a positron could be interacting $${\cal L}=g\bar\psi O_{\alpha\beta}\psi A^{\alpha}\Omega^{\beta},$$ $\cal L$ is the interaction Lagrangian.
[^8]: The results of the papers [@pg11aa; @pg11a] and [@pg12a] can be accounted as rough estimations only.
[^9]: The branching of the decay rates of a para- and an orthopositronium $\frac{\Gamma_2(p-Ps)}{\Gamma_3(o-Ps)}$ had been measuring in the experiments of 1952 and 1954 only. The presented results are recalculated by means of the first direct experimental value $\Gamma_3(o-Ps)=7.262(15)\,\mu s^{-1}$ [@pg11b].
[^10]: Let us mark, the additional contributions, which are similar to the ones of Ref. [@pg55a], appeared in the calculations of the hyperfine splitting (HFS) of the ground state of two-fermion system by the quasipotential method [@Log; @Kad] when taking into account the dependence of the interaction kernel on the relative energies [@Tyuk; @Boi]. For example, the $O(\alpha^2 ln\alpha)$ correction to the HFS, obtained from one-photon diagram, is equal to $$\Delta E^{HFS}_{tr}(\alpha^2 ln\alpha)=E_F\frac{\mu^2\alpha^2}{m_1
m_2}
(\frac{m_1}{m_2}+\frac{m_2}{m_1}+2)ln\alpha^{-1},$$ when using the version of the quasipotential approach based on the amplitude $\tau=(\hat{G}^+_0)^{-1}\widehat{G_0
TG_0}^{+}(\hat{G}^+_0)^{-1}$. Here $\hat{G}^{+}$ is the two-time Green’s function projected onto the positive-energy states, $m_1$ and $m_2$ are the masses of the constituent particles, $E_F$ is the Fermi energy. The index “[*tr*]{}” designates that the diagram of one-transversal-photon exchange is under consideration.
Using the kernel constructed from the on-shell physical amplitude we can find out that the last term in the brackets disappears. However, the total result to the HFS accounting all diagrams is the same in both of versions of quasipotential approach.
We can also come across with the similar situation in calculations of the anomalous corrections of $O(\alpha ln\alpha)$ order, which do not appear in the method on the mass shell. However, the version, based on the two-time Green’s function, shows the contraction of these anomalously large terms, taking into account two-photon exchange diagrams [@Boi].
|
---
author:
- 'Biel Roig-Solvas[^1]'
- 'Lee Makowski[^2]'
- 'Dana H. Brooks'
title: A Proximal Operator for Multispectral Phase Retrieval Problems
---
[^1]: Department of Electrical and Computer Engineering, Northeastern University, Boston, MA (, ).
[^2]: Department of Bioengineering, Northeastern University, Boston, MA ().
|
---
abstract: |
In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and $L^2$-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of ${{\mathcal I}}$-indexed categories where ${{\mathcal I}}$ is any small category admitting a finite ${{\mathcal I}}$-$CW$-model for its ${{\mathcal I}}$-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass–Serre graphs of groups to higher dimensions. In particular, we obtain necessary conditions for developability of a finite complex of groups from an action of a finite group on a finite category without loops.\
Key words: finiteness obstruction, Euler characteristic of a category, $L^2$-Euler characteristic, projective class group, homotopy colimits of categories, Grothendieck construction, spaces over a category, Grothendieck fibration, complex of groups, small category without loops.\
2010 *Mathematics Subject Classification*. Primary: 18F30, 19J05 ; Secondary: 18G10, 19A49, 55U35, 19A22, 46L10.
address:
- |
Thomas M. Fiore\
Department of Mathematics and Statistics\
University of Michigan-Dearborn\
4901 Evergreen Road\
Dearborn, MI 48128\
U.S.A.
- |
Wolfgang Lück\
Mathematisches Institut der Universität Bonn\
Endenicher Allee 60\
53115 Bonn\
Germany
- |
Roman Sauer\
Fakultät für Mathematik\
Universität Regensburg\
Universitätsstr. 31\
93053 Regensburg\
Germany
author:
- 'Thomas M. Fiore'
- Wolfgang Lück
- Roman Sauer
title: Euler characteristics of categories and homotopy colimits
---
Introduction and Statement of Results
=====================================
In our previous paper [@FioreLueckSauerFinObsAndEulCharOfCats(2009)], we presented a unified conceptual framework for Euler characteristics of categories in terms of finiteness obstructions and projective class groups. Many excellent properties of our invariants stem from the homological origins of our approach: the theory of modules over categories and the dimension theory of modules over von Neumann algebras provide us with an array of tools and techniques. In the present paper, we additionally draw upon the homotopy theory of diagrams to prove the compatibility of our invariants with homotopy colimits.
If ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ is a diagram of categories (or more generally a pseudo functor into the 2-category of small categories), then our invariants of the homotopy colimit can be computed in terms of the invariants of the vertex categories ${{\mathcal C}}(i)$. In particular, our Homotopy Colimit Formula, Theorem \[the:homotopy\_colimit\_formula\], states $$\label{equ:intro_hocolim}
\chi\bigl({\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}};R\bigr) = \sum_{n \geq 0} (-1)^n
\cdot \sum_{\lambda \in \Lambda_n} \chi({{\mathcal C}}(i_\lambda);R)$$ under certain hypotheses. The set $\Lambda_n$ indexes the ${{\mathcal I}}$-$n$-cells of a finite ${{\mathcal I}}$-$CW$-model $E{{\mathcal I}}$ for the ${{\mathcal I}}$-classifying space of ${{\mathcal I}}$, that is, we have a functor $E{{\mathcal I}}\colon {{\mathcal I}}^{{\operatorname{op}}} \to {{\EuR}{SPACES}}$ which is inductively built by gluing finitely many cells of the form ${\operatorname{mor}}_{{\mathcal I}}(-,i_\lambda)\times D^n$ for $\lambda \in \Lambda_n$, and moreover $E{{\mathcal I}}(i)\simeq *$ for all objects $i$ of ${{\mathcal I}}$. Similar formulas hold for the finiteness obstruction, the functorial Euler characteristic, the functorial $L^2$-Euler characteristic, and the $L^2$-Euler characteristic.
Motivation for such a formula is provided by the classical Inclusion-Exclusion Principle: if $A$, $B$, and $A \cap B$ are finite simplicial complexes, then one has $$\chi(A\cup B)=\chi(A)
+\chi(B)-\chi(A\cap B).$$ However, one cannot expect the Euler characteristic to be compatible with pushouts, even in the simplest cases. The pushout in ${{\EuR}{CAT}}$ of the discrete categories $$\{\ast\} \leftarrow \{y,z\} \rightarrow
\{\ast'\}$$ is a point, but $\chi(\text{point}) \neq 1+1-2$. On the other hand, their *homotopy* pushout in ${{\EuR}{CAT}}$ is the category whose objects and nontrivial morphisms are pictured below. $$\xymatrix{y \ar[r] \ar[d] & \ast' \\ \ast & z \ar[l] \ar[u]}$$ The classifying space of this category has the homotopy type of $S^1$, so that $$\chi(\text{homotopy
pushout})=\chi(\{\ast\})+\chi(\{\ast'\})-\chi(\{y,z\})$$ is true. In fact, the formula for homotopy pushouts is a special case of : the category ${{\mathcal I}}=\{1
\leftarrow 0 \rightarrow 2\}$ admits a finite model with $\Lambda_0=\{1,2\}$ and $\Lambda_1=\{0\}$, as constructed in Example \[exa:finite\_model\_for\_pushout\]. See Example \[exa:homotopy\_pushout\] for the homotopy pushout formulas of the other invariants.
The Homotopy Colimit Formula in Theorem \[the:homotopy\_colimit\_formula\] has many applications beyond homotopy pushouts. Other special cases are formulas for Euler characteristics of products, homotopy orbits, and transport groupoids. Our formulas also have ramifications for the developability of Haefliger’s *complexes of groups* in geometric group theory. If a group $G$ acts on an $M_\kappa$-polyhedral complex by isometries preserving cell structure, and if each $g \in G$ fixes each cell pointwise that $g$ fixes setwise, then the quotient space is also an $M_\kappa$-polyhedral complex, see Bridson–Haefliger [@Bridson-Haefliger(1999) page 534]. Let us call the quotient $M_\kappa$-polyhedral complex $Q$. To each face $\overline{\sigma}$ of $Q$, one can assign the stabilizer $G_\sigma$ of a chosen representative cell $\sigma$. This assignment, along with the various conjugated inclusions of groups obtained from face inclusions, is called the *complex of groups associated to the group action*. It is a pseudo functor from the poset of faces of $Q$ into groups. In the finite case, the Euler characteristic and $L^2$-Euler characteristic of the homotopy colimit can be computed in terms of the original complex and the order of the group. We prove this in Theorem \[thm:Euler\_characteristic\_of\_hocolim\_of\_quotient\_complex\]. Homotopy colimits of complexes of groups play a special role in Haefliger’s theory, see the discussion after Definition \[def:complex\_of\_groups\].
In Section \[sec:finiteness\_obstruction\_and\_Euler\_characteristics\], we review the notions and results from [@FioreLueckSauerFinObsAndEulCharOfCats(2009)] that we need in this sequel. Explanations of the finiteness obstruction, the functorial Euler characteristic, the Euler characteristic, the functorial $L^2$-Euler characteristic, the $L^2$-Euler characteristic, and the necessary theorems are all contained in Section \[sec:finiteness\_obstruction\_and\_Euler\_characteristics\] in order to make the present paper self-contained. Section \[sec:spaces\_over\_a\_category\] is dedicated to an assumption in the Homotopy Colimit Formula, namely the requirement that a finite ${{\mathcal I}}$-$CW$-model exists for the ${{\mathcal I}}$-classifying space of ${{\mathcal I}}$. We recall the notion of ${{\mathcal I}}$-$CW$-complex, present various examples, and prove that finite models are preserved under equivalences of categories. Homotopy colimits of diagrams of categories are recalled in Section \[sec:homotopy\_colimits\_of\_categories\]. The homotopy colimit construction in ${{\EuR}{CAT}}$ is the same as the Grothendieck construction, or the category of elements. Thomason proved that the homotopy colimit construction has the expected properties. We prove our main theorem, the Homotopy Colimit Formula, in Section \[sec:Homotopy\_colimit\_formula\], work out various examples in Section \[sec:examples\_of\_the\_homotopy\_colimit\_formula\], and derive the generalized Inclusion-Exclusion Principle in Section \[sec:Combinatorial\_Applications\_of\_the\_Homotopy\_Colimit\_Formula\]. We review the groupoid cardinality of Baez–Dolan and the Euler characteristic of Leinster in Section \[sec:comparison\_with\_Leinster\], and compare our Homotopy Colimit Formula with Leinster’s compatibility with Grothendieck fibrations in terms of weightings. We apply our results to Haefliger complexes of groups in Section \[sec:complexes\_of\_groups\] to prove Theorems \[thm:Euler\_characteristic\_of\_hocolim\_of\_quotient\_complex\] and \[the:extension\_of\_Haefligers\_corollary\], which express Euler characteristics of complexes of groups associated to group actions in terms of the initial data.
[**Acknowledgements.**]{} All three authors were supported by the Sonderforschungsbereich 878 – Groups, Geometry and Actions – and the Leibniz-Preis of Wolfgang Lück.
Thomas M. Fiore was supported at the University of Chicago by NSF Grant DMS-0501208. At the Universitat Autònoma de Barcelona he was supported by grant SB2006-0085 of the Spanish Ministerio de Educación y Ciencia under the Programa Nacional de ayudas para la movilidad de profesores de universidad e investigadores espa$\tilde{\text{n}}$oles y extranjeros. Thomas M. Fiore also thanks the Centre de Recerca Matemàtica in Bellaterra (Barcelona) for its hospitality during the CRM Research Program on Higher Categories and Homotopy Theory in 2007-2008, where he heard Tom Leinster speak about Euler characteristics. He also thanks the Max Planck Insitut für Mathematik for its hospitality and support during his stay in Summer 2010.
The Finiteness Obstruction and Euler Characteristics {#sec:finiteness_obstruction_and_Euler_characteristics}
====================================================
We quickly recall the main definitions and results needed from our first paper [@FioreLueckSauerFinObsAndEulCharOfCats(2009)] in order to make this article as self-contained as possible. See [@FioreLueckSauerFinObsAndEulCharOfCats(2009)] for proofs and more detail.
Throughout this paper, let $\Gamma$ be a category and $R$ an associative, commutative ring with identity. The first ingredient we need is the theory of modules over categories developed by Lück [@Lueck(1989)], and recalled in [@FioreLueckSauerFinObsAndEulCharOfCats(2009)]. An *$R\Gamma$-module* is a contravariant functor from $\Gamma$ into the category of left $R$-modules. For example, if $\Gamma$ is a group $G$ viewed as a one-object category, then an $R\Gamma$-module is the same as a right module over the group ring $RG$. An $R\Gamma$-module $P$ is *projective* if it is projective in the usual sense of homological algebra, that is, for every surjective $R\Gamma$-morphism $p \colon M \to N$ and every $R\Gamma$-morphism $f \colon P \to N$ there exists an $R\Gamma$-morphism $\overline{f}
\colon P \to M$ such that $p \circ \overline{f} = f$. An $R\Gamma$-module $M$ is *finitely generated* if there is a surjective $R\Gamma$-morphism $B(C) \to M$ from an $R\Gamma$-module $B(C)$ that is free on a collection $C$ of sets indexed by ${\operatorname{ob}}(\Gamma)$ such that $\coprod_{x \in {\operatorname{ob}}(\Gamma)} C_x$ is finite. Explicitly, the *free $R\Gamma$-module on the ${\operatorname{ob}}(\Gamma)$-set $C$* is $$\label{equ:free_RGamma_module}
B(C) := \bigoplus_{x \in {\operatorname{ob}}(\Gamma)} \bigoplus_{C_x} R{\operatorname{mor}}_\Gamma(?,x).$$ A contravariant $R\Gamma$-module may be tensored with a covariant $R\Gamma$-module to obtain an $R$-module: if $M\colon \Gamma^{{\operatorname{op}}}
\to R\text{-}{{\EuR}{MOD}}$ and $N\colon \Gamma \to R\text{-}{{\EuR}{MOD}}$ are functors, then the *tensor product* $M \otimes_{R\Gamma} N$ is the quotient of the $R$-module $$\bigoplus_{x \in {\operatorname{ob}}(\Gamma)} M(x) \otimes_R N(x)$$ by the $R$-submodule generated by elements of the form $$(M(f)m) \otimes n - m \otimes (N(f)n)$$ where $f:x \to y$ is a morphism in $\Gamma$, $m \in M(y)$, and $n
\in N(x)$.
Finite projective resolutions of the constant $R\Gamma$-module $\underline{R}$ play a special role in our theory of Euler characteristic for categories. A resolution $P_*$ of an $R\Gamma$-module $M$ is said to be *finite projective* if it has finite length and each $P_n$ is finitely generated and projective. We say that a category $\Gamma$ *is of type (FP$_R$)* if the constant $R\Gamma$-module $\underline{R} \colon
\Gamma^{{\operatorname{op}}} \to R\text{-}{{\EuR}{MOD}}$ with value $R$ admits a finite projective resolution. Categories in which every endomorphism is an isomorphism, the so-called *EI-categories*, provide important examples. Finite EI-categories in which $|{\operatorname{aut}}(x)|$ is invertible in $R$ for each object $x$ are of type (FP$_R$). Further examples of categories of type (FP$_R$) include categories $\Gamma$ which admit a finite $\Gamma$-$CW$-model for the classifying $\Gamma$-space $E\Gamma$ (see Section \[sec:spaces\_over\_a\_category\] and Examples \[exa:finite\_model\_for\_I\_with\_terminal\_object\], \[exa:finite\_model\_for\_parallel\_arrows\], \[exa:finite\_model\_for\_pushout\], and \[exa:finite\_model\_for\_q\_interior\]). In fact, such categories $\Gamma$ are even *of type (FF$_R$)*: the cellular chains on a finite $\Gamma$-$CW$-model for $E\Gamma$ provide a finite free resolution of $\underline{R}$. In general, if a category is of type (FF$_{{\mathbb Z}}$), then it is of type (FF$_R$) for any associative, commutative ring $R$ with identity.
A home for the finiteness obstruction of a category $\Gamma$ is provided by the *projective class group* $K_0(R\Gamma)$. The generators of this abelian group are the isomorphism classes of finitely generated projective $R\Gamma$-modules and the relations are given by expressions $[P_0] - [P_1] + [P_2] = 0$ for every exact sequence $0 \to P_0 \to P_1 \to P_2 \to 0$ of finitely generated projective $R\Gamma$-modules.
\[def:finiteness\_obstruction\_of\_a\_category\] Let $\Gamma$ be a category of type (FP$_R$) and $P_*$ a finite projective resolution of the constant $R\Gamma$-module $\underline{R}$. The *finiteness obstruction of $\Gamma$ with coefficients in $R$* is $$o(\Gamma;R) := \sum_{n \geq 0} (-1)^n \cdot [P_n] \; \in K_0(R\Gamma).$$ We also use the notation $[\underline{R}]$, or simply $[R]$, to denote the finiteness obstruction $o(\Gamma;R)$. The finiteness obstruction, when it exists, does not depend on the choice $P_*$ of finite projective resolution of $\underline{R}$.
The finiteness obstruction is compatible with most everything one could hope for. If $F \colon \Gamma_1 \to \Gamma_2$ is a right adjoint, and $\Gamma_1$ is of type (FP$_R$), then $\Gamma_2$ is of type (FP$_R$) and $F_*o(\Gamma_1;R)=o(\Gamma_2;R)$ (here the group homomorphism $F_*$ is induced by induction with $F$). Since an equivalence of categories is a right adjoint (and also a left adjoint), a particular instance of the previous sentence is: if $F
\colon \Gamma_1 \to \Gamma_2$ is an equivalence of categories, then $\Gamma_1$ is of type (FP$_R$) if and only if $\Gamma_2$ is, and in this case $F_*o(\Gamma_1;R)=o(\Gamma_2;R)$. The finiteness obstruction is also compatible with finite coproducts of categories, finite products of categories, restriction along admissable functors, and homotopy colimits, as we prove in Theorem \[the:homotopy\_colimit\_formula\]. If $G$ is a finitely presented group of type (FP$_\mathbb{Z}$), then Wall’s finiteness obstruction $o(BG)$ is the same as $o(\widehat{G};\mathbb{Z})$, which is the finiteness obstruction of $G$ viewed as a one-object category $\widehat{G}$ with morphisms $G$. The finiteness obstruction in Definition \[def:finiteness\_obstruction\_of\_a\_category\] is a special case of the finiteness obstruction of a finitely dominated $R\Gamma$-chain complex $C$, denoted $o(C)\in K_0(R\Gamma)$. The image of $o(C)$ in the reduced $K$-theory $\tilde{K}_0(R\Gamma)$ vanishes if and only if $C$ is $R\Gamma$-homotopy equivalent to a finite free $R\Gamma$-chain complex, see [@Lueck(1989) Chapter 11].
We will occasionally work with directly finite categories. A category is called *directly finite* if for any two objects $x$ and $y$ and morphisms $u \colon x \to y$ and $v \colon y \to x$ the implication $vu = {\operatorname{id}}_x \implies uv = {\operatorname{id}}_y$ holds. If $\Gamma_1$ and $\Gamma_2$ are equivalent categories, then $\Gamma_1$ is directly finite if and only if $\Gamma_2$ is directly finite. Examples of directly finite categories include groupoids, and more generally EI-categories.
A key result in the theory of modules over an EI-category is Lück’s splitting of the projective class group of $\Gamma$ into the projective class groups of the automorphism groups ${\operatorname{aut}}_\Gamma(x)$, one each isomorphism class of objects. We next recall the relevant maps and notation. For $x \in {\operatorname{ob}}(\Gamma)$, we denote $R{\operatorname{aut}}_\Gamma(x)$ by $R[x]$ for simplicity. The *splitting functor at $x \in {\operatorname{ob}}(\Gamma)$* $$\begin{aligned}
\label{equ:splitting_functor}
& S_x \colon {{\EuR}{MOD}}\text{-}R\Gamma \to {{\EuR}{MOD}}\text{-}R[x], &
\label{S_x}\end{aligned}$$ maps an $R\Gamma$-module $M$ to the quotient of the $R$-module $M(x)$ by the $R$-submodule generated by all images of $R$-module homomorphisms $M(u)\colon M(y) \to M(x)$ induced by all non-invertible morphisms $u\colon x \to y$ in $\Gamma$. The right $R[x]$-module structure on $M(x)$ induces a right $R[x]$-module structure on $S_xM$. Note that $S_xM$ is an $R[x]$-module, not an $R\Gamma$-module. The functor $S_x$ respects direct sums, sends epimorphisms to epimorphisms, and sends free modules to free modules. If $\Gamma$ is directly finite, then $S_x$ also preserves finitely generated and projective. The *extension functor at $x \in {\operatorname{ob}}(\Gamma)$* $$\begin{aligned}
\label{equ:extension_functor}
& E_x \colon {{\EuR}{MOD}}\text{-}R[x] \to {{\EuR}{MOD}}\text{-}R\Gamma&
\label{E_x}\end{aligned}$$ maps an $R[x]$-module $N$ to the $R\Gamma$-module $N \otimes_{R[x]} R{\operatorname{mor}}_\Gamma(?,x)$. The functor $E_x$ respects direct sums, sends epimorphisms to epimorphisms, sends free modules to free modules, and preserves finitely generated and projective. If $\Gamma$ is directly finite, and $P$ is a projective $R[x]$-module, then there is a natural isomorphism $P \cong S_xE_xP$ compatible with direct sums.
\[the\_splitting\_of\_K-theory\_for\_EI\_categories\] If $\Gamma$ is an EI-category, then the group homomorphisms $$\xymatrix{K_0(R\Gamma) \ar@<.4ex>[r]^-S & \ar@<.4ex>[l]^-E }{\operatorname{Split}}K_0(R\Gamma) :=
\bigoplus_{\overline{x} \in {\operatorname{iso}}(\Gamma)} K_0(R{\operatorname{aut}}_\Gamma(x))$$ defined by $$S[P]= \{[S_xP] \mid \overline{x} \in {\operatorname{iso}}(\Gamma)\}$$ and $$E \{[Q_x] \mid \overline{x} \in {\operatorname{iso}}(\Gamma)\}=
\sum_{\overline{x} \in {\operatorname{iso}}(\Gamma)} [E_xQ_x],$$ are isomorphisms and inverse to one another. They are covariantly natural with respect to functors between EI-categories.
If $\Gamma$ is not an EI-category, then the splitting homomorphism $S\colon K_0(R\Gamma) \to {\operatorname{Split}}K_0(R\Gamma)$ may not be an isomorphism. However, $S$ is covariantly natural with respect to functors between directly finite categories, see [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 3.15].
The splitting functors $S_x$ allow us to define the notion of $R\Gamma$-rank ${\operatorname{rk}}_{R\Gamma}$ for finitely generated $R\Gamma$-modules, which in turn allows the definition of the functorial Euler characteristic, as we explain next. We assume a fixed notion of a rank ${\operatorname{rk}}_R(N) \in {{\mathbb Z}}$ for finitely generated $R$-modules $N$ such that ${\operatorname{rk}}_R(R) = 1$ and ${\operatorname{rk}}_R(N_1) = {\operatorname{rk}}_R(N_0) + {\operatorname{rk}}_R(N_2)$ for any sequence $0 \to N_0 \to N_1 \to N_2 \to 0$ of finitely generated $R$-modules. If $R$ is a commutative principal ideal domain, we use ${\operatorname{rk}}_R(N) := \dim_F(F \otimes_R N)$, where $F$ is the quotient field of $R$. Let $U(\Gamma)$ be the free abelian group on the set of isomorphism classes of objects in $\Gamma$, that is $U(\Gamma)
:= {{\mathbb Z}}{\operatorname{iso}}(\Gamma).$ The augmentation homomorphism $\epsilon\colon U(\Gamma) \to {{\mathbb Z}}$ adds up the components of an element of $U(\Gamma)$.
\[def:rank\_of\_fin\_gen\_prof\_RGamma-module\] If $M$ is a finitely generated $R\Gamma$-module $M$, then its *$R\Gamma$-rank* is $${\operatorname{rk}}_{R\Gamma}(M) :=
\bigl\{{\operatorname{rk}}_R(S_xM \otimes_{R[x]} R) \mid \overline{x} \in
{\operatorname{iso}}(\Gamma)\bigr\} \quad \in U(\Gamma).$$
\[def:functorial\_Euler\_characteristic\_of\_a\_category\]\[def:Euler\_characteristic\_of\_a\_category\] Suppose that $\Gamma$ is of type (FP$_R$). The *functorial Euler characteristic of $\Gamma$ with coefficients in $R$* is the image of the finiteness obstruction $o(\Gamma;R) \in K_0(R\Gamma)$ under the homomorphism ${\operatorname{rk}}_{R\Gamma}\colon K_0(R\Gamma) \to U(\Gamma)$, that is $$\chi_f(\Gamma;R):={\operatorname{rk}}_{R\Gamma} o(\Gamma;R)=\left\{\sum_{n \geq 0
}(-1)^n{\operatorname{rk}}_R(S_xP_n \otimes_{R[x]} R) \mid \overline{x} \in
{\operatorname{iso}}(\Gamma) \right\} \quad \in U(\Gamma),$$ where $P_*$ is any finite projective $R\Gamma$-resolution of the constant $R\Gamma$-module $\underline{R}$. The *Euler characteristic of $\Gamma$ with coefficients in $R$* is the sum of the components of the functorial Euler characteristic, that is, $$\chi(\Gamma;R):=\epsilon(\chi_f(\Gamma;R))=\sum_{\overline{x} \in {\operatorname{iso}}(\Gamma)} \sum_{n \geq 0 }(-1)^n{\operatorname{rk}}_R(S_xP_n \otimes_{R[x]} R).$$
For example, if ${{\mathcal G}}$ is a finite groupoid, then $\chi_f({{\mathcal G}}) \in U({{\mathcal G}})$ is $(1,1,\dots,1)$, and $\chi({{\mathcal G}})$ counts the isomorphism classes of objects, or equivalently the connected components, of ${{\mathcal G}}$.
\[the:chi\_f\_determines\_chi\] Let $R$ be a Noetherian ring and $\Gamma$ a directly finite category of type (FP$_R$). Then the Euler characteristic and topological Euler characteristic of $\Gamma$ agree. That is, $H_n(B\Gamma;R)$ is a finitely generated $R$-module for every $n \geq 0$, there exists a natural number $d$ with $H_n(B\Gamma;R) = 0$ for all $n > d$, and $$\chi(\Gamma;R) = \chi(B\Gamma;R)= \sum_{n \geq 0} (-1)^n \cdot {\operatorname{rk}}_{R}(H_n(B\Gamma;R)) \in {{\mathbb Z}}.$$ Here $\chi(\Gamma;R)$ is defined in Definition \[def:Euler\_characteristic\_of\_a\_category\] and $B\Gamma$ denotes the geometric realization of the nerve of $\Gamma$.
The functorial Euler characteristic and Euler characteristic have many desirable properties. They are invariant under equivalence of categories and are compatible with finite products and finite coproducts. As we prove in Theorem \[the:homotopy\_colimit\_formula\], they are also compatible with homotopy colimits.
The $L^2$-Euler characteristic, which is in some sense the better invariant, can be defined similarly by taking $R={{\mathbb C}}$ and using the $L^2$-rank ${\operatorname{rk}}^{(2)}_\Gamma$ rather than the $R\Gamma$-rank. For this we need group von Neumann algebras and their dimension theory from Lück [@Lueck(1998a)] and [@Lueck(1998b)], as recalled in our first paper [@FioreLueckSauerFinObsAndEulCharOfCats(2009)] for the purpose of Euler characteristics. If $G$ is a group, its *group von Neumann algebra* $${{\mathcal N}}(G) = {{\mathcal B}}(l^2(G))^G$$ is the algebra of bounded operators on $l^2(G)$ that are equivariant with respect to the right $G$-action. If $G$ is finite, ${{\mathcal N}}(G)$ is the group ring ${{\mathbb C}}G$. In any case, the group ring ${{\mathbb C}}G$ embeds as a subring of ${{\mathcal N}}(G)$ by sending $g \in G$ to the isometric $G$-equivariant operator $l^2(G) \to l^2(G)$ given by left multiplication with $g$. In particular, we can view ${{\mathcal N}}(G)$ as a ${{\mathbb C}}G$-${{\mathcal N}}(G)$-bimodule and tensor ${{\mathbb C}}G$-modules on the right with ${{\mathcal N}}(G)$. If $G$ is the automorphism group of an object in $\Gamma$, then we write ${{\mathcal N}}(x)$ for ${{\mathcal N}}\bigl({\operatorname{aut}}_\Gamma(x)\bigr)$.
The *von Neumann dimension*, $\dim_{{{\mathcal N}}(G)}$, is a function that assigns to *every* right ${{\mathcal N}}(G)$-module $M$ a non-negative real number of $\infty$. It is the unique such function which satisfies Hattori-Stallings rank, additivity, cofinality, and continuity. If $G$ is a finite group, then ${{\mathcal N}}(G) = {{\mathbb C}}G$ and we get for a ${{\mathbb C}}G$-module $M$ the von Neumann dimension $$\dim_{{{\mathcal N}}(G)}(M) = \frac{1}{|G|} \cdot \dim_{{{\mathbb C}}}(M),$$ where $\dim_{{{\mathbb C}}}$ is the dimension of $M$ viewed as a complex vector space. A category $\Gamma$ is said to be *of type* ($L^2$) if for one (and hence every) projective ${{\mathbb C}}\Gamma$-resolution $P_*$ of the constant ${{\mathbb C}}\Gamma$-module $\underline{{{\mathbb C}}}$ we have $$\sum_{\overline{x} \in {\operatorname{iso}}(\Gamma)} \sum_{n \geq 0} \dim_{{{\mathcal N}}(x)} H_n\bigl(S_x P_*\otimes_{{{\mathbb C}}[x]}{{\mathcal N}}(x)\bigr) < \infty.$$ Note that the projective resolution $P_*$ of $\underline{{{\mathbb C}}}$ is not required to be of finite length, nor finitely generated. Examples of categories of type ($L^2$) include finite EI-categories, in particular finite posets and finite groupoids. Infinite categories can also be of type ($L^2$), for example any (small) groupoid with finite automorphism groups such that $$\label{equ:finiteness of aut-sum}
\sum_{\overline{x} \in {\operatorname{iso}}({{\mathcal G}})} \frac{1}{|{\operatorname{aut}}_{{{\mathcal G}}}(x)|}<\infty$$ holds is of type ($L^2$). The condition of type ($L^2$) is weaker than (FP$_{{\mathbb C}}$), since any directly finite category of type (FP$_{{\mathbb C}}$) is also of type ($L^2$).
\[def:functorial\_L2-Euler\_characteristic\_of\_a\_category\] \[def:L2-Euler\_characteristic\_of\_a\_category\] Suppose that $\Gamma$ is of type ($L^2$). Define $$U^{(1)}(\Gamma) :=
\left\{\sum_{\overline{x} \in {\operatorname{iso}}(\Gamma)} r_{\overline{x}} \cdot \overline{x}
\;\bigg|\;
r_{\overline{x}} \in {{\mathbb R}},
\sum_{\overline{x} \in {\operatorname{iso}}(\Gamma)} |r_{\overline{x}}| < \infty\right\}\subseteq\prod_{\bar{x}\in{\operatorname{iso}}(\Gamma)}{{\mathbb R}}.$$ The *functorial $L^2$-Euler characteristic of $\Gamma$* is $$\chi_f^{(2)}(\Gamma):=\left\{\sum_{n \geq 0} (-1)^n \dim_{{{\mathcal N}}(x)} H_n\bigl(S_x P_*\otimes_{{{\mathbb C}}[x]}{{\mathcal N}}(x)\bigr)\mid \bar{x} \in
{\operatorname{iso}}(\Gamma) \right\} \in U^{(1)}(\Gamma),$$ where $P_*$ is any projective ${{\mathbb C}}\Gamma$-resolution of the constant ${{\mathbb C}}\Gamma$-module $\underline{{{\mathbb C}}}$. The *$L^2$-Euler characteristic of $\Gamma$* is the sum over $\bar{x} \in
{\operatorname{iso}}(\Gamma)$ of the components of the functorial Euler characteristic, that is, $$\chi^{(2)}(\Gamma):=\sum_{\overline{x} \in {\operatorname{iso}}(\Gamma)} \sum_{n
\geq 0} (-1)^n \dim_{{{\mathcal N}}(x)} H_n\bigl(S_x P_*\otimes_{{{\mathbb C}}[x]}{{\mathcal N}}(x)\bigr).$$
If ${{\mathcal G}}$ is a groupoid such that holds, then the functorial $L^2$-Euler characteristic $\chi_f^{(2)}({{\mathcal G}}) \in \prod_{\overline{x} \in {\operatorname{iso}}({{\mathcal G}})} {{\mathbb R}}$ has at $\overline{x} \in {\operatorname{iso}}({{\mathcal G}})$ the value $1/|{\operatorname{aut}}_{{\mathcal G}}(x)|$. The $L^2$-Euler characteristic is $$\label{eq:L2_Euler_Characteristic_of_Groupoid}
\chi^{(2)}({{\mathcal G}}) = \sum_{\overline{x} \in {\operatorname{iso}}({{\mathcal G}})}
\frac{1}{|{\operatorname{aut}}_{{\mathcal G}}(x)|}.$$ See Lemma \[lem:chi(2)\_and\_chi\] for an explicit formula for $\chi^{(2)}(\Gamma)$ in the case of a finite, skeletal EI-category $\Gamma$ in which the left ${\operatorname{aut}}_\Gamma(y)$-action on ${\operatorname{mor}}_\Gamma(x,y)$ is free for every two objects $x,y \in
{\operatorname{ob}}(\Gamma)$.
\[def:L2rank\_of\_fin\_gen\_prof\_RGamma-module\] Let $M$ be a finitely generated ${{\mathbb C}}\Gamma$-module $M$. Its *$L^2$-rank* is $${\operatorname{rk}}_{\Gamma}^{(2)}(M) :=
\bigl\{\dim_{{{\mathcal N}}(x)}(S_xM \otimes_{{{\mathbb C}}[x]} {{\mathcal N}}(x))\mid \bar{x}
\in {\operatorname{iso}}(\Gamma)\bigr\}
\in U(\Gamma) \otimes_{{{\mathbb Z}}} {{\mathbb R}}= \bigoplus_{{\operatorname{iso}}(\Gamma)} {{\mathbb R}}.$$
\[the:comparing\_o\_and\_chi(2)\] Suppose that $\Gamma$ is a directly finite category of type (FP$_{{{\mathbb C}}}$). Then $\Gamma$ is of type ($L^2$) and the image of the finiteness obstruction $o(\Gamma;{{\mathbb C}})$ (see Definition \[def:finiteness\_obstruction\_of\_a\_category\]) under the homomorphism $${\operatorname{rk}}_{\Gamma}^{(2)} \colon K_0({{\mathbb C}}\Gamma) \to U(\Gamma) \otimes_{{{\mathbb Z}}} {{\mathbb R}}= \bigoplus_{\overline{x} \in {\operatorname{iso}}(\Gamma)} {{\mathbb R}}$$ is the functorial $L^2$-Euler characteristic $\chi_f^{(2)}(\Gamma)$.
The $L^2$-Euler characteristic agrees with the groupoid cardinality of Baez–Dolan [@Baez-Dolan(2001)] and the Euler characteristic of Leinster [@Leinster(2008)] in certain cases, see Lemma \[lem:chi(2)\_and\_chi\] and Section \[sec:comparison\_with\_Leinster\]. In particular, the Baez–Dolan groupoid cardinality of a groupoid satisfying is . However, the Baez–Dolan groupoid cardinality and Leinster’s Euler characteristic $\chi_L(\Gamma)$ only depend on the underlying graph of $\Gamma$, whereas our invariants truly depend on the category structure. For instance, $\chi_L$ is $\frac{1}{2}$ for both the two-element monoid $({{\mathbb Z}}/2,\times)$ and the two-element group $({{\mathbb Z}}/2,+)$, whereas $\chi^{(2)}$ is 1 respectively $\frac{1}{2}$. The distinction can already be seen on the level of the finiteness obstructions. The Euler characteristic $\chi(-)$ and topological Euler characteristic $\chi(B-)$ can also distinguish categories with the same underlying directed graph as in the following example. For $S=\{1,2,3,4\}$, $G_1=\langle(1234)\rangle$, $G_2=\langle (12),(34) \rangle$, and $k=1,2$, let $\Gamma_k$ be the EI-category with objects $x$ and $y$ and ${\operatorname{mor}}(x,y):=S$, ${\operatorname{mor}}(x,x):=\{{\operatorname{id}}_x\}$, ${\operatorname{mor}}(y,y):=G_k$, and ${\operatorname{mor}}(y,x)=\emptyset$. Composition in $\Gamma_k$ is the composition in $G_k$ and the left $G_k$-action on $S$, that is, $\Gamma_k$ is the EI-category associated to the respective $G_k$-$\{1\}$-biset $S$ as in Subsection 6.4 of Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009)]. Then $\Gamma_1$ and $\Gamma_2$ have the same underlying directed graph but $\chi(\Gamma_1;{{\mathbb Q}})=\chi(B\Gamma_1;{{\mathbb Q}})=1$ and $\chi(\Gamma_2;{{\mathbb Q}})=\chi(B\Gamma_2;{{\mathbb Q}})=0$ by Theorem 6.23 (iii) of Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009)]. An infinite example of categories with the same underlying graph but different Euler characteristics is provided by the groups ${{\mathbb Z}}$ and ${{\mathbb Z}}*{{\mathbb Z}}$, each of which admits a finite $\Gamma$-$CW$-model for its respective $\Gamma$-classifying space. The categories $\widehat{{{\mathbb Z}}}$ and $\widehat{{{\mathbb Z}}*{{\mathbb Z}}}$ have the same underlying directed graph, but we have $\chi^{(2)}(\widehat{{{\mathbb Z}}})=0\neq\chi^{(2)}(\widehat{{{\mathbb Z}}*{{\mathbb Z}}})$, and similarly for $\chi$. Typically, the Euler characteristic of a category $\Gamma_{\text{free}}$ free on a directed graph $(V,E)$ is the same as the Euler characteristic of the directed graph $(V,E)$. For the topological Euler characteristic this is clearly true, since $B\Gamma_{\text{free}}$ is homotopy equivalent to the topological realization $|(V,E)|$. If $\Gamma_{\text{free}}$ is directly finite and $R$ is Noetherian, then we also have $\chi(\Gamma_{\text{free}})=\chi(|(V,E)|)$ by Theorem \[the:chi\_f\_determines\_chi\]. For example for the directed graph with one vertex and one arrow we have $\chi(\widehat{{{\mathbb N}}})=0=\chi(S^1)$.
The functorial $L^2$-Euler characteristic and the $L^2$-Euler characteristic have many desirable properties. They are invariant under equivalence of categories and are compatible with finite products, finite coproducts, and isofibrations and coverings between finite groupoids. We prove in Theorem \[the:homotopy\_colimit\_formula\] the compatibility with homotopy colimits. In the case of a group $G$, the $L^2$-Euler characteristic of $\widehat{G}$ coincides with the classical $L^2$-Euler characteristic of $G$, which is $1/\vert G \vert$ when $G$ is finite. The $L^2$-Euler characteristic is also closely related to the geometry and topology of the classifying space for proper $G$-actions, namely the functorial $L^2$-Euler characteristic of the proper orbit category ${\underline{{{\EuR}{Or}}}(G)}$ is equal to the equivariant Euler characteristic of the classifying space $\underline{E}G$ for proper $G$-actions, whenever $\underline{E}G$ admits a finite $G$-$CW$-model.
The question arises: what are sufficient conditions for the Euler characteristic and $L^2$-Euler characteristic to coincide with the Euler characteristic of the classifying space? This is answered in the following Theorem.
\[the:coincidence\] Suppose $\Gamma$ is directly finite and of type (FF$_{{\mathbb Z}}$). Then the functorial $L^2$-Euler characteristic of Definition \[def:functorial\_L2-Euler\_characteristic\_of\_a\_category\] coincides with the functorial Euler characteristic of Definition \[def:functorial\_Euler\_characteristic\_of\_a\_category\] for any associative, commutative ring $R$ with identity $$\chi_f^{(2)}(\Gamma)=\chi_f(\Gamma;R)\in U(\Gamma) \subseteq
U^{(1)}(\Gamma),$$ and thus $\chi^{(2)}(\Gamma)=\chi(\Gamma;R)$ in Definition \[def:L2-Euler\_characteristic\_of\_a\_category\] and Definition \[def:Euler\_characteristic\_of\_a\_category\].
If $R$ is additionally Noetherian, then $$\label{equ:coincidence_for_type_FFZ_and_directly_finite}
\chi(B\Gamma;R)=\chi(\Gamma;R)=\chi^{(2)}(\Gamma).$$ Moreover, if $\Gamma$ is merely of type (FF$_{{\mathbb C}}$) rather than (FF$_{{\mathbb Z}}$), then equation holds for any Noetherian ring $R$ containing ${{\mathbb C}}$.
Any category $\Gamma$ which admits a finite $\Gamma$-$CW$-model in the sense of Section \[sec:spaces\_over\_a\_category\] is of type (FF$_R$) for any ring $R$, by an application of the cellular $R$-chain functor. Thus, Theorem \[the:coincidence\] applies to any directly finite category $\Gamma$ which admits a finite $\Gamma$-$CW$-model. For example, finite categories without loops are directly finite and admit finite models (Lemma \[lem:scwol\_directly\_finite\_EI\] and Theorem \[the:finite\_models\_for\_finite\_scwols\]), so equation holds for instance for $\{j \rightrightarrows k\}$, $\{k \leftarrow
j \to \ell\}$, and finite posets. The monoid ${{\mathbb N}}$ and group ${{\mathbb Z}}$, viewed as one-object categories $\widehat{{{\mathbb N}}}$ and $\widehat{{{\mathbb Z}}}$, are also directly finite and admit finite models (see Example \[exa:finite\_model\_for\_N\_and\_Z\]), so we have $$0=\chi(S^1;R)=\chi(B\widehat{{{\mathbb N}}};R)=\chi(\widehat{{{\mathbb N}}};R)=\chi^{(2)}(\widehat{{{\mathbb N}}})$$ and $$0=\chi(S^1;R)=\chi(B\widehat{{{\mathbb Z}}};R)=\chi(\widehat{{{\mathbb Z}}};R)=\chi^{(2)}(\widehat{{{\mathbb Z}}})$$ ($B\widehat{{{\mathbb N}}} \to B\widehat{{{\mathbb Z}}}\simeq S^1$ is a homotopy equivalence by Quillen’s Theorem A, see Rabrenovi[ć]{} [@Rabrenovic(2005) Proposition 10]). The equations $\chi(\widehat{{{\mathbb N}}};R)=0=\chi^{(2)}(\widehat{{{\mathbb N}}})$ and $\chi(\widehat{{{\mathbb Z}}};R)=0=\chi^{(2)}(\widehat{{{\mathbb Z}}})$ also follow from Example \[exa:homotopy\_hocolimit\_and\_trivial\_functor\], since the finite models for $\widehat{{{\mathbb N}}}$ and $\widehat{{{\mathbb Z}}}$ in Example \[exa:finite\_model\_for\_N\_and\_Z\] each have one ${{\mathcal I}}$-$0$-cell and one ${{\mathcal I}}$-$1$-cell.
We may use Theorem \[the:coincidence\] to obtain an explicit formula for Euler characteristics of finite categories without loops as follows. Let $\Gamma$ be a finite category without loops, and choose a skeleton $\Gamma'$. Let $c_n(\Gamma')$ denote the number of paths $$i_0 \to i_1 \to i_2 \to \cdots \to i_n$$ of $n$-many non-identity morphisms in $\Gamma'$. Then $c_n(\Gamma')$ is the number of $n$-cells in the $CW$-complex $B\Gamma'$, and we have $$\label{equ:Euler_characteristic_for_skeletal_finite_scwols}
\chi(\Gamma;R)=\chi^{(2)}(\Gamma)=\chi(B\Gamma;R)=\chi(B\Gamma';R)=\sum_{n
\geq 0} (-1)^n c_n(\Gamma').$$ See [@Leinster(2008) Corollary 1.5] for a different derivation of this formula for Leinster’s Euler characteristic $\chi_L(\Gamma)$ in the case $\Gamma$ was already skeletal. See also Examples \[exa:homotopy\_hocolimit\_and\_trivial\_functor\] and \[exa:Euler\_characteristics\_of\_finite\_scwols\] where skeletality of ${{\mathcal I}}$ is not required.
If $F:\Gamma_1 \to \Gamma_2$ is a functor such that $BF$ is a homotopy equivalence, and both $\Gamma_1$ and $\Gamma_2$ are of type (FP$_R$), and if $$\chi(\Gamma_1;R)=\chi(B\Gamma_1;R) \;\;\text{ and }\;\;\chi(\Gamma_2;R)=\chi(B\Gamma_2;R),$$ then clearly $\chi(\Gamma_1;R)=\chi(\Gamma_2;R)$. However, it is possible for two categories to be homotopy equivalent, one of which is (FP$_R$) and the other is not, so that one has a notion of Euler characteristic and the other does not. In Section 10 of Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009)] such an example is discussed.
Spaces over a Category {#sec:spaces_over_a_category}
======================
An important hypothesis in our Homotopy Colimit Formula involves the idea of a space over a category, see Davis–Lück [@Davis-Lueck(1998)]. Namely, we assume that the indexing category ${{\mathcal I}}$ for the diagram ${{\mathcal C}}$ of categories admits a finite ${{\mathcal I}}$-$CW$-model for its ${{\mathcal I}}$-classifying space. Essentially this means it is possible to functorially assign a contractible $CW$-complex $E{{\mathcal I}}(i)$ to each $i \in {\operatorname{ob}}({{\mathcal I}})$, and moreover, these local $CW$-complexes are constructed globally by gluing ${{\mathcal I}}$-$n$-cells of the form ${\operatorname{mor}}_{{\mathcal I}}(-,i_\lambda)\times
D^n$ onto the already globally constructed $(n-1)$-skeleton $E{{\mathcal I}}_n$. The Homotopy Colimit Formula then expresses the invariants of the homotopy colimit of ${{\mathcal C}}$ in terms of the invariants of the categories ${{\mathcal C}}(i_\lambda)$ at the base objects $i_\lambda$ for $E{{\mathcal I}}$.
The gluing described above takes place in the more general category of ${{\mathcal I}}$-spaces. A *(contravariant) ${{\mathcal I}}$-space* is a contravariant functor from ${{\mathcal I}}$ to the category ${{\EuR}{SPACES}}$ of (compactly generated) topological spaces. As usual, we will always work in the category of compactly generated spaces (see Steenrod [@Steenrod(1967)]). A *map between ${{\mathcal I}}$-spaces* is a natural transformation. Given an object $i \in
{\operatorname{ob}}({{\mathcal I}})$, we obtain an ${{\mathcal I}}$-space ${\operatorname{mor}}_{{\mathcal I}}(?,i)$ which assigns to an object $j$ the discrete space ${\operatorname{mor}}_{{\mathcal I}}(j,i)$.
The next definition is taken from Davis–Lück [@Davis-Lueck(1998) Definition 3.2], where an ${{\mathcal I}}$-$CW$-complex is called a free ${{\mathcal I}}$-$CW$-complex and we will omit the word free here. The more general notion of ${{\mathcal I}}$-$CW$-complex was defined by Dror Farjoun [@DrorFarjoun(1987) 1.16 and 2.1]. See also Piacenza [@Piacenza(1991)].
\[def:calc-CW-complex\] A *(contravariant) ${{\mathcal I}}$-$CW$-complex* $X$ is a contravariant ${{\mathcal I}}$-space $X$ together with a filtration $$\emptyset = X_{-1} \subset X_0 \subset X_1 \subset X_2 \subset\ldots
\subset X_n \subset \ldots \subset X = \bigcup_{n \geq 0} X_n$$ such that $X = {\operatorname{colim}}_{n \to \infty} X_n$ and for any $n \geq 0$ the *$n$-skeleton* $X_n$ is obtained from the $(n-1)$-skeleton $X_{n-1}$ by [*attaching ${{\mathcal I}}$-$n$-cells*]{}, i.e., there exists a pushout of ${{\mathcal I}}$-spaces of the form $${\begin{CD}
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal I}}}(-,i_\lambda) \times S^{n-1} @>>> X_{n-1}\\
@V{}VV @V{}VV\\
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal I}}}(-,i_\lambda) \times D^n @>>> X_n
\end{CD}
}$$ where the vertical maps are inclusions, $\Lambda_n$ is an index set, and the $i_\lambda$-s are objects of ${{\mathcal I}}$. In particular, $X_0=\coprod_{\lambda \in \Lambda_0} {\operatorname{mor}}_{{{\mathcal I}}}(-,i_\lambda)$.
We refer to the inclusion functor ${\operatorname{mor}}_{{{\mathcal I}}}(-,i_\lambda) \times (D^n-S^{n-1}) \to X$ as an *${{\mathcal I}}$-$n$-cell based at $i_\lambda$*.
An ${{\mathcal I}}$-$CW$-complex has *dimension $\le n$* if $X = X_n$. We call $X$ *finite dimensional* if there exists an integer $n$ with $X = X_n$. It is called *finite* if it is finite dimensional and $\Lambda_n$ is finite for every $n \geq 0$.
The definition of a *covariant ${{\mathcal I}}$-$CW$-complex* is analogous.
\[def:classifying\_calc-space\] A model for *the classifying ${{\mathcal I}}$-space* $E{{\mathcal I}}$ is an ${{\mathcal I}}$-$CW$-complex $E{{\mathcal I}}$ such that $E{{\mathcal I}}(i)$ is contractible for all objects $i$.
The universal property of $E{{\mathcal I}}$ is that for any ${{\mathcal I}}$-$CW$-complex $X$ there is up to homotopy precisely one map of ${{\mathcal I}}$-spaces from $X$ to $E{{\mathcal I}}$. In particular two models for $E{{\mathcal I}}$ are ${{\mathcal I}}$-homotopy equivalent (see Davis–Lück [@Davis-Lueck(1998) Theorem 3.4]). A model for the usual *classifying space* $B{{\mathcal I}}$ is given by $E{{\mathcal I}}\otimes_{{{\mathcal I}}} {\{\bullet\}}$ (see [@Davis-Lueck(1998) Definition 3.10]), where ${\{\bullet\}}$ is the constant covariant ${{\mathcal I}}$-space with value the one point space and $\otimes_{{{\mathcal I}}}$ denotes the tensor product of a contravariant and a covariant ${{\mathcal I}}$-space as follows (see [@Davis-Lueck(1998) Definition 1.4]).
\[def:tensor\_product\_for\_I-spaces\] Let $X$ be a contravariant ${{\mathcal I}}$-space and $Y$ a covariant ${{\mathcal I}}$-space. The *tensor product of $X$ and $Y$* is $$X \otimes_{{\mathcal I}}Y = \Biggl( \coprod_{i \in {{\mathcal I}}} X(i) \times Y(i) \Biggr) / \sim$$ where $(X(\phi)(x),y)\sim(x,Y(\phi)y)$ for all morphisms $\phi:i \to j$ in ${{\mathcal I}}$ and points $x \in X(j)$ and $y \in Y(i)$.
We present some examples of classifying ${{\mathcal I}}$-spaces for various categories ${{\mathcal I}}$.
\[exa:finite\_model\_for\_I\_with\_terminal\_object\] If ${{\mathcal I}}$ has a terminal object $t$, then a finite model for the classifying ${{\mathcal I}}$-space $E{{\mathcal I}}$ is simply ${\operatorname{mor}}_{{\mathcal I}}(-,t)$.
\[exa:finite\_model\_for\_parallel\_arrows\] Let ${{\mathcal I}}=\{j \rightrightarrows k\}$ be the category consisting of two objects and a single pair of parallel arrows between them. All other morphisms are identity morphisms. We obtain a finite model $X$ for the classifying ${{\mathcal I}}$-space $E{{\mathcal I}}$ as follows. The ${{\mathcal I}}$-$CW$-space $X$ has a single ${{\mathcal I}}$-$0$-cell based at $k$ and a single ${{\mathcal I}}$-1-cell based at $j$. The gluing map ${\operatorname{mor}}_{{\mathcal I}}(-,j) \times S^0 \to
{\operatorname{mor}}_{{\mathcal I}}(-,k)$ is induced by the two parallel arrows $j
\rightrightarrows k$. Then $X(j)=D^1\simeq \ast$ and $X(k)=\ast$.
\[exa:finite\_model\_for\_pushout\] Let ${{\mathcal I}}=\{k \leftarrow j \to
\ell \}$ be the category with objects $j$, $k$ and $\ell$, and precisely one morphism from $j$ to $k$ and one morphism from $j$ to $\ell$. All other morphisms are identity morphisms. A finite model for $E{{\mathcal I}}$ is given by the ${{\mathcal I}}$-$CW$-complex with precisely two ${{\mathcal I}}$-$0$-cells ${\operatorname{mor}}_{{\mathcal I}}(?,k)$ and ${\operatorname{mor}}_{{\mathcal I}}(?,\ell)$ and precisely one ${{\mathcal I}}$-$1$-cell ${\operatorname{mor}}_{{\mathcal I}}(?,j) \times D^1$ whose attaching map ${\operatorname{mor}}_{{\mathcal I}}(?,j)
\times S^0 \to {\operatorname{mor}}_{{\mathcal I}}(?,k) \amalg{\operatorname{mor}}_{{\mathcal I}}(?,\ell)$ is the disjoint union of the canonical maps ${\operatorname{mor}}_{{\mathcal I}}(?,j) \to {\operatorname{mor}}_{{\mathcal I}}(?,k)$ and ${\operatorname{mor}}_{{\mathcal I}}(?,j)\to
{\operatorname{mor}}_{{\mathcal I}}(?,\ell)$. The value of this $1$-dimensional ${{\mathcal I}}$-$CW$-complex at the objects $k$ and $\ell$ is a point and at the object $j$ is $D^1$. Hence it is a finite model for $E{{\mathcal I}}$.
\[exa:finite\_model\_for\_q\_interior\] Let ${{\mathcal I}}$ be the category with objects the non-empty subsets of $[q]=\{0,1, \dots, q\}$ and a unique arrow $J \to K$ if and only if $K \subseteq J$. In other words ${{\mathcal I}}$ is the [*opposite*]{} of the poset of non-empty subsets of $[q]$. Then ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model $X$ for the classifying ${{\mathcal I}}$-space $E{{\mathcal I}}$ as follows. The functor $X\colon {{\mathcal I}}^{{\operatorname{op}}} \to {{\EuR}{SPACES}}$ assigns to $L$ the space $|\Delta[L]|$, which is the geometric realization of the simplicial set which maps $[m]$ to the set of weakly order preserving maps $[m] \to L$. The space $|\Delta[L]|$ is homeomorphic to the standard simplex with $\text{card}(L)$ vertices. The $n$-skeleton $X_n$ of $X$ sends each $L$ to the $n$-skeleton of $|\Delta[L]|$. The ${{\mathcal I}}$-cells of $X$ are attached globally in the following way. The 0-skeleton is $$X_0=\coprod_{J \subseteq [q], |J|=1}{\operatorname{mor}}_{{\mathcal I}}(-,J).$$ For $n\leq q$, we construct $X_n$ out of $X_{n-1}$ as the pushout $$\xymatrix{\coprod_J {\operatorname{mor}}_{{\mathcal I}}(-,J) \times |\partial \Delta[n]| \ar[r] \ar[d] & X_{n-1} \ar[d]
\\ \coprod_J {\operatorname{mor}}_{{\mathcal I}}(-,J) \times |\Delta[n]| \ar[r] & X_n.}$$ The disjoint unions are over all $J \subseteq [q]$ with $|J|=n+1$. The $J$-component of the gluing map is induced by the $(n-1)$-face inclusion $$\xymatrix{|\Delta[K]| \ar[r] & \partial |\Delta[J]| \cong \partial |\Delta[n]|}$$ for all $K \subseteq J$ with $|K|=n$. Clearly $X$ is a finite ${{\mathcal I}}$-$CW$-complex. For each object $L$ of ${{\mathcal I}}$, we have $X(L)=|\Delta[L]| \simeq \ast$, so that $X$ is a finite model for $E{{\mathcal I}}$.
\[exa:finite\_model\_for\_N\_and\_Z\] Infinite categories may also admit finite models. Let ${{\mathcal I}}=\widehat{{{\mathbb N}}}$ be the monoid of natural numbers ${{\mathbb N}}$ viewed as a one-object category. A finite model $X$ for the $\widehat{{{\mathbb N}}}$-classifying space has $X_0(*)={\operatorname{mor}}_{\widehat{{{\mathbb N}}}}(*,*)={{\mathbb N}}$ and $X_1(*)=[0,\infty)$. This model has a single $\widehat{{{\mathbb N}}}$-$0$-cell ${\operatorname{mor}}_{\widehat{{{\mathbb N}}}}(-,*)$ and a single $\widehat{{{\mathbb N}}}$-$1$-cell ${\operatorname{mor}}_{\widehat{{{\mathbb N}}}}(-,*)\times D^1$. The gluing map ${{\mathbb N}}\times S^0 \to {{\mathbb N}}$ sends $(n,-1)$ and $(n,1)$ to $n$ and $n+1$ respectively. Similarly, the group of integers ${{\mathbb Z}}$ viewed as a one object category admits a finite model $Y$ with one $\widehat{{{\mathbb Z}}}$-$0$-cell and one $\widehat{{{\mathbb Z}}}$-$1$-cell, so that $Y_0(*)={{\mathbb Z}}$ and $Y_1(*)={{\mathbb R}}$.
Suppose a category ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model for $E{{\mathcal I}}$. Then the cellular $R$-chains of a finite model provide a finite free resolution of the constant $R{{\mathcal I}}$-module $\underline{R}$, so ${{\mathcal I}}$ is of type (FF$_R$). If ${{\mathcal I}}$ is additionally directly finite and $R$ is Noetherian, then $\chi(B{{\mathcal I}};R)=\chi({{\mathcal I}};R)=\chi^{(2)}({{\mathcal I}})$ by Theorem \[the:coincidence\].
\[rem:Ebarcalc\] There exists a functorial construction $E^{{\operatorname{bar}}}{{\mathcal I}}$ of $E{{\mathcal I}}$ by a kind of bar construction. Namely, the contravariant functor $E^{{\operatorname{bar}}}{{\mathcal I}}\colon {{\mathcal I}}\to {{\EuR}{SPACES}}$ sends an object $i$ to the space $B^{{\operatorname{bar}}}(i \downarrow {{\mathcal I}})$, which is the geometric realization of the nerve of the category of objects under $i$ (see Davis–Lück [@Davis-Lueck(1998) page 230] and also Bousfield–Kan [@Bousfield-Kan(1972) page 327]). An equivalent definition of the bar construction in terms of the tensor product in Definition \[def:tensor\_product\_for\_I-spaces\] is $$\label{equ:bar_construction_as_tensor}
E^{{\operatorname{bar}}}{{\mathcal I}}=\{*\} \otimes_{{\mathcal I}}B^{{\operatorname{bar}}} (?\downarrow {{\mathcal I}}\downarrow ??),$$ from which we prove that $E^{{\operatorname{bar}}}{{\mathcal I}}$ is an ${{\mathcal I}}$-$CW$-complex. The ${{\mathcal I}}\times {{\mathcal I}}^{{\operatorname{op}}}$-space $B^{{\operatorname{bar}}} (?\downarrow {{\mathcal I}}\downarrow ??)$ is an ${{\mathcal I}}\times
{{\mathcal I}}^{{\operatorname{op}}}$-$CW$-complex (see [@Davis-Lueck(1998) page 228]). For each path $$i_0 \to i_1 \to i_2 \to \cdots \to i_n$$ of $n$-many non-identity morphisms in ${{\mathcal I}}$, $B^{{\operatorname{bar}}}
(?\downarrow {{\mathcal I}}\downarrow ??)$ has an $n$-cell based at $(i_0,i_n)$, that is a cell of the form ${\operatorname{mor}}_{{\mathcal I}}(?,i_0) \times
{\operatorname{mor}}_{{\mathcal I}}(i_n,??) \times D^n$. By [@Davis-Lueck(1998) Lemma 3.19 (2)], the tensor product $E^{{\operatorname{bar}}}{{\mathcal I}}$ in is an ${{\mathcal I}}$-$CW$-complex: an $(m+n)$-cell based at $i$ is an $n$-cell of $B^{{\operatorname{bar}}}
(?\downarrow {{\mathcal I}}\downarrow ??)$ based at $(i,j)$ and an $m$-cell of the $CW$-complex $*(j)$ (see [@Davis-Lueck(1998) page 229]). More explicitly, for each path of $n$-many non-identity morphisms $$\label{equ:sequence_of_I_morphisms}
i_0 \to i_1 \to i_2 \to \cdots \to i_n$$ the ${{\mathcal I}}$-$CW$-complex $E^{{\operatorname{bar}}}{{\mathcal I}}$ has an $n$-cell based at $i_0$.
Though the bar construction is in general not a finite ${{\mathcal I}}$-$CW$-complex, it is in certain cases. For example, if ${{\mathcal I}}$ has only finitely many morphisms, no nontrivial isomorphisms, and no nontrivial endomorphisms, then there are only finitely many paths as in , and hence only finitely many ${{\mathcal I}}$-cells in $E^{{\operatorname{bar}}}{{\mathcal I}}$.
The bar construction is also compatible with induction. Given a functor $\alpha \colon {{\mathcal I}}\to {{\mathcal D}}$, we obtain a map of ${{\mathcal D}}$-spaces $$E^{{\operatorname{bar}}}\alpha \colon \alpha_*E^{{\operatorname{bar}}}{{\mathcal I}}\to E^{{\operatorname{bar}}}{{\mathcal D}},$$ where $\alpha_*$ denotes induction with the functor $\alpha$ (see [@Davis-Lueck(1998) Definition 1.8]). If $T \colon \alpha
\to \beta$ is a natural transformation of functors ${{\mathcal I}}\to
{{\mathcal D}}$, we obtain for any ${{\mathcal I}}$-space $X$ a natural transformation $T_* \colon \alpha_*X \to \beta_*X$ which comes from the map of ${{\mathcal I}}$-${{\mathcal D}}$-spaces ${\operatorname{mor}}_{{{\mathcal D}}}(??,\alpha(?)) \to
{\operatorname{mor}}_{{{\mathcal D}}}(??,\beta(?))$ sending $g \colon {??} \to \alpha(?)$ to $T(?) \circ g \colon {??} \to \beta(?)$.
\[lem:finite\_models\_and\_equivalences\_of\_categories\] Suppose ${{\mathcal I}}$ and ${{\mathcal J}}$ are equivalent categories. Then ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model for $E{{\mathcal I}}$ if and only if ${{\mathcal J}}$ admits a finite ${{\mathcal J}}$-$CW$-model for $E{{\mathcal J}}$. More precisely, if $F\colon {{\mathcal I}}\to {{\mathcal J}}$ is an equivalence of categories and $Y$ is a finite ${{\mathcal J}}$-$CW$-model for $E{{\mathcal J}}$, then the restriction ${\operatorname{res}}_F Y$ is a finite ${{\mathcal I}}$-$CW$-model for $E{{\mathcal I}}$.
For any functor $F\colon {{\mathcal I}}\to {{\mathcal J}}$, we have an adjunction $${\operatorname{ind}}_F\colon {{\mathcal I}}\text{-}{{\EuR}{SPACES}}\rightleftarrows {{\mathcal J}}\text{-}{{\EuR}{SPACES}}\colon {\operatorname{res}}_F$$ defined by $${\operatorname{ind}}_F(X):=X(?)\otimes_{{\mathcal I}}{\operatorname{mor}}_{{\mathcal J}}\bigl(??,F(?)\bigr) \;\;\;\;\; {\operatorname{res}}_F(Y):=Y \circ F(?).$$ The ${{\mathcal I}}$-space ${\operatorname{res}}_F(Y)$ is naturally homeomorphic to $Y(?)\otimes_{{\mathcal J}}{\operatorname{mor}}_{{\mathcal J}}\bigl(F(??),?\bigr)$. But since we are assuming $F$ is an equivalence of categories, it is a left adjoint in an adjoint equivalence $(F,G)$, and we have natural homeomorphisms of ${{\mathcal I}}$-spaces $$\begin{aligned}
{\operatorname{res}}_F(Y) & \cong & Y(?)\otimes_{{\mathcal J}}{\operatorname{mor}}_{{\mathcal J}}\bigl(F(??),?\bigr) \\
& \cong & Y(?)\otimes_{{\mathcal J}}{\operatorname{mor}}_{{\mathcal J}}\bigl(??,G(?)\bigr) \\
& \cong & {\operatorname{ind}}_G(Y).\end{aligned}$$ Since ${\operatorname{ind}}_G$ is a left adjoint, so is ${\operatorname{res}}_F$, and ${\operatorname{res}}_F$ therefore preserves pushouts. Note also $${\operatorname{res}}_F {\operatorname{mor}}_{{\mathcal J}}(?,j) = {\operatorname{mor}}_{{\mathcal J}}\bigl(F(?),j\bigr) \cong {\operatorname{mor}}_{{\mathcal I}}\bigl(?,G(j)\bigr).$$
If $Y$ is a finite ${{\mathcal J}}$-$CW$-model for $E{{\mathcal J}}$ with $n$-skeleton $${\begin{CD}
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal J}}}(-,j_\lambda) \times S^{n-1} @>>> Y_{n-1}\\
@V{}VV @V{}VV\\
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal J}}}(-,j_\lambda) \times D^n @>>> Y_n,
\end{CD}
}$$ then $X:={\operatorname{res}}_F Y$ is a finite ${{\mathcal I}}$-$CW$-complex with $n$-skeleton $${\begin{CD}
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal I}}}\bigl(-,G(j_\lambda)\bigr) \times S^{n-1} @>>> X_{n-1}\\
@V{}VV @V{}VV\\
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal I}}}\bigl(-,G(j_\lambda)\bigr) \times D^n @>>> X_n.
\end{CD}
}$$ Clearly, ${\operatorname{res}}_F Y$ is contractible at each object $i$, since ${\operatorname{res}}_F Y (i)=Y(F(i)) \simeq \ast$.
Homotopy Colimits of Categories {#sec:homotopy_colimits_of_categories}
===============================
\[def:homotopy\_colimit\] Let ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ be a covariant functor from some (small) index category ${{\mathcal I}}$ to the category of small categories. Its *homotopy colimit* $${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$$ is the following category. Objects are pairs $(i,c)$, where $i \in {\operatorname{ob}}({{\mathcal I}})$ and $c
\in {\operatorname{ob}}\bigl({{\mathcal C}}(i)\bigr)$. A morphism from $(i,c)$ to $(j,d)$ is a pair $(u,f)$, where $u \colon i \to j$ is a morphism in ${{\mathcal I}}$ and $f \colon {{\mathcal C}}(u)(c) \to d$ is a morphism in ${{\mathcal C}}(j)$. The composition of the morphisms $(u,f) \colon (i,c) \to (j,d)$ and $(v,g) \colon (j,d) \to (k,e)$ is the morphism $$(v,g) \circ (u,f) = (v \circ u, g \circ {{\mathcal C}}(v)(f)) \colon (i,c) \to (k,e).$$ The identity of $(i,c)$ is given by $({\operatorname{id}}_i,{\operatorname{id}}_c)$.
This homotopy colimit construction for functors is often called the *Grothendieck construction* or the *category of elements*.
In which sense is ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ a homotopy colimit? First, recall from [@Illusie(1972)] that the nerve functor induces an equivalence of categories $\text{Ho}\; {{\EuR}{CAT}}\to \text{Ho}\; {{\EuR}{SSET}}$, where $\text{Ho}\; {{\EuR}{CAT}}$ denotes the localization of ${{\EuR}{CAT}}$ with respect to nerve weak equivalences and $\text{Ho}\; {{\EuR}{SSET}}$ denotes the localization of ${{\EuR}{SSET}}$ with respect to the usual weak equivalences. In [@Thomason(1979)], Thomason proved that ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ in ${{\EuR}{CAT}}$ corresponds to the Bousfield–Kan construction in ${{\EuR}{SSET}}$ under this equivalence of categories. Consequently, ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ has a universal property in the form of a bijection $$\label{equ:hocolim_explanation}
\text{Ho}\; {{\EuR}{CAT}}({\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}, \Gamma) \cong \text{Ho} \;{{\EuR}{CAT}}^{{\mathcal I}}({{\mathcal C}}, \underline{\Gamma}),$$ for any category $\Gamma$. Here $\underline{\Gamma}$ indicates the ${{\mathcal I}}$-diagram that is constant $\Gamma$. In [@Thomason(1980)], Thomason proved that ${{\EuR}{CAT}}$ admits a cofibrantly generated model structure in which the weak equivalences are the nerve weak equivalences, so that the associated projective model structure on ${{\EuR}{CAT}}^{{\mathcal I}}$ exists. The model-theoretic construction of a homotopy colimit of the ${{\mathcal I}}$-diagram ${{\mathcal C}}$ in ${{\EuR}{CAT}}$ as a colimit of a cofibrant replacement of ${{\mathcal C}}$ in the projective model structure therefore works. This model-theoretic construction also has the universal property in , so is isomorphic to ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ in $\text{Ho}\; {{\EuR}{CAT}}$, i.e. weakly equivalent to ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ in ${{\EuR}{CAT}}$. A direct proof that ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ satisfies the universal property is in Grothendieck’s letter [@Grothendieck(1983)], see the article of Maltsiniotis[^1] [@Maltsiniotis(2005) Section 3.1].
\[rem:homotopy\_colimit\_of\_pseudo\_functor\] If ${{\mathcal C}}$ is merely a pseudo functor, then it of course still has a homotopy colimit. A *pseudo functor* ${{\mathcal C}}\colon {{\mathcal I}}\to
{{\EuR}{CAT}}$ is like an ordinary functor, but only preserves composition and unit up to specified coherent natural isomorphisms ${{\mathcal C}}_{v,u}\colon {{\mathcal C}}(v) \circ {{\mathcal C}}(u) \Rightarrow {{\mathcal C}}(v \circ
u)$ and ${{\mathcal C}}_i\colon 1_{{{\mathcal C}}(i)} \Rightarrow {{\mathcal C}}({\operatorname{id}}_i)$. Moreover, ${{\mathcal C}}_{v,u}$ is required to be natural in $v$ and $u$. The objects and morphisms of the *homotopy colimit* ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ are defined as in the strict case of Definition \[def:homotopy\_colimit\]. The composition in ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ is defined by the modified rule $$(v,g) \circ (u,f) = (v \circ u, g \circ ({{\mathcal C}}(v)(f))\circ {{\mathcal C}}_{v,u}^{-1}(c))$$ while the identity of the object $(i,c)$ is given by $$({\operatorname{id}}_i,{{\mathcal C}}_i^{-1}(c)).$$ The homotopy colimit of a pseudo functor ${{\mathcal C}}\colon {{\mathcal I}}\to
{{\EuR}{CAT}}$ is an ordinary 1-category with strictly associative and strictly unital composition.
\[rem:homotopy\_colimit\_is\_2-functor\] For a fixed category ${{\mathcal I}}$, the homotopy colimit construction ${\operatorname{hocolim}}_{{{\mathcal I}}}-$ is a strict 2-functor from the strict 2-category of pseudo functors ${{\mathcal I}}\to {{\EuR}{CAT}}$, pseudo natural transformations, and modifications into the strict 2-category ${{\EuR}{CAT}}$.
\[exa:hocolim\_constant\] If ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ is a constant functor, say constantly a category also called ${{\mathcal C}}$, then ${\operatorname{hocolim}}_{{{\mathcal I}}}
{{\mathcal C}}= {{\mathcal I}}\times {{\mathcal C}}$.
\[exa:homotopy\_hocolimit\_and\_when\_I\_has\_terminal\_object\] Suppose ${{\mathcal I}}$ has a terminal object $t$ and ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ is a strict covariant functor. Then ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ is homotopy equivalent to ${{\mathcal C}}(t)$ as follows. This is analogous to the familiar fact that ${{\mathcal C}}(t)$ is a colimit of ${{\mathcal C}}$. The components of the universal cocone $$\label{equ:cocone_for_I_with_terminal_object}
\pi \colon {{\mathcal C}}\Rightarrow \Delta_{{{\mathcal C}}(t)}$$ are ${{\mathcal C}}(i \to t)$. Applying ${\operatorname{hocolim}}_{{\mathcal I}}-$ to and composing with the projection gives us a functor $F$ $$\xymatrix@R=1pc@C=4pc{{\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}\ar[r]_-{{\operatorname{hocolim}}_{{\mathcal I}}\pi} \ar@/^1.5pc/[rr]^F & {{\mathcal I}}\times {{\mathcal C}}(t)
\ar[r]_-{{\operatorname{pr}}_{{{\mathcal C}}(t)}} & {{\mathcal C}}(t) \\ (i,c) \ar@{|->}[rr] & & {{\mathcal C}}(i \to
t)(c).}$$ The functor $G \colon {{\mathcal C}}(t) \to {\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$, $G(c)=(t,c)$ is a homotopy inverse, since $F\circ G ={\operatorname{id}}_{{{\mathcal C}}(t)}$ and we have a natural transformation ${\operatorname{id}}_{{\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}} \Rightarrow G\circ F$ with components $$\xymatrix{(i \to t, {\operatorname{id}}_{{{\mathcal C}}(i \to t)}) \colon (i,c) \ar[r] & (t,{{\mathcal C}}(i \to t) c).}$$
Let ${{\mathcal H}}$ denote the homotopy colimit of the ${{\mathcal I}}$-diagram of categories ${{\mathcal C}}$. We now construct an ${{\mathcal I}}$-diagram of ${{\mathcal H}}$-spaces $E^{{\mathcal H}}$ with the property that its tensor product with $E {{\mathcal I}}$ is ${{\mathcal H}}$-homotopy equivalent to a classifying ${{\mathcal H}}$-space for ${{\mathcal H}}$. This ${{\mathcal I}}$-diagram of ${{\mathcal H}}$-spaces $E^{{\mathcal H}}$ will play an important role in the inductive proof of the Homotopy Colimit Formula Theorem \[the:homotopy\_colimit\_formula\].
Let ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ be a strict covariant functor, and abbreviate ${{\mathcal H}}= {\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$. Define a functor $$\begin{aligned}
& E^{{{\mathcal H}}} \colon {{\mathcal I}}\to {{\mathcal H}}\text{-}{{\EuR}{SPACES}}&
\label{Ecalh}\end{aligned}$$ as follows. Given an object $i \in {{\mathcal I}}$, we have the functor $$\begin{aligned}
& \alpha(i) \colon {{\mathcal C}}(i) \to {{\mathcal H}}&
\label{alpha(i)}\end{aligned}$$ sending an object $c$ to the object $(i,c)$ and a morphism $f \colon
c \to d$ to the morphism $({\operatorname{id}}_i,f)$. We define $$E^{{{\mathcal H}}}(i) = \alpha(i)_* E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr).$$ Consider a morphism $u \colon i \to j$ in ${{\mathcal I}}$. It induces a natural transformation $T(u) \colon \alpha(i) \to \alpha(j) \circ
{{\mathcal C}}(u)$ from the functor $\alpha(i) \colon {{\mathcal C}}(i) \to {{\mathcal H}}$ to the functor $\alpha(j) \circ {{\mathcal C}}(u) \colon {{\mathcal C}}(i) \to {{\mathcal H}}$ by assigning to an object $c$ in ${{\mathcal C}}(i)$ the morphism $$(u,{\operatorname{id}}_{{{\mathcal C}}(u)(c)}) \colon \alpha(i)(c) = (i,c)
\to \alpha(j)\circ {{\mathcal C}}(u)(c) = (j,{{\mathcal C}}(u)(c)).$$ From Remark \[rem:Ebarcalc\] we obtain a map of ${{\mathcal H}}$-spaces $$T(u)_* \colon\alpha(i)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr)
\to \alpha(j)_*{{\mathcal C}}(u)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr)$$ and a map of ${{\mathcal C}}(j)$-spaces $$E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(u)\bigr) \colon {{\mathcal C}}(u)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr) \to E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(j)\bigr).$$ Finally, for the morphism $u$ in ${{\mathcal I}}$, we define $E^{{{\mathcal H}}}(u)
\colon E^{{{\mathcal H}}}(i) \to E^{{{\mathcal H}}}(j)$ by the composite $$\alpha(i)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr)
\xrightarrow{T(u)_*} \alpha(j)_*{{\mathcal C}}(u)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr)
\xrightarrow{\alpha(j)_*(E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(u)\bigr)}
\alpha(j)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(j)\bigr).$$
Define the homotopy colimit of the covariant functor $E^{{{\mathcal H}}}$ of to be the contravariant ${{\mathcal H}}$-space $$\begin{aligned}
{\operatorname{hocolim}}_{{{\mathcal I}}} E^{{{\mathcal H}}} & := & (i,c) \mapsto E{{\mathcal I}}\otimes_{{{\mathcal I}}} \left(E^{{{\mathcal H}}}(i,c)\right).
\label{hocolim_cali_ebarcon}\end{aligned}$$
\[lem:model\_for\_E(hocolim\_cali\_calc)\] Consider any model $E{{\mathcal I}}$ for the classifying ${{\mathcal I}}$-space of the category ${{\mathcal I}}$. Then the contravariant ${{\mathcal H}}$-space $E{{\mathcal I}}\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ of is ${{\mathcal H}}$-homotopy equivalent to the classifying ${{\mathcal H}}$-space $E{{\mathcal H}}$ of the category ${{\mathcal H}}:={\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$.
We first show that for any object $(i,c)$ in ${{\mathcal H}}$ the space $E{{\mathcal I}}\otimes_{{{\mathcal I}}} \left( E^{{{\mathcal H}}}(i,c) \right)$ is contractible. The covariant functor $ E^{{{\mathcal H}}}(i,c) \colon {{\mathcal I}}\to {{\EuR}{SPACES}}$ sends an object $j$ to $$\begin{aligned}
\alpha(j)_*\left(E^{{\operatorname{bar}}} {{\mathcal C}}(j) \right)(i,c) & = & \alpha(j)_*\left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right) (?) \otimes_{{{\mathcal H}}} {\operatorname{mor}}_{{{\mathcal H}}}\bigl((i,c),?)\bigr)
\\
& = &
\left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right)(?) \otimes_{{{\mathcal C}}(j)} {\operatorname{mor}}_{{{\mathcal H}}}\bigl((i,c),(j,?)\bigr)
\\
& = &
\left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right)(?) \otimes_{{{\mathcal C}}(j)}
\left(\coprod_{u \in {\operatorname{mor}}_{{{\mathcal I}}}(i,j)} {\operatorname{mor}}_{{{\mathcal C}}(j)}\bigl({{\mathcal C}}(u)(c),?\bigr)\right)
\\
& = &
\coprod_{u \in {\operatorname{mor}}_{{{\mathcal I}}}(i,j)}
\left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right)(?) \otimes_{{{\mathcal C}}(j)}{\operatorname{mor}}_{{{\mathcal C}}(j)}\bigl({{\mathcal C}}(u)(c),?\bigr)
\\
& = &
\coprod_{u \in {\operatorname{mor}}_{{{\mathcal I}}}(i,j)} \left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right)\bigl({{\mathcal C}}(u)(c)\bigr).\end{aligned}$$ Since $\left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right)\bigl({{\mathcal C}}(u)(c)\bigr)$ is contractible, the projection $$\coprod_{u \in {\operatorname{mor}}_{{{\mathcal I}}}(i,j)} \left(E^{{\operatorname{bar}}}{{\mathcal C}}(j)\right)\bigl({{\mathcal C}}(u)(c)\bigr)
\to {\operatorname{mor}}_{{{\mathcal I}}}(i,j)$$ is a homotopy equivalence. Hence the collection of these projections for $j \in {\operatorname{ob}}({{\mathcal I}})$ induces a map of ${{\mathcal I}}$-spaces $${\operatorname{pr}}\colon E^{{{\mathcal H}}}(i,c) \to {\operatorname{mor}}_{{{\mathcal I}}}(i,?)$$ whose evaluation at each object $j$ in ${\operatorname{ob}}({{\mathcal I}})$ is a homotopy equivalence. We conclude from Davis–Lück [@Davis-Lueck(1998) Theorem 3.11] that $$E{{\mathcal I}}\otimes_{{{\mathcal I}}} {\operatorname{pr}}\colon E{{\mathcal I}}\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}(i,c)
\xrightarrow{\simeq} E{{\mathcal I}}\otimes_{{{\mathcal I}}} {\operatorname{mor}}_{{{\mathcal I}}}(i,?).$$ is a homotopy equivalence. Since $E{{\mathcal I}}\otimes_{{{\mathcal I}}}
{\operatorname{mor}}_{{{\mathcal I}}}(i,?) = E{{\mathcal I}}(i)$ is contractible, this implies that for any object $(i,c)$ in ${{\mathcal H}}$ the space $E{{\mathcal I}}\otimes_{{{\mathcal I}}}
\left( E^{{{\mathcal H}}}(i,c) \right)$ is contractible, as we initially claimed.
It remains to show that $E{{\mathcal I}}\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ has the ${{\mathcal H}}$-homotopy type of an ${{\mathcal H}}$-$CW$-complex. It is actually an ${{\mathcal H}}$-$CW$-complex. The following argument, that $E{{\mathcal I}}\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ has the homotopy type of an ${{\mathcal H}}$-$CW$-complex, will be used again later.[^2]
We have a filtration of $E{{\mathcal I}}$ $$\emptyset = E{{\mathcal I}}_{-1} \subseteq E{{\mathcal I}}_{0} \subseteq E{{\mathcal I}}_{1}
\subseteq \ldots \subseteq E{{\mathcal I}}_{n} \subseteq \ldots \subseteq
E{{\mathcal I}}= \bigcup_{n \geq 0} E{{\mathcal I}}_{n}$$ such that $$E{{\mathcal I}}={\operatorname{colim}}_{n \to \infty} E{{\mathcal I}}_{n}$$ and for every $n \geq 0$ there exists a pushout of ${{\mathcal I}}$-spaces $$\begin{aligned}
&
{\begin{CD}
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal I}}}(-,i_\lambda) \times S^{n-1} @>>> E{{\mathcal I}}_{n-1}\\
@V{}VV @V{}VV\\
\coprod_{\lambda \in \Lambda_n} {\operatorname{mor}}_{{{\mathcal I}}}(-,i_\lambda) \times D^n @>>> E{{\mathcal I}}_n.
\end{CD}
}
&
\label{cali_CW_decomposition_of_Ecali}\end{aligned}$$ Since $- \otimes_{{{\mathcal I}}} Z$ has a right adjoint [@Davis-Lueck(1998) Lemma 1.9] we get $$E{{\mathcal I}}\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}
= {\operatorname{colim}}_{n \to \infty} E{{\mathcal I}}_n \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$$ as a colimit of ${{\mathcal H}}$-spaces. After an application of $-\otimes_{{\mathcal I}}E^{{{\mathcal H}}}$ to , we obtain pushouts of ${{\mathcal H}}$-spaces $$\begin{aligned}
&
{\begin{CD}
\coprod_{\lambda \in \Lambda_n} E^{{{\mathcal H}}}(i_\lambda) \times S^{n-1} @>f_{n-1}>> E{{\mathcal I}}_{n-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}\\
@V{}VV @V{}VV\\
\coprod_{\lambda \in \Lambda_n} E^{{{\mathcal H}}}(i_\lambda) \times D^n @>>> E{{\mathcal I}}_{n} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}
\end{CD}
}
&
\label{pushout_of_calh-spaces}\end{aligned}$$ where the left vertical arrow and hence the right vertical arrow are cofibrations of ${{\mathcal H}}$-spaces. By induction we may assume that $E{{\mathcal I}}_{n-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ has the homotopy type of an ${{\mathcal H}}$-$CW$-complex. Since the vertical maps are cofibrations, by replacing it with a homotopy equivalent ${{\mathcal H}}$-$CW$-complex we do not change the homotopy type of the pushout (the usual proof for spaces goes through for ${{\mathcal H}}$-spaces). Hence we may assume that $E{{\mathcal I}}_{n-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ is a ${{\mathcal H}}$-$CW$-complex. We may also assume that $f_{n-1}$ is cellular: since the vertical maps are cofibrations, by replacing $f_{n-1}$ by a homotopic cellular map, which exists by Davis–Lück [@Davis-Lueck(1998) cf. Theorem 3.7], we also do not change the homotopy type of the pushout. See Selick [@Selick(1997) Theorem 7.1.8] for a proof of this statement for spaces which translates verbatim to the setting of ${{\mathcal H}}$-spaces. If $f_{n-1}$ is cellular, diagram is a cellular pushout. Hence we completed the induction step, showing that $E{{\mathcal I}}_{n}
\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ has the homotopy type of an ${{\mathcal H}}$-$CW$-complex.
It remains to show that $E{{\mathcal I}}\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ has the homotopy type of a ${{\mathcal H}}$-$CW$-complex: choose ${{\mathcal H}}$-$CW$-complexes $Z_n$ and ${{\mathcal H}}$-homotopy equivalences $g_n:Z_n\to E{{\mathcal I}}_{n} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$. By iteratively replacing $Z_n$ by the mapping cylinder of $$Z_{n-1}\xrightarrow{g_{n-1}} E{{\mathcal I}}_{n-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}\to E{{\mathcal I}}_{n} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}\xrightarrow{\bar g_n}Z_n,$$ where $\bar g_n$ is a homotopy inverse of $g_n$, one finds a new sequence of homotopy equivalences $g_n':Z_n\to E{{\mathcal I}}_{n} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ (with the modified ${{\mathcal H}}$-$CW$-complexes $Z_n$) such that $g_n'\vert_{Z_{n-1}}=g_{n-1}'$.
Homotopy Colimit Formula for Finiteness Obstructions and Euler Characteristics {#sec:Homotopy_colimit_formula}
==============================================================================
In this section we prove the main theorem of this paper: the Homotopy Colimit Formula. It expresses the finiteness obstruction, the Euler characteristic, and the $L^2$-Euler characteristic of the homotopy colimit of a diagram in ${{\EuR}{CAT}}$ in terms of the respective invariants for the diagram entries at the base objects for cells in a finite model for the ${{\mathcal I}}$-classifying space of ${{\mathcal I}}$. Analogous formulas for the functorial counterparts of the Euler characteristic and $L^2$-Euler characteristic are included. The Homotopy Colimit Formula is initially stated and proved for strict functors ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$, but we prove that it also holds for pseudo functors ${{\mathcal D}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ in Corollary \[cor:homotopy\_colimit\_formula\_for\_pseudo\_functors\]. The full generality of pseudo functors is needed for the applications to complexes of groups in Section \[sec:complexes\_of\_groups\].
Homotopy Colimit Formula
------------------------
\[the:homotopy\_colimit\_formula\] Let ${{\mathcal I}}$ be a small category such that there exists a finite ${{\mathcal I}}$-$CW$-model for its classifying ${{\mathcal I}}$-space. Fix such a finite ${{\mathcal I}}$-$CW$-model $E{{\mathcal I}}$. Denote by $\Lambda_n$ the finite set of $n$-cells $\lambda =
{\operatorname{mor}}_{{\mathcal I}}(?,i_\lambda) \times D^n$ of $E{{\mathcal I}}$. Let ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ be a covariant functor. Abbreviate ${{\mathcal H}}= {\operatorname{hocolim}}_{{{\mathcal I}}}
{{\mathcal C}}$. Then:
1. \[the:homotopy\_colimit\_formula:directly\_finite\] If ${{\mathcal I}}$ is directly finite, and ${{\mathcal C}}(i)$ is directly finite for every object $i \in
{\operatorname{ob}}({{\mathcal I}})$, then the category ${{\mathcal H}}$ is directly finite;
2. \[the:homotopy\_colimit\_formula:(EI)\] If ${{\mathcal I}}$ is an EI-category, ${{\mathcal C}}(i)$ is an EI-category for every object $i \in
{\operatorname{ob}}({{\mathcal I}})$, and for every automorphism $u \colon i \xrightarrow{\cong} i$ the map ${\operatorname{iso}}({{\mathcal C}}(i)) \to {\operatorname{iso}}({{\mathcal C}}(i)), \; \overline{x} \mapsto
\overline{{{\mathcal C}}(u)(x)}$ is the identity, then the category ${{\mathcal H}}$ is an EI-category;
3. \[the:homotopy\_colimit\_formula:(FP)\] If for every object $i$ the category ${{\mathcal C}}(i)$ is of type (FP$_R$), then the category ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ is of type (FP$_R$);
4. \[the:homotopy\_colimit\_formula:(FF)\] If for every object $i$ the category ${{\mathcal C}}(i)$ is of type (FF$_R$), then the category ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ is of type (FF$_R$);
5. \[the:homotopy\_colimit\_formula:o\] If for every object $i$ the category ${{\mathcal C}}(i)$ is of type (FP$_R$), then we obtain for the finiteness obstruction $$o({{\mathcal H}};R) = \sum_{n \geq 0} (-1)^n \cdot
\sum_{\lambda \in \Lambda_n} \alpha(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R)),$$ where $\alpha(i_\lambda)_* \colon K_0(R{{\mathcal C}}(i_\lambda)) \to K_0(R{{\mathcal H}})$ is the homomorphism induced by the canonical functor $\alpha(i_\lambda) \colon {{\mathcal C}}(i_\lambda) \to {{\mathcal H}}$ defined in ;
6. \[the:homotopy\_colimit\_formula:chi\] Suppose that ${{\mathcal I}}$ is directly finite and ${{\mathcal C}}(i)$ is directly finite for every object $i \in
{\operatorname{ob}}({{\mathcal I}})$. If for every object $i$ the category ${{\mathcal C}}(i)$ is additionally of type (FP$_R$) then we obtain for the functorial Euler characteristic $$\chi_f({{\mathcal H}};R) =
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \alpha(i_\lambda)_*(\chi_f({{\mathcal C}}(i_\lambda);R)),$$ where $\alpha(i_\lambda)_* \colon U({{\mathcal C}}(i_\lambda)) \to U({{\mathcal H}})$ is the homomorphism induced by the canonical functor $\alpha(i_\lambda) \colon {{\mathcal C}}(i_\lambda) \to {{\mathcal H}}$ defined in . Summing up, we also have $$\chi\bigl({{\mathcal H}};R\bigr) = \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \chi({{\mathcal C}}(i_\lambda);R).$$ If $R$ is Noetherian, in addition to the direct finiteness and (FP$_R$) hypotheses, we obtain for the Euler characteristics of the classifying spaces $$\chi\bigl(B{{\mathcal H}};R\bigr) = \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \chi(B{{\mathcal C}}(i_\lambda);R);$$
7. \[the:homotopy\_colimit\_formula:chi(2)\] Suppose that ${{\mathcal I}}$ is directly finite and ${{\mathcal C}}(i)$ is directly finite for every object $i \in {\operatorname{ob}}({{\mathcal I}})$. If for every object $i$ the category ${{\mathcal C}}(i)$ is additionally of type ($L^2$), then ${{\mathcal H}}$ is of type ($L^2$) and we obtain for the functorial $L^2$-Euler characteristic $$\chi_f^{(2)}({{\mathcal H}}) =
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
\alpha(i_\lambda)_*\bigl(\chi_f^{(2)}({{\mathcal C}}(i_\lambda))\bigr),$$ where $\alpha(i_\lambda)_*
\colon U^{(1)}({{\mathcal C}}(i_\lambda)) \to U^{(1)}({{\mathcal H}})$ is the homomorphism induced by the canonical functor $\alpha(i_\lambda) \colon {{\mathcal C}}(i_\lambda) \to {{\mathcal H}}$ defined in , and we obtain for the $L^2$-Euler characteristic $$\chi^{(2)}({{\mathcal H}}) =
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \chi^{(2)}({{\mathcal C}}(i_\lambda)).$$
\[the:homotopy\_colimit\_formula:directly\_finite\] Consider morphisms $(u,f) \colon(i,c) \to (j,d)$ and $(v,g) \colon(j,d) \to
(i,c)$ in ${{\mathcal H}}$ with $(v,g) \circ (u,f) = {\operatorname{id}}_{(i,c)}$. This implies $vu = {\operatorname{id}}_i$ and $g \circ {{\mathcal C}}(v)(f) = {\operatorname{id}}_c$. Since ${{\mathcal I}}$ and ${{\mathcal C}}(i)$ are by assumption directly finite, we conclude $uv = {\operatorname{id}}_j$ and ${{\mathcal C}}(v)(f) \circ g =
{\operatorname{id}}_{{{\mathcal C}}(v)(d)}$. Hence $$\begin{gathered}
(u,f) \circ (v,g) = \bigl(uv,f \circ {{\mathcal C}}(u)(g)\bigr) = \bigl(uv,
{{\mathcal C}}(uv)(f) \circ {{\mathcal C}}(u)(g)\bigr) =
\bigl(uv,{{\mathcal C}}(u)({{\mathcal C}}(v)(f) \circ g)\bigr)
\\
= \bigl(uv,{{\mathcal C}}(u)({\operatorname{id}}_{{{\mathcal C}}(v)(d)})\bigr) =
\bigl({\operatorname{id}}_j,{\operatorname{id}}_{{{\mathcal C}}(u)\bigl({{\mathcal C}}(v)(d)\bigr)}\bigr) =({\operatorname{id}}_j,{\operatorname{id}}_d).
\end{gathered}$$\[the:homotopy\_colimit\_formula:(EI)\] Consider an endomorphism $(u,f) \colon (i,c) \to (i,c)$ in ${{\mathcal H}}$. Since ${{\mathcal I}}$ is an EI-category, $u\colon i \to i$ is an automorphism. Since $\overline{{{\mathcal C}}(u)(c)} = \overline{c}$ by assumption, we can choose an isomorphism $g \colon c
\xrightarrow{\cong} {{\mathcal C}}(u)(c)$. Hence $fg$ is an endomorphism in ${{\mathcal C}}(i)$. Since ${{\mathcal C}}(i)$ is an EI-category, and $g$ is an isomorphism, $f$ is also an isomorphism. Since $u$ and $f$ are isomorphisms, $(u,f)$ is an isomorphism.\
\[the:homotopy\_colimit\_formula:(FP)\] and \[the:homotopy\_colimit\_formula:o\]. We say that an $R{{\mathcal H}}$-chain complex $C_*$ is of type (FP$_R$) if it admits a *finite projective approximation*, i.e., there is a finite length chain complex $P_*$ of finitely generated, projective $R{{\mathcal H}}$-modules together with an $R{{\mathcal H}}$-chain map $f_*\colon P_* \to C_*$ such that $H_n(f_*(i,c))$ is bijective for all $n \geq 0$ and $(i,c) \in {\operatorname{ob}}({{\mathcal H}})$. If $C_*$ is of type (FP$_R$), define its finiteness obstruction $$o(C_*) := \sum_{n \geq 0} (-1)^n \cdot [P_n] \quad \in K_0(R{{\mathcal H}})$$ for any choice $P_*$ of finite projective approximation. This is independent of the choice of $P_*$ and the basic properties of it were studied by Lück [@Lueck(1989) Chapter 11]. If $0[\underline{R}]$ is the $R{{\mathcal H}}$-chain complex concentrated in dimension zero and given there by the constant $R{{\mathcal H}}$-module $\underline{R}$, then ${{\mathcal H}}$ is of type (FP$_R$) if and only if $0[\underline{R}]$ is of type (FP$_R$) and in this case $$o({{\mathcal H}};R) = o(0[\underline{R}]) \in K_0(R{{\mathcal H}}).$$
Consider a finite ${{\mathcal I}}$-$CW$-complex $X$. We want to show by induction over the dimension of $X$ that the $R{{\mathcal H}}$-chain complex $C_*(X \otimes_{{\mathcal I}}E^{{{\mathcal H}}})$ is of type (FP$_R$) and satisfies $$o\bigl(C_*(X \otimes_{{\mathcal I}}E^{{{\mathcal H}}})\bigr)
= \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
\alpha(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R)),$$ where $\Lambda_n$ denotes the set of $n$-cells of $X$ and $i_\lambda$ is the object at which the $n$-cell $\lambda = {\operatorname{mor}}_{{\mathcal I}}(?,i_\lambda) \times D^n$ of $X$ is based.
The induction beginning, where $X$ is the empty set, is obviously true. The induction step is done as follows. Let $d$ be the dimension of $X$. Then $X_d$ is obtained from $X_{d-1}$ by a pushout of ${{\mathcal I}}$-spaces $${\xymatrix
{\coprod_{\lambda \in \Lambda_d} {\operatorname{mor}}_{{{\mathcal C}}}(-,i_\lambda) \times
S^{d-1} \ar[r]^-{} \ar[d]_{} &
X_{d-1} \ar[d]^{} \\
\coprod_{\lambda \in \Lambda_d} {\operatorname{mor}}_{{{\mathcal C}}}(-,i_\lambda)
\times D^d\ar[r]^-{} &
X = X_d.
}
}$$ Applying $- \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ to it yields, because $E^{{{\mathcal H}}}(i) = \alpha(i)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i)\bigr)$, a pushout of ${{\mathcal H}}$-spaces with a cofibration as left vertical arrow $$\xymatrix{
\coprod_{\lambda \in \Lambda_d} \alpha(i_\lambda)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i_\lambda)\bigr) \times S^{d-1} \ar[d]
\ar[r] & X_{d-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}} \ar[d]
\\
\coprod_{\lambda \in \Lambda_d} \alpha(i_\lambda)_*E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i_\lambda)\bigr) \times D^{d} \ar[r] & X
\otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}.}$$ In the sequel we can assume without loss of generality that $X_{d-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ and $X \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}$ are ${{\mathcal H}}$-$CW$-complexes and the diagram above is a pushout of ${{\mathcal H}}$-$CW$-complexes, since this can be arranged by replacing them by homotopy equivalent ${{\mathcal H}}$-$CW$-complexes (see the proof of Lemma \[lem:model\_for\_E(hocolim\_cali\_calc)\]). We obtain an exact sequence of $R{{\mathcal H}}$-chain complexes $$0 \to C_*(X_{d-1} \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}) \to C_*(X \otimes_{{{\mathcal I}}} E^{{{\mathcal H}}}) \to
\bigoplus_{\lambda \in \Lambda_d} \Sigma^d C_*\bigl(\alpha(i_\lambda)_*E^{{\operatorname{bar}}}{{\mathcal C}}(i_\lambda)\bigr)
\to 0.$$ Consider $\lambda \in \Lambda_d$. Since ${{\mathcal C}}(i_\lambda)$ is of type (FP$_R$), we can find a finite projective $R{{\mathcal C}}(i_\lambda)$-chain complex $P_*$ whose homology is concentrated in dimension zero and given there by the constant $R{{\mathcal C}}(i_\lambda)$-module $\underline{R}$. Since $C_*(E^{{\operatorname{bar}}}{{\mathcal C}}(i_\lambda))$ is a projective $R{{\mathcal C}}(i_\lambda)$-chain complex with the same homology, there is an $R{{\mathcal C}}(i_\lambda)$-chain homotopy equivalence $f_* \colon P_* \xrightarrow{\simeq}
C_*\bigl(E^{{\operatorname{bar}}}{{\mathcal C}}(i_\lambda)\bigr)$ (see Lück [@Lueck(1989) Lemma 11.3 on page 213] and $$o({{\mathcal C}}(i_\lambda);R) = o(P_*) = \sum_{n \geq 0} (-1)^n \cdot [P_n] \in K_0(R{{\mathcal C}}(i_\lambda)).$$ Obviously $$\alpha(i_\lambda)_*f_* \colon \alpha(i_\lambda)_*P_* \xrightarrow{\simeq}
\alpha(i_\lambda)_*C_*\bigl(E^{{\operatorname{bar}}}\bigl({{\mathcal C}}(i_\lambda)\bigr)\bigr) =
C_*\bigl(\alpha(i_\lambda)_*E^{{\operatorname{bar}}}{{\mathcal C}}(i_\lambda)\bigr)$$ is an $R{{\mathcal H}}$-chain homotopy equivalence. Hence $C_*(\alpha(i_\lambda)_*E^{{\operatorname{bar}}}{{\mathcal C}}(i_\lambda))$ and, by the induction hypothesis, $C_*(X_{d-1} \otimes_{{\mathcal I}}E^{{{\mathcal H}}})$ are $R{{\mathcal H}}$-chain complexes of type (FP$_R$). We conclude from Lück [@Lueck(1989) Lemma 11.3 on page 213] that $C_*(X \otimes_{{\mathcal I}}E^{{{\mathcal H}}})$ is of type (FP$_R$) and $$o\bigl(C_*(X \otimes_{{\mathcal I}}E^{{{\mathcal H}}})\bigr)
= o\bigl(C_*(X_{d-1} \otimes_{{\mathcal I}}E^{{{\mathcal H}}})\bigr) + \sum_{\lambda \in \Lambda_d}
o\bigl(\Sigma^d \alpha(i_\lambda)_*C_*(E^{{\operatorname{bar}}}{{\mathcal C}}(i_\lambda))\bigr).$$ This implies together with the induction hypothesis applied to $X_{d-1}$ $$\begin{aligned}
\lefteqn{o\bigl(C_*(X \otimes_{{\mathcal I}}E^{{{\mathcal H}}})\bigr)}
& &
\\
& = &
\sum_{n = 0}^{d-1} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \alpha(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R))
+
\sum_{\lambda \in \Lambda_d} (-1)^d \cdot \alpha(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R))
\\
& = &
\sum_{n = 0}^d (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \alpha(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R)).
\end{aligned}$$ This finishes the induction step.
Assertions \[the:homotopy\_colimit\_formula:(FP)\] and \[the:homotopy\_colimit\_formula:o\] follow by taking $X =
E{{\mathcal I}}$.\
\[the:homotopy\_colimit\_formula:(FF)\] This proof is analogous to that of assertion \[the:homotopy\_colimit\_formula:(FP)\].\
\[the:homotopy\_colimit\_formula:chi\] By \[the:homotopy\_colimit\_formula:directly\_finite\] and \[the:homotopy\_colimit\_formula:(FP)\], the category ${{\mathcal H}}$ is directly finite and of type (FP$_R$). Then an application of ${\operatorname{rk}}_{R{{\mathcal H}}}$ to the formula for $o({{\mathcal H}};R)$ in \[the:homotopy\_colimit\_formula:o\] yields the formula for $\chi_f({{\mathcal H}};R)$ in \[the:homotopy\_colimit\_formula:chi\] by the naturality of ${\operatorname{rk}}_{R-}$ with respect to the functors $\alpha(i_\lambda)$ between directly finite categories, see Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 4.9].
An application of the augmentation homomorphism $\epsilon \colon
U({{\mathcal H}}) \to {{\mathbb Z}}$ to the formula for $\chi_f({{\mathcal H}};R)$ yields the formula for $\chi({{\mathcal H}};R)$. We also use the naturality of the augmentation homomorphism, that is, the commutativity of diagram (4.5) in [@FioreLueckSauerFinObsAndEulCharOfCats(2009)] for $F=\alpha(i_\lambda)$.
If $R$ is additionally Noetherian, then Theorem \[the:chi\_f\_determines\_chi\] applies, and the Euler characteristics of the categories agree with the Euler characteristics of the classifying spaces.\
\[the:homotopy\_colimit\_formula:chi(2)\] The proofs for the functorial $L^2$-Euler characteristic and the $L^2$-Euler characteristic are somewhat more complicated since the property ($L^2$) is more general than (FP$_R$), and the $L^2$-Euler characteristic comes from the finiteness obstruction only in the case (FP$_R$). The proofs are variations of the proofs for assertions \[the:homotopy\_colimit\_formula:(FP)\] and \[the:homotopy\_colimit\_formula:o\]. Instead of using Lück [@Lueck(1989) Lemma 11.3 on page 213], we now use the basic properties of $L^2$-Euler characteristics for chain complexes of modules over group von Neumann algebras [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 5.7]. For example, we use [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 5.7 (iv)], which says for any injective group homomorphism $i \colon H \to G$ and ${{\mathcal N}}(H)$-chain complex $C_*$, we have $\chi^{(2)}(C_*) = \chi^{(2)}({\operatorname{ind}}_{i_*}
C_*)$, provided the sum of the $L^2$-Betti numbers of $C_*$ is finite. The injectivity hypothesis is easily verified: for every object $i \in {\operatorname{ob}}({{\mathcal I}})$ and object $x \in {{\mathcal C}}(i)$ the functor $\alpha(i) \colon{{\mathcal C}}(i) \to {{\mathcal H}}$ clearly induces an injection ${\operatorname{aut}}_{{{\mathcal C}}(i)}(x) \to {\operatorname{aut}}_{{{\mathcal H}}}(i,x)$. This finishes the proof of Theorem \[the:homotopy\_colimit\_formula\].
\[cor:homotopy\_colimit\_formula\_for\_pseudo\_functors\] Theorem \[the:homotopy\_colimit\_formula\] on homotopy colimits holds for pseudo functors ${{\mathcal D}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$.
We first remark that the pseudo functor ${{\mathcal D}}\colon {{\mathcal I}}\to
{{\EuR}{CAT}}$ is equivalent to a strict functor ${{\mathcal C}}\colon {{\mathcal I}}\to
{{\EuR}{CAT}}$ in the following sense. As usual, we denote by ${\operatorname{Hom}}({{\mathcal I}},{{\EuR}{CAT}})$ the strict 2-category of pseudo functors ${{\mathcal I}}\to {{\EuR}{CAT}}$, pseudo natural transformations between them, and modifications. The pseudo functor ${{\mathcal D}}$ is equivalent to a strict functor ${{\mathcal C}}$ *as objects of the 2-category* ${\operatorname{Hom}}({{\mathcal I}},{{\EuR}{CAT}})$. For example, we may take ${{\mathcal C}}$ to be the strict functor $$i \mapsto {\operatorname{mor}}_{{\operatorname{Hom}}({{\mathcal I}},{{\EuR}{CAT}})}({{\mathcal I}}(i,-),{{\mathcal D}}).$$
The equivalence between ${{\mathcal C}}$ and ${{\mathcal D}}$ in ${\operatorname{Hom}}({{\mathcal I}},{{\EuR}{CAT}})$ has two useful consequences. Since $${\operatorname{hocolim}}_{{{\mathcal I}}}\colon{\operatorname{Hom}}({{\mathcal I}},{{\EuR}{CAT}}) \to {{\EuR}{CAT}}$$ is a strict 2-functor, it sends any equivalence between ${{\mathcal C}}$ and ${{\mathcal D}}$ to an equivalence in ${{\EuR}{CAT}}$ between the categories ${\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}}$ and ${\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal D}}$. Another consequence of the equivalence between ${{\mathcal C}}$ and ${{\mathcal D}}$ is that for every $i \in {{\mathcal I}}$, the categories ${{\mathcal C}}(i)$ and ${{\mathcal D}}(i)$ are equivalent. With these observations we reduce Corollary \[cor:homotopy\_colimit\_formula\_for\_pseudo\_functors\] to Theorem \[the:homotopy\_colimit\_formula\].\
\[the:homotopy\_colimit\_formula:directly\_finite\] Suppose ${{\mathcal D}}(i)$ is directly finite for every $i\in{\operatorname{ob}}({{\mathcal I}})$ and ${{\mathcal I}}$ is directly finite. Since direct finiteness is preserved under equivalence of categories by Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 3.2], and ${{\mathcal C}}(i)$ is equivalent to ${{\mathcal D}}(i)$, we see that ${{\mathcal C}}(i)$ is directly finite for every $i\in{\operatorname{ob}}({{\mathcal I}})$. Hence ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ is directly finite by Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:directly\_finite\]. Since ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}}$ is equivalent to ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$, it is also directly finite, again by [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 3.2].\
\[the:homotopy\_colimit\_formula:(EI)\] Suppose that ${{\mathcal I}}$ is an EI-category, ${{\mathcal D}}(i)$ is an EI-category for every $i \in
{\operatorname{ob}}({{\mathcal I}})$, and for every automorphism $u \colon i
\xrightarrow{\cong} i$ the map ${\operatorname{iso}}({{\mathcal D}}(i)) \to {\operatorname{iso}}({{\mathcal D}}(i)),
\; \overline{y} \mapsto \overline{{{\mathcal D}}(u)(y)}$ is the identity. Since EI is preserved under equivalence of categories [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 3.11], and ${{\mathcal C}}(i)$ is equivalent to ${{\mathcal D}}(i)$, we see ${{\mathcal D}}(i)$ is an EI-category.
We claim that for each automorphism $u$, the functor ${{\mathcal C}}(u)$ also induces the identity on isomorphism classes of objects of ${{\mathcal C}}(i)$. Let $\phi\colon {{\mathcal D}}\to {{\mathcal C}}$ be a pseudo equivalence, that is, an equivalence in the 2-category ${\operatorname{Hom}}({{\mathcal I}},{{\EuR}{CAT}})$. For $x \in {{\mathcal C}}(i)$, there is a $y \in
{{\mathcal D}}(i)$ and an isomorphism $x \cong \phi_i(y)$. We have isomorphisms $${{\mathcal C}}(u)(x)\cong{{\mathcal C}}(u) \phi_i(y) \cong \phi_i {{\mathcal D}}(u)(y)\cong \phi_i(y) \cong
x,$$ and ${{\mathcal C}}(u)$ induces the identity on isomorphism classes of objects of ${{\mathcal C}}(i)$. Then ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ is an EI-category by Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(EI)\], and so is ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}}$, again by [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 3.11].\
\[the:homotopy\_colimit\_formula:(FP)\] and \[the:homotopy\_colimit\_formula:(FF)\] similarly follow from Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(FP)\] and \[the:homotopy\_colimit\_formula:(FF)\], since property (FP$_R$), property (FF$_R$), and the finiteness obstruction are all invariant under equivalence of categories [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Theorem 2.8].\
\[the:homotopy\_colimit\_formula:o\] Suppose ${{\mathcal D}}(i)$ is of type (FP$_R$) for every $i\in{\operatorname{ob}}({{\mathcal I}})$. Then every ${{\mathcal C}}(i)$ is also of type (FP$_R$), since property (FP$_R$) is invariant under equivalence of categories [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Theorem 2.8]. As in , we have for each $i \in {{\mathcal I}}$ the functor $$\alpha^{{\mathcal D}}(i) \colon {{\mathcal D}}(i) \to {\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}}$$ which sends an object $d$ to the object $(i,d)$ and a morphism $g
\colon d \to d'$ to the morphism $({\operatorname{id}}_i,g \circ {{\mathcal D}}^{-1}_i(d))$. From a pseudo equivalence $\psi\colon {{\mathcal C}}\to {{\mathcal D}}$ we obtain a strictly commutative diagram $$\label{equ:naturality_for_alpha_to_hocolim}
\xymatrix@C=3pc{{{\mathcal C}}(i) \ar[d]_{\psi_i} \ar[r]^-{\alpha^{{\mathcal C}}(i)} & {\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}\ar[d]^{{\operatorname{hocolim}}_{{\mathcal I}}\psi}
\\ {{\mathcal D}}(i) \ar[r]_-{\alpha^{{\mathcal D}}(i)} & {\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}}}$$ for each $i \in {\operatorname{ob}}({{\mathcal I}})$. Since the finiteness obstruction is invariant under equivalence of categories [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Theorem 2.8], we may use Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:o\] for ${{\mathcal C}}$ to obtain $$\aligned
o({\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}};R) &=
({\operatorname{hocolim}}_{{\mathcal I}}\psi)_\ast(o({\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}};R)) \\
&=({\operatorname{hocolim}}_{{\mathcal I}}\psi)_\ast\left( \sum_{n \geq 0} (-1)^n \cdot
\sum_{\lambda \in \Lambda_n} \alpha^{{\mathcal C}}(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R))\right) \\
&= \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
({\operatorname{hocolim}}_{{\mathcal I}}\psi)_\ast\circ \alpha^{{\mathcal C}}(i_\lambda)_*(o({{\mathcal C}}(i_\lambda);R)) \\
&=\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
\alpha^{{\mathcal D}}(i_\lambda)_*\circ (\psi_{i_\lambda})_\ast(o({{\mathcal C}}(i_\lambda);R)) \\
&=\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \alpha^{{\mathcal D}}(i_\lambda)_*(
o({{\mathcal D}}(i_\lambda);R)).
\endaligned$$\
\[the:homotopy\_colimit\_formula:chi\] follows from \[the:homotopy\_colimit\_formula:directly\_finite\], \[the:homotopy\_colimit\_formula:(FP)\], and \[the:homotopy\_colimit\_formula:o\] in the same way that Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi\] follows from Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(EI)\], \[the:homotopy\_colimit\_formula:(FP)\], and \[the:homotopy\_colimit\_formula:o\].\
\[the:homotopy\_colimit\_formula:chi(2)\] Suppose that ${{\mathcal I}}$ is directly finite and ${{\mathcal D}}(i)$ is directly finite for every object $i
\in {\operatorname{ob}}({{\mathcal I}})$. Suppose also for every object $i \in {{\mathcal I}}$ the category ${{\mathcal D}}(i)$ is of type ($L^2$). By the proof of Corollary \[cor:homotopy\_colimit\_formula\_for\_pseudo\_functors\] \[the:homotopy\_colimit\_formula:directly\_finite\] above, the values of the strict functor ${{\mathcal C}}$ are directly finite categories. If $\Gamma_1$ and $\Gamma_2$ are equivalent categories, then $\Gamma_1$ is both directly finite and of type ($L^2$) if and only if $\Gamma_2$ is both directly finite and of type ($L^2$) [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 5.15 (i)]. Since each ${{\mathcal D}}(i)$ is directly finite, of type ($L^2$), and equivalent to ${{\mathcal C}}(i)$, we see that each ${{\mathcal C}}(i)$ is also directly finite and of type ($L^2$). So we may now apply Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:directly\_finite\] and \[the:homotopy\_colimit\_formula:chi(2)\] to ${{\mathcal C}}$ and conclude that ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ is directly finite and of type ($L^2$). Again using the preservation of the direct finiteness and ($L^2$) under equivalence [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 5.15 (i)], and the equivalence of ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ with ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}}$, we see ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}}$ is both directly finite and of type ($L^2$).
To prove the formulas for $\chi_f^{(2)}$ and $\chi^{(2)}$, we use [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 5.15 (ii)], which says: if $F \colon \Gamma_1 \to \Gamma_2$ is an equivalence of categories, and both $\Gamma_1$ and $\Gamma_2$ are both directly finite and of type ($L^2$), then $U^{(1)}(F)\chi^{(2)}_f(\Gamma_1)=\chi^{(2)}_f(\Gamma_2)$ and $\chi^{(2)}(\Gamma_1) = \chi^{(2)}(\Gamma_2).$ We apply this to the equivalences $\psi_i$ and ${\operatorname{hocolim}}_{{\mathcal I}}\psi$, and use the commutativity of diagram . For readability, we write $({\operatorname{hocolim}}_{{\mathcal I}}\psi)_*$ for $U({\operatorname{hocolim}}_{{\mathcal I}}\psi)$ and $\alpha(i_\lambda)_*$ for $U^{(1)}(\alpha(i_\lambda))$, et cetera. $$\aligned
\chi^{(2)}_f({\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal D}})&=({\operatorname{hocolim}}_{{\mathcal I}}\psi)_* \chi^{(2)}_f({\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}) \\
&=({\operatorname{hocolim}}_{{\mathcal I}}\psi)_*\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
\alpha(i_\lambda)_*\bigl(\chi_f^{(2)}({{\mathcal C}}(i_\lambda))\bigr) \\
&=\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
({\operatorname{hocolim}}_{{\mathcal I}}\psi)_\ast\circ \alpha^{{\mathcal C}}(i_\lambda)_*\bigl(\chi_f^{(2)}({{\mathcal C}}(i_\lambda))\bigr) \\
&=\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
\alpha^{{\mathcal D}}(i_\lambda)_*\circ (\psi_{i_\lambda})_\ast\bigl(\chi_f^{(2)}({{\mathcal C}}(i_\lambda))\bigr)\\
&=\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda\in \Lambda_n} \alpha^{{\mathcal D}}(i_\lambda)_* \bigl(\chi_f^{(2)}({{\mathcal D}}(i_\lambda))\bigr).\\
\endaligned$$ The formula for $\chi^{(2)}$ follows by summing up the components of the functorial $L^2$-Euler characteristics.
The Case of an Indexing Category of Type (FP$_R$) {#subsec:The_case_of_an_indexing_categegory_of_type_(FP)}
-------------------------------------------------
The Homotopy Colimit Formula of Theorem \[the:homotopy\_colimit\_formula\] can be extended to the case, where ${{\mathcal I}}$ is of type (FP$_R$) and not necessarily of type (FF$_R$) as follows (recall that the existence of a finite ${{\mathcal I}}$-$CW$-model for $E{{\mathcal I}}$ implies ${{\mathcal I}}$ is of type (FF$_R$), since cellular chains then provide a finite free resolution of $\underline{R}$. ). The evaluation of the covariant functor $$E^{{{\mathcal H}}} \colon {{\mathcal I}}\to {{\mathcal H}}\text{-}{{\EuR}{SPACES}}$$ of at every object $i \in {{\mathcal I}}$ is an ${{\mathcal H}}$-$CW$-complex. Composing it with the cellular chain complex functor yields a covariant functor $$C_*(E^{{{\mathcal H}}}) \colon {{\mathcal I}}\to R{{\mathcal H}}\text{-}{{\EuR}{CHCOM}}$$ whose evaluation at every object in ${{\mathcal I}}$ is a free $R{{\mathcal H}}$-chain complexes. Since by assumption ${{\mathcal C}}(i)$ is of type (FP$_R$), $C_*(E^{{{\mathcal H}}})(i)$ is $R{{\mathcal H}}$-chain homotopy equivalent to a finite projective $R{{\mathcal H}}$-chain complex for every object $i \in
{{\mathcal I}}$. Since $R{\operatorname{mor}}_{{{\mathcal I}}}(?,i) \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})$ is $R{{\mathcal H}}$-isomorphic to $C_*(E^{{{\mathcal H}}})$, we conclude for every finitely generated projective $R\Gamma$-module $P$ that $P
\otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})$ is $R{{\mathcal H}}$-chain homotopy equivalent to finite projective $R{{\mathcal H}}$-chain complex and in particular possesses a finiteness obstruction $o\bigl(P
\otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}}\bigr) \in K_0(R{{\mathcal H}})$ (see Lück [@Lueck(1989) Theorem 11.2 on page 212]). Because of Lück [@Lueck(1989) Theorem 11.2 on page 212] we obtain a homomorphism $$\alpha_{{{\mathcal C}}} \colon K_0(R{{\mathcal I}}) \to K_0(R{{\mathcal H}}),
\quad [P] \mapsto o\bigl(P \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})\bigr).$$ The chain complex version of the proof of Lemma \[lem:model\_for\_E(hocolim\_cali\_calc)\] shows that the $R{{\mathcal H}}$-chain complex $C_*({{\mathcal I}}) \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})$ is a projective $R{{\mathcal H}}$-resolution of the constant $R\Gamma$-module $\underline{R}$. Choose a finite projective $R{{\mathcal I}}$-chain complex $P_*$ and an $R{{\mathcal I}}$-chain homotopy equivalence $f_* \colon P_*
\xrightarrow{\simeq} C_*({{\mathcal I}})$. Then $f_*\otimes_{R{{\mathcal I}}} {\operatorname{id}}\colon P_* \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}}) \to C_*({{\mathcal I}})
\otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})$ is an $R\Gamma$-chain homotopy equivalence of $R\Gamma$-chain complexes and $P_* \otimes_{R{{\mathcal I}}}
C_*(E^{{{\mathcal H}}})$ is is $R{{\mathcal H}}$-chain homotopy equivalent to finite projective $R{{\mathcal H}}$-chain complex by Lück [@Lueck(1989) Theorem 11.2 on page 212]. This implies $$o(\Gamma;R) = o\bigl(P_* \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})\bigr).$$ We conclude from [@Lueck(1989) Theorem 11.2 on page 212] $$o\bigl(P_* \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})\bigr) =
\sum_{n \geq p} (-1)^n \cdot o\bigl(P_n \otimes_{R{{\mathcal I}}} C_*(E^{{{\mathcal H}}})\bigr)$$ Since $o({{\mathcal I}};R)$ is $\sum_{n \geq p} (-1)^n \cdot [P_n]$, this implies
\[the:the\_homotopy\_colimit\_formula\_for\_an\_indexing\_category\_of\_type\_(FP)\] We obtain under the conditions above $$\alpha_{{{\mathcal C}}} \bigl(o({{\mathcal I}};R)\bigr)
=
o({{\mathcal H}};R).$$
See Section \[sec:comparison\_with\_Leinster\] for a comparison with Leinster’s Euler characteristic and his results.
Examples of the Homotopy Colimit Formula {#sec:examples_of_the_homotopy_colimit_formula}
========================================
We now present several examples of the Homotopy Colimit Formula Theorem \[the:homotopy\_colimit\_formula\]. These include the cases: ${{\mathcal I}}$ with a terminal object, the constant functor, the trivial functor, homotopy pushouts, homotopy orbits, and the transport groupoid. For the transport groupoid in the finite case, see also Example \[exa:transport\_groupoid\_finite\_case\].
\[exa:hocolim\_formula\_for\_I\_with\_terminal\_object\] Suppose that ${{\mathcal I}}$ has a terminal object $t$ and ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ is a functor. Then ${\operatorname{mor}}_{{\mathcal I}}(-,t)$ is a finite ${{\mathcal I}}$-CW model for $E{{\mathcal I}}$. If every category ${{\mathcal C}}(i)$ is of type (FP$_R$), then $o({{\mathcal H}};R)=\alpha(t)_*(o({{\mathcal C}}(t);R)$. If ${{\mathcal I}}$ and ${{\mathcal C}}$ additionally satisfy the hypotheses of Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi\], then $\chi_f({{\mathcal H}};R)=\chi_f({{\mathcal C}}(t);R)$ and $\chi({{\mathcal H}};R)=\chi({{\mathcal C}}(t);R)$, as we anticipated in Example \[exa:homotopy\_hocolimit\_and\_when\_I\_has\_terminal\_object\]. Similar statements hold for $\chi^{(2)}_f$ and $\chi^{(2)}$ in the $L^2$ case.
\[exa:homotopy\_hocolimit\_formula\_of\_constant\_functor\] Consider the situation of Theorem \[the:homotopy\_colimit\_formula\] in the special case where the covariant functor ${{\mathcal C}}\colon {{\mathcal I}}\to
{{\EuR}{CAT}}$ is constant ${{\mathcal C}}\in {{\EuR}{CAT}}$. Suppose that ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model for $E{{\mathcal I}}$. Then we may draw various conclusions about the homotopy colimit ${{\mathcal H}}={{\mathcal I}}\times {{\mathcal C}}$. If ${{\mathcal I}}$ and ${{\mathcal C}}$ are of type (FP$_R$), then so is ${{\mathcal I}}\times {{\mathcal C}}$. If ${{\mathcal I}}$ and ${{\mathcal C}}$ are of type (FF$_R$), then so is ${{\mathcal I}}\times {{\mathcal C}}$. The statements in Theorem \[the:homotopy\_colimit\_formula\] provide us with formulas in terms of ${{\mathcal C}}$ for $o({{\mathcal I}}\times {{\mathcal C}};R)$, $\chi_f({{\mathcal I}}\times {{\mathcal C}};R)$, $\chi({{\mathcal I}}\times {{\mathcal C}};R)$, $\chi_f^{(2)}({{\mathcal I}}\times {{\mathcal C}})$, and $\chi^{(2)}({{\mathcal I}}\times
{{\mathcal C}})$. We recall that the invariants $o$, $\chi_f$, $\chi$, $\chi_f^{(2)}$, and $\chi^{(2)}$ are multiplicative, see Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Theorems 2.17, 4.22, and 5.17].
\[exa:homotopy\_hocolimit\_and\_trivial\_functor\] Consider the situation of Theorem \[the:homotopy\_colimit\_formula\] in the special case where the covariant functor ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ is constantly the terminal category, which consists of a single object and its identity morphism. Then ${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ agrees with ${{\mathcal I}}$, as we see from Example \[exa:hocolim\_constant\]. Obviously ${{\mathcal C}}(i)$ is of type (FF$_R$), its finiteness obstruction is $[R] \in K_0(R) = K_0(R{{\mathcal C}}(i))$ and both its Euler characteristic and $L^2$-Euler characteristic equals $1$. We obtain from Theorem \[the:homotopy\_colimit\_formula\] $$\begin{array}{lcll}
o({{\mathcal I}};R)
& = &
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} [R{\operatorname{mor}}_{{{\mathcal I}}}(?,i_\lambda)] & \in K_0(R{{\mathcal I}});
\\
\chi_f({{\mathcal I}};R)
& = &
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \overline{i_\lambda} & \in U(\Gamma);
\\
\chi({{\mathcal I}};R)
& = &
\sum_{n \geq 0} (-1)^n \cdot |\Lambda_n| & \in {{\mathbb Z}};
\\
\chi_f^{(2)}({{\mathcal I}})
& = &
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \overline{i_\lambda} & \in U^{(1)}({{\mathcal I}});
\\
\chi^{(2)}({{\mathcal I}})
& = &
\sum_{n \geq 0} (-1)^n \cdot |\Lambda_n| & \in {{\mathbb R}}.
\end{array}$$
\[exa:homotopy\_pushout\] Let ${{\mathcal I}}$ be the category with objects $j$, $k$ and $\ell$ such that there is precisely one morphism from $j$ to $k$ and from $j$ to $\ell$ and all other morphisms are identity morphisms. $${{\mathcal I}}=\{\xymatrix{k & j \ar[l]_{g} \ar[r]^{h} & \ell}\}$$ By Example \[exa:finite\_model\_for\_pushout\], the category ${{\mathcal I}}$ admits a finite model for the classifying ${{\mathcal I}}$-space $E{{\mathcal I}}$.
A covariant functor ${{\mathcal C}}\colon {{\mathcal I}}\to {{\EuR}{CAT}}$ is the same as specifying three categories ${{\mathcal C}}(j)$, ${{\mathcal C}}(k)$ and ${{\mathcal C}}(\ell)$ and two functors ${{\mathcal C}}(g) \colon {{\mathcal C}}(j) \to {{\mathcal C}}(k)$ and ${{\mathcal C}}(h)
\colon {{\mathcal C}}(j) \to {{\mathcal C}}(\ell)$. Let ${{\mathcal H}}= {\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}$ be the homotopy colimit. Let $\alpha(i) \colon {{\mathcal C}}(i) \to {{\mathcal H}}$ be the canonical functor for $i = j,k,\ell$. Then we obtain a square of functors which commutes up to natural transformations $$\xymatrix{{{\mathcal C}}(j) \ar[r]^{{{\mathcal C}}(g)} \ar[d]_{{{\mathcal C}}(h)} \ar[dr]^{\alpha(j)}
& {{\mathcal C}}(k) \ar[d]^{\alpha(k)}
\\
{{\mathcal C}}(\ell) \ar[r]_{\alpha(\ell)} & {{\mathcal H}}.}$$ It induces diagrams which do [**not**]{} commute in general $$\xymatrix{K_0(R{{\mathcal C}}(j)) \ar[r]^{{{\mathcal C}}(g)_*} \ar[d]_{{{\mathcal C}}(h)_*} \ar[dr]^{\alpha(j)_*}
& K_0(R{{\mathcal C}}(k)) \ar[d]^{\alpha(k)_*}
\\
K_0(R{{\mathcal C}}(\ell)) \ar[r]_{\alpha(\ell)_*} & K_0({{\mathcal H}})}$$ and $$\xymatrix{U({{\mathcal C}}(j)) \ar[r]^{{{\mathcal C}}(g)_*} \ar[d]_{{{\mathcal C}}(h)_*} \ar[dr]^{\alpha(j)_*}
& U(R{{\mathcal C}}(k)) \ar[d]^{\alpha(k)_*}
\\
U(R{{\mathcal C}}(\ell)) \ar[r]_{\alpha(\ell)_*} & U({{\mathcal H}}).}$$ Suppose that ${{\mathcal C}}(i)$ is of type (FP$_R$) for $i = j,k,\ell$. We conclude from Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(FP)\] that ${{\mathcal H}}$ is of type (FP$_R$) and $$\begin{array}{lcll}
o({{\mathcal H}};R)
& = &
\alpha(k)_*\bigl(o({{\mathcal C}}(k);R)) + \alpha(\ell)_*\bigl(o({{\mathcal C}}(\ell);R))
- \alpha(j)_*\bigl(o({{\mathcal C}}(j;R));
& \in K_0(R{{\mathcal H}});
\\
\chi_f({{\mathcal H}};R)
& = &
\alpha(k)_*\bigl(\chi_f({{\mathcal C}}(k);R)\bigr) + \alpha(\ell)_*\bigl(\chi_f({{\mathcal C}}(\ell);R)\bigr) -
\alpha(j)_*\bigl(\chi_f({{\mathcal C}}(j);R)\bigr);
& \in U({{\mathcal H}});
\\
\chi({{\mathcal H}};R)
& = &
\chi({{\mathcal C}}(k);R) + \chi({{\mathcal C}}(\ell);R) - \chi({{\mathcal C}}(j);R);
& \in {{\mathbb Z}};
\\
\chi_f^{(2)}({{\mathcal H}})
& = &
\alpha(k)_*\bigl(\chi^{(2)}_f({{\mathcal C}}(k)\bigr) + \alpha(\ell)_*\bigl(\chi^{(2)}_f({{\mathcal C}}(\ell))\bigr) -
\alpha(j)_*\bigl(\chi^{(2)}_f({{\mathcal C}}(j))\bigr);
& \in U^{(1)}({{\mathcal H}});
\\
\chi^{(2)}({{\mathcal H}})
& = &
\chi^{(2)}({{\mathcal C}}(k)) + \chi^{(2)}({{\mathcal C}}(\ell)) - \chi^{(2)}({{\mathcal C}}(j));
& \in {{\mathbb R}}.
\end{array}$$
\[exa:homotopy\_colimit\_and\_group\_actions\] Suppose that a group $G$ acts on a category ${{\mathcal C}}$ from the left. This can be viewed as a covariant functor $\widehat{G} \to
{{\EuR}{CAT}}$ whose source is the groupoid $\widehat{G}$ with one object and $G$ as its automorphism group. Let ${{\mathcal H}}= {\operatorname{hocolim}}_{\widehat{G}} {{\mathcal C}}$ be its homotopy colimit, also called the *homotopy orbit*. Notice that ${{\mathcal H}}$ and ${{\mathcal C}}$ have the same set of objects.
Suppose there is a finite model for $BG$ of the group $G$, or equivalently, a finite model for the $\widehat{G}$-classifying space $E\widehat{G}$ of the category $\widehat{G}$. Let $\chi(BG) \in {{\mathbb Z}}$ be its Euler characteristic. Let $\alpha \colon {{\mathcal C}}\to {{\mathcal H}}$ be the canonical inclusion. Suppose that ${{\mathcal C}}$ is of type (FP$_R$). Then we conclude from Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(FP)\] that ${{\mathcal H}}$ is of type (FP$_R$) and we have $$\begin{array}{lcll}
o({{\mathcal H}};R) & = & \chi(BG) \cdot \alpha_*\bigl(o({{\mathcal C}};R)\bigr) & \in K_0(R{{\mathcal H}});
\\
\chi_f({{\mathcal H}};R) & = & \chi(BG) \cdot \alpha_*\bigl(\chi_f({{\mathcal C}};R)\bigr)
& \in U({{\mathcal H}});
\\
\chi({{\mathcal H}};R) & = & \chi(BG) \cdot \chi({{\mathcal C}};R) & \in {{\mathbb Z}};
\\
\chi_f^{(2)}({{\mathcal H}};R) & = & \chi(BG) \cdot \alpha_*\bigl(\chi_f^{(2)}({{\mathcal C}};R)\bigr)
& \in U^{(1)}({{\mathcal H}});
\\
\chi^{(2)}({{\mathcal H}};R) & = & \chi(BG) \cdot \chi^{(2)}({{\mathcal C}};R) & \in {{\mathbb R}}.
\end{array}$$
\[exa:transport\_groupoid\] Let $G$ be a group and let $S$ be a left $G$-set. Its *transport groupoid* ${{\mathcal G}}^G(S)$ has $S$ as its set of objects. The set of morphisms from $s_1$ to $s_2$ is $\{g \in G \mid gs_1 = s_2\}$. The composition is given by the multiplication in $G$. Denote by $\underline{S}$ the category whose set of objects is $S$ and which has no morphisms besides the identity morphisms. The group $G$ acts from the left on $\underline{S}$. One easily checks that ${{\mathcal G}}^G(S)$ is the homotopy orbit of $\underline{S}$ defined in Example \[exa:homotopy\_colimit\_and\_group\_actions\].
Recall from Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 6.15 (iv)]: if $\Gamma$ is a quasi-finite EI-category and for any morphism $f
\colon x \to y$ in $\Gamma$, the order of the finite group $\{g \in
{\operatorname{aut}}(x) \mid f \circ g = f\}$ is invertible in $R$, then $\Gamma$ is of type (FP$_R$) if and only if ${\operatorname{iso}}(\Gamma)$ is finite and for every object $x \in {\operatorname{ob}}(\Gamma)$ the trivial $R[x]$-module $R$ is of type (FP$_R$). Thus, category $\underline{S}$ is of type (FP$_R$) if and only if $S$ is finite. Suppose that $\underline{S}$ is of type (FP$_R$) and there is a finite model for $BG$. Obviously $o(\underline{S};R)$ is given in $K_0(R\underline{S}) = \oplus_S
K_0(R)$ by the collection $\{[R] \in K_0(R) \mid s \in S\}$.
Suppose for simplicity that $G$ acts transitively on $S$. Fix an element $s \in S$. Let $G_s$ be its isotropy group. Since $S$ is finite, $G_s$ is a subgroup of $G$ of finite index, namely $[G:G_s] = |S|$. The transport groupoid ${{\mathcal G}}^G(S)$ is connected and the automorphism group of $s$ is $G_s$. Hence evaluation at $s$ induces an isomorphism $${\operatorname{ev}}\colon K_0(R{{\mathcal G}}^G(S)) \xrightarrow{\cong} K_0(R[G_s]).$$ The composition $$K_0(R\underline{S}) \xrightarrow{\alpha_*} K_0(R{{\mathcal G}}^G(S))
\xrightarrow{\cong} K_0(R[G_s])$$ sends $o(\underline{S};R)$ to $|S| \cdot [RG_s]$, where $\alpha \colon \underline{S} \to {{\mathcal G}}^G(S)$ is the obvious inclusion. Hence Example \[exa:homotopy\_colimit\_and\_group\_actions\] implies $${\operatorname{ev}}\bigl(o({{\mathcal G}}^G(S);R)\bigr)
= \chi(BG) \cdot |S| \cdot [RG_S] \quad \in K_0(RG_s).$$ Since $BG$ has a finite model, $BG_s$ as a finite covering of $BG$ has a finite model. The cellular $RG_s$-chain complex of $EG_s$ yields a finite free resolution of the trivial $RG_s$-module $R$. This implies $${\operatorname{ev}}\bigl(o({{\mathcal G}}^G(S);R)\bigr)
= \chi(BG_s) \cdot [RG_s] \quad \in K_0(RG_s).$$ Hence we obtain the equality in $K_0(RG_s)$ $$\chi(BG_s) \cdot [RG_s] = \chi(BG) \cdot |S| \cdot [RG_S]
= \chi(BG) \cdot [G:G_s] \cdot [RG_s].$$ This is equivalent to the equality of integers $$\chi(BG_s) = \chi(BG) \cdot [G:G_s].$$ This equation is compatible with the well-know fact that for a $d$-sheeted covering $\overline{X} \to X$ of a finite $CW$-complex $X$ the total space $\overline{X}$ is again a finite $CW$-complex and we have $\chi(\overline{X}) = d \cdot \chi(X)$.
For the transport groupoid in the finite case, see also Example \[exa:transport\_groupoid\_finite\_case\].
Combinatorial Illustrations of the Homotopy Colimit Formula {#sec:Combinatorial_Applications_of_the_Homotopy_Colimit_Formula}
===========================================================
The classical Inclusion-Exclusion Principle follows from the Homotopy Colimit Formula Theorem \[the:homotopy\_colimit\_formula\]. We can also easily calculate the cardinality of a coequalizer in ${{\EuR}{SETS}}$ in certain cases. These are different proofs of Examples 3.4.d and 3.4.b of Leinster’s paper [@Leinster(2008)].
\[the:inclusion\_exclusion\] Let $X$ be a set and $S_0,
\dots, S_q$ finite subsets of $X$. Then $$|S_0 \cup S_1 \cup \cdots \cup S_q|=
\sum_{\emptyset \neq J \subseteq [q]}(-1)^{|J|-1} \cdot
\left|\bigcap_{j \in J} S_j\right|.$$
Let ${{\mathcal I}}$ be the category in Example \[exa:finite\_model\_for\_q\_interior\] and consider the finite ${{\mathcal I}}$-$CW$-model for its classifying ${{\mathcal I}}$-space constructed there. We define a functor ${{\mathcal C}}: {{\mathcal I}}\to {{\EuR}{SETS}}$ by ${{\mathcal C}}(J):=\bigcap_{j\in J} S_j$. The functor $${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}\xymatrix{\ar[r] &} {\operatorname{colim}}_{{{\mathcal I}}} {{\mathcal C}}= S_0 \cup S_1 \cup \cdots \cup S_q$$ is an equivalence of categories, since it is surjective on objects and fully faithful. We have $$\aligned |S_0 \cup S_1 \cup \cdots \cup S_q| &=\chi(S_0 \cup S_1 \cup \cdots \cup S_q)
\\ &= \chi({\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}})
\\ &= \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \chi ({{\mathcal C}}(i_\lambda))
\\ &= \sum_{n \geq 0} (-1)^n \cdot \sum_{J \subseteq [q] \;\text{and}\;
|J|=n+1} \chi({{\mathcal C}}(J))
\\ &= \sum_{n \geq 0} (-1)^n \left( \sum_{J \subseteq [q] \;\text{and}\;
|J|=n+1} \left|\bigcap_{j \in J} S_j \right| \right)
\\&= \sum_{\emptyset \neq J \subseteq [q]} \left( (-1)^{|J|-1} \left|\bigcap_{j \in J} S_j \right|
\right).
\endaligned$$
\[the:cardinality\_of\_coequalizer\] Let ${{\mathcal I}}$ be the category $$\xymatrix{a \ar@<.5ex>[r]^f \ar@<-.5ex>[r]_g & b}$$ and ${{\mathcal C}}:{{\mathcal I}}\to {{\EuR}{SETS}}$ a functor such that:
1. the maps ${{\mathcal C}}f$ and ${{\mathcal C}}g$ are injective,
2. the images of the maps ${{\mathcal C}}f$ and ${{\mathcal C}}g$ are disjoint, and
3. the sets ${{\mathcal C}}a$ and ${{\mathcal C}}b$ are finite.
Then the coequalizer ${\operatorname{colim}}{{\mathcal C}}$ has cardinality $|{{\mathcal C}}b|-|{{\mathcal C}}a|$.
The assumptions that ${{\mathcal C}}f$ and ${{\mathcal C}}g$ are injective and have disjoint images imply that the functor $${\operatorname{hocolim}}_{{{\mathcal I}}} {{\mathcal C}}\xymatrix{\ar[r] &} {\operatorname{colim}}_{{{\mathcal I}}} {{\mathcal C}}$$ is fully faithful. Clearly it is also surjective on objects, and hence an equivalence of categories. The category ${{\mathcal I}}$ has a finite ${{\mathcal I}}$-$CW$-model for its classifying ${{\mathcal I}}$-space, constructed explicitly in Example \[exa:finite\_model\_for\_parallel\_arrows\]. By Theorem \[the:homotopy\_colimit\_formula\], we have $$\aligned
\chi({\operatorname{colim}}_{{{\mathcal I}}} {{\mathcal C}})&=\chi({\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}})
\\ &= \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \chi ({{\mathcal C}}(i_\lambda))
\\ &= \chi({{\mathcal C}}b) - \chi({{\mathcal C}}a)
\\ &= |{{\mathcal C}}b| - |{{\mathcal C}}a|.
\endaligned$$
Comparison with Results of Baez–Dolan and Leinster {#sec:comparison_with_Leinster}
==================================================
We recall Baez–Dolan’s groupoid cardinality [@Baez-Dolan(2001)] and Leinster’s Euler characteristic for certain finite categories [@Leinster(2008)], compare our Homotopy Colimit Formula with his result on compatibility with Grothendieck fibrations, prove an analogue for indexing categories ${{\mathcal I}}$ that admit finite ${{\mathcal I}}$-$CW$-models for their classifying ${{\mathcal I}}$-spaces, and finally mention a Homotopy Colimit Formula for Leinster’s invariant in a restricted case.
Review of Leinster’s Euler Characteristic
-----------------------------------------
Let $\Gamma$ be a category with finitely many objects and finitely many morphisms. A *weighting* on $\Gamma$ is a function $q^{\bullet} \colon
{\operatorname{ob}}(\Gamma) \to {{\mathbb Q}}$ such that for all objects $x \in {\operatorname{ob}}(\Gamma)$, we have $$\sum_{y \in {\operatorname{ob}}(\Gamma)} |{\operatorname{mor}}_\Gamma(x,y)| \cdot q^y =
1.$$ A *coweighting* $q_{\bullet}$ on $\Gamma$ is a weighting on $\Gamma^{{\operatorname{op}}}$. If a finite category admits both a weighting $q^{\bullet}$ and a coweighting $q_{\bullet}$, then $\sum_{y \in
{\operatorname{ob}}(\Gamma)} q^y = \sum_{x \in {\operatorname{ob}}(\Gamma)} q_x$. For a discusion of which matrices have the form $\left(|{\operatorname{mor}}_\Gamma(x,y)| \right)_{x,y
\in {\operatorname{ob}}(\Gamma)}$ for some finite category $\Gamma$, see Allouch [@Allouch2008] and [@Allouch2010].
As proved in [@FioreLueckSauerFinObsAndEulCharOfCats(2009)], free resolutions of the constant $R\Gamma$-module $\underline{R}$ give rise to weightings on $\Gamma$.
\[the:weighting\_from\_free\_resolution\] Let $\Gamma$ be a finite category. Suppose that the constant $R \Gamma$-module $\underline{R}$ admits a finite free resolution $P_*$. If $P_n$ is free on the finite ${\operatorname{ob}}(\Gamma)$-set $C_n$, that is $$\label{the:weighting_from_free_resolution:equation}
P_n=B(C_n)=\bigoplus_{y \in {\operatorname{ob}}(\Gamma)} \bigoplus_{C^y_n}
R{\operatorname{mor}}_\Gamma(?,y),$$ then the function $q^{\bullet} \colon {\operatorname{ob}}(\Gamma) \to {{\mathbb Q}}$ defined by $$q^y:=\sum_{n \geq 0}(-1)^n \cdot |C^y_n|$$ is a weighting on $\Gamma$.
\[cor:weighting\_from\_finite\_model\] Let ${{\mathcal I}}$ be a finite category. Suppose that ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model $X$ for the classifying ${{\mathcal I}}$-space. Then the function $q^{\bullet} \colon {\operatorname{ob}}({{\mathcal I}}) \to {{\mathbb Q}}$ defined by $$q^y:=\sum_{n \geq 0}(-1)^n(\text{number of $n$-cells of $X$ based at $y$})$$ is a weighting on ${{\mathcal I}}$.
As explained in Section 7.5 of [@FioreLueckSauerFinObsAndEulCharOfCats(2009)], we use this Corollary to obtain several of Leinster’s weightings in [@Leinster(2008)] from ${{\mathcal I}}$-$CW$-models for ${{\mathcal I}}$-classifying spaces. If ${{\mathcal I}}$ has a terminal object, then we obtain from the finite model in Example \[exa:finite\_model\_for\_I\_with\_terminal\_object\] the weighting which is 1 on the terminal object and 0 otherwise. The category ${{\mathcal I}}=\{j \rightrightarrows k\}$ in Example \[exa:finite\_model\_for\_parallel\_arrows\] has weighting $(q^j,q^k)=(-1,1)$. The category ${{\mathcal I}}=\{k \leftarrow j \to \ell\}$ in Example \[exa:finite\_model\_for\_pushout\] has weighting $(q^j,q^k,q^\ell)=(-1,1,1)$. Lastly, the category in Example \[exa:finite\_model\_for\_q\_interior\] has weighting $q^J:=(-1)^{|J|-1}$.
Weightings and coweightings play a key role in Leinster’s notion of Euler characteristic. See also Berger–Leinster [@Berger+Leinster(2008)].
\[def:Leinsters\_Euler\_characteristic\] A finite category $\Gamma$ *has an Euler characteristic in the sense of Leinster* if it admits both a weighting and a coweighting. In this case, its *Euler characteristic in the sense of Leinster* is defined as $$\chi_L(\Gamma) := \sum_{y \in {\operatorname{ob}}(\Gamma)} q^y = \sum_{x \in {\operatorname{ob}}(\Gamma)} q_x$$ for any choice of weighting $q^{\bullet}$ or coweighting $q_{\bullet}$.
The Euler characteristic of Leinster agrees with the *groupoid cardinality* of Baez–Dolan [@Baez-Dolan(2001)] in the case of a finite groupoid ${{\mathcal G}}$, namely they are both $$\sum_{\overline{x}
\in {\operatorname{iso}}({{\mathcal G}})} \frac{1}{|{\operatorname{aut}}_{{{\mathcal G}}}(x)|}.$$ The Euler characteristic of Leinster agrees with our $L^2$-Euler characteristic in some cases, as in the following Lemma.
\[lem:chi(2)\_and\_chi\] Let $\Gamma$ be a finite EI-category which is skeletal, i.e., if two objects are isomorphic, then they are equal. Suppose that the left ${\operatorname{aut}}_\Gamma(y)$-action on ${\operatorname{mor}}_\Gamma(x,y)$ is free for every two objects $x,y
\in {\operatorname{ob}}(\Gamma)$.
Then $\Gamma$ is of type (FP$_{{\mathbb C}}$) and of type ($L^2$), and has an Euler characteristic in the sense of Leinster. Furthermore, the $L^2$-Euler characteristic $\chi^{(2)}(\Gamma)$ of Definition \[def:L2-Euler\_characteristic\_of\_a\_category\] coincides with Leinster’s Euler characteristic $\chi_L(\Gamma)$ of Definition \[def:Leinsters\_Euler\_characteristic\]: $$\chi^{(2)}(\Gamma) = \chi_L(\Gamma).$$ Moreover, these are both equal to $$\sum_{l \ge 0} (-1)^l \cdot \sum_{x_0,x_l \in {\operatorname{ob}}(\Gamma)} \sum
\frac{1}{|{\operatorname{aut}}(x_{l})| \cdot |{\operatorname{aut}}(x_{l-1})| \cdot \cdots \cdot
|{\operatorname{aut}}(x_{0})|},$$ where the inner sum is over all paths $x_0 \to x_1 \to \cdots \to
x_l$ from $x_0$ to $x_l$ such that $x_0, \ldots, x_l$ are all distinct [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Example 6.33].
This concludes the review of Leinster’s and Baez–Dolan’s invariants and how they relate to our $L^2$-Euler characteristic. Next we turn to a comparison of homotopy colimit results.
Comparison with Leinster’s Proposition 2.8
------------------------------------------
Leinster’s result on homotopy colimits, rephrased in our notation to make the comparison more apparent, is below.
Let ${{\mathcal I}}$ be a category with finitely many objects and finitely many morphisms, and ${{\mathcal C}}:{{\mathcal I}}\to {{\EuR}{CAT}}$ a pseudo functor. Assume that ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ has finitely many objects and finitely many morphisms. Let $q^{\bullet}$ be a weighting on ${{\mathcal I}}$ and suppose that ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ and all ${{\mathcal C}}(i)$ have Euler characteristics. Then $$\chi_L({\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}) =\sum_{i \in {\operatorname{ob}}({{\mathcal I}})} q^i \chi_L({{\mathcal C}}(i)).$$
For example, if ${{\mathcal I}}=\{k \leftarrow j \to \ell\}$, then ${{\mathcal I}}$ admits the weighting $(q^j,q^k,q^\ell)=(-1,1,1)$ as discussed above. If ${{\mathcal C}}:{{\mathcal I}}\to {{\EuR}{CAT}}$ is a pseudo functor, and the homotopy pushout has finitely many objects and finitely many morphisms, and ${\operatorname{hocolim}}_{{\mathcal I}}{{\mathcal C}}$ and all ${{\mathcal C}}(i)$ have Euler characteristics, then Leinster’s result says that the homotopy pushout has the Euler characteristic $\chi_L({{\mathcal C}}(k))+\chi_L({{\mathcal C}}(\ell))-\chi_L({{\mathcal C}}(j))$.
Leinster’s Proposition 2.8 tells us how the Euler characteristic is compatible with Grothendieck fibrations. We can remove the hypothesis of finite from that Proposition, at the expense of requiring a finite model, as in the following theorem for our invariants.
\[the:hocolim\_weighting\] Let ${{\mathcal I}}$ be a finite category. Suppose that ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model $X$ for the classifying ${{\mathcal I}}$-space of ${{\mathcal I}}$. Let $q^{\bullet} \colon {\operatorname{ob}}({{\mathcal I}}) \to {{\mathbb Q}}$ be the ${{\mathcal I}}$-Euler characteristic of $X$, namely $$q^i:=\sum_{n \geq 0}(-1)^n(\text{number of $n$-cells of $X$ based at $i$}).$$ Let ${{\mathcal C}}:{{\mathcal I}}\to {{\EuR}{CAT}}$ be a functor such that for every object $i$ the category ${{\mathcal C}}(i)$ is of type (FP$_R$). Suppose that ${{\mathcal I}}$ is directly finite and ${{\mathcal C}}(i)$ is directly finite for all $i \in {\operatorname{ob}}({{\mathcal I}})$. Then $$\chi({\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}};R)=\sum_{i\in{\operatorname{ob}}({{\mathcal I}})}q^i\chi({{\mathcal C}}(i);R).$$ If each ${{\mathcal C}}(i)$ is of type ($L^2$) rather than (FP$_R$), we have $$\chi^{(2)}({\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}})=\sum_{i\in{\operatorname{ob}}({{\mathcal I}})}q^i\chi^{(2)}({{\mathcal C}}(i)).$$
By Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi\], we have $$\aligned
\chi({\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}}; R)&=\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda
\in \Lambda_n} \chi ({{\mathcal C}}(i_\lambda); R)
\\ &=\sum_{n \geq 0} (-1)^n \cdot \sum_{i\in{\operatorname{ob}}({{\mathcal I}})}(\text{number of $n$-cells of $X$ based at $i$})\chi ({{\mathcal C}}(i);R)
\\ &=\sum_{i\in{\operatorname{ob}}({{\mathcal I}})}\sum_{n \geq 0} (-1)^n(\text{number of $n$-cells of $X$ based at $i$})\chi ({{\mathcal C}}(i);R)
\\ &=\sum_{i\in{\operatorname{ob}}({{\mathcal I}})}q^i\chi({{\mathcal C}}(i);R).
\endaligned$$ The statement for $\chi^{(2)}$ is proved similarly from Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi(2)\].
Whenever $\chi({\operatorname{colim}}_{{{\mathcal I}}}{{\mathcal C}};R)=\chi({\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}};R)$, Theorem \[the:homotopy\_colimit\_formula\] and Theorem \[the:hocolim\_weighting\] can be used to calculate the Euler characteristic of a colimit. Indeed, the hypotheses of Examples \[the:inclusion\_exclusion\] and \[the:cardinality\_of\_coequalizer\] guaranteed the equivalence of the colimit and the homotopy colimit, and this equivalence was a crucial ingredient in those proofs. For example, under Leinster’s hypothesis of familial representability on $\mathcal{C}$, each connected component of ${\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}}$ has an initial object, so $$\chi({\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}};R)=\chi({\operatorname{colim}}_{{{\mathcal I}}}{{\mathcal C}};R)$$ (recall that ${\operatorname{colim}}_{{{\mathcal I}}}{{\mathcal C}}$ is the set of connected components of ${\operatorname{hocolim}}_{{{\mathcal I}}}{{\mathcal C}}$ whenever ${{\mathcal C}}$ takes values in ${{\EuR}{SETS}}$). This is the role of familial representability in his Examples 3.4.
As a corollary to our Homotopy Colimit Formula for the $L^2$-Euler characteristic, we have a Homotopy Colimit Formula for Leinster’s Euler characteristic when they agree.
Let ${{\mathcal I}}$ be a skeletal, finite, EI-category such that the left ${\operatorname{aut}}_{{\mathcal I}}(y)$-action on ${\operatorname{mor}}_{{\mathcal I}}(x,y)$ is free for every two objects $x,y
\in {\operatorname{ob}}({{\mathcal I}})$. Assume there exists a finite ${{\mathcal I}}$-$CW$-model for the ${{\mathcal I}}$-classifying space of ${{\mathcal I}}$. Let ${{\mathcal C}}\colon
{{\mathcal I}}\to {{\EuR}{CAT}}$ be a covariant functor such that for each $i \in {\operatorname{ob}}({{\mathcal I}})$, the category ${{\mathcal C}}(i)$ is a skeletal, finite, EI and the left ${\operatorname{aut}}_{{{\mathcal C}}(i)}(d)$-action on ${\operatorname{mor}}_{{{\mathcal C}}(i)}(c,d)$ is free for every two objects $c,d
\in {\operatorname{ob}}({{\mathcal C}}(i))$. Assume for every object $i \in {\operatorname{ob}}({{\mathcal I}})$, for each automorphism $u \colon i \to i$ in ${{\mathcal I}}$, and each $\overline{x} \in {\operatorname{iso}}({{\mathcal C}}(i))$ we have $\overline{{{\mathcal C}}(u)(x)} =
\overline{x}$.
Then ${{\mathcal H}}:={\operatorname{hocolim}}_{i \in I} {{\mathcal C}}$ is again a skeletal, finite, EI-category such that the left ${\operatorname{aut}}_{{{\mathcal H}}}(h)$-action on ${\operatorname{mor}}_{{{\mathcal H}}}(g,h)$ is free for every two objects $g,h \in {\operatorname{ob}}({\operatorname{hocolim}}_{i \in I} {{\mathcal C}})$, and $$\chi_L({{\mathcal H}}) =
\sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n} \chi_L({{\mathcal C}}(i_\lambda); R).$$
The category ${{\mathcal H}}$ is an EI-category by Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(EI)\]. Skeletality and finiteness of ${{\mathcal H}}$ follow directly from the skeletality and finiteness of ${{\mathcal I}}$ and ${{\mathcal C}}(i)$, and the definition of ${{\mathcal H}}$. The hypotheses on ${{\mathcal C}}(i)$ imply that $\chi^{(2)}({{\mathcal C}}(i))=\chi_L({{\mathcal C}}(i))$ by Theorem \[lem:chi(2)\_and\_chi\], and similarly $\chi^{(2)}({{\mathcal H}})=\chi_L({{\mathcal H}})$. Finally, Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi(2)\], which is the Homotopy Colimit Formula for the $L^2$-Euler characteristic $\chi^{(2)}$, implies the formula is also true for Leinster’s Euler characteristic $\chi_L$ in the special situation of the Corollary.
Scwols and Complexes of Groups {#sec:complexes_of_groups}
==============================
As an illustration of the Homotopy Colimit Formula, we consider Euler characteristics of small categories without loops (*scwols*) and complexes of groups in the sense of Haefliger [@Haefliger(1991)], [@Haefliger(1992)] and Bridson–Haefliger [@Bridson-Haefliger(1999)]. One-dimensional complexes of groups are the classical Bass–Serre graphs of groups [@Serre(1980)]. For finite scwols, the Euler characteristic, $L^2$-Euler characteristic, and Euler characteristic of the classifying space all coincide, essentially because finite scwols admit finite models for their classifying spaces. The Euler characteristic of a finite scwol is particularly easy to find: one simply chooses a skeleton, counts the paths of non-identity morphisms of length $n$, and then computes the alternating sum of these numbers.
Scwols and complexes of groups are combinatorial models for polyhedral complexes and group actions on them. The poset of faces of a polyhedral complex is a scwol. Suppose a group $G$ acts on an $M_\kappa$-polyhedral complex by isometries preserving cell structure, and suppose each group element $g \in G$ fixes each cell pointwise that $g$ fixes setwise. In this case, the quotient is also an $M_\kappa$-polyhedral complex, say $Q$, and we obtain a pseudo functor from its scwol of faces into groups. Namely, to a face $\overline{\sigma}$ of $Q$, one associates the stabilizer $G_\sigma$ for a selected representative $\sigma$ of $\overline{\sigma}$. Inclusions of subfaces of $Q$ then correspond to inclusions of stabilizer subgroups up to conjugation. This pseudo functor is the complex of groups associated to the group action.
However, it is sometimes easier to work directly with the combinatorial model rather than with the original $M_\kappa$-polyhedral complex, and consider instead appropriate group actions on the associated scwol, as in Definition \[def:group\_action\_on\_a\_scwol\]. Then the quotient category of a scwol is again a scwol, and the associated pseudo functor on the quotient scwol is called the *complex of groups associated to the group action*. Any group-valued pseudo functor on a scwol that arises in this way is called *developable*.
Our main results in this section concern the Euler characteristics of homotopy colimits of complexes of groups associated to group actions in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. Theorem \[thm:Euler\_characteristic\_of\_hocolim\_of\_quotient\_complex\], concludes that the Euler characteristic and $L^2$-Euler characteristic of the homotopy colimit are $\chi({{\mathcal X}}/G)$ and $\chi^{(2)}({{\mathcal X}})/\vert G \vert$ respectively, $G$ and ${{\mathcal X}}$ are finite. These formulas provide necessary conditions for developability. That is, if $F$ is a pseudo functor from a scwol ${{\mathcal Y}}$ to groups, one may ask if there are a scwol ${{\mathcal X}}$ and a group $G$ such that ${{\mathcal Y}}$ is isomorphic to ${{\mathcal X}}/G$ and $F$ is the associated complex of groups. To obtain conditions on $\chi({{\mathcal X}})$, $\chi^{(2)}({{\mathcal X}})$, and $\vert G \vert$, one forms the homotopy colimit of $F$, calculates its Euler characteristic and $L^2$-Euler characteristic, and then compares with the formulas of Theorem \[thm:Euler\_characteristic\_of\_hocolim\_of\_quotient\_complex\]. A simple case is illustrated in Example \[exa:necessary\_conditions\_for\_developability:single\_arrow\]. Another application of the formulas is the computation of the Euler characteristic and $L^2$-Euler characteristic for the transport groupoid of a finite left $G$-set, as in Example \[exa:transport\_groupoid\_finite\_case\]. We finish with Theorem \[the:extension\_of\_Haefligers\_corollary\], which extends Haefliger’s formula for the Euler characteristic of the classifying space of the homotopy colimit of a complex of groups in terms of Euler characteristics of lower links and groups.
One novel aspect of our approach is that we do not require scwols to be skeletal. We prove in Theorem \[the:reduction\_to\_skeletal\_case\] that any scwol with a $G$-action in the sense of Definition \[def:group\_action\_on\_a\_scwol\] can be replaced by a skeletal scwol with a $G$-action, and this process preserves quotients, stabilizers, complexes of groups, and homotopy colimits. Moreover, if the initial $G$-action was free on the object set, then so is the $G$-action on the object set of the skeletal replacement.
We begin by recalling the notions in Chapter III.${{\mathcal C}}$ of Bridson–Haefliger [@Bridson-Haefliger(1999)], rephrased in the conceptual framework of 2-category theory.
\[not:2-category\_of\_groups\] We denote by ${{\EuR}{GROUPS}}$ the 2-category of groups. Objects are groups and morphisms are group homomorphisms. The 2-cells are given by conjugation: a 2-cell $(g,a)$ $$\xymatrix{H \ar@/^1.5pc/[rr]_{\quad}^{a}="1"
\ar@/_1.5pc/[rr]_{a'}="2" & & G
\ar@{}"1";"2"|(.2){\,}="7" \ar@{}"1";"2"|(.8){\,}="8" \ar@{=>}"7"
;"8"^{(g,a)} }$$ is an element $g \in G$ such that $ga(h)g^{-1}=a'(h)$ for all $h \in H$. The vertical composition is $(g_2,a_2)\odot (g_1,a_1)=(g_2g_1,a_1)$ and the horizontal composition of $$\xymatrix@C=3pc{ H \ar@/^2pc/[rr]_{\quad}^{a}="1" \ar@/_2pc/[rr]_{a'}
="2" & & \ar@{}"1";"2"|(.2){\,}="7" \ar@{}"1";"2"|(.8){\,}="8"
\ar@{=>}"7" ;"8"^{(g,a)} G
\ar@/^2pc/[rr]_{\quad}^{b}="1"
\ar@/_2pc/[rr]_{b'}="2" & & \ar@{}"1";"2"|(.2){\,}="7"
\ar@{}"1";"2"|(.8){\,}="8" \ar@{=>}"7" ;"8"^{(k,b)}
K}$$ is $(kb(g),ba)$.
A *scwol*[^3] is a [**s**]{}mall [**c**]{}ategory [**w**]{}ith[**o**]{}ut [**l**]{}oops, that is, a small category in which every endomorphism is trivial.
The categories $\{j \rightrightarrows k\}$ and ${{\mathcal I}}=\{k \leftarrow
j \to \ell\}$ of Examples \[exa:finite\_model\_for\_parallel\_arrows\] and \[exa:finite\_model\_for\_pushout\] are scwols. Every partially ordered set is a scwol, for example, the set of simplices of a simplicial complex, ordered by the face relation, is a scwol. The poset of non-empty subsets of $[q]$, and its opposite category in Example \[exa:finite\_model\_for\_q\_interior\], are scwols. The opposite category of a scwol is also a scwol.
\[lem:scwol\_directly\_finite\_EI\] Every scwol is an EI-category and consequently also directly finite.
Every endomorphism in a scwol is trivial, and therefore an automorphism, so every scwol is an EI-category. By Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Lemma 3.13], every EI-category is also directly finite.
For a direct proof of direct finiteness: if $u \colon x \to y$ and $v \colon y \to x$ are morphisms in a scwol, then $vu $ and $uv$ are automorphisms, and hence both $vu = {\operatorname{id}}_x$ and $uv = {\operatorname{id}}_y$ hold automatically.
\[the:finite\_models\_for\_finite\_scwols\] Suppose ${{\mathcal I}}$ is a finite scwol. Then ${{\mathcal I}}$ admits a finite ${{\mathcal I}}$-$CW$-model for its ${{\mathcal I}}$-classifying space in the sense of Definition \[def:classifying\_calc-space\].
By Lemma \[lem:finite\_models\_and\_equivalences\_of\_categories\], we may assume that ${{\mathcal I}}$ is skeletal.
Since ${{\mathcal I}}$ has only finitely many morphisms, no nontrivial isomorphisms, and no nontrivial endomorphisms, there are only finitely many paths of non-identity morphisms. Thus the bar construction of $E^{{\operatorname{bar}}}{{\mathcal I}}$ Remark \[rem:Ebarcalc\] has only finitely many ${{\mathcal I}}$-cells.
\[cor:finite\_scwols\_FF\] Any finite scwol ${{\mathcal I}}$ is of types (FF$_R$) and (FP$_R$) for every associative, commutative ring $R$ with identity. Moreover, any finite scwol is also of type ($L^2$).
The cellular $R$-chains of the finite model in Theorem \[the:finite\_models\_for\_finite\_scwols\] provide a finite, free resolution of the constant module $\underline{R}$. By Theorem \[the:comparing\_o\_and\_chi(2)\], any directly finite category of type (FP$_{{\mathbb C}}$) is of type ($L^2)$. Scwols are directly finite by Lemma \[lem:scwol\_directly\_finite\_EI\].
\[exa:Euler\_characteristics\_of\_finite\_scwols\] Let ${{\mathcal I}}$ be any finite scwol. Then by Corollary \[cor:finite\_scwols\_FF\] it is of type (FF$_R$) for any associative, commutative ring with identity, and by Theorems \[the:chi\_f\_determines\_chi\] and \[the:coincidence\], we have $$\chi({{\mathcal I}};R)=\chi(B{{\mathcal I}};R)=\chi^{(2)}({{\mathcal I}}).$$ If $\Gamma$ is any skeleton of ${{\mathcal I}}$, then by , $$\label{equ:Euler_characteristic_given_by_paths}
\chi(\Gamma;R)=\sum_{n \geq 0}
(-1)^n c_n(\Gamma),$$ where $c_n(\Gamma)$ is the number of paths of $n$-many non-identity morphisms in $\Gamma$. But by Fiore–Lück–Sauer [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Theorem 2.8 and Corollary 4.19], type (FF$_R$) and the Euler characteristic are invariant under equivalence of categories between directly finite categories, so $\chi({{\mathcal I}};R)=\chi(\Gamma;R)$ and all three invariants $\chi({{\mathcal I}};R)$, $\chi(B{{\mathcal I}};R)$, $\chi^{(2)}({{\mathcal I}})$ are given by .
We now arrive at the main notion of this section: a complex of groups. We will apply our Homotopy Colimit Formula to complexes of groups.
\[def:complex\_of\_groups\] Let ${{\mathcal Y}}$ be a scwol. A *complex of groups over ${{\mathcal Y}}$* is a pseudo functor $F\colon {{\mathcal Y}}\to {{\EuR}{GROUPS}}$ such that $F(a)$ is injective for every morphism $a$ in ${{\mathcal Y}}$. For each object $\sigma$ of ${{\mathcal Y}}$, the group $F(\sigma)$ is called the *local group at $\sigma$*.
In 2.5 and 2.4 of [@Haefliger(1991)] and [@Haefliger(1992)] respectively, Haefliger denotes by $CG(X)$ the homotopy colimit of a complex of groups $G(X)\colon C(X) \to {{\EuR}{GROUPS}}$. Bridson–Haefliger use the notation $CG({{\mathcal Y}})$ in [@Bridson-Haefliger(1999) III.${{\mathcal C}}$.2.8]. The fundamental group of a complex of groups $G(X)$ in the sense of [@Bridson-Haefliger(1999) Definition 3.5 on p. 548] equals the fundamental group of the geometric realization of $CG(X)$ [@Bridson-Haefliger(1999) Appendix A.12 on p. 578 and Remark A.14 on p. 579]. Categories which are homotopy colimits of complexes of groups are characterized by Haefliger on page 283 of [@Haefliger(1992)]. From the homotopy colimit $CG(X)$, Haefliger reconstructs the category $C(X)$ and the complex of groups $G(X)$ up to a coboundary on pages 282-283 of [@Haefliger(1992)]. Every aspherical realization [@Haefliger(1992) Definition 3.3.4] of $G(X)$ has the homotopy type of the geometric realization of the homotopy colimit, denoted $BG(X)$ [@Haefliger(1992) page 296]. The homotopy colimit also plays a role in the homology and cohomology of complexes of groups [@Haefliger(1992) Section 4]; a left $G(X)$-module is a functor $CG(X)\to {{\EuR}{ABELIAN}\text{-}{\EuR}{GROUPS}}$.
We return to our recollection of complexes of groups and examples that arise from group actions.
A *morphism from a complex of groups $F$ to a group $G$* is a pseudo natural transformation $F \Rightarrow \Delta_G$, where $\Delta_G$ indicates the constant 2-functor ${{\mathcal Y}}\to {{\EuR}{GROUPS}}$ with value $G$.
A typical example of a complex of groups equipped with a morphism to a group $G$ arises from an action of a group $G$ on a scwol, as we now explain.
\[def:group\_action\_on\_a\_scwol\] An *action of a group $G$ on a scwol ${{\mathcal X}}$* is a group homomorphism from $G$ into the group of strictly invertible functors ${{\mathcal X}}\to {{\mathcal X}}$ such that
1. \[def:group\_action\_on\_a\_scwol:(i)\] For every nontrivial morphism $a$ of ${{\mathcal X}}$ and every $g \in G$, we have $gs(a) \neq t(a)$,
2. \[def:group\_action\_on\_a\_scwol:(ii)\] For every nontrivial morphism $a$ of ${{\mathcal X}}$ and every $g \in G$, if $gs(a)=s(a)$, then $ga=a$.
\[exa:Z2\_action\_on\_circle\] The group $G={{\mathbb Z}}_2$ acts in the sense of Definition \[def:group\_action\_on\_a\_scwol\] on the scwol ${{\mathcal X}}$ pictured below. $$\xymatrix{x \ar[r]^{h} \ar[d]_{g} & z \\ y & x' \ar[l]^{g'}
\ar[u]_{h'}}$$ The group ${{\mathbb Z}}_2$ permutes respectively $x$ and $x'$, $g$ and $g'$, and $h$ and $h'$. The objects $y$ and $z$ are fixed. This action of ${{\mathbb Z}}_2$ on ${{\mathcal X}}$ is a combinatorial model for a reflection action on $S^1$.
\[exa:Z2\_Z\_action\_on\_line\] Consider the scwol ${{\mathcal X}}$ pictured below. The group $G=\{\pm 1
\}\ltimes {{\mathbb Z}}$ acts on ${{\mathcal X}}$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\] where $-1 \cdot m:=-m$ and $n\cdot m:=m +2n$. $$\xymatrix@C=2pc{ \cdots \ar[r] & -2 & -1 \ar[r] \ar[l] & 0 & 1
\ar[r] \ar[l] & 2 & \cdots \ar[l] }$$ This action of $\{\pm 1 \}\ltimes {{\mathbb Z}}$ on ${{\mathcal X}}$ is a combinatorial model for the reflection and translation action on $\mathbb{R}$.
\[lem:consequences\_of\_group\_action\] If a group $G$ acts on a scwol ${{\mathcal X}}$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\], then the following statements hold.
1. \[lem:consequences\_of\_group\_action:(i)\] If $\sigma$ is an object of ${{\mathcal X}}$ and $g, h \in G$, then $g \sigma \cong h \sigma$ implies $g \sigma = h \sigma$.
2. \[lem:consequences\_of\_group\_action:(ii)\] If $a$ is a morphism in ${{\mathcal X}}$ and $g,h\in G$, then $gs(a)=hs(a)$ implies $ga=ha$.
3. \[lem:consequences\_of\_group\_action:(iii)\] If $\sigma\cong\tau$, then the stabilizers $G_\sigma$ and $G_\tau$ are equal.
For statement \[lem:consequences\_of\_group\_action:(i)\], $g \sigma \cong h \sigma$ implies $\sigma \cong (g^{-1}h) \sigma$, so $\sigma = (g^{-1}h) \sigma$ by Definition \[def:group\_action\_on\_a\_scwol\] part \[def:group\_action\_on\_a\_scwol:(i)\], and $g\sigma =h \sigma$.
For statement \[lem:consequences\_of\_group\_action:(ii)\], $gs(a)=hs(a)$ implies $(h^{-1} g) s(a) = s(a)$ and $(h^{-1} g) a = a$ by Definition \[def:group\_action\_on\_a\_scwol\] part \[def:group\_action\_on\_a\_scwol:(ii)\], and finally $ga=ha$.
For statement \[lem:consequences\_of\_group\_action:(iii)\], suppose $\sigma\cong\tau$ and $g\sigma=\sigma$. We have $$\tau \cong \sigma = g\sigma \cong g\tau.$$ Then $\tau=g\tau$ by \[lem:consequences\_of\_group\_action:(i)\], and $G_\sigma \subseteq G_\tau$. The proof is symmetric, so we also have $G_\tau \subseteq G_\sigma$.
\[def:quotient\_scwol\] If a scwol ${{\mathcal X}}$ is equipped with a $G$-action as above, then the *quotient scwol* ${{\mathcal X}}/ G$ has objects and morphisms $${\operatorname{ob}}({{\mathcal X}}/G ) := ({\operatorname{ob}}({{\mathcal X}})) /G$$ $${\operatorname{mor}}({{\mathcal X}}/G ) := ({\operatorname{mor}}({{\mathcal X}})) /G.$$ Composition and identities are induced by those of ${{\mathcal X}}$.
\[rem:projection\_functor\_for\_group\_action\] The projection functor $p\colon {{\mathcal X}}\to {{\mathcal X}}/ G$ induces a bijection $$\label{equ:projection_covering_1} \xymatrix{\{a \in
{\operatorname{mor}}({{\mathcal X}}) \vert sa=x\} \ar[r] & \{b \in {\operatorname{mor}}({{\mathcal X}}/ G)\vert sb=p(x) \}}$$ for each $x \in {{\mathcal X}}$. If $G / {{\mathcal X}}$ is connected and the action of $G$ on ${\operatorname{ob}}({{\mathcal X}})$ is free, then $p$ is a *covering of scwols*. That is, in addition to the bijection , $p$ induces a bijection $$\label{equ:projection_covering_2} \xymatrix{\{a \in
{\operatorname{mor}}({{\mathcal X}}) \vert ta=x\} \ar[r] & \{b \in {\operatorname{mor}}({{\mathcal X}}/ G)\vert tb=p(x) \}}$$ for each $x \in {{\mathcal X}}$.
\[lem:quotient\_scwol\_of\_skeletal\_scwol\_is\_skeletal\] If ${{\mathcal X}}$ is a skeletal scwol, and a group $G$ acts on ${{\mathcal X}}$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\], then the quotient scwol ${{\mathcal X}}/ G$ is also skeletal.
Suppose $\overline{\sigma}$ is isomorphic to $\overline{\tau}$ in ${{\mathcal X}}/ G$. We show $\overline{\sigma}$ is actually equal to $\overline{\tau}$. If $\overline{a}\colon \overline{\sigma} \to
\overline{\tau}$ is an isomorphism with inverse $\overline{b}$, then there are lifts $a\colon \sigma \to \tau$ and $b \colon \tau \to
\sigma'$ in ${{\mathcal X}}$, and an element $g \in G$ such that $g(ba)={\operatorname{id}}_\sigma$. Since $g$ fixes the source of $ba$, the group element $g$ fixes also $ba$, so $ba={\operatorname{id}}_\sigma$ and $\sigma'=\sigma$. Since $ab$ is an endomorphism of $\tau$, it is therefore ${\operatorname{id}}_\tau$. By the skeletality of ${{\mathcal X}}$, we have $\sigma
= \tau$, and also $\overline{\sigma}=\overline{\tau}$.
\[lem:cells\_of\_quotient\_scwol\_are\_quotient\_of\_scwol\_cells\] Suppose ${{\mathcal X}}$ is a scwol equipped with an action of a group $G$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. Let $\Lambda_n({{\mathcal X}})$ respectively $\Lambda_n({{\mathcal X}}/G)$ denote the set of paths of $n$-many non-identity composable morphisms in ${{\mathcal X}}$ respectively ${{\mathcal X}}/G$. Give $\Lambda_n({{\mathcal X}})$ the induced $G$-action. Then the function $$\Lambda_n({{\mathcal X}}) \to \Lambda_n({{\mathcal X}}/G)$$ $$(a_1,\dots,a_n) \mapsto (\overline{a}_1, \dots, \overline{a}_n)$$ induces a bijection $\Lambda_n({{\mathcal X}})/G\to \Lambda_n({{\mathcal X}}/G)$.
Remark \[rem:projection\_functor\_for\_group\_action\] implies that a path $(a_1,\dots,a_n)$ in ${{\mathcal X}}$ consists entirely of non-identity morphisms if and only if the projection $(\overline{a}_1, \dots,
\overline{a}_n)$ in ${{\mathcal X}}/G$ consists entirely of non-identity morphisms, so from now on we work only with non-identity morphisms. Note $$(g_1a_1,g_2a_2,\dots,g_na_n)=(g_1a_1,g_1a_2,\dots,g_1a_n)$$ by Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(ii)\]. For injectivity, we have $(\overline{a}_1, \dots,
\overline{a}_n)=(\overline{b}_1, \dots, \overline{b}_n)$ if and only if for some $g_i \in G$ $$(g_1a_1,g_2a_2,\dots,g_na_n)=(b_1,\dots,b_n),$$ which happens if and only if for some $g \in G$ $$(ga_1,ga_2,\dots,ga_n)=(b_1,\dots,b_n),$$ (take $g=g_1$). For the surjectivity, we can lift any path $(\overline{a}_1, \dots, \overline{a}_n)$ by first lifting $\overline{a}_1$ to $a_1$, then $\overline{a}_2$ to $a_2$, and so on using Remark \[rem:projection\_functor\_for\_group\_action\].
\[def:complex\_of\_groups\_from\_a\_group\_action\] Let $G$ be a group and ${{\mathcal X}}$ a scwol upon which $G$ acts in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. Let $p \colon
{{\mathcal X}}\to {{\mathcal X}}/G$ denote the quotient map.
Haefliger and Bridson–Haefliger define a pseudo functor $F\colon
{{\mathcal X}}/G \to {{\EuR}{GROUPS}}$ as follows. In the procedure choices are made, but different choices lead to isomorphic complexes of groups. For each object $\overline{\sigma}$ of ${{\mathcal X}}/G$, choose an object $\sigma$ of ${{\mathcal X}}$ such that $p(\sigma)=\overline{\sigma}$ (our overline convention is the opposite of that in [@Bridson-Haefliger(1999)]). Then $F(\overline{\sigma})$ is defined to be $G_\sigma$, the isotropy group of $\sigma$ under the $G$-action.
If $\overline{a}\colon \overline{\sigma} \to \overline{\tau}$ is a morphism in ${{\mathcal X}}/G$, then there exists a unique morphism $a$ in ${{\mathcal X}}$ such that $p(a)=\overline{a}$ and $sa=\sigma$, as in . For $\overline{a}$ we choose an element $h_{\overline{a}} \in G$ such that $h_{\overline{a}} \cdot
ta$ is the object $\tau$ of ${{\mathcal X}}$ chosen above so that $p(\tau)=\overline{\tau}$. An injective group homomorphism $F(\overline{a})\colon G_\sigma \to G_\tau$ is defined by $$F(\overline{a})(g):=h_{\overline{a}} g h_{\overline{a}}^{-1}.$$
Suppose $\overline{a}$ and $\overline{b}$ are composable morphisms of ${{\mathcal X}}/G$. We define a 2-cell in ${{\EuR}{GROUPS}}$ $$F_{\overline{b},\overline{a}}\colon F(\overline{b}) \circ F(\overline{a}) \Rightarrow
F(\overline{b} \circ \overline{a})$$ to be $(h_{\overline{b}\overline{a}}h_{\overline{a}}^{-1}
h_{\overline{b}}^{-1} ,F(\overline{b}) \circ F(\overline{a}))$ as in Notation \[not:2-category\_of\_groups\].
The pseudo functor $F\colon {{\mathcal X}}/G \to {{\EuR}{GROUPS}}$ is called the [*complex of groups associated to the group action of $G$ on the scwol ${{\mathcal X}}$*]{}. This complex of groups comes equipped with a morphism to the group $G$, that is, a pseudo natural transformation $F \Rightarrow \Delta_G$. The inclusion of each isotropy group $F(\overline{\sigma})=G_\sigma$ into $G$ provides the components of the pseudo natural transformation.
\[exa:Z2\_complex\_of\_groups\] The quotient scwols for the actions in Examples \[exa:Z2\_action\_on\_circle\] and \[exa:Z2\_Z\_action\_on\_line\] are both $\{k \leftarrow j \rightarrow \ell \}$, and the associated complexes of groups are both $$\xymatrix{{{\mathbb Z}}_2 & \{0\} \ar[l] \ar[r] &{{\mathbb Z}}_2.}$$
\[rem:finite\_group\_finite\_scwol\_imply\_hocolim\_hypotheses\_satisfied\] If a group $G$ acts on a scwol in the sense of Definition \[def:group\_action\_on\_a\_scwol\], each object stabilizer is finite, and the quotient scwol is finite, then the associated complex of groups $F\colon {{\mathcal X}}/G \to {{\EuR}{GROUPS}}$ satisfies all of the hypotheses of the Homotopy Colimit Formula in Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi(2)\] and in Corollary \[cor:homotopy\_colimit\_formula\_for\_pseudo\_functors\] \[the:homotopy\_colimit\_formula:chi(2)\]. If, in addition, $R$ is a ring such that the order $\vert H\vert $ of each object stabilizer $H\subset G$ is invertible in $R$, then $F\colon {{\mathcal X}}/G \to {{\EuR}{GROUPS}}$ also satisfies all of the hypotheses of the Homotopy Colimit Formula in Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi\] and in Corollary \[cor:homotopy\_colimit\_formula\_for\_pseudo\_functors\] \[the:homotopy\_colimit\_formula:chi\]. See Examples \[exa:Z2\_action\_on\_circle\], \[exa:Z2\_Z\_action\_on\_line\], and \[exa:Z2\_complex\_of\_groups\].
Even without finiteness assumptions, it is possible to replace scwols with skeletal scwols and preserve much of the accompanying structure, as Theorem \[the:reduction\_to\_skeletal\_case\] explains.
\[the:reduction\_to\_skeletal\_case\] Let $G$ be a group acting on a scwol ${{\mathcal X}}$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. Let $\Gamma$ be any skeleton of ${{\mathcal X}}$, $i\colon \Gamma \to {{\mathcal X}}$ the inclusion, and $r \colon {{\mathcal X}}\to \Gamma$ a functor equipped with a natural isomorphism $ir \cong {\operatorname{id}}_{{\mathcal X}}$, and satisfying $r i = {\operatorname{id}}_\Gamma$. Then there is a $G$-action on the scwol $\Gamma$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\] such that following hold.
1. \[the:reduction\_to\_skeletal\_case:r\_equivariant\] The functor $r$ is $G$-equivariant.
2. \[the:reduction\_to\_skeletal\_case:r\_induces\_equivalence\_of\_quotients\] The induced functor $\overline{r}$ on quotient categories is an equivalence of categories compatible with the quotient maps, that is, the diagram below commutes. $$\label{equ:the:reduction_to_skeletal_case:rbar_compatible_with_projections}
\xymatrix{{{\mathcal X}}\ar[r]^r \ar[d]_{p^{{\mathcal X}}} & \Gamma \ar[d]^{p^{\Gamma}} \\
{{\mathcal X}}/G \ar[r]_{\overline{r}} & \Gamma/G}$$
3. \[the:reduction\_to\_skeletal\_case:inclusion\_preserves\_stabilizers\] The inclusion $i\colon \Gamma \to {{\mathcal X}}$ preserves stabilizers, that is $G_{i\gamma}=G_{\gamma}$ for all $\gamma \in {\operatorname{ob}}(\Gamma)$. Note that the inclusion may not be $G$-equivariant.
4. \[the:reduction\_to\_skeletal\_case:complexes\_of\_groups\_agree\] Choices can be made in the definitions of $F^{{\mathcal X}}$ and $F^\Gamma$ (the complexes of groups associated to the $G$-actions on ${{\mathcal X}}$ and $\Gamma$ in Definition \[def:complex\_of\_groups\_from\_a\_group\_action\]), so that the diagram below strictly commutes. $$\label{equ:the:reduction_to_skeletal_case:complexes_of_groups_agree}
\xymatrix{{{\mathcal X}}/G \ar[rr]^{\overline{r}} \ar[dr]_{F^{{\mathcal X}}} & &
\Gamma/G \ar[dl]^{F^\Gamma} \\ & {{\EuR}{GROUPS}}& }$$
5. \[the:reduction\_to\_skeletal\_case:homotopy\_colimits\_equivalent\] The functor $(\overline{r},{\operatorname{id}})$ is an equivalence of categories $$\xymatrix{(\overline{r},{\operatorname{id}})\colon {\operatorname{hocolim}}_{{{\mathcal X}}/G} F^{{\mathcal X}}\ar[r] & {\operatorname{hocolim}}_{\Gamma/G} F^\Gamma.}$$
6. \[the:reduction\_to\_skeletal\_case:free\_implies\_free\] If $G$ acts freely on ${\operatorname{ob}}({{\mathcal X}})$, then $G$ acts freely on ${\operatorname{ob}}(\Gamma)$.
To define the group action, let ${\operatorname{Aut}}({{\mathcal X}})$ and ${\operatorname{Aut}}(\Gamma)$ denote the strictly invertible endofunctors on ${{\mathcal X}}$ and $\Gamma$ respectively, and consider the monoid homomorphism $$\label{eqn:action_on_skeleton}
\varphi \colon {\operatorname{Aut}}({{\mathcal X}}) \to {\operatorname{End}}(\Gamma), \;\; F \mapsto r \circ F \circ i.$$ This is strictly multiplicatively because the natural isomorphism of functors $$\begin{aligned}
r \circ G \circ F \circ i & = & r \circ G \circ {\operatorname{id}}_{{{\mathcal X}}} \circ F \circ i \\
& \cong & (r \circ G \circ i) \circ (r \circ F \circ i),\end{aligned}$$ and skeletality of $\Gamma$ imply $\varphi(GF)$ agrees with $\varphi(G)\varphi(F)$ on objects of $\Gamma$, so each component $\varphi(GF)(\gamma)\cong\varphi(G)\varphi(F)(\gamma)$ is an endomorphism in the scwol $\Gamma$, and is therefore trivial. By naturality, $\varphi(GF)$ and $\varphi(G)\varphi(F)$ agree on morphisms also. Consequently, $\varphi$ takes values in ${\operatorname{Aut}}(\Gamma)$ and is a homomorphism $\varphi \colon {\operatorname{Aut}}({{\mathcal X}}) \to {\operatorname{Aut}}(\Gamma)$.
We define a $G$-action on $\Gamma$ as the composite of the action $G
\to {\operatorname{Aut}}({{\mathcal X}})$ with $\varphi$ in . We indicate the action of $g$ on $\Gamma$ by $\varphi(g)\gamma$ and the action of $g$ on ${{\mathcal X}}$ by $gx$. For simplicity, we suppress $i$ from the notation when indicating the $G$-action in ${{\mathcal X}}$ on objects and morphisms of $\Gamma$, so for example, if $a$ is morphism in $\Gamma$, then $gs(a)$ actually means $gis(a)$ throughout.
To verify Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(i)\] for $\Gamma$, suppose $a$ is a nontrivial morphism in $\Gamma$ and $\varphi(g)s(a)=t(a)$, that is $rgs(a)=t(a)$. Then $gs(a)
\cong t(a)$ in ${{\mathcal X}}$, but $gs(a) \neq t(a)$ (for if $gs(a)=t(a)$, then $a$ must be trivial by Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(i)\] for ${{\mathcal X}}$). Let $b\colon t(a) \to gs(a)$ be an isomorphism in ${{\mathcal X}}$ and consider the composite $ba \colon s(a) \to t(a)
\to gs(a)$. Then $gs(ba)=gs(a)=t(ba)$, so $ba$ must be trivial by Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(i)\] for ${{\mathcal X}}$. Consequently $a=b^{-1}$ is a nontrivial *iso*morphism in $\Gamma$, and we have a contradiction to either skeletality or the no loops requirement. Thus $\varphi(g)s(a)\neq t(a)$, and Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(i)\] holds for $\Gamma$. The verification of Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(ii)\] is shorter: if $a$ is a nontrivial morphism in $\Gamma$ and $\varphi(g)s(a)=s(a)$, that is $rgs(a)=s(a)$, then $gs(a)\cong s(a)$, and $gs(a)=s(a)$ by Lemma \[lem:consequences\_of\_group\_action\] \[lem:consequences\_of\_group\_action:(i)\] for ${{\mathcal X}}$. Finally, $ga=a$ by Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(ii)\] for ${{\mathcal X}}$, $rga=a$ as $a$ is in $\Gamma$, and $\varphi(g)a=a$. The action of $G$ on $\Gamma$ satisfies Definition \[def:group\_action\_on\_a\_scwol\] and we may form the quotient scwol $\Gamma/G$ as in Defition \[def:quotient\_scwol\], which is skeletal by Lemma \[lem:quotient\_scwol\_of\_skeletal\_scwol\_is\_skeletal\].\
\[the:reduction\_to\_skeletal\_case:r\_equivariant\] For the $G$-equivariance of $r$, let $f\colon x \to y$ be a morphism in ${{\mathcal X}}$ and consider the naturality diagram. $$\xymatrix@C=6pc{rgirx \ar[r]^{rgirf=\varphi(g)r(f)} \ar[d]_\cong & rgiry \ar[d]^\cong \\ rgx \ar[r]_{rgf} & rgy}$$ The vertical morphisms must be identities by skeletality of $\Gamma$ and the no loops condition, so $\varphi(g)r(f)=r(gf)$. Equivariance on objects then follows by taking $f={\operatorname{id}}_x$.\
\[the:reduction\_to\_skeletal\_case:r\_induces\_equivalence\_of\_quotients\] Diagram commutes by definition of $\overline{r}$. The functor $\overline{r}$ is surjective on objects because $p^\Gamma r$ and $p^{{\mathcal X}}$ are. The functor $\overline{r}$ is fully faithful since the equivariant bijection $r(x,y)\colon {\operatorname{mor}}_{{\mathcal X}}(x,y) \to {\operatorname{mor}}_\Gamma(r(x),r(y))$ induces the equivariant bijection $\overline{r}(p^{{\mathcal X}}x, p^{{\mathcal X}}y)$.\
\[the:reduction\_to\_skeletal\_case:inclusion\_preserves\_stabilizers\] Let $\gamma \in {\operatorname{ob}}(\Gamma)$, and suppose $gi\gamma=i\gamma$. Then $$\begin{aligned}
\varphi(g)\gamma & \overset{\text{def}}{=} & r(g i\gamma) \\
& = & r (i \gamma) \\
& = & \gamma\end{aligned}$$ and $G_{i\gamma} \subseteq G_\gamma$. Now suppose $\varphi(g)\gamma =\gamma$. Then $r(gi\gamma)=\gamma$ by definition, and $g i \gamma
\cong i\gamma$ in ${{\mathcal X}}$, which says $g\cdot i \gamma = i\gamma$ by Lemma \[lem:consequences\_of\_group\_action\] \[lem:consequences\_of\_group\_action:(i)\], and $G_\gamma \subseteq G_{i\gamma} $.\
\[the:reduction\_to\_skeletal\_case:complexes\_of\_groups\_agree\] We claim that choices can be made in the definitions of the associated complexes of groups $F^{{\mathcal X}}$ and $F^\Gamma$ (see Definition \[def:complex\_of\_groups\_from\_a\_group\_action\]) so that diagram strictly commutes. First choose a skeleton ${{\mathcal Q}}$ of the quotient ${{\mathcal X}}/G$, define $F^{{\mathcal X}}$ on object in the skeleton ${{\mathcal Q}}$, and then extend to all objects in ${{\mathcal X}}/G$. For every $\overline{q} \in
{\operatorname{ob}}({{\mathcal Q}})$, select a $q \in {\operatorname{ob}}({{\mathcal X}})$ such that $p^{{\mathcal X}}(q)=\overline{q}$ and define $F^{{\mathcal X}}(\overline{q})=G_q$. We remain with the choice of the selected preimage $q$ of $\overline{q}$ throughout. If $\overline{\sigma} \in {\operatorname{ob}}({{\mathcal X}}/G)$ and $\overline{a}\colon \overline{q} \cong \overline{\sigma}$ is an isomorphism in ${{\mathcal X}}/G$, then also define $F^{{\mathcal X}}(\overline{\sigma})=G_q$. This is allowed, since $\overline{a}\colon \overline{q}\cong \overline{\sigma}$ implies existence of morphisms $a\colon q \to g_\sigma \sigma$ and $b \colon
\sigma \to g_qq$ in ${{\mathcal X}}$, and the composite $$\xymatrix{q \ar[r]^{a} & g_\sigma\sigma \ar[r]^{g_\sigma b} &
g_\sigma g_q q}$$ is trivial by Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(i)\]. The opposite composite is also trivial, as it is a loop, and we have $q \cong g_\sigma \sigma$ in ${{\mathcal X}}$. Then by Lemma \[lem:consequences\_of\_group\_action\] \[lem:consequences\_of\_group\_action:(iii)\], $G_q = G_{g_\sigma \sigma}$ and we may define $F^{{\mathcal X}}(\overline{\sigma})=G_q$ because $p^{{{\mathcal X}}}(g_\sigma
\sigma)=\overline{\sigma}$. In particular, the selected preimage of $\overline{\sigma}$ in ${{\mathcal X}}$ is $g_\sigma \sigma$ and we select $h_{\overline{a}}=e_G$ for $\overline{a}\colon \overline{q}\cong
\overline{\sigma}$ in Definition \[def:complex\_of\_groups\_from\_a\_group\_action\], so $F^{{\mathcal X}}(\overline{a})={\operatorname{id}}_{G_q}$. We remark that the isomorphism $\overline{a}$ is the only morphism $\overline{q} \to \overline{\sigma}$ because there are no loops in ${{\mathcal X}}/G$, so the element $g_\sigma \sigma$ is uniquely defined as the target of the unique morphism $a$ with source $q$ and $p^{{\mathcal X}}$-image $\overline{a}$.
We next define $F^\Gamma$ on objects of $\Gamma/G$ using the equivalence $\overline{r}$ and the definition of $F^{{\mathcal X}}$ on objects of ${{\mathcal Q}}$. For $\overline{q} \in {\operatorname{ob}}({{\mathcal Q}})$, we also define $F^\Gamma(\overline{r}(\overline{q}))=G_q$. This is allowed: for $\overline{r}(\overline{q})=\overline{r(q)}$ we choose $r(q)$ as the selected preimage in ${\operatorname{ob}}(\Gamma)$, and $ir(q) \cong q$ in ${{\mathcal X}}$, so $G_{r(q)}=G_{ir(q)}=G_q$ by \[the:reduction\_to\_skeletal\_case:inclusion\_preserves\_stabilizers\] and Lemma \[lem:consequences\_of\_group\_action\] \[lem:consequences\_of\_group\_action:(iii)\]. Every $\overline{\gamma} \in {\operatorname{ob}}(\Gamma/G)$ is of the form $\overline{r}(\overline{q})$ for a unique $\overline{q} \in {{\mathcal Q}}$, so $F^\Gamma$ is now defined on all objects of $\Gamma/G$, and we have $F^\Gamma \circ \overline{r}=F^{{\mathcal X}}$ on all objects of ${{\mathcal X}}/G$.
We must now define $F^{{\mathcal X}}$ and $F^\Gamma$ on morphisms so that $F^\Gamma \circ \overline{r}=F^{{\mathcal X}}$ for morphisms also. The idea is to first define $F^{{\mathcal X}}$ on morphisms in the skeleton ${{\mathcal Q}}$, then extend to all of ${{\mathcal X}}/G$, and then define $F^\Gamma$ on morphisms of $\Gamma/G$. If $\overline{a}\colon \overline{q}_1 \to \overline{q}_2$ is a morphism in ${{\mathcal Q}}$, then there is a unique morphism $a$ in ${{\mathcal X}}$ with source $q_1$ and $p^{{\mathcal X}}(a)=\overline{a}$. Select any $h_{\overline{a}}$ such that $h_{\overline{a}}ta=q_2$. Then we define an injective group homomorphism $F(\overline{a})\colon
G_{q_1} \to G_{q_2}$ by $$F(\overline{a})(g):=h_{\overline{a}} g h_{\overline{a}}^{-1}.$$ If $\overline{b}\colon \overline{\sigma}_1 \to \overline{\sigma}_2$ is any morphism in ${{\mathcal X}}/G$, then there exists a unique $\overline{a}$ in ${{\mathcal Q}}$ and a unique commutative diagram with vertical isomorphisms as below. $$\xymatrix{\overline{q}_1 \ar[r]^{\overline{a}} \ar[d]_{\cong} &
\overline{q}_2 \ar[d]^{\cong} \\ \overline{\sigma}_1
\ar[r]_{\overline{b}} & \overline{\sigma}_2 }$$ Then we choose $h_{\overline{b}}$ to be $h_{\overline{a}}$, and we consequently have $F(\overline{a})=F(\overline{b})$. If $\overline{c}\colon \overline{r}(\overline{q}_1) \to
\overline{r}(\overline{q}_2)$ is a morphism in $\Gamma/G$, then there is a unique $\overline{a}\colon \overline{q}_1 \to
\overline{q}_2$ in ${{\mathcal Q}}$ with $\overline{r}(\overline{a})=\overline{c}$ and we choose $h_{\overline{c}}$ to be $h_{\overline{a}}$. Manifestly, we have $F^\Gamma \circ \overline{r}=F^{{\mathcal X}}$. The coherences of $F^{{\mathcal X}}$ and $F^\Gamma$ are also compatible, since they are determined by the $h_{\overline{a}}$’s.\
\[the:reduction\_to\_skeletal\_case:homotopy\_colimits\_equivalent\] From \[the:reduction\_to\_skeletal\_case:r\_induces\_equivalence\_of\_quotients\] we know $\overline{r}$ is a surjective-on-objects equivalence of categories and from \[the:reduction\_to\_skeletal\_case:complexes\_of\_groups\_agree\] we have $F^{{\mathcal X}}=F^\Gamma \circ \overline{r}$. From this, one sees $$\xymatrix{(\overline{r},{\operatorname{id}})\colon {\operatorname{hocolim}}_{{{\mathcal X}}/G} F^{{\mathcal X}}= {\operatorname{hocolim}}_{{{\mathcal X}}/G} F^\Gamma \circ \overline{r} \ar[r] & {\operatorname{hocolim}}_{\Gamma/G} F^\Gamma}$$ is an equivalence of categories.\
\[the:reduction\_to\_skeletal\_case:free\_implies\_free\] If the action of $G$ on ${\operatorname{ob}}({{\mathcal X}})$ is free, then for each $\gamma \in {\operatorname{ob}}(\Gamma)$, the group $G_\gamma=G_{i\gamma}$ (see \[the:reduction\_to\_skeletal\_case:inclusion\_preserves\_stabilizers\]) is trivial, and $G$ acts freely on ${\operatorname{ob}}(\Gamma)$.
In Theorem \[the:reduction\_to\_skeletal\_case\], it is even possible to select a skeleton so that the inclusion is $G$-equivariant, though we will not need this. See Section \[sec:appendix\].
In [@FioreLueckSauerFinObsAndEulCharOfCats(2009) Theorems 5.30 and 5.37], we proved the compatibility of the $L^2$-Euler characteristic with coverings and isofibrations of finite connected groupoids. Theorem \[the:compatibility\_of\_chi\_with\_free\_group\_action\_on\_finite\_scowls\] is an analogue for scwols (see Remark \[rem:projection\_functor\_for\_group\_action\]).
\[the:compatibility\_of\_chi\_with\_free\_group\_action\_on\_finite\_scowls\] Let $G$ be a finite group acting on a finite scwol ${{\mathcal X}}$. If $G$ acts freely on ${\operatorname{ob}}({{\mathcal X}})$, then $$\chi({{\mathcal X}}/G;R)=\frac{\chi({{\mathcal X}};R)}{\vert G \vert} \;\; \text{ and }
\; \; \chi^{(2)}({{\mathcal X}}/G)=\frac{\chi^{(2)}({{\mathcal X}})}{\vert G \vert}.$$ Recall $\chi(-;R)$ and $\chi^{(2)}(-)$ agree for finite scwols by Example \[exa:Euler\_characteristics\_of\_finite\_scwols\].
By Theorem \[the:reduction\_to\_skeletal\_case\] \[the:reduction\_to\_skeletal\_case:r\_equivariant\], \[the:reduction\_to\_skeletal\_case:r\_induces\_equivalence\_of\_quotients\], and \[the:reduction\_to\_skeletal\_case:free\_implies\_free\], we may assume ${{\mathcal X}}$ is skeletal.
A consequence of Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(ii)\] (independent of skeletality) is that an element $g \in G$ fixes a path $a=(a_1,\dots,a_n)$ in ${{\mathcal X}}$ if and only if $g$ fixes $sa_1$, so $G_{sa_1}=G_a$. Then $G$ acts freely on $\Lambda_n({{\mathcal X}})$, since it acts freely on ${\operatorname{ob}}({{\mathcal X}})$.
The scwol ${{\mathcal X}}/G$ is skeletal by Lemma \[lem:quotient\_scwol\_of\_skeletal\_scwol\_is\_skeletal\], and by Example \[exa:Euler\_characteristics\_of\_finite\_scwols\] and Lemma \[lem:cells\_of\_quotient\_scwol\_are\_quotient\_of\_scwol\_cells\] we have $$\begin{aligned}
\chi^{(2)}({{\mathcal X}}/G) & = & \sum_{n \geq 0} (-1)^n c_n({{\mathcal X}}/G) \\
& = & \sum_{n \geq 0} (-1)^n \vert\Lambda_n({{\mathcal X}}/G) \vert \\
& = & \sum_{n \geq 0} (-1)^n \vert\Lambda_n({{\mathcal X}})/G \vert \\
& = & \sum_{n \geq 0} (-1)^n \frac{\vert\Lambda_n({{\mathcal X}})\vert}{\vert G\vert} \\
& = & \frac{1}{\vert G\vert}\sum_{n \geq 0} (-1)^n \vert\Lambda_n({{\mathcal X}})\vert \\
& = & \frac{1}{\vert G\vert}\sum_{n \geq 0} (-1)^n c_n({{\mathcal X}}) \\
& = & \frac{\chi^{(2)}({{\mathcal X}})}{\vert G \vert}.\end{aligned}$$
A complex of groups is called *developable* if it is isomorphic to a complex of groups associated to a group action. A classical theorem of Bass–Serre says that every complex of groups on a scwol with maximal path length 1 is developable. The following gives a necessary condition of developability of a complex of groups from a scwol and group of specified Euler characteristics.
\[thm:Euler\_characteristic\_of\_hocolim\_of\_quotient\_complex\] Let $G$ be a finite group that acts on a finite scwol ${{\mathcal X}}$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. Let $F\colon {{\mathcal X}}/G
\to {{\EuR}{GROUPS}}$ be the associated complex of groups. Then $$\chi^{(2)}({\operatorname{hocolim}}_{{{\mathcal X}}/G} F )= \frac{\chi^{(2)}({{\mathcal X}})}{\vert
G\vert} = \frac{\chi({{\mathcal X}};{{\mathbb C}})}{\vert G\vert} =
\frac{\chi(B{{\mathcal X}};{{\mathbb C}})}{\vert G\vert}.$$ If $R$ is a ring such that the orders of subgroups $H\subset G$ are invertible in $R$, then we also have $$\chi({\operatorname{hocolim}}_{{{\mathcal X}}/G} F;R)=\chi({{\mathcal X}}/G;R).$$
By Theorem \[the:reduction\_to\_skeletal\_case\] \[the:reduction\_to\_skeletal\_case:r\_equivariant\], \[the:reduction\_to\_skeletal\_case:r\_induces\_equivalence\_of\_quotients\], \[the:reduction\_to\_skeletal\_case:complexes\_of\_groups\_agree\], and \[the:reduction\_to\_skeletal\_case:homotopy\_colimits\_equivalent\], we may assume ${{\mathcal X}}$ is skeletal. Then ${{\mathcal X}}/G$ is also skeletal by Lemma \[lem:quotient\_scwol\_of\_skeletal\_scwol\_is\_skeletal\].
Let $\Lambda_n({{\mathcal X}})$ respectively $\Lambda_n({{\mathcal X}}/G)$ denote the set of paths of $n$-many non-identity composable morphisms in ${{\mathcal X}}$ respectively ${{\mathcal X}}/G$. Then by Lemma \[lem:cells\_of\_quotient\_scwol\_are\_quotient\_of\_scwol\_cells\], the sets $\Lambda_n({{\mathcal X}})/G$ and $\Lambda_n({{\mathcal X}}/G)$ are in bijective correspondence.
We will also use the fact that an element $g \in G$ fixes a path $a=(a_1,\dots,a_n)$ in ${{\mathcal X}}$ if and only if $g$ fixes $sa_1$, so $G_{sa_1}=G_a$. This is a consequence of Definition \[def:group\_action\_on\_a\_scwol\] \[def:group\_action\_on\_a\_scwol:(ii)\].
By Theorem \[the:finite\_models\_for\_finite\_scwols\], $E^{{\operatorname{bar}}}{{\mathcal X}}$ and $E^{{\operatorname{bar}}}( {{\mathcal X}}/ G )$ are finite models for the skeletal scwols ${{\mathcal X}}$ and ${{\mathcal X}}/G$, and the $n$-cells are indexed by $\Lambda_n({{\mathcal X}})$ and $\Lambda_n({{\mathcal X}}/G)$, respectively. For each path $(a_1,\dots,a_n)$ in ${{\mathcal X}}$, there is an $n$-cell in $E^{{\operatorname{bar}}}{{\mathcal X}}$ based at $sa_1$. A similar statement holds for ${{\mathcal X}}/G$ and $E^{{\operatorname{bar}}}( {{\mathcal X}}/ G )$.
Now we may apply the Homotopy Colimit Formula to the associated complex of groups $F:{{\mathcal X}}/ G \to {{\EuR}{GROUPS}}$ by Remark \[rem:finite\_group\_finite\_scwol\_imply\_hocolim\_hypotheses\_satisfied\]. For the Euler characteristic, we have $$\begin{aligned}
\chi({\operatorname{hocolim}}_{{{\mathcal X}}/G} F;R) & = & \sum_{n \geq 0} (-1)^n \cdot \left(
\sum_{\overline{a} \in \Lambda_n({{\mathcal X}}/G)} \chi(F(s\overline{a}_1);R)
\right) \\ & = & \sum_{n \geq 0} (-1)^n \cdot \left(
\sum_{\overline{a} \in \Lambda_n({{\mathcal X}}/G)} 1\right)
\\ & = & \sum_{n \geq 0} (-1)^n \vert \Lambda_n({{\mathcal X}}/G) \vert
\\ & = & \sum_{n \geq 0} (-1)^n c_n({{\mathcal X}}/G)
\\ & = & \chi({{\mathcal X}}/G;R).\end{aligned}$$ For the $L^2$-Euler characteristic on the other hand, we have $$\begin{aligned}
\chi^{(2)}({\operatorname{hocolim}}_{{{\mathcal X}}/G} F) &=& \sum_{n \geq 0} (-1)^n \cdot \left( \sum_{\overline{a} \in \Lambda_n({{\mathcal X}}/G)} \chi^{(2)}(F(s\overline{a}_1)) \right) \\
& = & \sum_{n \geq 0} (-1)^n \cdot \left( \sum_{\overline{a} \in \Lambda_n({{\mathcal X}}/G)} \frac{1}{\vert G_{sa_1} \vert} \right) \\
& = & \sum_{n \geq 0} (-1)^n \cdot \left( \sum_{\overline{a} \in \Lambda_n({{\mathcal X}})/G} \frac{1}{\vert G_{a} \vert} \right) \\ & = & \sum_{n \geq 0} (-1)^n \cdot \left( \sum_{\overline{a} \in \Lambda_n({{\mathcal X}})/G} \frac{\vert \text{orbit}(a) \vert}{\vert G \vert} \right) \\ & = & \frac{1}{\vert G \vert} \sum_{n \geq 0} (-1)^n \cdot \left( \sum_{\overline{a} \in \Lambda_n({{\mathcal X}})/G} \vert \text{orbit}(a) \vert \right) \\ & = & \frac{1}{\vert G \vert} \sum_{n \geq 0} (-1)^n \vert \Lambda_n({{\mathcal X}}) \vert
\\ & = & \frac{1}{\vert G \vert} \sum_{n \geq 0} (-1)^n c_n({{\mathcal X}})
\\ & = & \frac{\chi^{(2)}({{\mathcal X}})}{\vert G \vert}.\end{aligned}$$
\[exa:necessary\_conditions\_for\_developability:single\_arrow\] By the classical theorem of Bass–Serre, any injective group homomorphism $$\label{equ:one_arrow_complex_of groups}
G_0 \to G_1$$ is a developable complex of groups. The $L^2$-Euler characteristic of the homotopy colimit of is $1/\vert G_1 \vert$ by Example \[exa:hocolim\_formula\_for\_I\_with\_terminal\_object\]. Theorem \[thm:Euler\_characteristic\_of\_hocolim\_of\_quotient\_complex\] then says we must have $$\frac{\vert G \vert}{\vert G_1 \vert} = \chi^{(2)}({{\mathcal X}})
=\chi(B{{\mathcal X}};{{\mathbb C}})$$ if is to be developable from a scwol ${{\mathcal X}}$ by an action of $G$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. Thus is not developable from any scwol ${{\mathcal X}}$ whose geometric realization has Euler characteristic 0, such as $\{j \rightrightarrows k\}$. Nor can be developed from any scwol ${{\mathcal X}}$ with $\chi(B{{\mathcal X}};{{\mathbb C}})$ negative. The integer $\vert G \vert
$ must also be divisible by $\vert G_1 \vert$, since $\chi(B{{\mathcal X}};{{\mathbb C}})$ is always an integer. Moreover, the Euler characteristic of ${{\mathcal X}}$ must be less than or equal to $\vert G
\vert$. This trivial example illustrates how one can find necessary conditions on ${{\mathcal X}}$ and $G$ if a given complex of groups is to be developable from ${{\mathcal X}}$ and $G$.
\[exa:transport\_groupoid\_finite\_case\] Let $X$ be a finite set and $G$ a finite group acting on $X$. Let $R$ be a ring such that the orders of subgroups of $G$ are invertible in $R$. Considering $X$ as a scwol, we clearly have an action in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. The associated complex of groups $F:X/G \to {{\EuR}{GROUPS}}$ assigns to $\text{orbit}(\sigma)$ the stabilizer $G_\sigma$. The homotopy colimit ${\operatorname{hocolim}}_{X/G} F$ is equivalent to the transport groupoid ${{\mathcal G}}^G(X)$ of Example \[exa:transport\_groupoid\], so $$\chi\left({{\mathcal G}}^G(X);R\right)=\chi({\operatorname{hocolim}}_{X/G} F;R)=\chi(X/G;R)=\vert
X/G \vert.$$ For the $L^2$-Euler characteristic, on the other hand, we have $$\chi^{(2)}\left({{\mathcal G}}^G(X)\right)=\chi^{(2)}({\operatorname{hocolim}}_{X/G}
F)=\frac{\chi^{(2)}(X)}{\vert G \vert}=\frac{\vert X\vert}{\vert G
\vert},$$ a formula obtained by Baez–Dolan [@Baez-Dolan(2001)].
We also generalize the following formula of Haefliger for the Euler characteristic of the homotopy colimit of a (not necessarily developable) complex of groups.
\[the:Haefligers\_corollary\] Let $G(X)$ be a complex of groups over a finite ordered simplicial cell complex $X$. Assume that each $G_\sigma$ is the fundamental group of a finite aspherical cell complex. Then $BG(X)$ has the homotopy type of a finite complex and its Euler-Poincaré characteristic is given by[^4] $$\chi(BG(X))=\sum_{\sigma \in {\operatorname{ob}}(C(X))}
(1-\chi(Lk^\sigma))\chi(G_\sigma).$$
The terms in Haefliger’s theorem have the following meanings. An [*ordered simplicial cell complex*]{} $X$ is by definition the nerve of a skeletal scwol, denoted $C(X)$. The notation $BG(X)$ signifies the geometric realization of the nerve of the homotopy colimit of the pseudo functor $G(X)\colon C(X) \to {{\EuR}{GROUPS}}$. An [*aspherical*]{} cell complex is one for which all homotopy groups beyond the fundamental group vanish. The [*lower link $Lk^\sigma$ of the object $\sigma$*]{} is the full subcategory of the scwol $\sigma \downarrow C(X)$ on all objects except $1_\sigma$.
\[the:extension\_of\_Haefligers\_corollary\] Let ${{\mathcal I}}$ be a finite skeletal scwol and $F\colon {{\mathcal I}}\to
{{\EuR}{GROUPS}}$ a complex of groups such that for each object $i$ of ${{\mathcal I}}$, the group $F(i)$ is of type (FF$_{{\mathbb Z}}$). Then $$\chi(B {\operatorname{hocolim}}_{{\mathcal I}}F )=\sum_{i \in {\operatorname{ob}}({{\mathcal I}})}
(1-\chi(BLk^i))\chi(B F(i)),$$ where $B$ indicates geometric realization composed with the nerve functor.
All hypotheses of Theorem \[the:homotopy\_colimit\_formula\]\[the:homotopy\_colimit\_formula:chi\] are satisfied. The skeletal scwol ${{\mathcal I}}$ is directly finite by Lemma \[lem:scwol\_directly\_finite\_EI\] and admits a finite ${{\mathcal I}}$-$CW$-model for its ${{\mathcal I}}$-classifying space by Theorem \[the:finite\_models\_for\_finite\_scwols\]. Each group ${{\mathcal C}}(i)$ is automatically directly finite, and assumed to be of type (FF$_{{\mathbb Z}}$). The bar construction model $E^{{\operatorname{bar}}}{{\mathcal I}}$ in Remark \[rem:Ebarcalc\] has an $n$-cell based at $i$ for each path of $n$-many non-identity morphisms in ${{\mathcal I}}$ $$i \to i_1 \to i_2 \to \cdots \to i_n.$$ Each such path in ${{\mathcal I}}$ corresponds uniquely to a path of $(n-1)$-many non-identity morphisms in the scwol $Lk^{i}$ beginning at the object $i \to i_1$. Thus $$\begin{aligned}
1-\chi(BLk^{i}) & = & 1- \sum_{m \geq 0} (-1)^m c_m(Lk^{i}) \\
& = & 1- \sum_{m \geq 0} (-1)^m \text{card}\{\text{$(m+1)$-paths in
${{\mathcal I}}$ beginning at $i$}\} \\
& = & 1- \sum_{n \geq 1} (-1)^{n-1} \text{card}\{\text{$n$-paths in
${{\mathcal I}}$ beginning at $i$}\} \\
& = & 1+ \sum_{n \geq 1} (-1)^{n} \text{card}\{\text{$n$-paths in
${{\mathcal I}}$ beginning at $i$}\} \\
& = & \sum_{n \geq 0} (-1)^{n} \text{card}\{\text{$n$-paths in
${{\mathcal I}}$ beginning at $i$}\}. \\\end{aligned}$$ Then by Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:directly\_finite\], Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:(FF)\], Theorem \[the:coincidence\], and Theorem \[the:homotopy\_colimit\_formula\] \[the:homotopy\_colimit\_formula:chi\], we have $$\begin{aligned}
\chi(B {\operatorname{hocolim}}_{{\mathcal I}}F ) & = & \chi({\operatorname{hocolim}}_{{\mathcal I}}F ) \\
& = & \sum_{n \geq 0} (-1)^n \cdot \sum_{\lambda \in \Lambda_n}
\chi(F(i_\lambda)) \\
& = & \sum_{i \in {\operatorname{ob}}({{\mathcal I}})}\left( 1-\chi(BLk^{i}) \right) \cdot
\chi(F(i)) \\
& = & \sum_{i \in {\operatorname{ob}}({{\mathcal I}})}\left( 1-\chi(BLk^{i}) \right) \cdot
\chi(BF(i)).\end{aligned}$$
The assumptions in our Theorem \[the:extension\_of\_Haefligers\_corollary\] on the groups $F(i)$ are related to the assumptions in Theorem \[the:Haefligers\_corollary\] on the groups $G_\sigma$ in that any finitely presentable group of type (FF$_{{\mathbb Z}}$) admits a finite model for its classifying space.
Appendix {#sec:appendix}
========
Let $G$ be a group acting on a scwol ${{\mathcal X}}$ in the sense of Definition \[def:group\_action\_on\_a\_scwol\]. In connection with Theorem \[the:reduction\_to\_skeletal\_case\], we remark that it is possible to choose a skeleton $\Gamma_0$ of ${{\mathcal X}}$, a $G$-equivariant functor $r\colon {{\mathcal X}}\to \Gamma_0$, and a natural isomorphism $\eta\colon ir \cong {\operatorname{id}}_{{\mathcal X}}$ so that
- the inclusion $i_0\colon \Gamma_0 \to {{\mathcal X}}$ is $G$-equivariant,
- $ri_0={\operatorname{id}}_{\Gamma_0}$, and
- for every object $x \in {\operatorname{ob}}({{\mathcal X}})$ and each $g \in G$, we have $\eta_{gx}=g\eta_x$.
To prove this, we first choose the object set of $\Gamma_0$ via an equivariant section of the projection $\pi\colon {\operatorname{ob}}({{\mathcal X}}) \to
{\operatorname{iso}}({{\mathcal X}})$, which assigns to each object of ${{\mathcal X}}$ its isomorphism class of objects. Let $\Theta$ denote the set of $G$-orbits of ${\operatorname{iso}}({{\mathcal X}})$. For each $G$-orbit $\theta \in \Theta$, we use the axiom of choice to select an element $\overline{x}_\theta
\in \theta$. For each $\theta$, select then a $\pi$-preimage $s(\overline{x}_\theta):=x_\theta$ of $\overline{x}_\theta$. On the orbit of each $\overline{x}_\theta$ we define the section $s$ by $s(g\overline{x}_\theta):=gx_\theta$. This is well defined, for if $g_1\overline{x}_\theta=g_2\overline{x}_\theta$, then $g_1x_\theta
\cong g_2x_\theta$, and $g_1x_\theta = g_2x_\theta$ by Lemma \[lem:consequences\_of\_group\_action\] \[lem:consequences\_of\_group\_action:(i)\]. Define the skeleton $\Gamma_0$ to be the full subcategory of ${{\mathcal X}}$ on the objects in the image of the equivariant section $s\colon
{\operatorname{iso}}({{\mathcal X}}) \to {\operatorname{ob}}({{\mathcal X}})$.
For each $\overline{x}_\theta$, and each $x \in
\overline{x}_\theta$, choose an isomorphism $\eta_x \colon x_\theta
\to x$. For $gx$, we define $\eta_{gx}$ as $g\eta_x$. Next, we define a functor $r\colon {{\mathcal X}}\to \Gamma_0$ on objects $x \in
{\operatorname{ob}}({{\mathcal X}})$ by $r(x):=s\pi(x)$ and on morphisms $f\colon x \to y$ by $r(f):= \eta_y \circ f \circ \eta_{x}^{-1}$. Then $\eta$ is clearly a natural isomorphism, the inclusion $i_0\colon \Gamma_0 \to {{\mathcal X}}$ is $G$-equivariant, and $ri_0={\operatorname{id}}_{\Gamma_0}$.
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[^1]: We thank George Maltsiniotis for clarifying these points about homotopy colimits in ${{\EuR}{CAT}}$.
[^2]: This is a well-known standard argument, which we present only so that the reader easily sees that it works in the setting of ${{\mathcal H}}$-spaces.
[^3]: Bridson–Haefliger additionally require scwols to be skeletal [@Bridson-Haefliger(1999) page 574]. However, we do not require scwols to be skeletal, since we prove in Theorem \[the:reduction\_to\_skeletal\_case\] that general statements about scwols can be reduced to the skeletal case.
[^4]: Haefliger’s original formula has, instead of the lower link $L^\sigma$, the upper link $L_\sigma$, which is the full subcategory of the scwol $C(X)
\downarrow \sigma$ on all objects except $1_\sigma$. However, this is merely a typo, for if we use the upper link $Lk_\sigma$ and consider the example $C(X)=\{k \leftarrow j \to \ell\}$ with pseudo functor $G(X)(\ell):={{\mathbb Z}}$ and $G(X)(j):=G(X)(k):=\{0\}$, then $\chi(BG(X))=\chi(S^1)=0$ but $\sum(1-\chi(Lk_\sigma))\chi(G_\sigma)=1$.
|
---
abstract: 'We present a theoretical study of the dissociative tunneling ionization process. Analytic expressions for the nuclear kinetic energy distribution of the ionization rates are derived. A particularly simple expression for the spectrum is found by using the Born-Oppenheimer (BO) approximation in conjunction with the reflection principle. These spectra are compared to exact non-BO *ab initio* spectra obtained through model calculations with a quantum mechanical treatment of both the electronic and nuclear degrees freedom. In the regime where the BO approximation is applicable imaging of the BO nuclear wave function is demonstrated to be possible through reverse use of the reflection principle, when accounting appropriately for the electronic ionization rate. A qualitative difference between the exact and BO wave functions in the asymptotic region of large electronic distances is shown. Additionally the behavior of the wave function across the turning line is seen to be reminiscent of light refraction. For weak fields, where the BO approximation does not apply, the weak-field asymptotic theory describes the spectrum accurately.'
author:
- Jens Svensmark
- 'Oleg I. Tolstikhin'
- Lars Bojer Madsen
title: Theory of dissociative tunneling ionization
---
Introduction
============
Currently a number of intense mid-infrared light sources are being developed [@Popmintchev1287; @Silva15; @PhysRevX.5.021034; @Duval15], spurred on by their uses in sub-attosecond pulse generation [@PhysRevLett.98.013901; @PhysRevLett.111.033002], strong-field holography [@Huismans61; @PhysRevA.84.043420] and laser-induced electron diffraction [@PhysRevX.5.021034; @Blaga12]. The low frequency and high intensity of these new sources mean that the tunneling picture is an appropriate framework for describing how these light sources interact with atoms and molecules. In this work we deal with the process of dissociative tunneling ionization in molecules, where a static electric field tunnel ionizes an electron, after which the nuclei dissociate. To our knowledge this is the first work on the theory of dissociative tunneling ionization. In the theory we treat the nuclear and electronic degrees freedom on an equal footing and fully quantum mechanically.
The reflection principle [@PhysRev.32.858; @:/content/aip/journal/jcp/58/9/10.1063/1.1679721; @PhysRevLett.108.073202; @PhysRevA.90.063408] is often used to describe the process of dissociative ionization. This principle can be applied within the framework of the Born-Oppenheimer (BO) approximation to relate the nuclear kinetic energy release (KER) spectrum to the nuclear wave function. It was formulated as early as 1928 [@PhysRev.32.858], and later put on a more rigorous foundation [@:/content/aip/journal/jcp/58/9/10.1063/1.1679721] (see also references therein for a list of early uses). In time-dependent cases where the time-scale of the electric field is shorter than that of nuclear motion, one assumes that the electrons make an instantaneous Frank-Condon transition to a dissociative electronic state. The probability distribution in the new electronic state is then the absolute value squared of the initial nuclear wave function times some dipole coupling factor. In case this dipole coupling factor is almost constant, the wave packet that enters the dissociative state is practically identical to the initial nuclear wave function. Classical energy conservation then dictates that the nuclear KER spectrum can be obtained by reflecting the nuclear wave function in the dissociative potential curve. This is the regime considered mostly in the literature. In the time-independent tunneling case, the electronic ionization rate takes the same role as the dipole coupling factor in the time-dependent case, that is, it multiplies the nuclear wave function before it enters the continuum. However, this electronic rate has an exponential dependence on the internuclear coordinate and can by no means be considered constant (as was also pointed out in Ref. [@PhysRevLett.92.163004]). It is therefore essential to consider the effect of this additional factor on the KER spectrum; the spectrum cannot be found simply by reflection of the nuclear wave function.
Imaging of the nuclear wave function is made possible through the reflection principle, by applying it in reverse on a measured KER spectrum [@PhysRevLett.82.3416; @PhysRevLett.108.073202; @PhysRevA.58.426]. This is often referred to as Coulomb explosion imaging. In the tunneling case the exponential dependence of the electronic rate on the internuclear coordinate means that the product of the electronic rate and the nuclear wave function is essentially different from the bare nuclear wave function, and the electronic rate therefore needs to be included to image the nuclear wave function based on the KER spectrum. In Ref. [@rtd] it was demonstrated that the BO approximation breaks down for weak fields. In this case the weak-field asymptotic theory (WFAT) [@PhysRevA.84.053423; @PhysRevA.89.013421] provides us with accurate results for the KER spectrum.
The paper is organized as follows. In Sec. \[sec:theory\] the theory for dissociative tunneling ionization of homonuclear molecules is outlined. We derive an exact expression for the KER spectrum and a corresponding expression in the BO approximation. Section \[sec:1d-calculation\] exemplifies the theory with numerical reduced dimensionality calculations. Numerically exact KER spectra are compared to KER spectra obtained in the BO approximation using the reflection principle. Imaging of the nuclear part of the wave function from the KER spectrum is demonstrated. Section \[sec:conclusion-outlook\] concludes the paper. Atomic units $\hbar=m_e=e=1$ are used throughout.
Theory {#sec:theory}
======
Basic equations
---------------
We consider a three-body system consisting of two heavy nuclei with masses $m_1=m_2$ and charges $q_1=q_2$, and one electron with mass $m_3=1$ and charge $q_3=-1$. In the center-of-mass frame these have coordinates ${\mathbf{r}}_1,{\mathbf{r}}_2$ and ${\mathbf{r}}_3$ fulfilling $m_1({\mathbf{r}}_1 +{\mathbf{r}}_2)+{\mathbf{r}}_3=0$. Let us introduce the reduced masses $$\begin{aligned}
M & = \frac{m_1}{2}, &
m & = \frac{2m_1}{2m_1+1} , \end{aligned}$$ effective charge $$\begin{aligned}
{q}& = \left(\frac{q_1}{m_1}+1\right) m, \end{aligned}$$ and Jacobi coordinates $$\begin{aligned}
{\mathbf{R}}&={\mathbf{r}}_2-{\mathbf{r}}_1,& {\mathbf{r}}&=\frac{{\mathbf{r}}_3}{m}.\end{aligned}$$ We assume that the orientation of the internuclear axis ${\mathbf{R}}$ is fixed in space. We also assume that the field is directed along the $z$-direction, ${\mathbf{F}}=F\hat{{\mathbf{z}}}$ and choose to consider $F\geq 0$ for definiteness. Due to the azimuthal symmetry of the molecule only the polar angle $\beta$ between ${\mathbf{R}}$ and ${\mathbf{F}}$ matters. This $\beta$ angle takes the role as an external parameter, and we omit explicit reference to it in the following. With these assumptions we can write the time-independent Schrödinger equation (SE) within the single-active-electron approximation as $$\begin{aligned}
\left[- \frac{1}{2M} {\frac{\partial^2 }{\partial {R} ^2}} - \frac{1}{2m} \Delta_{{\mathbf{r}}} + U(R) + V\left({\mathbf{r}}{;}R\right) + {q}zF - E(F)\right] \Psi({\mathbf{r}},R) & = 0,\label{eq:schrodinger}\end{aligned}$$ where the effective $U(R)$ potential describes how the nuclei interact with each other and the effective $V({\mathbf{r}}{;}R)$ potential describes how the electron interacts with the nuclei. For a system with several electrons the $U(R)$ potential represents the BO potential of the system without the active electron. In this work, we assume that $U(R)$ is monotonically decreasing, i.e., it corresponds to a purely dissociative BO curve.
We assume that the nuclei cannot pass through each other. This gives the boundary condition $$\begin{aligned}
\Psi({\mathbf{r}},R=0)=0,\label{eq:zero_bc}\end{aligned}$$ and we consider Eq. (\[eq:schrodinger\]) in the interval $0\leq R$. We also impose outgoing-wave boundary conditions in the electronic coordinate ${\mathbf{r}}$, the exact form of these will be specified below. With these boundary conditions the wave function we seek as a solution of Eq. (\[eq:schrodinger\]) is a Siegert state [@PhysRev.56.750; @PhysRevLett.79.2026; @starka], with a complex energy $E(F)=\mathcal{E}(F)-\frac{i}{2}\Gamma(F)$, where $\Gamma(F)$ is the ionization rate, and it is normalized by $$\begin{aligned}
\int d^3{\mathbf{r}} \int_0^\infty dR\ \Psi^2({\mathbf{r}},R) & = 1.\end{aligned}$$ The outgoing-wave boundary condition in the electronic coordinate means that the solution we seek to Eq. (\[eq:schrodinger\]) describes tunneling of the electron. This tunneling is followed by dissociation of the nuclei for the considered class of strictly dissociative potentials $U(R)$. In the following we will describe the energy distribution of the dissociated nuclei.
Energy distribution of dissociated nuclei {#sec:asymptotics}
-----------------------------------------
Our aim is to describe the energy distribution of the nuclei, i.e., the KER spectrum, after the molecule is ionized by tunneling of the electron. To this end we need to consider the problem in the $r\to\infty$ limit, where the electron is far away from the nuclei. In this limit we assume that the electron-nuclear interaction potential takes the form $$\begin{aligned}
V({\mathbf{r}}{;}{R})|_{r\to\infty} & = -\frac{Z}{r}+O(r^{-2}),\end{aligned}$$ where $Z=2q_1$ is the total charge of the remaining core system. This assumption makes our problem separable in electron and nuclear coordinates in this asymptotic region. By seeking the partial solutions in the form $\Psi({\mathbf{r}},R)=f({\mathbf{r}},k)g(R,k)$, Eq. (\[eq:schrodinger\]) can be written as the separated equations
$$\begin{aligned}
\left[ - \frac{1}{2m} \Delta_{{\mathbf{r}}} -\frac{Z}{r} + {q}Fz - E_{{\mathbf{r}}}\right] f({\mathbf{r}},k) & = 0,\label{eq:as_x_eq}\\
\left[- \frac{1}{2M} {\frac{ d^2 }{d {R} ^2}} + {U}(R) - E_{R}\right] g(R,k) & = 0,\label{eq:as_R_eq}
\end{aligned}$$
with separation constants given by
$$\begin{aligned}
E(F) & = E_{{\mathbf{r}}} + E_{R},\\
E_{R} & = \frac{k^2}{2M},\end{aligned}$$
where we assume $U(R)|_{R\to\infty}=0$ and $k\geq 0$ is the wave number for the state $g(R,k)$. Equation (\[eq:zero\_bc\]) amounts to $$\begin{aligned}
g(R=0,k) & = 0.\label{eq:zero_bc_g}\end{aligned}$$ We choose the continuum solutions of Eq. (\[eq:as\_R\_eq\]) to be real and normalized by $$\begin{aligned}
\int_0^\infty g(R,k)g(R,k') dR & = 2\pi\delta(k-k').\label{eq:g_norm}\end{aligned}$$ The conditions Eqs. (\[eq:zero\_bc\_g\])-(\[eq:g\_norm\]) completely specify the nuclear problem Eq. (\[eq:as\_R\_eq\]).
The electronic problem Eq. (\[eq:as\_x\_eq\]) has a potential consisting of a Coulomb term and a linear field term. This problem is separable in parabolic coordinates [@landau1977quantum], which we will therefore use. First we introduce mass-scaled quantities
$$\begin{aligned}
\tilde{{\mathbf{r}}} & = \sqrt{m} {\mathbf{r}},\\
\tilde{F} & = \frac{q}{\sqrt{m}}F\\
\tilde{Z} & = \sqrt{m}Z.\end{aligned}$$
Then the following form of the parabolic coordinates is introduced (as in Ref. [@PhysRevA.84.053423])
\[eq:parab\_coord\] $$\begin{aligned}
\xi & = \tilde{r} + \tilde{z}, &0\leq \xi <\infty\\
\eta & = \tilde{r} - \tilde{z}, &0\leq \eta <\infty\\
\varphi & = \arctan\frac{ \tilde{y}}{ \tilde{x}}, &0\leq \varphi <2\pi.
\end{aligned}$$
With this choice of coordinates a potential barrier forms in the $\eta$ coordinates and therefore $\eta$ takes the role as the ’tunneling coordinate’.
In the asymptotic region $\eta\to\infty$ Eq. (\[eq:as\_x\_eq\]) has a solution that is a linear combination of partial solutions of the form [@PhysRevA.84.053423] $$\begin{aligned}
f({\mathbf{r}},k)|_{\eta\to\infty} & = \eta^{-1/2}f(\eta)\Phi_\nu(\xi,\varphi),\end{aligned}$$ where the outgoing-wave $f(\eta)$ is given by $$\begin{aligned}
f(\eta) &= \frac{2^{1/2}}{(\tilde{F}\eta)^{1/4}}\exp\left(i\left[\frac{\tilde{F}^{1/2}}{3}\eta^{3/2}+\frac{E_{{\mathbf{r}}}}{\tilde{F}^{1/2}}\eta^{1/2}\right]\right),\label{eq:f_eta}\end{aligned}$$ $\Phi_\nu(\xi,\varphi)$ is the ionization channel function defined by $$\begin{aligned}
\left[{\frac{\partial }{\partial \xi}} \xi{\frac{\partial }{\partial \xi}} + \frac{1}{4\xi} {\frac{\partial^2 }{\partial {\varphi} ^2}}+\tilde{Z}+\frac{E_{{\mathbf{r}}}\xi}{2}-\frac{\tilde{F}\xi^2}{4} \right] \Phi_\nu(\xi,\varphi) & = \beta_\nu \Phi_\nu(\xi,\varphi),\label{eq:xi_phi_ad_eq}\end{aligned}$$ and $\nu=(n_\xi,m)$ is a set of parabolic quantum numbers labeling the different ionization channels, see Fig. \[fig:parab\_coord\]. With our choice of $F\geq 0$ the potential in Eq. (\[eq:xi\_phi\_ad\_eq\]) goes to infinity as $\xi$ goes to infinity, so the parabolic channels $\nu$ are purely discrete.
![(Color online) The blue/red surface is a paraboloid showing a surface of constant $\eta$. The gray paraboloid is the same for a smaller value of $\eta$. The electron is ionized in the negative $z$ direction due to its negative charge, given that the electric field points in the positive $z$-direction. The $\Phi_\nu(\xi,\varphi)$ states \[Eq. (\[eq:xi\_phi\_ad\_eq\])\] ’live’ in the constant $\eta$ paraboloids. The colors in the blue/red surface illustrates an example of the nodal structure of such a $\Phi_\nu(\xi,\varphi)$ state. The curvature of the paraboloids means that the $\Phi_\nu(\xi,\varphi)$ states are bound. This means that $\eta$ is the only coordinate where we have to consider the wave function at infinity, i.e., $\eta$ is the ’tunneling’ coordinate. For large $\eta$ the polar angle $\beta$, which specifies the orientation of the molecule, does not matter for the asymptotic form of the wave function in parabolic coordinates, though it matters for the size of the coefficients \[Eq. (\[eq:spectrum\_ampl\_def\])\].[]{data-label="fig:parab_coord"}](fig1.pdf)
The full wave function can be expressed as a linear combination, discrete in $\nu$, continuous in $k$, of the $f({\mathbf{r}},k)g(R,k)$ products, $$\begin{aligned}
\Psi({\mathbf{r}},R)|_{\eta\to\infty} & = \sum_\nu \int_0^\infty C_\nu(k)\eta^{-1/2}f(\eta)\Phi_\nu(\xi,\varphi) g(R,k) \frac{dk}{2\pi},\label{eq:wf_expansion}\end{aligned}$$ where the asymptotic expansion coefficient $C_\nu(k)$ can be calculated by $$\begin{aligned}
C_\nu(k) & = \frac{1}{ \eta^{-1/2}f(\eta)} \int_0^\infty g(R,k) \Braket{\Phi_{\nu}(\xi,\varphi)|\Psi({\mathbf{r}},R)}_{(\xi,\varphi)} dR|_{\eta\to\infty}.\label{eq:spectrum_ampl_def}\end{aligned}$$ $\Braket{{}\cdot{}}_{(\xi,\varphi)}$ indicates integration w.r.t. the coordinates $\xi$ and $\varphi$ over their full range. Note that the polar angle $\beta$, which we suppressed in the notation, only enters Eq. (\[eq:spectrum\_ampl\_def\]) through the wave function $\Psi({\mathbf{r}},R)$. The KER dissociation spectrum into the channel $\nu$ is defined in terms of these expansion coefficients by $$\begin{aligned}
P_\nu(k) & ={\left| C_\nu(k) \right|}^2 .\label{eq:spectrum_def}\end{aligned}$$ This is the main observable of interest. By inserting Eq. (\[eq:f\_eta\]) and Eq. (\[eq:spectrum\_ampl\_def\]) and assuming $E_{\mathbf{r}}$ to be real, which is approximately the case for small $F$, we obtain $$\begin{aligned}
P_\nu(k)
& =\frac{\tilde{F}^{1/2}}{ 2\eta^{1/2} } {\left| \int g(R,k) \Braket{\Phi_{\nu}(\xi,\varphi)|\Psi({\mathbf{r}},R)}_{(\xi,\varphi)} dR \right|}^2 _{\eta\to\infty}.\label{eq:spec_expr}\end{aligned}$$ The exact KER spectrum in the channel $\nu$ can thus be obtained by projecting the wave function on the channel state $\Phi_\nu(\xi,\varphi)$, and further projecting this on the continuum states $g(R,k)$ of the $U(R)$ potential. The total KER spectrum can then be obtained by summing over all the channels $$\begin{aligned}
P(k) & = \sum_\nu P_\nu(k).\end{aligned}$$ In the $F\to 0$ limit the total rate can be obtained by $$\begin{aligned}
\Gamma & = \int_0^{\infty} P(k) \frac{dk}{2\pi}.\end{aligned}$$
Born-Oppenheimer approximation {#sec:bo}
------------------------------
Now that we have a recipe for finding the exact KER spectrum, we consider some approximations for ease of predictions and gain in physical insight. We first consider the BO approximation, which appears in the limit $m_1=m_2\to\infty$. In this limit $m=1=q$, and the wave function takes the form $\Psi_\text{BO}({\mathbf{r}},R)=\psi_e({\mathbf{r}},R)\chi(R)$. The electronic and nuclear part of BO wave function fulfills the BO equations
$$\begin{aligned}
\left[ - \frac{1}{2} \Delta_{{\mathbf{r}}} + V({\mathbf{r}}{;}R) + Fz - E_e(R;F)\right] \psi_e({\mathbf{r}};R) & = 0,\label{eq:BO_elec_eq}\\
\left[- \frac{1}{2M} {\frac{ d^2 }{d {R} ^2}} + U(R)+E_e(R;F) - E_{\text{BO}}(F)\right] \chi(R) & = 0.\label{eq:BO_nuc_eq}
\end{aligned}$$
We impose zero boundary condition for the nuclear wave function $$\begin{aligned}
\chi(R=0) & = 0\end{aligned}$$ and the following normalizations $$\begin{aligned}
\int_0^\infty dR\ {\chi^2(R)} & = 1,\\
\int d^3{\mathbf{r}}\ \psi_e^2({\mathbf{r}};R) & = 1.\end{aligned}$$
In the asymptotic limit $\eta\to\infty$ the electronic Eq. (\[eq:BO\_elec\_eq\]) takes the same form as Eq. (\[eq:as\_x\_eq\]), and it can be written in parabolic coordinates in the same manner. The electronic wave function then takes the outgoing-wave form $$\begin{aligned}
\psi_e({\mathbf{r}};R)|_{\eta\to\infty} & = \eta^{-1/2}f(\eta) \sum_\nu f_\nu(R) \Phi_\nu(\xi,\varphi) ,\end{aligned}$$ where $f(\eta)$ is from Eq. (\[eq:f\_eta\]) and $\Phi_\nu(\xi,\varphi)$ are solutions of Eq. (\[eq:xi\_phi\_ad\_eq\]), with $E_{\mathbf{r}}$ replaced by $E_e(R;F)$ in both. The asymptotic coefficient $f_\nu(R)$ defines the ionization amplitude in channel $\nu$ [@PhysRevA.84.053423]. The partial electronic ionization rate is given by $$\begin{aligned}
\Gamma_{e,\nu}(R) & = {\left| f_\nu(R) \right|}^2\label{eq:partial_elec_rate}.\end{aligned}$$ By considering the flux of the electron probability through a surface at large negative $z$, one can show that in the weak field limit the total electronic rate $\Gamma_e(R)=-2\operatorname{Im}(E_e(R;F))$ is given as a sum over $\nu$ of all the partial electronic rates.
By inserting the BO wave function into Eq. (\[eq:spec\_expr\]) we obtain $$\begin{aligned}
P_\nu(k)
& = {\left| \int_0^\infty g(R,k) f_{\nu}(R)\chi(R)dR \right|}^2 .\label{eq:Spec_BO}\end{aligned}$$ The separation of electronic and nuclear coordinates in the BO approximation means that this expression for the KER spectrum does not contain any explicit reference to electronic coordinates, as opposed to the expression for the exact spectrum Eq. (\[eq:spec\_expr\]). Equation (\[eq:Spec\_BO\]) is similar to a result previously put forward in the literature \[Eq. (1) of Ref. [@PhysRevLett.92.163004]\], except that the correct complex ionization amplitude $f_\nu(R)$ was taken as $\sqrt{\Gamma_{e,\nu}(R)}$. In the cases we have considered, the phase variations of $f_\nu(R)$ are sufficiently small that they can be safely neglected, explaining the successful use of the aforementioned replacement in Ref. [@PhysRevLett.92.163004], but this is not generally true.
To evaluate the integral in Eq. (\[eq:Spec\_BO\]) we will use the reflection principle [@PhysRev.32.858; @:/content/aip/journal/jcp/58/9/10.1063/1.1679721]. At the heart of the reflection principle lies an important mathematical component which we denote the reflection approximation [@:/content/aip/journal/jcp/58/9/10.1063/1.1679721]. This approximation amounts to setting $$\begin{aligned}
g(R,k) & = \sqrt{-2\pi{\frac{\partial R_t}{\partial k}}}\delta\left(R-R_t\right),\label{eq:g_as_Delt}\end{aligned}$$ which is exact in the $M\to\infty$ limit. In Eq. (\[eq:g\_as\_Delt\]) $R_t$ is the classical turning point [^1] for the $g(R,k)$ function defined by $$\begin{aligned}
{U}(R_t)=E_R=\frac{k^2}{2M}.\end{aligned}$$ In order to determine the derivative ${\frac{\partial R_t}{\partial k}}$ the form of the dissociative $U(R)$ potential must be known. Inserting Eq. (\[eq:g\_as\_Delt\]) in Eq. (\[eq:Spec\_BO\]) yields $$\begin{aligned}
P_\nu(k)
& = 2\pi{\left| {\frac{\partial R_t}{\partial k}} \right|}\Gamma_{e,\nu}(R_t) {\left| \chi(R_t) \right|}^2 .\label{eq:Spec_result_BO}\end{aligned}$$ This result shows that using the reflection approximation in conjunction with the BO approximation we obtain a KER spectrum that is expressed as a product of a Jacobian factor, the electronic rate and the field-dressed nuclear wave function \[Eq. (\[eq:BO\_nuc\_eq\])\]. This is a lot simpler to calculate than evaluating either integrals in Eqs. (\[eq:spec\_expr\]) or (\[eq:Spec\_BO\]), and is easily reversed to give a way to image the field-dressed nuclear wave function, and it is applicable to any molecule with a dissociative BO curve.
Weak-field limit within the Born-Oppenheimer approximation {#sec:elec_wfat}
----------------------------------------------------------
The exact electronic rate $\Gamma_{e,\nu}(R)$ is often not available, since finding it requires solving the electronic problem Eq. (\[eq:BO\_elec\_eq\]), which is a highly non-trivial task for many systems. In such cases the weak-field asymptotic theory (WFAT) [@PhysRevA.89.013421; @PhysRevA.84.053423; @PhysRevA.87.043426] can be employed to obtain the rate. WFAT is an analytic theory which expresses the ionization rate in terms of properties of the field-free state. It is applicable in the weak-field limit.
Let $\beta^{(0)}_\nu(R)$ and $\Phi_\nu^{(0)}(\xi,\varphi;R)$ denote the adiabatic eigenvalues and eigenfunctions solving Eq. (\[eq:xi\_phi\_ad\_eq\]) for $F=0$ with the field-free electronic energy $E_e(R;F=0)$ replacing $E_{\mathbf{r}}$. Ref. [@PhysRevA.84.053423] provides analytic expressions for these quantities. In terms of these the asymptotic field-free electronic wave function can be written $$\begin{aligned}
\psi_{e,0}({\mathbf{r}};R)|_{\eta\to\infty}& = \sum_\nu f_\nu(R,F=0) \eta^{\beta_\nu^{(0)}(R)/\varkappa(R)-1/2}e^{-\varkappa(R)\eta/2}\Phi_\nu^{(0)}(\xi,\varphi;R),\label{eq:field_free_elec_wf}\end{aligned}$$ where $$\begin{aligned}
\varkappa(R) & = \sqrt{-2E_e(R;F=0)}.\end{aligned}$$ The electronic WFAT rate is then given by [@PhysRevA.84.053423] $$\begin{aligned}
\Gamma_{e,\nu}^\text{WFAT}(R) & = {\left| f_\nu(R,F=0) \right|}^2 W_\nu(R)\label{eq:rate_WFAT_elec}\end{aligned}$$ where the field factor $W_\nu(R)$ is defined by $$\begin{aligned}
W_\nu(R)= \frac{{\varkappa(R)}}{2}\left(\frac{4{\varkappa^2(R)}}{ F }\right)^{2\frac{\beta_\nu^{(0)}(R) }{{\varkappa(R)}}} \exp\left(-\frac{{2\varkappa^3(R)}}{3F } \right),\label{eq:field_factor_elec}\end{aligned}$$ and the asymptotic coefficients $f_\nu(R,F=0)$ can be found from the electronic wave function by inversion of Eq. (\[eq:field\_free\_elec\_wf\]) $$\begin{aligned}
f_\nu(R,F=0) & = \left.\frac{ \Braket{\Phi_\nu^{(0)}(\xi,\varphi;R)|\psi_{e,0}({\mathbf{r}};R)}_{(\xi,\varphi)}}{ \eta^{\beta_\nu^{(0)}(R)/\varkappa(R)-1/2}e^{-\varkappa(R)\eta/2}}\right|_{\eta\to\infty}.\end{aligned}$$
Weak-field asymptotic theory {#sec:full_wfat}
----------------------------
WFAT can also be applied for the exact state, and not just in the BO approximation as above. In this section we will give the pertaining formulas. Let, as before, $\beta^{(0)}_\nu$ and $\Phi_\nu^{(0)}(\xi,\varphi)$ denote the adiabatic eigenvalues and eigenfunctions solving Eq. (\[eq:xi\_phi\_ad\_eq\]) for $F=0$ now with the field-free energy $E(F=0)$. In terms of these the asymptotic field-free wave function can be written [@PhysRevA.84.053423] $$\begin{aligned}
\Psi_0({\mathbf{r}},R)|_{\eta\to\infty}& = \sum_\nu \int_0^\infty C_\nu(k,F=0) g(R,k) \eta^{\beta_\nu^{(0)}/\varkappa(k)-1/2}e^{-\varkappa(k)\eta/2}\Phi_\nu^{(0)}(\xi,\varphi)\frac{dk}{2\pi}\label{eq:field_free_wf}\end{aligned}$$ where $$\begin{aligned}
\varkappa(k) & = \sqrt{2\left(\frac{1}{2M}k^2-E(F=0)\right)}.
\end{aligned}$$ The WFAT [@PhysRevA.84.053423] yields the following expression for the KER spectrum $$\begin{aligned}
P_{\nu}^{\text{WFAT}}(k) & = {\left| C_\nu(k,F=0) \right|}^2W_\nu(k),\label{eq:spec_WFAT}\end{aligned}$$ where the field factor $W_\nu(k)$ is given by $$\begin{aligned}
W_\nu(k)= \frac{{\varkappa(k)}}{2}\left(\frac{4\sqrt{m}{\varkappa^2(k)}}{ qF }\right)^{2\frac{\beta_\nu^{(0)} }{{\varkappa(k)}}} \exp\left(-\frac{2{\sqrt{m}\varkappa^3(k)}}{3qF } \right),\label{eq:full_field_factor}\end{aligned}$$ and the field-free asymptotic coefficients $C_\nu(k,F=0)$ can be found by inversion of Eq. (\[eq:field\_free\_wf\]) $$\begin{aligned}
C_\nu(k,F=0) & = \left.\frac{ \int_0^\infty g(R,k)\Braket{\Phi_\nu^{(0)}(\xi,\varphi)| \Psi_0({\mathbf{r}},R)}_{(\xi,\varphi)} dR}{ \eta^{\beta_\nu^{(0)}/\varkappa(k)-1/2}e^{-\varkappa(k)\eta/2}}\right|_{\eta\to\infty}.\end{aligned}$$
Illustrative 1D calculations {#sec:1d-calculation}
============================
Solving Eq. (\[eq:schrodinger\]) in 3D is a computationally heavy task, so we have used a 1D model to illustrate our central points. In this section we compare exactly calculated KER spectra with those obtained through the BO approximation, Eq. (\[eq:Spec\_result\_BO\]), and the WFAT, Eq. (\[eq:spec\_WFAT\]), within this 1D model. In the following we will consider a model of H$_2^+$ as an example. The potentials we consider are thus
$$\begin{aligned}
U(R)& = \frac{1}{R},\\
V({z}{;}R)& = -\sum_{\pm}\frac{1}{\sqrt{\left({z}\pm\frac{R}{2}\right)^2+a(R)}},
\end{aligned}$$
with $m_1=m_2=1836$ and $q_1=q_2=1$. The interaction between the nuclei and the electrons $V({z}{;}R)$ is described by a soft-core Coulomb potential. The function $a(R)$ is chosen in such a way that the BO potential of this potential reproduces the BO potential energy curve of 3D H$_2^+$ [@PhysRevA.53.2562; @PhysRevA.67.043405; @PhysRevLett.98.253003]. We use the method described in Ref. [@PhysRevA.91.013408] to solve the 1D equivalent of Eq. (\[eq:schrodinger\]) given by $$\begin{aligned}
\left[- \frac{1}{2M} {\frac{\partial^2 }{\partial {R} ^2}} - \frac{1}{2m} {\frac{\partial^2 }{\partial {z} ^2}} + U(R) + V\left(z{;}R\right) + {q}zF - E(F)\right] \Psi(z,R) & = 0.\label{eq:schrodinger_1D}\end{aligned}$$ In the 1D model the index $\nu$, which describes what happens in the paraboloids of constant $\eta$ ’transversal’ to $z$, is of no meaning, and it hence does not appear in any of the 1D equivalents of the 3D equations. The 1D equivalent of the exact KER spectrum Eq. (\[eq:spec\_expr\]) is $$\begin{aligned}
P(k)
& =\frac{(2Fq{\left| z \right|})^{1/2}}{ m^{1/2} } {\left| \int g(R,k) \Psi(z,R) dR \right|}^2_{z\to -\infty} .\label{eq:spec_expr_1D}\end{aligned}$$ Equations (\[eq:Spec\_BO\]) and (\[eq:Spec\_result\_BO\]) apply to the 1D case with appropriately redefined quantities. The WFAT expressions Eqs. (\[eq:rate\_WFAT\_elec\]), (\[eq:field\_factor\_elec\]) and (\[eq:spec\_WFAT\]), (\[eq:full\_field\_factor\]) are the same as in the 1D case, but the asymptotic coefficients are now found from $$\begin{aligned}
f(R)&=\left.\frac{ \psi_{e,0}(z;R)}{{\left| z \right|}^{Z/\varkappa(R)}e^{-\varkappa(R){\left| z \right|}}}\right|_{z\to -\infty},\label{eq:1D_asymp_coeff}\end{aligned}$$ and $$\begin{aligned}
C(k)&=\left.\frac{\int_0^\infty g(R,k) \Psi_{0}(z,R)dR}{{\left| z \right|}^{mZ/\varkappa(k)}e^{-\varkappa(k){\left| z \right|}}}\right|_{z\to -\infty}.\end{aligned}$$ In Eq. (\[eq:1D\_asymp\_coeff\]), $\psi_{e,0}(z;R)$ denotes the field-free 1D electronic BO wave function.
From wave function to KER spectrum
----------------------------------
![(Color online) The field dressed nuclear wave function ${\left| \chi(R) \right|}^2$ (light blue shaded area in the lower $U(R)+E_e(R;F)$ BO curve) is multiplied by the electronic rate $\Gamma_e(R)$ (dashed purple line) and reflected in the dissociative $U(R)$ BO curve to give a KER spectrum (solid blue line in upper right corner, \[Eq. (\[eq:Spec\_result\_BO\])\]), using the relation $U(R)=\frac{k^2}{2M}$ to translate $k$ into $R$. This is compared to the exact KER spectrum $P(k)$ (red dashed line, \[Eq. (\[eq:spec\_expr\_1D\])\]). A field strength of $F=0.034$ was used for this calculation. The solid gray line in the lower part of the figure shows the field-free nuclear wave function ${\left| \chi_0(R) \right|}^2$. The surface plot in the upper part of the figure shows the continuum states $g(R,k)$ of the $U(R)$ potential, these are solutions of Eq. (\[eq:as\_R\_eq\]).[]{data-label="fig:spec_1D_ground"}](fig2.pdf)
Figure \[fig:spec\_1D\_ground\] illustrates how the BO approximation can be used in conjunction with the reflection principle to determine the KER spectrum. The figure shows a calculation for the ground state of the H$_2^+$ model at $F=0.034$. The field dressed nuclear wave function ${\left| \chi(R) \right|}^2$ is multiplied by the electronic rate $\Gamma_e(R)$. The exponential dependence of the electronic rate $\Gamma_e(R)$ on the internuclear coordinate means that the product $\Gamma_e(R){\left| \chi(R) \right|}^2$ (see \[Eq. (\[eq:Spec\_result\_BO\])\]) has its maximum at a value of $R\approx 3$, which is significantly different from the maximum of the bare nuclear wave function at $R_0\approx 2$. This in turn means that the transition to the continuum which is determined by the product $\Gamma_e(R){\left| \chi(R) \right|}^2$ and not the bare nuclear wave function is far from ’vertical’ in $R$ with respect to the initial nuclear wave function, and the spectrum peaks at a lower energy around $1/R\approx 0.33$ and not at $1/R_0\approx 0.5$.
Using WFAT within the BO approximation we can make a statement about in which direction the maximum of the spectrum shifts when the field is varied. In these approximations the main dependence of the electronic rate on the field is contained in the exponent $-\frac{2\varkappa^3(R)}{3F}$, see Eq. (\[eq:field\_factor\_elec\]). The electronic energy $E_e(R;F=0)$, in terms of which $\varkappa(R)$ is defined, generally depends very much on the system considered. In the case of H$_2^+$ it is a monotonically increasing function of $R$, since when the two potential wells around each of the nuclei start to overlap the electron is more tightly bound. This in turn means that the electronic rate is an increasing function of $R$, as can also be seen in Fig. \[fig:spec\_1D\_ground\]. When the strength of the field increases the exponent $-\frac{2\varkappa^3(R)}{3F}$ grows, but at the same time the slope of this exponent with respect to $R$ decreases, since $\varkappa^3(R)$ is multiplied by a smaller number. The smaller slope means that the location of the maximum of the product $\Gamma_e(R){\left| \chi(R) \right|}^2$ is shifted less from the maximum of ${\left| \chi(R) \right|}^2$ as the field strength increases, and conversely, as the field strength is decreased the maximum of the product $\Gamma_e(R){\left| \chi(R) \right|}^2$ is shifted more towards larger $R$. These shifts are directly reflected in the spectrum, which is given as the reflection of the $\Gamma_e(R){\left| \chi(R) \right|}^2$ product in the BO and reflection approximations.
Figure \[fig:spec\_1D\] shows KER spectra obtained using as initial state the first vibrationally exited state of H$_2^+$. We have chosen to show these results as they are for the lowest state with a non-trivial nodal structure in $R$. In the figure two different field strengths are considered. In the top panel we see that the nodal structure of the nuclear wave function is reflected in the KER spectrum, although one peak is a lot larger than the other. This asymmetry can be understood in the BO approximation, see Eq. (\[eq:Spec\_result\_BO\]), as due to the fact that the electronic rate $\Gamma_e(R)$ has an exponential dependence on $R$. In the WFAT it can be understood as resulting from the exponential dependence of the field factor \[Eq. (\[eq:full\_field\_factor\])\] on $k$. For the lower field strength the structures at $E_R>0.4$ are not visible as the KER spectrum falls below the numerical precision limit of our calculation.
![(Color online) KER Spectra normalized to their maxima. Solid (red) line: $P(k)$ \[Eq. (\[eq:spec\_expr\_1D\])\]. Dashed dotted (blue) line: BO combined with reflection principle \[Eq. (\[eq:Spec\_result\_BO\])\]. Short dashed (green) line: WFAT \[1D equivalent of Eq. (\[eq:spec\_WFAT\])\]. The insets show the normalized KER spectra on a linear scale. The critical field for use of BO \[Eq. (\[eq:F\_BO\])\] is for H$_2^+$: $F_\text{BO}=0.0315$. (a) $P_\text{max}^\text{BO}/P_\text{max}^\text{Exact}=2.32$ and $P_\text{max}^\text{WFAT}/P_\text{max}^\text{Exact}=0.29$. (b) $P_\text{max}^\text{BO}/P_\text{max}^\text{Exact}=95.0$ and $P_\text{max}^\text{WFAT}/P_\text{max}^\text{Exact}=0.54$. []{data-label="fig:spec_1D"}](fig3.pdf)
For the large field strength \[Fig. \[fig:spec\_1D\](a)\] we see that the BO KER spectrum has a shape much closer to the exact KER spectrum than for the lower field strength. Also the maximum value of the BO KER spectrum is more than an order of magnitude closer to the maximum value of the exact KER spectrum for the larger field strength. This can be understood on the basis of the retardation argument provided in Ref. [@rtd]: The BO approximation is expected to hold as long as the electron is close enough to the nuclei that the time it takes for the electron to go to its present location from the nuclei is shorter than the time it takes for the nuclei to move. A typical electron velocity can be estimated as $\varkappa_e=\sqrt{-2E_e(R_0;0)}$, where $R_0$ is the equilibrium internuclear distance, which for H$_2^+$ is $R_0=2$. A typical time scale for the nuclear motion can be estimated as $T=\frac{1}{2\omega_e}$, where $\omega_e$ is obtained by expanding the BO potential around $R_0$ to second order $U(R)+E_e(R;0)\approx U(R_0)+E_e(R_0;0)+\frac{1}{2}M\omega_e^2(R-R_0)^2$. Using these estimates Ref. [@rtd] defines a critical distance $$\begin{aligned}
{z}_\text{BO} & = \varkappa_e T=\frac{\varkappa_e}{2\omega_e},\label{eq:z_BO}\end{aligned}$$ such that for ${\left| {z}\right|}<{z}_\text{BO}$ we expect BO to work well, while for ${\left| {z}\right|}>{z}_\text{BO}$ we expect it to break down. Since the magnitude of the wave function is essentially unchanged after the tunneling, the BO approximation is expected to work well when the outer turning point is within this ${z}_\text{BO}$ distance, so a critical field $$\begin{aligned}
F_\text{BO} & = 2 \varkappa_e\omega_e \label{eq:F_BO}\end{aligned}$$ can be estimated, such that the BO approximation is expected to give good results for larger fields, but fail for smaller fields. The two field strengths of Fig. \[fig:spec\_1D\] lies on either side of this critical field, which for the system under consideration is $F_\text{BO}=0.0315$. As we increase the field strength further the BO gives even better results. For the lower field strength where BO fails we can apply the WFAT, see Sec. \[sec:full\_wfat\]. In Fig. \[fig:spec\_1D\] we see that the shape of the WFAT KER spectrum indeed is closer to the exact KER spectrum than the BO KER spectrum for the weaker field strength, and it is also closer in magnitude to the maximum value. For the larger field strength the WFAT KER spectrum is further from the exact KER spectrum in both shape and magnitude.
From KER spectrum back to wave function
---------------------------------------
![(Color online) From the exact KER spectrum $P(k)$ (upper right corner, \[Eq. (\[eq:spec\_expr\_1D\])\]) at $F=0.034$ the magnitude of the asymptotic wave function has been found by reversing the reflection principle, giving $P(k)/(2\pi{\left| \frac{dR}{dk} \right|})$, using the relation $U(R)=\frac{k^2}{2M}$ to translate $k$ into $R$. From this, the field-dressed nuclear wave function has been imaged by dividing with the electronic rate $\Gamma_e(R)$ and normalizing. In the lowest part of the plot, the short dashed (purple) line shows this imaging using the exact electronic rate $\Gamma_e(R)=-2\operatorname{Im}(E_e(R;F))$, the long dashed (red) line shows it using the BO WFAT approximation $\Gamma_e^\text{WFAT}(R)$ \[Eq. (\[eq:rate\_WFAT\_elec\])\]. The solid gray line shows the field-free nuclear wave function ${\left| \chi_0(R) \right|}^2$. The shaded (light blue) area shows the field-dressed nuclear wave function ${\left| \chi(R) \right|}^2$. The surface plot in the upper part of the figure shows the continuum states $g(R,k)$.[]{data-label="fig:Reconstructed_chi"}](fig4.pdf)
The field dressed nuclear wave function can be imaged from a measurement of the KER spectrum by inverting Eq. (\[eq:Spec\_result\_BO\]) for fields sufficiently large that the BO approximation applies. To demonstrate this we have taken the exact KER spectrum from our calculation at $F=0.034$ for the first vibrationally exited state and divided it by the Jacobian factor and the electronic rate to obtain an image of the nuclear density. Since an experimental KER spectrum is typically not known on an absolute scale, we have then normalized this quantity. In a calculation on a more complicated system than the one considered here the exact electronic rate is often not available, so we also show the result using the WFAT approximation for the electronic rate \[Eq. (\[eq:rate\_WFAT\_elec\])\]. The results are compared to the nuclear wave function known from the calculation in our model in Fig. \[fig:Reconstructed\_chi\]. They do not agree perfectly, but the nodal structure is correctly reproduced.
For smaller field strengths where the BO is not applicable this type of imaging is not possible. The KER spectrum, however, does give us access to the asymptotic wave function, as it is the norm square of the expansion coefficients of this, see Eq. (\[eq:wf\_expansion\]). For the cases we have looked at, the phase of the asymptotic coefficient $C(k)$ varies very little over the range where it has support. In our model we have access to the full wave function, and this we show in Fig. \[fig:wf\]. The imaging through the 1D equivalent of Eq. (\[eq:wf\_expansion\]) would only give access to the part at large negative ${z}$.
In the classically allowed region at large negative ${z}$ the maximum of the wave function follows a classical trajectory. This is a prediction of the WKB theory, which applies as long we are not too close to the turning line. The classical trajectories can be found using Newton’s second law
$$\begin{aligned}
m \ddot{z} +{\frac{\partial }{\partial z}}V(z;R) + Fq & = 0,\\
M \ddot{R} +{\frac{\partial }{\partial R}}V(z;R) + {\frac{\partial }{\partial R}}U(R) & = 0.
\end{aligned}$$
\[eq:clas\_traj\_newton\]
A tempting choice of initial condition for the differential Eqs. (\[eq:clas\_traj\_newton\]) would be to choose the $({z},R)$ values at the intersection of the outer turning line and the maximum ridge of the wave function, with zero velocity in both ${z}$ and $R$ direction. However, the WKB fails near the turning line, and therefore we cannot expect the wave function to follow a classical trajectory here. Instead we have chosen as initial condition some point at the maximum of the wave function at a large negative ${z}$ value away from the turning line. The influence of the $V(z;R)$ potential can be neglected for sufficiently large negative ${z}$, in this region we can write the separated energy conservation equations
$$\begin{aligned}
\frac{1}{2}m \dot{{z}}^2 + Fq{z}& = E-\frac{1}{2M}k^2,\\
\frac{1}{2}M \dot{R}^2 + {U}(R) & = \frac{1}{2M}k^2.
\end{aligned}$$
\[eq:clas\_traj\_energy\]
The initial velocities have then been determined from Eqs. (\[eq:clas\_traj\_energy\]), using the real part of the total (quantum) energy for $E$ and the $k$ at which the KER spectrum $P(k)$ \[Eq. (\[eq:spec\_expr\_1D\])\] peaks. The classical trajectories shown in Fig. \[fig:wf\] were found using such initial conditions, and then propagated inwards.
From Fig. \[fig:wf\] it can be seen that contrary to the exact wave function, the position of the ridge of the BO wave function in $R$ does not change with ${z}$. This is expected as the BO approximation appears in the limit of infinite nuclear mass, so classical motion in the nuclear coordinate is not possible. The asymptotic wave function that we can image using Eq. (\[eq:wf\_expansion\]) is therefore a non-BO wave function.
It might seem strange that the BO is able to give the correct KER spectrum when the spectrum is the norm square of the expansion coefficients of the asymptotic wave function, and the BO gives a wrong description of this asymptotic wave function. However, the fact that the BO wave function does not obtain a probability current (or velocity in the classical picture) in the $R$-direction does not alter its projection on the continuum states. The important point is whether the BO wave function is similar to the exact wave function as it emerges at the outer turning line after tunneling, and this is the case if the turning line is within the critical BO distance ${z}_\text{BO}$ \[Eq. (\[eq:z\_BO\])\].
In Fig. \[fig:wf\] we also see that for the larger field strength the tunneling is completed before the critical BO distance is reached, contrary to at the smaller field strength. We see that for the large field strength the electronic and full turning lines agree quite well in the region where most of the wave function is localized, but for the smaller field strength they do not.
![(Color online) Absolute value of wave function normalized with the electron density $\rho({z})=\int {\left| \Psi({z},R) \right|}^2 dR$ for $F=0.035$. Solid purple lines: Full turning lines $\operatorname{Re}(E(F))=V({z}{;}R)+U(R)+Fq{z}$. The long dashed red line shows for each $z$ the $R$ at which the wave function ${\left| \Psi(z,R) \right|}$ has its maximum. The solid pink line shows a classical trajectory \[Eq. (\[eq:clas\_traj\_newton\])\]. The black dot at the end of the classical trajectory is the exit point $(z_{k_\text{max}},R_{k_\text{max}})$ determined from the maximum of the spectrum $k_\text{max}$ (see main text). The short dashed lines are the simple straight line estimates for the tunneling and initial classical motion described around Eq. (\[eq:directions\]).[]{data-label="fig:wf_max"}](fig6.pdf)
A refraction phenomenon
-----------------------
One can notice that a phenomenon reminiscent of light refraction occurs for the wave function around the turning line in Fig. \[fig:wf\]. It is evident, that the direction in which the maximum of the wave function ’moves’ changes noticeably at the turning line, when the wave function escapes from the classically forbidden tunneling region into the classically allowed region. The change of direction is due to the two different types of ’motion’ involved. When the wave function emerges from the tunneling region it has essentially zero average velocity in the $R$ direction. This means that we can apply the reflection principle in reverse on the spectrum to find the $R_{k_\text{max}}$ coordinate at which the maximum of the wave function emerges from the tunneling region by the relation $U(R_{k_\text{max}})=\frac{k_\text{max}^2}{2M}$, where $k_\text{max}$ is the value of $k$ for which the spectrum $P(k)$ has its maximum. The $z$ value corresponding to this $R_{k_\text{max}}$ can then by found by considering the turning line $V(z_{k_\text{max}},R_{k_\text{max}})+U(R_{k_\text{max}})+Fqz_{k_\text{max}}=\operatorname{Re}E$.
In Fig. \[fig:wf\_max\] we see that near the turning line the location of the wave function ridge differs from the classical trajectory. This is expected, since the prediction that the wave function ridge should follow a classical trajectory comes from WKB theory, which fails near the turning line. Nevertheless, we can roughly describe the dissociative tunneling ionization process in two steps. First the system tunnels from the central region around $z=0$ to the exit point $(z_{k_\text{max}},R_{k_\text{max}})$. This motion can roughly be described by a straight line from the maximum of the nuclear wave function ${\left| \chi(R) \right|}^2$ that has the largest $R$ value, since this is the maximum that will dominate the tunneling, to the exit point. Notice that this tunneling is not simply the electron tunneling out, but a correlated process involving both the electronic and nuclear degrees of freedom. In the classically allowed region the initial direction of the wave function from the exit point can be found from the classical trajectory: The initial slope of the classical trajectory that starts at the exit point $(z_{k_\text{max}},R_{k_\text{max}})$ with zero velocity in both $z$ and $R$ directions can be found to be $$\begin{aligned}
\left.\frac{\dot{z}}{\dot{R}}\right|_{\text{exit}}=\left.\frac{M}{m}\frac{{\frac{\partial }{\partial z}}V(z;R) + Fq}{{\frac{\partial }{\partial R}}V(z;R)+{\frac{\partial }{\partial R}}U(R)}\right|_{(z_{k_\text{max}},R_{k_\text{max}})}.\label{eq:directions}\end{aligned}$$ This is not exactly the trajectory that describes the motion of the wave function ridge, but it is quite close. These two directions are different as they come from different types of motion, and hence we see the refraction-like phenomenon at the turning line.
Conclusion {#sec:conclusion-outlook}
==========
We have formulated theory for the dissociative tunneling ionization process, and derived exact formulas for the KER spectrum, as well as approximations in the framework of the BO and reflection approximations. We have demonstrated that the reflection principle can be used in conjunction with the BO approximation to image the field-dressed nuclear wave function from the KER spectrum. For weaker fields, where the BO approximation fails, the WFAT can be used to find the KER spectrum. We have also demonstrated a qualitative difference between asymptotic BO and exact wave functions, as the latter shows classical motion in the nuclear coordinate, whereas the former does not move at all due to the infinite nuclear mass of the BO approximation. Around the turning line the wave function exhibits a behavior similar to refraction of light.
This work was supported by the ERC-StG (Project No. 277767-TDMET), and the VKR center of excellence, QUSCOPE. The numerical results presented in this work were performed at the Centre for Scientific Computing, Aarhus <http://phys.au.dk/forskning/cscaa/>. O. I. T. acknowledges support from the Ministry of Education and Science of Russia (State Assignment No. 3.679.2014/K).
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[^1]: We consider monotonically decreasing $U(R)$ potentials, so there is only one classical turning point.
|
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abstract: 'The Galactic supernova remnant W49B has one of the most impressive X-ray emission line spectra obtained with the [*Advanced Satellite for Cosmology and Astronomy (ASCA)*]{}. We use both plasma line diagnostics and broadband model fits to show that the Si and S emission lines require multiple spectral components. The spectral data do not necessarily require individual elements to be spatially stratified, as suggested by earlier work, although when line images are considered, it is possible that Fe is stratified with respect to Si and S. Most of the X-ray emitting gas is from ejecta, based on the element abundances required, but is surprisingly close to being in collisional ionization equilibrium. A high ionization age implies a high internal density in a young remnant. The fitted emission measure for W49B indicates a minimum density of 2 cm$^{-3}$, with the true density likely to be significantly higher. W49B probably had a Type Ia progenitor, based on the relative element abundances, although a low-mass Type II progenitor is still possible. We find persuasive evidence for Cr and possibly Mn emission in the spectrum—the first detection of these elements in X-rays from a cosmic source.'
author:
- 'Una Hwang (1,2), Robert Petre (1), John P. Hughes (3)'
title: 'The X-ray Line Emission from the Supernova Remnant W49B'
---
Introduction
============
When the Galactic supernova remnant W49B (G43.3-0.2) was first detected as an X-ray source by the [*Einstein Observatory*]{}, its centrally-bright morphology raised speculations that its X-ray emission is produced by nonthermal synchrotron processes (Pye et al. 1984), as in the Crab. The subsequent discovery of a prominent Fe K emission line blend with [*EXOSAT*]{} showed that the X-ray emission is thermal, and probably dominated by supernova ejecta (Smith et al. 1985). The [*Advanced Satellite for Cosmology and Astronomy (ASCA)*]{} has since provided the highest quality X-ray spectral data yet available for W49B, revealing a spectacular array of prominent emission lines from the elements Si, S, Ar, Ca, and Fe.
Fujimoto (1995) analyzed the intensity ratios of the Ly$\alpha$ and He$\alpha$ lines (n=2$\rightarrow$n=1 transitions in the H- and He-like ions, respectively) using the data, and concluded that the elements Si, S, Ar, and Ca cannot have the same average ionization age for a given temperature, with Si and S having the largest discrepancies. With the ionization age defined as the product $n_et$ of the ambient electron density $n_e$ and the time $t$ since the gas was shock-heated (Gorenstein, Harnden & Tucker 1974), this implies that each element either has a different density or was shocked at a different time. This result is important, if true, since all the elements Si, S, Ar, and Ca occupy essentially the same spatial zones in supernova models, while Fe initially occupies a spatial zone interior to these elements (e.g., Nomoto 1997a, Thielemann, Nomoto, & Hashimoto 1990). The X-ray images of the remnant in emission lines of Si, S, and Fe (with a low spatial resolution of FWHM $>1'$) suggest that the Si and S emission is distributed outside more centrally peaked continuum (4$-$6 keV) and Fe K emission. These results are interpreted as evidence for stratification of the supernova ejecta, with the Fe ejecta lying interior to the Si and S ejecta.
In this paper, we re-examine the data for W49B, carrying out a more complete and careful analysis of the X-ray line emission. We examine evidence for emission from less abundant elements, and interpret the spectrum using both line intensity ratios and simple broadband model fits. We also present the previously unpublished High Resolution Imager (HRI) data for W49B—essentially a Si emission line image through the combined effect of the narrow 0.2$-$2.2 keV bandpass of the HRI and interstellar absorption of the W49B spectrum at energies below $\sim$1 keV. On the basis of our new results, we suggest emission from multiple thermal components as an alternative to the stratification scenario of Fujimoto As discussed later, however, our results do still allow for the possible spatial separation of Fe from other elements.
After the submission of this paper, we were made aware of a similar analysis of these data reported by Sun & Wang (1999). Our independent analyses lead to similar general conclusions.
For completeness, we note that the remnant displays a sharply defined, but incomplete radio shell of radius $\sim100''$, with high surface brightness and very low polarization (Moffett & Reynolds 1994, and references therein). Based on HI absorption measurements in the radio, the distance to the remnant is 8 $\pm$ 2 kpc (Radhakrishnan et al. 1972; adjusted for a Galactocentric radius of 8 kpc, following Moffett & Reynolds 1994). To our knowledge, no optical emission has been reported for this remnant, and it is likely that any such emission is highly absorbed because of W49B’s large distance through the Galactic plane. Absorption is virtually negligible for infrared emission, however, and IRAS detected strong emission from the vicinity of the remnant (Saken, Fesen, & Shull 1992). Though W49B is probably a young remnant, its infrared colors are more consistent with an older rather than a young remnant. It is possible that that this is due in part to source confusion in a crowded region.
Data Reduction
==============
(Tanaka 1994) features two Solid-State Imaging Spectrometers (SIS0 and SIS1) and two Gas Imaging Spectrometers (GIS2 and GIS3), each with a dedicated mirror. The point-spread function (PSF) of each of the X-ray mirrors has a narrow core of about $1'$ FWHM, and a half-power diameter of $3'$. Each SIS is a square array of four CCD chips, of which 1, 2, or all 4 may be exposed at a time (1-, 2-, or 4-CCD mode). The SIS provides moderate resolution spectra between the energies 0.5 $-$ 10 keV, with the best instrument performance in 1-CCD mode ($\Delta \rm{E} \sim 1/\sqrt{E} \sim$ 2% at 6 keV at satellite launch in 1993). Because we focus on the line emission in this paper, and the GIS provides lower spectral and spatial resolution data than does the SIS, we do not present the GIS data.
There are ten observations of W49B, all taken in 1993 during the Performance Verification phase of the mission. The source was observed in a variety of instrument modes and detector positions because it was used as a gain calibrator for the mission. We examine five of the observations in detail; we do not present the other five because they have exposure times shorter than 3 ks, and in all but one, the source is also either off or near the edge of the SIS field of view. Our observations are in different SIS instrument modes (1-, 2-, and 4-CCD), as noted in Table 1, with the source at different positions on the detector.
Since the instrument performance and calibration for the early observations are good, we use the REV2 cleaned data taken directly from the HEASARC archive. Interested readers may consult the Data Reduction Guide (1997) for technical information on the data processing procedures. We focus on the 20 ks 2-CCD mode observation because it has the longest exposure time at a single source position. For fitting line intensities, we also consider a combined SIS spectrum in which all the pulse-height spectra, response matrices, and effective areas are weighted by the exposure time and averaged. We optimize the signal in the high energy spectrum by taking all the spectra from a 3.2$'$ radius region centered on the source. This region contains roughly 75% of the total flux from the source.
The background is taken from blank sky fields provided by the Guest Observer Facility. Since W49B is in the Galactic Ridge, we have also verified that the results are not sensitive to the exact background used. We have compared the SIS0 2-CCD mode results using the blank sky field with those using a local background taken away from the source at the edges of the SIS, and find very good consistency. The only substantial differences are between the best-fit line intensities of the weak Fe features beyond the strong Fe K blend, but the 90% confidence error ranges are in good agreement. Roughly 80% of the total SIS counts at energies above 1.5 keV come from the source using either background subtraction.
The High Resolution Imager has a bandpass of 0.2$-$2.2 keV and provides a spatial resolution of 5$''$ at the center of its 38$'$ wide field of view. A 34 ks observation ( sequence rh500098n00) centered on W49B was processed with software provided by Snowden & Kuntz (1998) to correct for vignetting, and to model and subtract the particle background (Snowden 1998). The image shown in Figure 1 is smoothed using an adaptive filter with a varying spatial scale corresponding to 50 counts per beam. W49B is a weak HRI source ($\sim$0.05 HRI counts/s) because the high column density effectively cuts off the spectrum below 1 keV. Compared to the [*Einstein*]{} HRI, the HRI has higher sensitivity and lower background, but a narrower bandpass. The two bright spots that dominate the [*Einstein*]{} HRI image (Pye 1984, Seward 1990) are clearly seen in the image, but the image also shows the surrounding diffuse emission more clearly. The diffuse emission appears to be interior to the radio shell, and shows no sign of limb-brightening.
Spectral Results
================
We discuss the spectral results for W49B in this section, treating the line intensities and plasma diagnostics first, followed by the fitting of simple broadband spectral models.
Line Spectrum
-------------
W49B boasts one of the most impressive emission line spectra obtained by , featuring prominent He$\alpha$ blends (n=2$\rightarrow$n=1 in the He-like ion) and Ly$\alpha$ transitions (n=2$\rightarrow$n=1 in the H-like ion) of Si, S, Ar, and Ca, clearly discernable higher level transitions ($1s3p\rightarrow 1s^2\ [{\rm hereafter}\ 3p]$, $1s4p\rightarrow 1s^2\ [{\rm hereafter}\ 4p]$), and a prominent Fe K blend. The line fluxes may be measured by modelling the continuum and fitting narrow gaussian functions for each distinct emission feature. Figure 2 shows the spectrum combining all the SIS data with the best-fit line model (to be described below).
Line blending can be a significant issue when measuring line fluxes with a moderate resolution spectrometer like the SIS. Two important instances of line blending that we address involve the Ly$\beta$ and the $3p$ and $4p$ transitions. At the SIS resolution, Ly$\beta$ lines are often blended with the He$\alpha$ emission of a higher atomic weight element, and can have a non-negligible flux if the corresponding Ly$\alpha$ lines are strong. At the temperature of the W49B continuum ($kT \sim$ 2 keV), the ratio of Ly$\beta$ to Ly$\alpha$ intensity is between 0.11 and 0.14 for Si, S, and Ar in the models of Raymond & Smith (1977, hereafter RS). For example, in W49B, this means that S Ly$\beta$ is responsible for about 25% of the flux that would have been attributed to Ar He$\alpha$. In our fits, we therefore include the Ly$\beta$ lines of Si, S, and Ar at their expected energies with an appropriately fixed intensity relative to the Ly$\alpha$ line. The $4p$ transition is also sometimes blended with other lines, but fortunately, its intensity relative to the $3p$ transition does not change strongly with temperature and is constant with ionization age. We therefore model the the $4p$ line with an intensity relative to $3p$ fixed at 0.6, as is appropriate for temperatures between 1$-$2 keV in the RS models. For Fe, we note that $3p$ is blended with the He$\alpha$ transitions of Ni at the resolution, and that its true flux is overestimated since we don’t model the Ni lines.
The energy scale of the W49B spectrum is established by its strong Ly$\alpha$ lines, since they are effectively single transitions at known energies. The measured Ly$\alpha$ energies are in excellent agreement with the expected values, with formal 90% confidence errors that are well within the nominal estimated 0.5% accuracy of the SIS gain; for Ca Ly$\alpha$, the formal error is comparable to the nominal gain accuracy. In our final fits, we fixed the energies of the Ly$\alpha$ lines at their expected values and adjusted the overall gain of the spectrum by 0.25%. The line energies of the He$\alpha$ blends are always freely fitted because they depend on the ionization state of the gas: not only do the relative intensities within the triplet of the He-like ion depend on the ionization state, but ions less ionized than the He-like stage may also make a significant contribution to this blend through lines with lower energies. This centroid is thus potentially a valuable diagnostic.
We fitted both the 2CCD-mode SIS0 spectrum and the spectrum combining all the SIS data at energies between 0.6 and 10 keV with a model for the continuum and the line features. Lines of Si, S, Ar, Ca, and Fe were included as described above and their fitted energies (for He$\alpha$) and intensities are given for the combined SIS data in Table 2 with their 90% confidence ($\Delta\chi^2$ = 2.71) errors. The continuum was modelled as two bremsstrahlung components, with temperature about 1.7 keV for the dominant component, and fixed at 0.2 keV for the second component. The inclusion of a second continuum component gives a better fit, gives more consistent line diagnostics results, and is consistent with the results of the broadband fits to be described below. The fitted column density is about $5 \times
10^{22}\ \rm{cm}^{-2}$ in the double continuum model, which is significantly higher than the value of $3 \times 10^{22}\
\rm{cm}^{-2}$ when only one continuum component is modelled; the Galactic column density measured in the radio is 1.8 $\times 10^{22}\
\rm{cm}^{-2}$ (Dickey & Lockman 1990). The X-ray measured absorption column density is significantly higher than the radio value, but this appears to be typical for neutral H column densities in excess of several times $10^{20}$ cm$^{-2}$ because of the presence of interstellar molecular gas (Arabadjis & Bregman 1999). The absolute Si line intensities are therefore significantly different depending how the continuum is modelled, but the Si line intensity ratios discussed below are much less so.
Plasma Diagnostics
------------------
Measured line intensity ratios and He$\alpha$ centroid energies can provide joint constraints on the emission-averaged temperature and ionization age if the lines are chosen to eliminate or minimize other dependences, such as element abundances or interstellar absorption. The ionization age—defined as $n_e t$, where $n_e$ is the ambient electron density, and $t$ is the time since the gas was shock-heated—parameterizes the ionization state of the gas (Gorenstein, Harnden & Tucker 1974). Gas that is suddenly heated by the passage of the supernova shock wave is ionized slowly through electron-ion collisions on a timescale $10^4/n_e$ yr, where $n_e$ is in cm$^{-3}$. This timescale may be comparable to, or larger than, the known or deduced age of the remnant for a typical ambient density of 0.1 cm$^{-3}$; most remnants are therefore expected not to be in ionization equilibrium. Since nonequilibrium ionization (NEI) affects the ion population, and line intensities are directly proportional to the population of the relevant ion, it can strongly affect X-ray line intensities. We calculate line emissivity ratios to compare with the observations for a grid of temperatures and ionization ages, using the updated code of Raymond & Smith (1977, hereafter RS) for the X-ray emission, and the matrix diagonalization code of Hughes & Helfand (1985) for the nonequilibrium ion fractions.
In Figure 3, we show the constraints on $kT$ and $nt$ based on 90% confidence limits for both the Ly$\alpha$ to He$\alpha$ and the He$3p+4p$ to He$\alpha$ intensity ratios. Separate panels show the results for each of the elements Si, S, Ar, Ca, and Fe. The observed line intensity ratios are taken from the fit to the combined SIS data with 90% limits determined from the two-dimensional $\Delta\chi^2$=4.61 contours. These limits are consistent with, but significantly tighter than, those from the 2-CCD mode SIS0 fits. The values of $kT$ and $nt$ allowed for Ar, Ca, and Fe are consistent with each other near ionization equilibrium at kT = 2.2$-$2.7 keV, with the exception that the Ca He$3p+4p$/He$\alpha$ ratio is marginally low. Considering that we have not included systematic uncertainties in our errors, the agreement between these three elements is very good, as shown by the overlaid contours in the last panel of Figure 3. The temperature is somewhat higher than the temperature of 1.7 keV inferred for the bremsstrahlung continuum. For the elements Si and S, however, the ratios are inconsistent with each other as well as with the results for Ar, Ca, and Fe. Since it is not possible for any single spectral component to simultaneously reproduce the line ratios for either Si or S, the emission from these elements must be more complex. For Si, this conclusion is supported by the Si He$\alpha$ line centroid. Its constraints are inconsistent with those from the ratio Si Ly$\alpha$/He$\alpha$, although a 0.5% systematic error on the line energy allows a small region of overlap near collisional ionization equilibrium (CIE; see Figure 4). The systematic errors on the other centroids are too large to confirm the results of the S line intensity diagnostics or to provide any new constraints for the other elements.
If we assume that one spectral component has a temperature of 2.2 keV and is at CIE, we may then infer parameters characterizing the second component. While the 2.2 keV CIE component gives the wrong Si centroid, the $kT$ and $nt$ values that do give the observed Si centroid also give low Ly$\alpha$ and He$3p+4p$ intensities relative to the He$\alpha$ intensity. Thus, the second component could reproduce the Si He$\alpha$ centroid and provide nearly all of the Si He$\alpha$ intensity, while contributing relatively little to Ly$\alpha$ or He$3p+4p$. Because of its low temperature, this component contributes less to the S emission, and almost nothing to the emission of the other elements. The broadband fits described below confirm that the single-temperature CIE model that accounts for most of the W49B spectrum is deficient in Si He$\alpha$ flux.
Our results may be compared to the earlier results of Fujimoto et al. (1995), who carried out a similar analysis for W49B using the same data. They use primarily the Ly$\alpha$ to He$\alpha$ line intensity ratios for Si, S, Ar, and Ca and, as noted earlier, conclude that each of these elements has a different ionization age for a given temperature, with the Si and S parameters being the most disjoint. We are able to reproduce their results by modelling the spectrum with one continuum component, and not including weaker lines in the model. When we account for the Ly $\beta$ lines, however, we find very good consistency between the parameters for Ar, Ca, and Fe. We also show that the Si and S emission requires multiple spectral components: their line ratios (and the Si centroid) are strongly inconsistent with a single temperature and ionization state. We conclude that the spectral data do not necessarily require stratification of the individual elements, as Fujimoto et al. concluded, but that they do reveal the spectral complexity of the emission from the elements Si and S.
Evidence for Emission from Cr and Mn
------------------------------------
As the spectral resolution and efficiency of X-ray spectrometers continue to improve, it becomes feasible to search for emission from elements other than the dozen most abundant ones that are now routinely included in spectral models. In Figure 5, the combined SIS spectrum of W49B between the energies 5.0 and 6.4 keV shows line-like features at energies of $\sim$ 5.7 and 6.1 keV. There are no emission lines in the RS model within the 90% confidence errors of the centroids, but these models do not include the elements Cr and Mn. Emission lines of Cr and Mn do have energies near those of the observed line features. The forbidden and resonance transitions of He-like Cr are at 5.655 and 5.682 keV, respectively, and those of Mn at 6.151 and 6.181 keV; the Ly$\alpha$ transitions of Cr and Mn are at 5.917 and 6.424 keV, respectively.
The fluxes and centroids of these features were determined by modelling each as a gaussian component along with a bremsstrahlung continuum for the portion of the spectrum between energies of 4.5 and 6.4 keV. The lower energy feature, which we tentatively attribute to Cr, appears at a line energy of 5.685$^{+0.020}_{-0.027}$ keV and has a flux of 3.0$^{+0.8}_{-1.1}\times 10^{-5} \rm{ph/cm}^2 \rm{/s}$, corresponding to an equivalent width of about 90 eV. For the other feature, attributed tentatively to Mn, the line energy and flux are 6.172 $^{+0.047}_{-0.049}$ keV and 1.3 $^{+1.4}_{-0.6} 10^{-5}$ cm$^{-3}$ s$^{-1}$, or an equivalent width of 60 eV. The reduction in $\chi^2$ from adding the “Cr” feature is more than 20 for 61 degrees of freedom, while the reduction in adding “Mn” is an additional 10. An F-statistic of greater than 10 for 2 and 60 degrees of freedom gives a probability greater than 99% that these features are real.
There are no atomic data currently available for the calculation of emissivities for either Cr or Mn. To check if the strengths of the observed features are consistent with our interpretation of them, we have carried out the tests summarized in Figure 6. We calculate the intrinsic He$\alpha$ emissivity for Si (Z=14), S (Z=16), Ar (Z=18), Ca (Z=20), Fe (Z=26), and Ni (Z=28) for the 2 keV CIE component without including factors for the element abundances. The crosses in the top panel of the figure shows these calculated emissivities plotted against atomic number, overlaid with a spline fit through the individual points. The calculated emissivities are then used with the measured He$\alpha$ line fluxes to estimate the element abundances for Si, S, Ar, Ca, Fe, and Ni (we use the Fe $3p+4p$ flux as an upper limit for the Ni He$\alpha$ flux). These abundances are plotted in the bottom panel of the figure as crosses with error bars after normalizing to the Ar abundance. Ar was chosen for the normalization as the lowest atomic number element for which most of the emission comes from the CIE component. We expect both Si and S to have additional emission from a soft component; the high abundances of Si and S (Si is off the scale of the plot) are spurious since their He$\alpha$ emission arises in part from a cooler NEI component (see the following section). For the elements Cr (Z=24) and Mn (Z=25), we interpolate the emissivities using the curve in the top panel of the figure, and attribute all the measured flux at 5.7 and 6.1 keV to the Cr and Mn He$\alpha$ blends, respectively. The resulting abundances are also plotted in the figure as crosses. They are seen to be consistent with the solar photospheric ratios (Anders & Grevesse 1989), which are plotted for each element as open circles in the same panel. As Cr, Mn, and Ni are the next most abundant elements with K-shell emission lines at energies above $\sim$ 4 keV, they are thus the atomic species most likely to be detected next.
Broadband Spectral Fitting
--------------------------
We turn next to fitting of the broadband spectrum of W49B. We use collisional ionization equilibrium (CIE) models of RS, nonequilibrium ionization (NEI) models for Sedov hydrodynamics of Hamilton, Sarazin, & Chevalier (1983; hereafter HSC) and single temperature, single-ionization age models of RS with ion fractions calculated according to Hughes & Helfand (1985)—the model used for the line diagnostics (§3.2). We expect that at least two spectral components will be required, and first verify that no single spectral component provides an adequate fit. An RS model with temperature near 1.7 keV does best, but severely underpredicts the flux of the Si He$\alpha$ blend and gives $\chi^2$ per degree of freedom greater than 3. A comparable fit is obtained with an NEI model with a similar temperature and an ionization age near 8$\times 10^{11}\ \rm{cm}^{-3}$ s. Most of the X-ray emitting plasma in W49B is clearly at or near CIE, as was first suggested by Smith (1985) based on the Fe K centroid and the Ly$\alpha$ to K$\alpha$ intensity ratio for the measured continuum temperature of 1.8 keV. The HSC Sedov models do much worse than the other NEI models because the grid of models available to us does not extend sufficiently close to CIE for the relevant temperatures.
We obtain a satisfactory fit to the overall spectrum by combining a CIE RS component with a low temperature NEI component. Using a HSC Sedov model for the NEI component gives the results shown in Table 3 and Figure 7. The HSC component contributes significantly to the emission lines of Si and S (providing nearly all of the Si He$\alpha$ blend) and to the surrounding continuum, but little at higher energies. As a Sedov model, it is itself intrinsically multi-temperature. If, instead, a single $kT$, single $nt$ model is used for the NEI component, it contributes primarily to the Si He $\alpha$ blend, and very little to other lines. Its contribution to the spectrum falls off more quickly with energy than for the Sedov component and its parameters are less well-constrained. Models with low temperatures between roughly 0.2 and 0.5 keV give good fits for a range of ionization ages consistent with those required to give the observed Si He$\alpha$ centroid. If two NEI components are used to model the spectrum, CIE is still favored for the dominant component, with a $\Delta\chi^2=2.7$ range for $nt > 10^{12}$ cm$^{-3}$ s.
Although the absolute abundances of the elements are model-dependent, it is worth examining the abundance results of our simple model fits. To limit the number of free parameters in the models, we linked element abundances together where possible. The CIE component contributes strongly to the line emission from Si, S, Ar, Ca, and Fe, so the abundances of these elements are fitted freely for this component. The Mg and Ni abundances are also fitted because Mg and Fe L emission are blended together at energies near 1.3 keV, with the Fe L atomic data having known deficiencies (Liedahl et al. 1995), while Ni and Fe emission are blended together at energies near 7.8 keV. The abundances of C, N, O, and Ne are fixed at their solar value—a sufficient assumption since this spectrum is strongly attenuated by absorption at the relevant energies below about 1 keV. The data cannot constrain a second full set of abundances for the NEI component so the C, N, O, Ne, and Mg abundances are fixed at their solar value, Si through Ca tied in their solar ratios, and Fe tied to Ni. The abundance results are summarized in Table 3. The primary CIE component requires abundances for Si, S, Ar, Ca, and Fe that are comparable to each other and significantly enhanced above the solar value. The Mg abundance is formally zero, but has a large error. The second, NEI component suggests an enhanced Si abundance and a rather low Fe abundance, but these abundances are nearly consistent when their errors are considered (see Table 3).
Discussion
==========
The results of the broadband fits indicate that the X-ray emission from W49B is dominated by its ejecta, and that the ejecta are at or very near CIE. Ejecta-dominated remnants are expected to be dynamically young, so a high ionization age requires a high density. We obtain a lower limit for the internal density of the X-ray emitting gas of 2 cm$^{-3}$ by using the X-ray emission measure for the CIE component and assuming that it fills a sphere with the radius of the radio shell. The same value was obtained by Smith (1985) from the EXOSAT X-ray data. Moffett & Reynolds (1994) also cite this value as the minimum density required for significant Faraday depolarization of the radio emission. The high density required to explain the X-ray emission thus also allows a possible explanation for the unusually low polarization in the radio. The actual density would need to be even higher since the X-ray emission does not appear to fill the radio shell and is clearly centrally concentrated. The high density places constraints on the properties of the progenitor. Massive progenitors of type O through B0 clear a surrounding radius of 15 pc during their main sequence lifetime through their fast stellar winds and photoionizing radiation (e.g., see Chevalier 1990). These progenitors do not seem plausible for W49B, which has a radius of only about 5 pc for a distance of 8 kpc, unless it is still interacting with dense circumstellar material ejected by a slow wind if the progenitor underwent a red giant phase. The element abundances are compared with the calculated abundances for various supernova progenitors in Figure 8. We focus on the relative abundances because we find that they are more consistent among the various models than are the absolute abundances. The 90% confidence limits for the measured abundances of Mg, S, Ar, Ca, Fe, and Ni relative to the abundance of Si in the CIE component are determined from the two-dimensional $\Delta\chi^2$=4.6 contours. The calculated abundances for two Type Ia models—the standard W7 model and a delayed detonation model (WDD2)—plus Type II models for progenitor masses of 13 and 15 M$_\odot$, are all taken from Nomoto (1997ab), and normalized to the Si abundance. All abundances are given relative to the solar photospheric abundances of Anders & Grevesse (1989). As can be seen from the Figure, the 90% confidence limit for the Mg abundance relative to Si is consistent with all the Type Ia and low mass Type II models that we considered. More massive Type II progenitors would produce too much Mg relative to Si to be consistent with the data. All the models underpredict Ar relative to Si, but the Type II models also underpredict S and Ca relative to Si. The WDD2 Type Ia model appears to be favored overall for the CIE component, but is still far from being consistent with all the observations.
Other possible nucleosynthesis diagnostics are the relative abundances of less abundant elements, such as Ti, Cr, and Mn. According to the nucleosynthesis models used for Figure 8, the Ti abundance relative to Mn and Cr is depressed relative to solar in Type Ia explosions, while the three are comparable in low-mass Type II explosions. We demonstrated above that Cr and Mn are in roughly their solar ratio relative to Ar. Our 90% confidence limits on the Ti He$\alpha$ flux is $2\times 10^{-5}$ cm$^{-2}$ s$^{-1}$ at an energy $\sim$ 4.97 keV (the He-like resonance and forbidden transitions of Ti are at 4.966 and 4.977 keV, and overlap the He $3p$ and $4p$ transitions of Ca at the resolution). Using Figure 6, the 90% upper limit for the Ti abundance relative to Ar is about 5.7 dex, or a few times the solar value. Thus our upper limit does not distinguish between the models, but future measurements of these quantities may improve enough to make such diagnostics useful. Foremost is the need for calculated emissivities for these elements so that measured line fluxes can be converted into reliable element abundances. Currently, the codes that calculate X-ray emission do not include the elements Ti, Cr, or Mn.
The total mass of the ejecta may formally be calculated by assuming a geometry for the remnant. The image is essentially a Si plus continuum image, showing two bright spots plus a shelf of fainter diffuse emission, each contributing roughly half of the image counts. Since the Fe K and hard continuum images of Fujimoto et al. (1995) are less extended than the Si and S images, we conclude that the soft spectral component is more spatially extended and that the hard component comes from the two bright spots. If we assume that the CIE component comes from the two bright lobes and model them as spheres of diameter 0.5$'$, the total mass of the CIE component is 1.6 M$\odot$. This mass is marginally consistent with the ejecta mass from a Type Ia explosion, but is based on uncertain assumptions about the geometry and the relative abundances of elements that do not exhibit strong emission lines. Our simple mass calculation therefore does not unambiguously differentiate between Type Ia and Type II progenitors. The calculated Si and Fe ejecta masses, at 0.02 and 0.04 M$\odot$ respectively, are lower than the Si and Fe masses for either Type Ia or Type II explosions. The calculated masses increase if the CIE component is more extended than assumed, and also if the soft NEI component in W49B is associated with ejecta.
Our broadband spectral fit results indicate that the Si and S lines have a significant contribution from the NEI component, and that Si He$\alpha$ in particular comes almost entirely from it. Meanwhile, the Ar, Ca, and Fe K emission comes almost entirely from the hotter CIE component. The images of Fe K and Si from Fujimoto show a distinct difference in morphology that can be explained naturally if, as we have assumed earlier, the two spectral components that we have identified in W49B have different spatial distributions: a compact component for Fe K and a more extended component for Si. We then expect that the Si Ly$\alpha$ emission should be morphologically similar to the Fe K emission (since both come from the CIE component), while Si He$\alpha$ should be morphologically different (since it comes from the other, cool NEI component). Fujimoto show that the radial profiles for these two Si lines actually look similar. Since the PSF is not much smaller than the remnant itself, and photon statistics are limited for an image of a single line feature, more sensitive and higher spatial resolution observations with Chandra and XMM will be necessary to resolve these discrepancies. If the distribution of the Si Ly$\alpha$ emission is truly similar to Si He$\alpha$ and distinct from Fe K, our spectral results would require both cool and hot extended Si ejecta, and compact hot Fe ejecta, implying spatial segregation of the Si and Fe ejecta.
It is also possible that the soft spectral component represents a cool blast wave, as suggested by Smith (1985). If so, our Sedov parameters require the explosion to have been very weak (E $< 10^{50}$ ergs), and the remnant age to be about 4000 yr—conclusions that are qualitatively similar to those reached by Smith We cannot be certain that the soft component is actually the blast wave, however. The soft emission, while extended, is not limb-brightened, and does not appear to fill the radio shell in currently available X-ray images. We also obtained better fits with enriched Si abundance than with the element abundances in solar ratios, suggesting that the soft NEI component may be from ejecta, though the models used are in need of updated atomic physics. In any case, forthcoming observations of the distribution of the Si emission will shed much light on this issue. The blast wave may be too faint to have been detected yet.
The X-ray observations have provided a wealth of new information about the unusual supernova remnant W49B. It has a rich and complex spectrum including previously undetected emission lines of the elements Cr and Mn. The remnant shows evidence for spatial differences between the X-ray emission of different elements that are not yet understood: they may be due to ejecta plus the blast wave, which has not yet been positively identified in X-rays, or to ejecta of different densities and spatial distributions. W49B bears the strong imprint of its progenitor, in that its X-ray emission is still dominated by its ejecta, but it is also clearly the product of its complex environment. It is an unusual example of an ejecta-dominated remnant wherein much of the gas is very near collisional ionization equilibrium, indicating that this young remnant has a very dense environment—whether interstellar or circumstellar. The current data and models do not conclusively resolve whether this was a Type Ia or Type II explosion, nor whether the soft spectral component is from a cool blast wave or from ejecta, but have raised tantalizing puzzles and hints of new discoveries that may be solved with future observations. These are fortunately soon forthcoming.
This work was based on archival X-ray data provided by the HEASARC at NASA Goddard Space Flight Center. We thank Steve Snowden for providing software to process the data. UH acknowledges partial support through the NASA Astrophysics Data Program, and JPH acknowledges support through NASA grants NAG 5-4794, NAG 5-4871, and NAG 5-6419.
Anders, L., & Grevesse, N. 1989, GeCoA, 53, 197 Arabadjis, J. S., & Bregman, J. N. 1999, , 510 806 Chevalier, R. A. 1990, in Supernovae (ed. Petschek, A. G.), New York: Springer-Verlag, 91 Dickey, J. M., & Lockman F. J. 1990, ARAA, 28, 215 Fujimoto, R., 1995, PASJ, 47, L31 1997, AJ, 114, 2058 Gorenstein, P., Harnden, R., & Tucker, W. 1974 , 192, 661 Hamilton, A. J. S., Sarazin, C. L., & Chevalier, R. A. 1983, ApJS, 51, 115 Hughes, J. P., & Helfand, D. J. 1985, , 291, 544 Liedahl, D. A., Osterheld, A. L., & Goldstein, W. H. 1994, ApJ, 438, L115 Moffett, D. A., & Reynolds, S. P. 1994, ApJ, 437, 705 Nomoto, K., Hashimoto, M., Tsujimoto, T., Thielemann, F.-K., Kishimoto, N., Kubo, Y., & Nakasato, N. 1997a, Nuclear Physics A., 616, 79 Nomoto, K., Iwamoto, K., Nakasato, N., Thielemann, F.-K., Brachwitz, F., Tsujimoto, T., Kubo, Y., & Kishimoto, N. 1997b, Nuclear Physics A, 621, 467 Pye, J. P., Becker, R. H., Seward, F. D., & Thomas, H. 1984, MNRAS, 207, 649 Raymond, J. C., & Smith, B. W. 1977 , 35, 419 Saken, J. M., Fesen, R. A., & Shull, M. 1992, ApJS, 81, 715 Seward, F. D. 1990, ApJS, 73, 781 Smith, A., Jones, L. R., Peacock, A., & Pye, J. P. 1985, ApJ, 296, 469 Snowden, S. L. 1998 , 117, 233 Snowden, S. L., & Kuntz, K. 1998 ftp://legacy.gsfc.nasa.gov/rosat/software/fortran/sxrb Sun, M., & Wang, Z. R. 1999, Adv. Space Res., in press Thielemann, F.-K., Nomoto, K., & Hashimoto, M. 1990, Supernovae (ed. Audoze, J., Bludman, S., Mochkovitch, R., & Zinn-Justin, J.), New York: Elsevier White, R. L., & Long, K. S. 1991, ApJ, 373, 543
[llcl]{} 1993 Apr 24 & 50005000 & 4 & 20.1\
& & 2 & 21.8\
1993 Oct 16 & 50005010 & 4 & 9.2\
1993 Oct 17 & 50005020 & 4 & 13.7\
1993 Nov 3 & 10020000 & 1 & 11.1\
1993 Nov 3 & 10020010 & 1 & 17.3\
& & & 93.2 total\
[llll]{} Si He $\alpha$ & 1.848 (1.846$-$1.850) & 6.7 (6.5$-$6.9)\
Si Ly $\alpha$ & 2.006 & 3.2 (3.0$-$3.4)\
Si He $3p+4p$ & 2.185, 2.294 & 0.15 (0.05$-$0.22)\
S He $\alpha$ & 2.455 (2.453$-$2.457) & 2.1 (2.0$-$2.25)\
S Ly $\alpha$ & 2.623 & 1.3 (1.2$-$1.35)\
S He $3p+4p$ & 2.884, 3.033 & 0.16 (0.11$-$0.20)\
Ar He $\alpha$ & 3.136 (3.131$-$3.141) & 0.32 (0.30$-$0.35)\
Ar Ly $\alpha$ & 3.323 & 0.19 (0.17$-$0.21)\
Ar He $3p+4p$ & 3.685, 3.875 & 0.067 (0.043$-$0.090)\
Ca He $\alpha$ & 3.880 (3.874$-$3.887) & 0.22 (0.20$-$0.25)\
Ca Ly $\alpha$ & 4.105 & 0.069 (0.057$-$0.080)\
Ca He $3p+4p$ & 4.582, 4.818 & 0.011 ($<$ 0.023)\
Fe K $\alpha$ & 6.658 (6.656$-$6.661) & 0.83 (0.81$-$0.86)\
Fe Ly $\alpha$ & 6.965 & 0.024 (0.013 $-$ 0.037)\
Fe He $3p+4p$ & 7.798, 8.216 & 0.074 (0.048$-$0.096)\
[ll]{} $\chi^2$, DOF & 219.7, 163\
$N_H\ (10^{22}\ \rm{cm}^{-2})$ & 5.0 (4.8$-$5.3)\
\
$kT$ (keV) & 2.0 (1.9$-$2.1)\
$EM$ ($n_e n_H V/4\pi d^2$ in cm$^{-5}$) & $4.9 \times 10^{12}$\
Unabsorbed Flux (0.5-10. keV in $10^{-10}$ ergs/cm$^2$/s) & 1.5\
Mg ($\odot$) & 0 ($<$1.7)\
Si ($\odot$) & 5.0 (3.7$-$7.0)\
S ($\odot$) & 6.4 (4.9$-$8.5)\
Ar ($\odot$) & 7.1 (5.4$-$9.5)\
Ca ($\odot$) & 7.0 (5.5$-$9.0)\
Fe ($\odot$) & 4.8 (4.0$-$5.8)\
Ni ($\odot$) & 14.9 (7.4$-$23)\
\
$<kT>$ (keV) & 0.15\
$<nt>$ (10$^{11} \rm{cm}^{-3}$ s) & 5.3\
Unabsorbed Flux (0.5-10. keV in $10^{-10}$ ergs/cm$^2$/s) & 90\
Si ($\odot$) & 2.8 (2.1$-$3.9)\
Fe ($\odot$) & 0.9 (0.2$-$1.6)\
|
---
author:
- 'Y. Zolotaryuk'
- 'M. M. Osmanov'
date: 'Received: date / Revised version: date'
title: Directed motion of domain walls in biaxial ferromagnets under the influence of periodic external magnetic fields
---
Introduction {#sec1}
============
One-dimensional ferromagnetic models are currently of considerable experimental and theoretical interest [@kik90pr; @ms91adp]. A large portion of this interest has been directed towards the domain wall (DW) response to an ac magnetic field. One of the important problems here is the development of different ways for obtaining a net DW drift under the influence of unbiased perturbations.
The ratchet effect [@jap97rmp; @r02pr; @hmn05annpl; @hm09rmp] has been shown to be an efficient tool to control the motion of particles and particle-like excitations. The mechanism of this effect is based on the breaking of all symmetries that connect two solutions with specular velocities [@fyz00prl]. For topological solitons this phenomenon has been investigated both theoretically [@m96prl; @sq02pre; @sz02pre; @fzmf02prl; @m-mqsm06c; @zs06pre] and experimentally [@cc01prl; @uckzs04prl; @bgnskk05prl] in continuous and discrete Klein-Gordon-type systems. It has been shown [@sz02pre; @fzmf02prl] that a [*biharmonic*]{} external field, consisting of a sinusoidal signal and its [*even*]{} overtone can yield a directed soliton motion. Similar biharmonic external field has been used in [@fo01pa] to control the dynamics of an individual spin.
It should be mentioned that spatial asymmetry can be used for controlling the domain wall motion. This can be achieved by creating a sawtooth-like asymmetric pattern on the magnetic film [@srn05njp; @srn06prb]. Experimental observation of this phenomenon was reported in [@hokonms05jap]. Asymmetric pinning potential that consists of triangular holes has been proposed and experimentally implemented in [@ap-jr-rvmkaapm09jpd]. Observations have shown that this asymmetry favours certain direction of the domain wall propagation.
On the other hand, since the work of Schlomann [@sm74ieeetm; @s75ieeetm] the problem of a DW drift under the influence of an oscillating magnetic field, polarised either in the plane containing the easy axis [@bgd90jetp; @clp91epl; @k07pmm] or in the plane perpendicular to it [@sm74ieeetm; @s75ieeetm; @bgd90jetp], has been studied in the literature. Despite a certain number of papers devoted to this problem, an interesting and important question arises: what are the necessary conditions which one has to impose on the unbiased external periodic magnetic field, such that a unidirectional DW motion will arise as a result? Also, we would like to point out the need of a unifying approach that would join together different ways of driving a unidirectional DW motion. In this paper, we show that the symmetry approach [@sz02pre; @fzmf02prl] should be a perfect tool for this task.
Thus, the aim of this work is to investigate in detail the possibility of the unidirectional motion of magnetic topological solitons (domain walls). In particular, we formulate the necessary conditions which have to be imposed on the external unbiased magnetic field in such a way that this motion will take place.
The paper is organised as follows. In the next section, we describe the equations of motion for the biaxial ferromagnet. In Section 3, the symmetries of the Landau-Lifshitz (LL) equation are discussed. The average domain wall velocity is computed analytically in Section 4. The numerical solution of the LL equation is given in Section 5. Conclusions and a final discussion are presented in the last section.
Model and equations of motion {#sec2}
=============================
The dynamics of the one-dimensional chain of classical spins in the continuum limit is described by the well-known Landau-Lifshitz (LL) equation $$\label{1}
\partial_t {\bf S}=-[{\bf S}\times (\partial_x^2{\bf S} + {\hat J}{\bf S})]+\epsilon
{\bf f}({\bf S},t),$$ where ${\bf S}(x,t)=(S_x,S_y,S_z)^T$ is a three-component dimensionless magnetisation vector. Without loss of generality, we assume the following normalisation condition: $S_x^2+S_y^2+S_z^2=1$. The matrix ${\hat J}=\mbox{diag}(J_x,J_y,J_z)$ contains the information about the anisotropy constants ($\beta_1\equiv J_x-J_y$, $\beta_3\equiv J_z-J_y$), so that the total energy of the magnet is given by $$\label{energy}
E=\frac{1}{2}\int_{-\infty}^{+\infty} [(\partial_x {\bf S})^2
-{\bf S}{\hat J}{\bf S}]~dx=
\int_{-\infty}^{+\infty} {\cal E}[{\bf S}(x,t)] dx.$$ Note that if $\beta_1 <0$, $\beta_3>|\beta_1|$, the OZ axis is the easiest axis, so that we have an easy axis ferromagnet with XY being the anisotropic hard plane. Here ${\cal E}({\bf S})$ is the energy density function. The perturbative term $\epsilon {\bf f}$ contains the external magnetic field and the phenomenological Landau-Gilbert damping $$\label{2}
\epsilon {\bf f}=-[{\bf S} \times {\bf H}(t)]+\lambda [{\bf S}\times {\bf S}_t].$$ Here the periodic external magnetic field ${\bf H}(t)={\bf H}(t+T)$ has zero mean value $\langle {\bf H}(t) \rangle_t=0$ and $\lambda$ is a damping constant. For the most of magnetic materials $\lambda \sim 0.008 \div 0.01$ [@aharoni].
In this paper, we are interested in computing the average DW velocity as a function of system parameters. But before embarking on this task, it is necessary to investigate the symmetry properties of the LL equation.
Symmetries of the Landau-Lifshitz equation {#sec3}
==========================================
According to the previous work [@fzmf02prl; @sz02pre], we state that the necessary condition for the occurrence of the directed DW motion is the breaking of all the symmetries that relate two solitons with the same
topological charge and with specular velocities: $$\widehat{\cal S}\; {\bf S}(x,t;v)= {\bf S}(x,t;-v).$$ The unperturbed ($\epsilon =0$) DW solution of the LL equation is well known [@s79lomi]. It is a topological soliton ${\bf S}^{(0)}(x,t;\phi)$ $=(\cos {\phi}~ \mbox{sech}z,
~\sin {\phi}~ \mbox{sech}z,~Q \tanh z)$, where $z= \xi (x+ x_0) - Q \zeta t$=$\xi(x-x_0-vt)$, $\xi= \sqrt{\beta_3-\beta_1 \cos^2{\phi}}$, $\zeta= \beta_1 \sin \phi~ \cos \phi$. Here $Q$ is the topological charge of the DW soliton with $Q=1$ corresponding to the kink (soliton) and $Q=-1$ to the antikink (antisoliton) solution. The azimuthal angle $\phi$ describes the direction of the projection of the magnetisation vector ${\bf S}$ on the XY plane, and thus the handedness or polarity of the DW changes depending on the interval to which the value of $\phi$ belongs: $0<\phi<\pi$ or $-\pi<\phi<0$. The DW velocity is defined by the value of $\phi$, moreover, $v(\phi)=v(\phi+\pi)$ and $v(-\phi)=-v(\phi)$.
Taking into account the properties of the unperturbed soliton and assuming that the perturbation is weak enough not to distort the soliton shape, one can define the soliton velocity and the center of mass as $$\label{Xc}
v=\frac{dX_c}{dt} = \frac{1}{E}\int_{-\infty}^{+\infty} x
\frac{\partial}{\partial t}
{\cal E}[{\bf S}(x,t)] dx.$$ It is easy to see that there exist only two types of symmetries that can relate two arbitrary solutions with opposite velocities and the same topological charge. These operations must include either a space reflection and a time shift or, vice versa, a time reflection and a space shift: $$\begin{aligned}
\nonumber
\widehat{\cal S}_{1\alpha}:&& x \to -x+x', t \to t+ \tau_\alpha,
S_{\alpha} \to -S_\alpha,\\
&& S_{z} \to -S_z;~~ \alpha=x,y, \label{S1}\\
\nonumber
\widehat{\cal S}_{2\alpha}:&& x \to x+x', t \to -t+ t',
S_{\alpha} \to -S_\alpha; \\
&& \alpha=x,y, \label{S2}\end{aligned}$$ where $x'$ and $t'$ are arbitrary constants and $\tau_\alpha=nT/2$, $n=0,1,2$. For the sake of clarity, let us briefly discuss the symmetries ${\widehat {\cal S}_{1x}}$ and ${\widehat {\cal S}_{1y}}$. The symmetry ${\widehat {\cal S}_{1x}}$ acts on the unperturbed solution by turning a DW ${\bf S}^{(0)}(x,t;\phi)$ into a solution ${\bf S}^{(0)}(x,t;\pi-\phi)$, while the symmetry ${\widehat {\cal S}_{1y}}$ turns a solution with $\phi$ into a solution with $-\phi$. In these cases, $v(\pi-\phi)=-v(\phi)$ and $v(-\phi)=-v(\phi)$, respectively. Therefore, the abovementioned symmetries connect two DW solutions with opposite velocities, while the other DW properties, in particular, the topological charge $Q$ or the width $\xi^{-1}$, remain unchanged. Note that although the symmetry ${\widehat {\cal S}}_{1y}$ changes the polarity of the DW, we still consider these two solutions as the same because the polarity is a local characteristic of a DW, in contrast to the topological charge.
Application of the perturbation ([\[2\]]{}) can or cannot destroy the above symmetries. Note that when the magnetic field is applied, the energy density (\[energy\]) must be complemented with the term $-({\bf S},{\bf H}(t))$. The symmetries $\widehat{\cal S}_{1\alpha}$ are present if there exists such $\tau$ that the following equalities are satisfied for the magnetic field ${\bf H}(t)$: $$\begin{aligned}
\nonumber
H_\alpha(t+\tau_\alpha)=-H_\alpha(t),~H_\beta(t+\tau_\alpha)=H_\beta(t),~\\
H_z(t+\tau_\alpha)=-H_z(t); ~~\alpha=x,y,~~ \beta\neq \alpha~.\label{CS1}\end{aligned}$$ One should stress that for each symmetry $\widehat{\cal S}_{1\alpha}$ there is a corresponding value of $\tau_\alpha$.
The symmetries $\widehat{\cal S}_{2\alpha}$ are always violated in the presence of dissipation ($\lambda \neq 0$), however, if $\lambda=0$, they are present if there exists such $t'$ that the following equalities take place: $$\begin{aligned}
\nonumber
H_\alpha(-t+t')=-H_\alpha(t),\;
H_\beta(-t+t')=H_\beta(t),\\
H_z(-t+t')=H_z(t);~~\alpha=x,y,~\beta \neq \alpha .\label{CS2}\end{aligned}$$ Thus, in a general case $\lambda \neq 0$, in order to obtain the directed soliton motion one has to apply a magnetic field for which in both (for $\alpha=x$ and $\alpha=y$) the sets of the equalities (\[CS1\]) at least one equation in each set does not hold. In the dissipationless case, one has to violate both the sets (\[CS1\]) and (\[CS2\]).
Consider now the oscillating magnetic field directed along one of the coordinate axes. If ${\bf H}(t)||$OZ, in order to break the symmetries, $\widehat{\cal S}_{1\alpha}$ one has to choose the respective component $H_z(t)$ in such a way that it satisfies the inequality $H_z(t+T/2) \neq -H_z(t)$. In the cases ${\bf H}(t)||$OX and ${\bf H}(t)||$OY, one of the symmetries $\widehat{\cal S}_{1\alpha}$ will always be present. Indeed, if ${\bf H}(t)=(H_x(t),0,0)^T$, the symmetry $\widehat{\cal S}_{1x}$ is present if there exists such $\tau_x$ that $H_x(t+\tau_x)=-H_x(t)$. Obviously, this equation holds if $\tau_x=T/2$. The symmetry $\widehat{\cal S}_{1y}$ is present if there exists such $\tau_y$ that the equality $H_x(t+\tau_y)=H_x(t)$ holds. Since the external magnetic field is periodic, this condition is automatically fulfilled for $\tau_y=T$. Similarly, if we consider ${\bf H}(t)=(0,H_y(t),0)^T$, the symmetry $\widehat{\cal S}_{1y}$ is present if $H_y(t+T/2)=-H_y(t)$, but for the presence of $\widehat{\cal S}_{1x}$ it is sufficient to guarantee the periodicity of the function $H_y(t)$. Thus, it is not possible to obtain a directed soliton motion by applying magnetic field only along OX or OY axis, at least, for arbitrary small perturbation. On the other hand, it is possible to obtain a directed motion in the case ${\bf H}(t)||$OZ, since one can violate the equality $H_z(t+T/2)=-H_z(t)$ by various choices of the magnetic field, for instance, by choosing it in the following biharmonic form : $$\label{mag-field}
{\bf H}(t)={\bf e}_z H(t) = {\bf e}_z [H_1 \cos (\omega t)+
H_2 \cos (m\omega t+\theta)],$$ with $m=2,3,\ldots$ . If $H_2 \neq 0$ and $m$ is even, the abovementioned equality is always violated, while for odd $m$’s it is always satisfied. In the dissipationless case $\lambda=0$, another set of symmetries, namely ${\widehat {\cal S}_{2\alpha}}$, must be broken. For a given choice of the magnetic field direction, this situation occurs if there does not exist such $t'$ that $H_z(-t+t')= H_z(t)$. For the function (\[mag-field\]) this means that the symmetries ${\widehat {\cal S}_{2\alpha}}$ are violated if $m$ is even and $H_2\neq 0$, $\theta \neq 0,\pm \pi$.
Computation of the average DW velocity using the perturbation theory
====================================================================
The perturbation theory in the first order implies that the perturbation (\[2\]) is too small to distort the soliton shape and it influences only the temporal evolution of the following soliton parameters: its center of mass $X(t)$ and the azimuthal angle $\phi(t)$. The equations that describe the parameter evolution are obtained in accordance with the papers [@p86jetp; @k89pd]: $$\begin{aligned}
\frac{d\phi}{dt}&=&-\frac{\epsilon}{2} \int_{-\infty}^{+\infty}
\frac{f_-(z)}{\cosh z} dz, \\
\nonumber
\frac{dX}{dt} &=& \frac{Q \zeta }{\xi}-\frac{\epsilon}{2\xi}
\int_{-\infty}^{+\infty}\left [
\frac{\zeta}{\xi^2} \frac{zf_-(z)}{\cosh z} +Q f_z\right ] dz,\\
f_{-}(z)&=&f_x \sin \phi-f_y \cos \phi,\end{aligned}$$ where $f_\alpha$ ($\alpha=x,y,z$) are the components of the perturbation vector (\[2\]), and the soliton parameters $\xi(t)= \sqrt{\beta_3-\beta_1 \cos^2{\phi(t)}}$ and $\zeta(t)= \beta_1 \sin \phi(t) \cos \phi(t)$ now depend on time.
Consider the case of the magnetic field (\[mag-field\]) directed along the OZ axis. In this case, the equations for the time evolution of the soliton parameters $X(t)$ and $\phi(t)$ have the following form: $$\begin{aligned}
\label{teq1} \frac{d\phi}{dt}&=&H(t)+\Gamma \sin 2\phi,
~~\Gamma=-\frac{\lambda \beta_1}{2},\\
\frac{d X}{dt}&=& -Q \frac{\beta_1 \sin 2\phi }{2\sqrt{\beta_3-
\beta_1 \cos^2 \phi}}.
\label{teq2}\end{aligned}$$ If one assumes that the soliton shape changes only insignificantly due to the perturbation, the vibrations of the azimuthal angle $\phi(t)$ around its equilibrium position appear to be small. Next, we assume the amplitudes $H_{1,2}$ to be small parameters. Then the oscillating solution of the equation (\[teq1\]) should be sought in the form $\phi(t)=\pm \pi/2+H_1 \phi_1(t)+H_1^2 \phi_2(t)+H_1^3 \phi_3(t)+{\cal O}(H_1^4)$. The initial value $\pm \pi/2$ comes from the fact that the Bloch wall configuration is energetically most favourable. After representing the external magnetic field in the form $H(t) \equiv H_1 h(t)= H_1 [\cos {\omega t}+ H_2/H_1 \cos {(2\omega t+\theta)}]
$ and substituting the expansion for $\phi(t)$ into equation (\[teq1\]), one finds an approximate expression for $\phi$ which contains only the $\phi_1(t)$ and $\phi_3(t)$ terms. Then it remains only to substitute the expression for $\phi(t)$ into equation (\[teq2\]) and to average it over one oscillation period. In the expansion of the r.h.s of the equation (\[teq2\]) into the Taylor series with respect to the parameter $H_1$, we have limited ourselves with the term of the order ${\cal O}(H_1^3)$. As a result, the following expression for the average DW velocity is obtained: $$\begin{aligned}
\nonumber
&&\langle v \rangle \simeq
%\lim_{t\to\infty}\frac{1}{t}
% =\int_{t}^{t+T}\frac{dX}{dt}dt \simeq
-QA(H_{1,2},\omega,\Gamma)
\sin(\theta-\theta_0),\\
&& A(H_{1,2},\omega,\Gamma)=\frac{3\beta_1^2}{16\beta_3^{3/2}}
\nonumber
\frac{H_2H_1^2}{(4\Gamma^2+\omega^2)\sqrt{\Gamma^2+\omega^2}},\\
&&\theta_0=2 \arctan(2\Gamma/\omega)-\arctan(\Gamma/\omega).\label{v1} \end{aligned}$$ We would like to stress that this expression is the same for the initial angles $\phi=\pi/2$ and $\phi=-\pi/2$.
In the case of odd $m$’s, we obtain $\langle v \rangle=0$. The expression (\[v1\]) clearly confirms the validity of the symmetry approach. The average soliton velocity becomes zero if $H_2=0$ and this signals the restoration of the symmetries ${\widehat {\cal S}_{1\alpha}}$. In the dissipationless limit ($\lambda \to 0$), $\theta_0 \to 0$ and thus $\langle v \rangle \propto \sin \theta$. In this limit, the symmetries ${\widehat {\cal S}_{2\alpha}}$ are restored if $H(t)=H(-t)$. The average DW velocity becomes zero at the values $\theta=0,\pm \pi$, so that they are precisely those values at which the function $H(t)$ is symmetric.
Numerical simulations {#sec4}
=====================
In order to verify the symmetry approach developed in Section \[sec3\], the initial LL equation has been discretized in the spatial dimension with the step $h=0.05$ and the resulting system of coupled ordinary differential equations has been integrated numerically using the fourth order Runge-Kutta method. It is convenient to solve numerically the LL equation in the following form: $$\begin{aligned}
\nonumber
&& -\partial_t {\bf S}=\frac{1}{1+\lambda^2} \left [{\bf S}
\times {\bf H}_* \right ] +
\frac{\lambda}{1+\lambda^2} \left [{\bf S} \times \left [{\bf S} \times
{\bf H}_* \right] \right],\\
&&{\bf H}_*=\partial^2_x {\bf S} + {\hat J} {\bf S}+ {\bf H}(t),
\label{LL2}\end{aligned}$$ which is equivalent to equation (\[1\]) with the perturbation (\[2\]). The validity of this method has been checked by monitoring the energy conservation in the purely Hamiltonian case $\lambda=0$ and ${\bf H}(t)=0$.
It should be emphasized that in order to compute the mean DW velocity, one has to average over the set of initial conditions: phase $-\pi \le \phi\le \pi $, initial time $0<t_0<T$, and time $t$. In the numerical simulations, we consider only the dissipative case $\lambda>0$, therefore we are interested in attractor(s) that correspond to moving DWs. If the perturbation (\[2\]) is small, then the phase space of the LL equation have to consist of the basin(s) of attraction of [*periodic*]{} attractor(s) \[limit cycle(s)\] that are locked to the frequency $\omega$ of the external magnetic field. Breaking the respective symmetries should manifest itself in [*desymmetrization*]{} of the basins of attraction that correspond to DWs moving with opposite velocities. Below we demonstrate that actually in the case of broken symmetries there exists only one attractor that corresponds to the directed DW motion. In this case, it is sufficient to compute only the average velocity on the attractor: $\langle v \rangle \to_{t \to \infty} [X(t+T)-X(t)]/T$, where $X(t)$ is the DW center of mass.
First, we consider the case ${\bf H}(t)||$OZ and the expression for the Z-component of the magnetic field given by equation (\[mag-field\]). The time evolution of the DW center computed as $X_{max}=\mbox{max}_{x \in (-\infty,+\infty)}{\cal E}[{\bf S}(x,t)]$ (shown in Figure \[fig1\]) clearly demonstrates the validity of the symmetry approach.
Curve 1 in this figure corresponds to the case of the single harmonic drive ($H_2=0$), where no directed DW motion is seen. Next, no directed DW motion is observed in the case of mixing two odd harmonics ($m=3$) as shown by curve 2. Curves 3-5 illustrate the evolution of the DW center for the case of two mixed harmonics with $m=2$. It is easy to notice that on the time scale $t > \lambda^{-1}$, the system settles on a periodic attractor (the periodicity can be observed from the insets in Figure \[fig1\]) that corresponds to the motion in the direction, defined by the phase shift $\theta$. The simulations have been performed for different initial values of $\phi$ and initial times $t_0$, and in all the cases the system settles on the same attractor (compare, for example, curves 4 and 5).
The dependence $\langle v \rangle (\theta)$ of the average DW velocity on the phase shift between the harmonics, $\theta$, is shown in Figure \[fig2\].
It appears to have a sinusoidal shape, as predicted by the perturbation theory result (\[v1\]). Another demonstration of the validity of the symmetry approach is the behavior of the points where $\langle v \rangle (\theta)=0$. In Figure \[fig2\] along with the data for $\lambda=0.01$, the data for $\lambda=0.1$ and $\lambda=0.3$ have been plotted as well. Although these values of damping do not correspond to realistic values for magnets, these results are very instructive for the illustration of the restoration of the symmetries ${\widehat {\cal S}_{2\alpha}}$. Indeed, when $\lambda \to 0$, the values of $\theta$, at which the DW velocity becomes zero, gradually shift to $0, \pm \pi$. As shown already in Section \[sec3\], both the symmetries ${\widehat {\cal S}_{2\alpha}}$ are restored if $H_z(-t)=H_z(t)$. This happens if $\theta=0,\pm \pi$ \[see equation (\[mag-field\])\]. We would like also to point out good correspondence between the perturbation theory results (shown by the solid line) given by equation (\[v1\]) and the results of direct integration of the LL equation.
Next, we consider the case of the magnetic field directed along the OX axis \[${\bf H}(t)={\bf e}_xH(t)$\], but which has the same functional dependence (\[mag-field\]). According to the symmetry approach, this periodic drive does not yield the directed DW motion. The numerical simulations of this case are illustrated by Figure \[fig3\]. The basins of attraction of two DW solutions that have opposite polarities and opposite velocities appear to be symmetric with respect to the value $\phi=0$. Indeed, if the initial value of the azimuthal angle is positive (see curves 1,3, and 5), the dynamics of the system settles on the attractor with the positive DW velocity ($v=0.00031$), while for negative values of $\phi$ (see curves 2,4 and 6) it tends to the solution with the DW velocity of opposite sign ($v=-0.00031$).
In the case of ${\bf H}(t)||$OY, the same scenario is observed.
Discussion and conclusions
==========================
In this paper, the symmetry approach for the analysis of the [*unidirectional*]{} domain wall motion has been developed. The main objective was to demonstrate that a proper choice of the oscillating unbiased magnetic field can yield a net directed motion of the domain wall. With the help of the symmetry approach, we have obtained the necessary conditions to be imposed on the magnetic field in order to obtain the unidirectional motion. When the magnetic field is applied along a certain coordinate axis, this motion turns out to be possible only if this axis coincides with the easy axis, say OZ. The symmetry arguments prohibit the directed DW motion if the magnetic field is applied along any of the hard axes. The necessary condition for the directed DW motion in the presence of dissipation is given by $H_z(t)\neq -H_z(t+T/2)$. Next, we demonstrate the cubic dependence $\langle v \rangle \propto H_1^2H_2$ of the average DW velocity on the magnetic field amplitude. These results appear to be similar to those for topological solitons in the ac driven sine-Gordon (SG) equation [@sz02pre]. Very recently, [@qca-n10pre] a rigorous proof of the universality of these results for a wide range of nonlinear systems driven by the biharmonic signals of the type (\[mag-field\]) has been obtained.
However there are certain differences in the directed soliton motion in the LL and SG cases. Due to the fact that the LL equation is three-component, while SG is scalar, there are more ways to apply the external field to the system, namely along any of three coordinate axes. But only in the case of magnetic field applied along the easy axis, the directed motion is possible. Here it should be stressed that we are interested in the average net motion which is independent from the the local properties of the DW such as handedness (polarity). The application of any ac signal along non-easy axes drives DWs with opposite polarities into opposite directions. The basins of respective attractors of the LL equation are symmetric with respect to the $\pi$-shift of the initial azimuthal angle. Therefore, a weak noise which is inevitable in realistic systems will lead to exploration of the whole phase space and eventually to zero net motion.
Another difference with respect to the SG case caused by the multicomponentness of the LL equation is a wider range of possibilities to drive a DW by external oscillating fields. Let us briefly discuss the case when two of the magnetic field components are nonzero. Consider first the case of magnetic field polarised in the easy plane XY \[${\bf H}(t)=(H_x(t),H_y(t),0)^T$\]: $H_x(t)=H_0^{(x)} \cos (m_x\omega t+\theta_x)$, $H_y(t)=H_0^{(y)} \cos (m_y \omega t+\theta_y)$. Here $m_x, m_y=1,2,3,\ldots$, and they do not have a common divisor. Note that such a way to control the DW motion has been suggested in [@bgd90jetp] for the particular case of $m_x=m_y=1$. The symmetry $\widehat{\cal S}_{1x}$ requires the simultaneous fulfilment of the equalities $H_x(t+T/2)=-H_x(t)$ and $H_y(t+T/2)=H_y(t)$. Similarly, the $\widehat{\cal S}_{1y}$ symmetry is present if both the equalities $H_x(t+T/2)=H_x(t)$ and $H_y(t+T/2)=-H_y(t)$ hold. If both $m_x$ and $m_y$ are odd, we have $H_\alpha(t+T/2)=-H_\alpha(t) \neq H_\alpha(t)$. Therefore none of the sets of equations (\[CS1\]) can be satisfied and thus both symmetries $\widehat{\cal S}_{1\alpha}$ are broken. If $m_x$ is even and $m_y$ is odd, we have $H_x(t+T/2)=H_x(t) \neq -H_x(t)$, thus $\widehat{\cal S}_{1x}$ is broken. But, $H_y(t+T/2)=-H_y(t)$, therefore $\widehat{\cal S}_{1y}$ is present. Similarly, if $m_y$ is even and $m_x$ is odd, the symmetry $\widehat{\cal S}_{1x}$ is satisfied and $\widehat{\cal S}_{1y}$ is broken. Therefore if one of $m_{x,y}$ is even and another one is odd, we expect no directed motion, whereas this motion must occur if both $m_{x,y}$ are odd.
Another way to drive unidirectionally a domain wall is to apply the oscillating magnetic field polarised in the plane that contains the easy axis [@k07pmm]:\
$H_y(t)=H_0^{(y)} \cos (m_y\omega t+\theta_y)$, $H_z(t)=H_0^{(z)} \cos (m_z \omega t+\theta_z)$; $m_y, m_z=1,2,3,\ldots$. Alternatively, one can consider the field polarised in the XZ plane. If both $m_y$ and $m_z$ are odd, it is impossible to break simultaneously the set of equalities (\[CS1\]) and (\[CS2\]). In this case, the breaking of $\widehat{\cal S}_{1x}$ takes place because $H_z(t+T/2)=-H_z(t)$ but $H_y(t+T/2)=-H_y(t)\neq H_y(t)$. However, the symmetry $\widehat{\cal S}_{1y}$ is still present. If $m_y$ is even and $m_z$ is odd, we obtain $H_z(t+T/2)=-H_z(t)$ but $H_y(t+T/2)=H_y(t) \neq -H_y(t)$, therefore $\widehat{\cal S}_{1x}$ is present while $\widehat{\cal S}_{1y}$ is broken. If $m_y$ is odd and $m_z$ is even both the symmetries $\widehat{\cal S}_{1x,y}$ are broken because $H_z(t+T/2)=H_z(t) \neq -H_z(t)$. Therefore this is the only way to obtain the directed DW motion with the help of magnetic field polarised in the YZ plane.
Finally, we would like to outline the future directions of applications of the symmetry approach. It is of interest to consider topological magnetic excitations in two- and three-dimensional systems, where alongside the directed translational motion a unidirectional rotation can take place as well, as shown previously for particles [@dzfy08prl]. Another question is how to apply the symmetry approach to the problem of DW directed motion in more complicated magnetic systems such as antiferromagnets, ferrites, magnetoelastic systems, and others. In this direction, some progress has already being accomplished in the papers [@ggg-d94prb; @gs95jmmm].
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---
abstract: 'We study the out-of-equilibrium dynamics of $p$-wave superconducting quantum wires with long-range interactions, when the chemical potential is linearly ramped across the topological phase transition. We show that the heat produced after the quench scales with the quench rate $\delta$ according to the scaling law $\delta^\theta$, where the exponent $\theta$ depends on the power law exponent of the long-range interactions. We identify the parameter regimes where this scaling can be cast in terms of the universal equilibrium critical exponents and can thus be understood within the Kibble-Zurek framework. When the electron hopping decays more slowly in space than pairing, it dominates the equilibrium scaling. Surprisingly, in this regime the dynamical critical behaviour arises only from paring and, thus, exhibits anomalous dynamical universality unrelated to equilibrium scaling. The discrepancy from the expected Kibble-Zurek scenario can be traced back to the presence of multiple universal terms in the equilibrium scaling functions of long-range interacting systems close to a second order critical point.'
author:
- Nicolò Defenu
- Giovanna Morigi
- 'Luca Dell’Anna'
- Tilman Enss
bibliography:
- 'KZM\_LR\_KC.bib'
title: 'Universal dynamical scaling of long-range topological superconductors'
---
One of the major challenges of contemporary physics is the identification of quantum phases of matter, which can serve as platforms for quantum computers. In this perspective, topological superconductors[@Hasan2010; @Bernevig2013] are promising constituents for quantum devices[@Nayak2008; @Terhal2015; @Kraus2013a; @Mazza2013], thanks to the presence of gapless Majorana modes, the so-called Majorana zero modes (MZM), which are localised at the chain edges and topologically protected. Since the first theoretical evidence of MZMs in superconducting wires[@Kitaev2001], several experimental platforms have revealed consistent signatures of Majorana physics both in one-dimensional[@Mourik2012; @Deng2012; @Das2012; @Albrecht2016] and two-dimensional[@Wang2012; @He2014; @Sun2016; @He2017] geometries. More recently, models of $p$-wave superconducting wires with long-range (LR) deformations have shown more robust topological properties[@Viyuela2015; @Viyuela:2018fpv], while strong enough LR pairing effects alter the nature of the topological phase[@Vodola2014; @Lepori2015; @Lepori2017; @Lepori2017add; @Alecce2017] and the spreading of correlations[@Foss-Feig2015; @Cevolani2015; @Vodola2016]. Experimental realisations of LR topological superconductors employ one-dimensional arrays of magnetic impurities on top of a conventional superconducting substrate[@Nadj-Perge2014; @Pawlak2016; @Ruby2017], leading to the realisation of an effective Kitaev Hamiltonian with both LR pairing and LR hopping[@Pientka2013; @Klinovaja2013; @Pientka2014; @Neupert2016]. In this context, understanding slow variations (quenches) of control fields in quantum systems is fundamental to adiabatic protocols [@Nielsen2000], since Majorana excitations cannot be realised by sudden manipulations of the system[@Perfetto2013]. These investigations constitute a fundamental contribution towards the understanding of dynamical scaling for quenches across topological phase transitions[@Ueda2010].
 (a) The energy spectrum of a one-dimensional topological superconductor for chemical potential $|\mu|<\mu_{c}$ with bulk topological invariant $w=1$. The blue solid line represents the degenerate groundstate, which hosts MZMs. (b) For $|\mu|>\mu_{c}$ there is a single ground state with $w=0$.](Fig1a.pdf)
In this Letter we characterise the out-of-equilibrium dynamics of a $p$-wave superconducting quantum wire with long-range interactions, whose chemical potential is linearly ramped across the equilibrium critical point. We determine the density of defects produced after the ramp and show that it scales as a power of the quench rate. We then connect the power-law exponent with the equilibrium critical properties and topological features and determine the phase diagram for the dynamics as a function of the decay exponent of the hopping and pairing terms.
![\[Fig1b\] Phase diagram of the $p$-wave superconducting Hamiltonian as a function of the exponents of the power-law decay of hopping, $\alpha$, and pairing, $\beta$. The labels indicate (i) short-range (SR) universality, (ii) hopping dominated long-range universality (LR$_{\alpha}$) for $\alpha<\beta$, (iii) pairing dominated long-range universality (LR$_{\beta}$) for $\beta<\alpha$. The scaling of defect generation is described by the exponent $\theta$, see Eq. . In the green region $\beta<\min(\alpha,2)$, dominant pairing determines both equilibrium exponents and standard Kibble-Zurek dynamical scaling. In the red region $\alpha<\min(\beta,2)$, instead, dominant hopping determines equilibrium exponents but dynamics is governed by LR pairing, giving rise to anomalous universal scaling (an. universality) $\theta=1/2$ for $\beta>2$.](Fig1b.pdf){width=".35\textwidth"}
We consider spinless electrons hopping across the $N$ sites of a linear chain in presence of $p$-wave pairing. The Hamiltonian reads $$\begin{aligned}
\label{h_klr}
\hat H=-\sum_{i}\Bigl[\sum_{r>0}\left(j_{r}\hat c^{\dagger}_{i}\hat c_{i+r}+\Delta_{r}\hat c^{\dagger}_{i}\hat c^{\dagger}_{i+r}+{\rm H.c.}\right)+\mu\hat c^{\dagger}_{i}\hat c_{i}\Bigr]+\mathcal C,\end{aligned}$$ where operators $c^{\dagger}_{i}$ create a fermion at site $i$ and fulfill the anticommutation relations $\{c_i,c_j^\dagger\}=\delta_{ij}$. Here, $\mu$ denotes the chemical potential, $\mathcal C=N\mu/2$ is an energy offset, $j_{r}$ and $\Delta_{r}$ are the hopping and pairing amplitudes, respectively, and depend on the intersite distance $r$ according to the power laws (reported here for open boundary conditions): $$\begin{aligned}
j_{r}^{\alpha}&=\frac{J}{N_{\alpha}}\,\frac{1}{r^{\alpha}},&\Delta_{r}^{\beta}&=\frac{d}{N_{\beta}}\,\frac{1}{r^{\beta}}\,,
\label{dr}\end{aligned}$$ with the hopping exponent $\alpha>1$, the pairing exponent $\beta>1$, the coefficients $J,d>0$, and $N_\gamma=2\sum_{r=1}^{N/2} r^{-\gamma}$ the Kac scaling, which guarantees extensivity of the energy[@Campa2014]. For sufficiently fast decaying interaction and hopping terms the system possesses two different phases separated by the quantum critical point $\mu_c=2J$[@Kitaev2001]. In the thermodynamic limit the two topological phases can be distinguished by the bulk topological invariant $w$: For $|\mu|>\mu_{c}$ the ground state is nondegenerate and $w=0$; in the nontrivial phase $|\mu|<\mu_{c}$ the bulk topological invariant $w=1$, and the ground state is doubly degenerate and can host MZMs, see Fig.\[Fig1\]. At finite size $N$ the spectrum is always gapped and for open boundary conditions the MZMs remain localized at the edges of the chain. The presence of the LR pairing and hopping terms in Eq. does not alter this phase diagram nor the values of the bulk topological invariant as long as $\alpha,\beta>1$[@Vodola2014; @Viyuela:2018fpv; @Alecce2017]. Nonetheless, LR connectivity modifies the universal critical behaviour of the model by changing the critical exponents. The resulting phases are displayed in Fig. \[Fig1b\]. Note that the equilibrium phase diagram of the long-range Kitaev chain radically differs from the one of the long-range quantum Ising model[@Defenu2016; @Defenu2017a].
In the following, we analyse the dynamics during slow variations of the chemical potential across the critical value according to $$\begin{aligned}
\label{ramp}
\mu=\mu_{c}-\delta\cdot t\,,\end{aligned}$$ where time varies in the interval $[-\mu_c/\delta,\mu_c/\delta]$, i.e., from the topologically trivial phase $\mu=2\mu_c$ deep into the nontrivial phase at $\mu=0$. We note that the time-dependent dynamics have been solved for an Ising model in transverse field[@Kolodrubetz2012], which can be mapped to the Kitaev model for $\alpha,\beta\to\infty$[@Fradkin1989]. Below we derive an exact solution which is valid for general $\alpha,\beta >1$ and in the thermodynamic limit. This solution allows us to determine the thermodynamic functions after the ramp. For this purpose we rewrite the Hamiltonian using momentum-space operators $\hat c_{k}=e^{i\pi/4}\sum_{r\in\mathbb{Z}}c_{r}e^{ikr}/\sqrt{N}$ with $k\in[-\pi,\pi)$. Using the spinor representation $\hat\psi_k= (\hat c_{k},\hat c_{-k}^{\dagger})^T$, the Hamiltonian is the sum of $2\times 2$ block matrices $\mathcal H_k\equiv \boldsymbol{h}_{k}(t)\cdot\boldsymbol{\hat\sigma}$, $$\begin{aligned}
\label{h_klr_compact}
\hat H(t)=\sum_{k} \hat\psi_k^\dagger\hat{\mathcal H}_k(t)\hat\psi_k \,,\end{aligned}$$ where $\boldsymbol{\hat\sigma}$ is the vector of the Pauli matrices $\hat\sigma^{j=1,2,3}$ and $\boldsymbol{h}_{k}(t)=(\Delta_{\beta}(k),0,\varepsilon_{\alpha}(k,t))$ is the pseudo-spin vector. Its elements depend on $\varepsilon_{\alpha}(k,t)=\mu(t)/2-j_{\alpha}(k)$ and on the momentum-space hopping and pairing coefficients: $$\begin{aligned}
j_{\alpha}(k)&=J\,\text{Re}\left[\operatorname{Li}_{\alpha}(e^{ik})\right]/\zeta(\alpha)\,,\label{funcj}\\
\Delta_{\beta}(k)&=d\,\text{Im}\left[\operatorname{Li}_{\beta}(e^{ik})\right]/\zeta(\beta)\,,\label{funcd}\end{aligned}$$ where $\operatorname{Li}_\alpha(z)$ denotes the polylogarithm and $\zeta(\alpha)$ the Riemann zeta function[@Abramowitz1964]. The Hamiltonian is diagonalized in terms of the fermionic quasiparticle operators $\hat \gamma_k(t)=u_k(t)\hat c_k+v_{-k}^*(t)\hat c_{-k}^\dagger$ to obtain $\hat H=\sum_{k}\omega_{k}(t)\bigl(\hat \gamma^{\dagger}_{k}(t)\hat\gamma_{k}(t)-\frac{1}{2}\bigr)$ with the quasiparticle spectrum $\omega_{k}(t)=2\sqrt{\varepsilon_{\alpha}(k,t)^{2}+\Delta_{\beta}(k)^{2}}$. The pseudo-spin vector $\boldsymbol{h}_{k}(t)$ identifies a direction in the two-dimensional plane of the Hamiltonian space. At a given instant of time, integrating the angle $\theta_{k}={\rm atan}(h_k^1/h^3_k)={\rm atan}(\Delta_\beta(k)/\varepsilon_\alpha(k))$ over the Brillouin zone yields the bulk topological invariant $w=\oint d\theta_{k}/(2\pi)$ of the corresponding equilibrium phase.
The dynamics of the Kitaev chain can be exactly described by the Heisenberg equations of motion for the original creation and annihilation operators, ${\rm i}d\hat {c}_{k}/d t=[\hat c_{k},\hat H]$. These equations can be cast into a matrix evolution for the Bogolyubov coefficients, $$\begin{aligned}
\label{dyn_sys}
i\frac{d}{dt}\begin{pmatrix}u_{k}\\v_{k}\end{pmatrix}=\begin{pmatrix}\varepsilon_{\alpha}(k,t) & -\Delta_{\beta}(k)\\
\Delta_{\beta}(k) & \varepsilon_{\alpha}(k,t)
\end{pmatrix}\begin{pmatrix}u_{k}\\v_{k}\end{pmatrix},\end{aligned}$$ which can be mapped into the Landau-Zener form[@dziarmaga2005; @dziarmaga2010; @Dutta2017], see App.\[lz\]. The excitation probability $p_k(t)$ can be computed exactly[@Damski2005a; @Bialonczyk:2018sbn], see App.\[defect\_sc\]. For a slow quench to the final time $\tau_{f}=\mu_c/\delta$, the excitation probability is well approximated by the Landau-Zener formula $$\begin{aligned}
\label{exc_prob}
p_{k}\simeq\exp\left(-\frac{\pi \Delta_{\beta}(k)^{2}}{\delta}\right)\,,\end{aligned}$$ which becomes exact for $k\ll\frac{\pi}{2}$ in the slow ramp limit $\delta\to0$. From Eq. we find population inversion $p_k\ge 1/2$ when $|k|<k_{\rm th}$. The threshold value $k_{\rm th}$ is determined analytically from a low-momentum expansion and reads $k_{\rm th}=[\delta\log(2)/(\pi c)]^{\theta}$ with $$\begin{aligned}
c=\begin{cases}\left(\frac{\cos(\beta\pi/2)\Gamma(1-\beta)}{\zeta(\beta)}\right)^{2} & \text{for}\,\,1<\beta<2,\\
\left(\frac{\zeta(\beta-1)}{\zeta(\beta)}\right)^{2} & \text{for}\,\,\beta>2.
\end{cases}\end{aligned}$$ It is remarkable that $k_{\mathrm{th}}\to\infty$ for $\beta\to 1^+$, corresponding to negative temperatures. In the small $\delta$ limit, this effect is only visible very close to the singular limit $\beta\gtrsim1$, while for intermediate $\beta$’s the tendency is reversed, see Fig.\[Fig2\]. These results have been numerically verified taking the full $k$ dependence of $\Delta_\beta(k)$ into account. We note that stable athermal distributions are generally expected in systems with diverging long-range interactions[@Kastner2011], and the case we analyse here seems to be no exception.
Using Eq. we can derive several thermodynamic properties for asymptotically slow drive $\delta\to0$. We discuss here the excitation density[@zurek2005; @chandran2012; @dziarmaga2010; @polkovnikov2005], $$\begin{aligned}
n_{\mathrm{exc}}=\frac1N \sum_{k}\langle\gamma_{k}^{\dagger}\gamma_{k}\rangle=\frac1N\sum_{k}p_{k}\,,
\label{eq:nt}\end{aligned}$$ which we compute in the thermodynamic limit, thus replacing the sum by an integral over the interval $k\in[-\pi,\pi)$. In the $\delta\to 0$ limit the exponent of $p_k$ in Eq. diverges and the total contribution to the integral only comes from the saddle point. Expanding around the vanishing effective frequency, we find the scaling law $$\begin{aligned}
\label{exc_scaling}
\lim_{\delta\to 0}n_{\mathrm{exc}}\sim \delta^\theta\quad\mathrm{with}\quad
\theta=
\begin{cases}
(2\beta-2)^{-1}& \text{for }\beta\leq 2, \\
1/2& \text{for}\beta>2.
\end{cases}\end{aligned}$$ At the border $\beta=2$ we find $n_{\rm ex}\propto \sqrt{\delta}/\log\delta$. These scalings are valid irrespectively of the value of $\alpha$, since the defect density solely depends on $\beta$. The corresponding dynamical phase diagram is depicted in Fig. \[Fig1b\]. Remarkably, the dynamical phases for $\alpha>2$ correspond to the regions of the equilibrium phase diagram, but in the hopping dominated regime $\alpha<2$ a different universal dynamical scaling arises. Such universal dynamical scaling with $\beta$ cannot be related to the equilibrium critical exponents, which involve $\alpha$, as generally happens in Fermi systems[@dziarmaga2010; @polkovnikov2005; @de_grandi2010] and its appearance can be traced back to the violation of the equilibrium scaling hypotheses due to the LR nature of the interactions.
In order to verify our analytical prediction, we numerically integrated Eq.. Initially at $t=t_{i}=-\mu_{c}/\delta$ the system is at equilibrium with Bogolyubov coefficients $u^{i}_{k},v^{i}_{k}=\cos\frac{\theta_{i}(k)}{2},
\sin\frac{\theta_{i}(k)}{2}$ where $\tan\theta_{i}(k)=\frac{\Delta_{\beta}(k)}{\varepsilon_{\alpha}(k)}$ and $\mu(t_{i})=2\mu_{c}=4J$. The Bogolyubov coefficients are then evolved numerically according to Eq. for a grid of $k$ points in the interval $[-\pi,\pi)$. Due to non-adiabatic effects arising during the critical stage of the dynamics $t\simeq0$, the resulting amplitudes at the final time $t_{f}=\mu_{c}/\delta$ differ from the ones of the equilibrium Hamiltonian with $\mu(t_{f})=0$. In order to quantify these deviations, we consider the excitation probability of each state $k$, after the slow ramp, with respect to its equilibrium ground state, $$\begin{aligned}
\label{expl_exc_prob}
p_{k}=1-|u^{f}_{k}u^{*}_{k}(t_{f})+v^{f}_{k}v^{*}_{k}(t_{f})|^{2},\end{aligned}$$ where $(u^{f}_{k},v^{f}_{k})$ are the equilibrium Bogolyubov amplitudes at $\mu=0$, while $(u_{k}(t_{f}),v_{k}(t_{f}))$ are the ones at the end of the dynamical evolution.
The excitation probability at the end of the slow quench, calculated according to Eq., is shown in Fig.\[Fig2\] as a function of the momentum $k$ for $\delta=0.5,\,0.05$ in panels (a) and (b), respectively. As the dynamical protocol crosses the $\mu_{c}=2J$ critical point, only low momentum modes $k\approx 0$ become soft during the dynamics. Indeed, Eq. only applies to excitations modes with $k<\pi/2$, as follows from the Landau-Zener mapping, see App.\[lz\], while high energy modes $k>\pi/2$ remain adiabatic and their excitation probability is not shown. Numerical points for the Bogolyubov modes excitation probability for $\alpha=(\infty,1.75,1.50, 1.25)$ are shown by squares, crosses, circles and triangles respectively (see legend of panel b), while the different values of $\beta=(\infty, 1.75,1.5,1.25)$ are reported respectively from top to bottom (gray, green, blue and red). In Fig.\[Fig2b\] we observe almost perfect agreement with the predictions of Eq. in the slow ramp case $\delta=0.05$. Indeed, corrections from finite and slightly asymmetric endpoints $t_{i}$ and $t_{f}$ do not influence the universal behaviour obtained in the $\delta\to 0$ limit at small momenta and can be safely discarded, see App.\[fr\_corr\].
The result of Eq. contradicts the result found using adiabatic perturbation theory, which produces the Kibble-Zurek relation between the universal slow dynamics and the equilibrium critical exponents $\theta=\nu/(1+z\nu)$[@dziarmaga2010; @polkovnikov2005; @de_grandi2010]. Here, the critical exponents $z\nu$ and $z$ describe the scaling of the spectrum at the critical point, $\omega_{k=0}\propto |\mu-\mu_{c}|^{z\nu}$ and $\omega_{k\to 0}\propto k^{z}$. In particular, at lowest order in the adiabatic expansion, the excitation probability $p_{k}=|\alpha_{k}|^{2}$ of the Bogolyubov quasi-particle states $|k\rangle=\hat{\gamma}_{k}^{\dagger}|0\rangle$ are given by the squared transition amplitudes induced by the perturbation operator $\hat{\partial}_{\mu}=\partial \hat{H}(\mu)/\partial\mu$ over the Bogolyubov vacuum $|0\rangle$ integrated over the whole dynamical trajectory $$\begin{aligned}
\label{ad_dens_exc}
\alpha_{k}\approx \int\langle k|\hat{\partial}_{\mu}|0\rangle e^{\frac{i}{\delta}\int^{\mu}\left(E_{k}(\mu')-E_{0}(\mu')\right)d\mu'}d\mu,\end{aligned}$$ where $E_{k}(\mu)$ is the energy of the state $|k\rangle$. In the $\delta\to 0$ limit, the saddle-point approximation holds and the integral only receives contribution from the vanishing gap region of the trajectory, i.e. the critical point. There, one can employ the universal scaling relations[@dziarmaga2010; @polkovnikov2005; @de_grandi2010] $$\begin{aligned}
\label{scaling_forms}
E_{k}(\mu)-E_{0}(\mu)&=\omega_{k}\approx \Delta\,F\left(\Delta/k^{z}\right)\\
\label{scaling_forms2}
\langle k|\hat{\partial}_{\mu}|0\rangle&\approx\frac{\Delta}{|\mu-\mu_{c}|k^{z}}G\left(\Delta/k^{z}\right)\end{aligned}$$ where $\Delta$ is the minimal gap $\Delta\propto |\mu-\mu_{c}|^{z\nu}$. Inserting Eqs.and into Eq. and making the integration dimensionless, one finds the universal scaling variables $\eta=k\delta^{-\frac{\nu}{1+z\nu}}$ and $\zeta=k^{1/\nu}(\mu-\mu_{c})$. Rephrasing the adiabatic perturbation theory expression for the defect density $n_{\mathrm{exc}}\approx\int dk |\alpha_{k}|^{2}/(2\pi)$ in terms of the universal variables $\eta$ and $\zeta$ immediately leads to the Kibble-Zurek result $\theta=\nu/(1+z\nu)$, see Eq. and Refs.[@dziarmaga2010; @polkovnikov2005; @de_grandi2010]. Since for the p-wave superconducting Hamiltonian in Eq. one has $z\nu=1$ and $z=\phi-1$, where $\phi=\mathrm{min}(\alpha,\beta)$, we can conclude that the scaling exponent $\theta$ in Eq. is inconsistent with the Kibble-Zurek scaling in the region $\alpha<\beta$.
We refer to this unexpected behaviour as *anomalous universal scaling*. We report its extent in Fig.\[Fig2c\] where the example case of LR hopping $\alpha=1.25$ and short range pairing $\beta=\infty$, well inside the anomalous universal scaling region, is studied. The excitation probability is reported as a function of the universal scaling variable $\eta=k\delta^{-\frac{\nu}{1+z\nu}}$ for several $\delta$ values. Remarkably, for this scaling the curves do not collapse, see Fig.\[Fig2c\]. Instead, universality is recovered when one considers the proper dynamical exponent $z_{d}=\nu_{d}^{-1}=1$, for nearest neighbour pairing which is the only responsible for the dynamics Fig.\[Fig2c\]. Indeed, perfect collapse of the excitation probabilities for various $\delta$ is observed in terms of the correct scaling variable $k/\delta^{\nu_{d}/(1+z_{d}\nu_{d})}=k/\delta^{1/2}$, see the inset in Fig.\[Fig2c\].
In conclusion, we have demonstrated that long-range coupling terms can lead to a novel scaling behaviour of heat produced by slow quenches in critical topological superconductors. We have shown that this behaviour cannot be understood within the framework of the Kibble-Zurek scaling. In particular, the introduction of long-range hopping terms which decay slower than the pairing couplings modifies the equilibrium critical properties but not the dynamical critical exponent $\theta$. In the traditional case, for dominant pairing $\alpha>\beta$, the MZMs are gapped in the broken phase and, approaching the critical point, the gap closes and couples them into a single Dirac mode at $\mu=\mu_{c}$. The Dirac mode arises at the topological phase transition and dominates the low energy spectrum of the system, being also responsible for the universal slow dynamics[@Vodola2014; @Dutta2017].
Instead, for dominant hopping term $\alpha<\beta$, the critical Dirac mode is not relevant in the low energy spectrum, since the pairing term is not the leading operator in the zero momentum limit, and the equilibrium low energy theory at the critical point does not show any trace of the topological order found in the broken phase ($w=1$). However, the subleading pairing term turns out to be dangerously irrelevant and signatures of the topological order are found in the dynamics, which is always governed by the sub-leading pairing term, which is responsible for the topological transition. It is worth noting that the discrepancy between the traditional scaling argument and the *anomalous universal scaling* is not related to the inapplicability of the adiabatic perturbation theory expression, as it may occur in Bose systems due to diverging occupations[@Polkovnikov2008; @Bachmann2017; @Defenu2018]. Rather, the *anomalous universal scaling* is the consequence of deviations from the universal scaling hypotheses, see Eq., occurring in LR systems. Similar deviations were already noticed in LR classical systems[@Flores-Sola2015; @Flores-Sola2016], but their consequences appear to be much more striking in the dynamics of quantum systems.
Our results can be straightforwardly generalised to higher dimensional cases. Moreover, we expect them to be generally valid for most of the interacting $p$-wave Hamiltonians[@Sau2010; @Jason2010; @Lutchyn2010], which reduce to the quadratic form of Eq. in the Bogolyubov approximation. These investigations are of fundamental importance in current technological applications, since slow dynamical manipulations of the Hamiltonian are necessary to realise MZMs[@Perfetto2013]. Finally, due to the possibility of experimentally measuring both the equilibrium and the dynamical critical scaling, the *anomalous universal scaling* can be used as a diagnostic for the existence of long-range tails in the hopping matrix and of topological excitations in superconducting systems.\
*Acknowledgements.* ND is grateful to S. Ruffo, A. Trombettoni and G. Gori for useful discussions at the early stages of this work. ND and TE acknowledge financial support by Deutsche Forschungsgemeinschaft (DFG) via Collaborative Research Centre SFB 1225 (ISOQUANT) and under Germany’s Excellence Strategy EXC-2181/1-390900948 (Heidelberg STRUCTURES Excellence Cluster). LD acknowledges financial support from the BIRD2016 project of the University of Padova. GM is grateful for financial support by the DFG Priority Program no. 1929 GiRyd and by the German Ministry of Education and Research (BMBF) via the Quantera project “NAQUAS”. Project NAQUAS has received funding from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Programme.
Bogolyubov Transformation {#bg_trans}
=========================
The Bogolyubov transformation which diagonalises the Kitaev hamiltonian is described here in details. Our starting point is the real space Hamiltonian for $N$ spinless fermions, Eq. of the main text, which reads $$\begin{aligned}
\label{h_klr}
H=-\sum_i\sum_{r>0}\left(j_{r}c^{\dagger}_{i}c_{i+r}+\Delta_{r}c^{\dagger}_{i}c^{\dagger}_{i+r}+{\rm H.c.}\right)-\mu\sum_{i}\left(c^{\dagger}_{i}c_{i}-\frac{1}{2}\right)\,,\end{aligned}$$ where the $c^{\dagger}_{i}$s are Fermionic creation operators which fulfil the anticommutation relations $\{c_j,c_\ell\}=\delta_{j\ell}$. We consider power law couplings for the hopping and pairing terms: $$\begin{aligned}
j_{r}^{\alpha}&=\frac{J}{N_{\alpha}}\,\frac{1}{\bar r^{\alpha}}\,,\label{jr}\\
\Delta_{r}^{\beta}&=\frac{d}{N_{\beta}}\,\frac{1}{\bar r^{\beta}}\,,\label{dr}\end{aligned}$$ where $\bar r=\min(r,N-r)$ and we have considered periodic boundary conditions. The exponents of the power laws can be different and take values $\alpha>1$ and $\beta>1$, which warrant a well defined ferromagnetic state energy. The normalisation coefficients $N_\gamma$ ($\gamma=\alpha,\beta$) garantee that the energy is extensive. They read $$\begin{aligned}
N_{\gamma}=2\sum_{r=1}^{N/2}\frac{1}{r^{\gamma}}\to2\zeta(\gamma)\,,\end{aligned}$$ where the expression on the right is exact in the thermodynamic limit and $\zeta(\gamma)$ is the Riemann $\zeta$-function [@Abramowitz1964].
Hamiltonian is quadratic and can be explicitly integrated in momentum space via a Bogolyubov transformation[@Vodola2014]. For this purpose we introduce the Fourier Space transformations of operators $c_j$, $$\begin{aligned}
\label{f_trans}
c_{k}=\frac{1}{\sqrt{N}}e^{-i\frac{\pi}{4}}\sum_{j\in\mathbb{Z}}c_{j}e^{ikj}\end{aligned}$$ where $k\in[-\pi,\pi)$ and it takes continous values in the thermodynamic limit. Using the Fourier representation in the Kitaev hamiltonian one finds $$\begin{aligned}
\label{h_klr}
H=\sum_k\left[(c^{\dagger}_{k}c_{k}
-c_{-k}c^{\dagger}_{-k})\varepsilon_\alpha(k)+(c^{\dagger}_{k}c^{\dagger}_{-k}+c_{-k}c_{k})\Delta_\beta(k)\right]\,,\end{aligned}$$ where the coefficients are a function of $k$ and read: $$\begin{aligned}
\varepsilon_{\alpha}(k)&=-\frac{\mu}{2}-j_{\alpha}(k)\,,\nonumber\\
j_{\alpha}(k)&=\sum_{r>0}j^{\alpha}_{r}\cos(kr)\,,\nonumber\\
\Delta_{\beta}(k)&=\sum_{r>0}\Delta^{\beta}_{r}\sin(kr)\,.\nonumber\end{aligned}$$
It is convenient to employ a Bogolyubov transformation in order to diagonalise the static Hamiltonian. We choose $$\begin{aligned}
c_{k}=u_{k}\gamma_{k}+v^{*}_{-k}\gamma^{\dagger}_{-k}\,,\end{aligned}$$ where $u_k$, $v_k$ are the Bogolyubov coefficients and $\gamma_k$ satisfy the fermionic anticommutation relations $\{\gamma_k,\gamma_k'^\dagger\}=\delta_{k,k'}$. With this transformation the Hamiltonian takes the diagonal form $$\begin{aligned}
H=2\sum_{k}\omega_{k}\left(\gamma^{\dagger}_{k}\gamma_{k}-\frac{1}{2}\right)\,,\end{aligned}$$ where the eigenfrequencies read $$\begin{aligned}
\label{spec_eq}
\omega_{k}=\sqrt{\varepsilon_{\alpha}(k)^{2}+\Delta_{\beta}(k)^{2}}\,.\end{aligned}$$ The diagonal form is found with the Bogolyubov coefficients $$\begin{aligned}
(u_{k},v_{k})=\left(\cos\frac{\theta_{k}}{2}, \sin\frac{\theta_{k}}{2}\right)\,,\end{aligned}$$ such that $$\begin{aligned}
\tan\theta_{k}=\frac{\Delta_{\beta}(k)}{\varepsilon_{\alpha}(k)}\,.\end{aligned}$$ This is the solution of the equilibrium model.
Taylor Expansion of the Polylogarithm
-------------------------------------
At lowest order in $k$ (namely, for $|k|\ll \pi$) , we expand the $k$-dependent coefficients and obtain the expressions: $$\begin{aligned}
j_{\alpha}(k)/J&=1+\sin(\alpha\pi/2)\frac{\Gamma(1-\alpha)}{\zeta(\alpha)}k^{\alpha-1}-\frac{\zeta(\alpha-2)}{2\zeta(\alpha)}k^{2}+O(k^{3})\quad\mathrm{if}\,\,\alpha<3,\label{j_exp1}\\
j_{\alpha}(k)/J&=1+\frac{2\log(k)-3}{4\zeta(3)}k^{2}+O(k^{3})\quad\mathrm{if}\,\,\alpha=3,\label{j_exp2}\\
j_{\alpha}(k)/J&=1-\frac{\zeta(\alpha-2)}{2\zeta(\alpha)}k^{2}+O(k^{\alpha-1})\quad\mathrm{if}\,\,\alpha>3,\label{j_exp3}\end{aligned}$$ and $$\begin{aligned}
\Delta_{\beta}(k)/d&=\cos(\beta\pi/2)\frac{\Gamma(1-\beta)}{\zeta(\beta)}k^{\beta-1}+\frac{\zeta(\beta-1)}{\zeta(\beta)}k +O(k^{3})\quad\mathrm{if}\,\,\beta<2,\label{d_exp1}\,.\\
\Delta_{\beta}(k)/d&=\frac{6(1-\log(k))}{\pi^{2}}k +O(k^{3})\quad\mathrm{if}\,\,\beta=2,\label{d_exp2}\,.\\
\Delta_{\beta}(k)/d&=\frac{\zeta(\beta-1)}{\zeta(\beta)}k+O(k^{\beta-1})\quad\mathrm{if}\,\,\beta>2,\label{d_exp3}\,.\end{aligned}$$ These expressions are valid for all exponents $\alpha>1$, once the analytic continuation of the Reimann $\zeta$-function is considered for the cases $\alpha\leq 2$ and $\beta<2$. For $\beta>2$ and $\alpha>3$ the non analytic terms in Eqs. and become sub-leading with respect to further analytic corrections and they can safely be discarded. Now we have all the necessary information to derive a full phase diagram for the extended Kitaev chain.
The scaling of the defect density {#defect_sc}
=================================
According to the solution of the effective LZ problem the excitation probability of each low momentum mode for an infinitely slow ramp is $$\begin{aligned}
p_{k}\approx e^{-\frac{\pi}{\delta^{2}}\Delta_{\beta}(k)^{2}}\end{aligned}$$ and the defect density can be computed integrating the excitation probability along $k$ $$\begin{aligned}
n_{\mathrm{exc}}\approx\int e^{-\frac{\pi}{\delta^{2}}\Delta_{\beta}(k)^{2}}\end{aligned}$$ in the infinitely slow ramp limit $\delta\to 0$ the above integral has to be computed using the saddle point method. Indeed, the integral remains not negligible only on an infinitesimal neighborhood of the saddle point $k=0$, where the pairing term $\Delta_{\beta}(k)$ vanishes. According to the low momentum expansions reported in the above section for $\beta>2$ one has $$\begin{aligned}
n_{\mathrm{exc}}\approx\int e^{-\frac{\pi}{\delta}\frac{\zeta(\beta-1)^{2}}{\zeta(\beta)^{2}}k^{2}}\approx \frac{\zeta(\beta)}{\zeta(\beta-1)}\sqrt{\delta}\propto \sqrt{\delta}\end{aligned}$$ as it shall be for a short-range system. In the long-range regime $\beta<2$ the saddle point approximation is less straightforward due to the divergence of the Hessian in the exponent. Considering the low momentum expansion in this regime the integration reads $$\begin{aligned}
n_{\mathrm{exc}}\approx\int e^{-\frac{\pi}{\delta}\left(\cos\left(\frac{\beta\pi}{2}\right)\frac{\Gamma(1-\beta)}{\zeta(\beta)}\right)^{2}k^{2(\beta-1)}}.\end{aligned}$$ It is convenient to define $\theta=2(\beta-1)^{-1}$ and $c=\pi\left(\cos\left(\frac{\beta\pi}{2}\right)\frac{\Gamma(1-\beta)}{\zeta(\beta)}\right)^{2}$, then we shall consider the transformation $$\begin{aligned}
k&=(\delta\,s/c)^{\theta}\\
dk&=\delta^{\theta}\frac{\theta}{c^{\theta}}
s^{{\theta}-1}\end{aligned}$$ the integral then reduces to $$\begin{aligned}
n_{\mathrm{exc}}\approx\int e^{-\frac{c}{\delta}\,k^{1/\theta}}=\frac{\theta\delta^{\theta}}{c^{\theta}}\int s^{\theta-1}e^{-s}ds=\frac{\theta\Gamma(\theta)}{c^{\theta}}\delta^{\theta}\propto\delta^{\theta}\end{aligned}$$ as argued in the main text.
At $\beta=2$ the low energy behavior for $\Delta_{\beta}(k)$ acquires logarithmic corrections and it shall then be treated separately. The low momentum limit in this case reads $$\begin{aligned}
\label{b2_exp}
\Delta_{2}(k)=-\frac{6}{\pi^{2}}k\log(k)+O(k^{2})\end{aligned}$$ leading to the excitation probability $$\begin{aligned}
n_{\mathrm{exc}}\approx\int e^{-\frac{36}{\delta\pi^{3}}k^{2}\log(k^{2})}.\end{aligned}$$ In this case one shall introduce a more complicated transformation $$\begin{aligned}
\label{log_trans}
s&=k\log(k)\\
k&=e^{W(s)}\\
dk&=\frac{ds}{1+\log(k)}=\frac{ds}{1+W(s)}\end{aligned}$$ where $W(s)$ is the Lambert function. The integral has now been reduced to $$\begin{aligned}
n_{\mathrm{exc}}\approx\int\frac{e^{-\frac{36}{\pi^{3}}\frac{s^{2}}{\delta}}ds}{1+W(s)}.\end{aligned}$$ The integration boundary has to be treated with care since the transformation is not univocal. However, one shall consider that we are interested only in a small neighbourhood of $k=0$, where the expansion in Eq. is valid. In this regime, it is sufficient to consider the lower branch of the Lambert function $W_{-1}(s)$, which is real in the interval $s\in[0,-1/e]$, leading to the momentum interval $k\in[0,1/e]$. In the $s\to 0^{-}$ limit $W_{-1}(s)$ obeys the asymptotic expansion[@Corless1996] $$\begin{aligned}
W_{-1}(s)=\log(-s)+O(\log\log(-s))\end{aligned}$$ Therefore our integral can be finally approximated with $$\begin{aligned}
n_{\mathrm{exc}}\approx\int_{0}^{-1/e} \frac{e^{-\frac{36}{\delta\pi^{3}}s^{2}}}{\log(-s)}=-\int_{0}^{1/e} \frac{e^{-\frac{36}{\delta\pi^{3}}s^{2}}}{\log(s)},\end{aligned}$$ the reduction of the integral boundaries to $k\in[0,1/e]$ is valid for $\delta\ll\frac{1}{e^{2}}$ and becomes exact in the $\delta\to 0$ limit. In order to proceed further, it is convenient to introduce the limit representation of the logarithm $$\begin{aligned}
\log(s)=\lim_{h\to0}\frac{s^{h}-1}{h}\end{aligned}$$ which in turns leads to $$\begin{aligned}
\frac{1}{\log(s)}=\lim_{h\to 0}\sum_{n=1}^{\infty}h\,s^{hn}.\end{aligned}$$ Once latter expression is plugged into the integral one obtains $$\begin{aligned}
\label{b2_exc_den}
n_{\mathrm{exc}}\approx-\lim_{h\to0}\sum_{n=1}^{\infty}h\int_{0}^{1/e} s^{hn}e^{-\frac{36}{\delta\pi^{3}}s^{2}}\approx-\lim_{h\to0}\sum_{n=1}^{\infty}\int_{0}^{+\infty} s^{hn}e^{-\frac{36}{\delta\pi^{3}}s^{2}}=-\frac{\sqrt{\pi^{3}\delta}}{6}\lim_{h\to0}h\sum_{n=1}^{\infty}\left(\frac{\sqrt{\delta}}{6}\right)^{h\,n}\frac{\Gamma\left(\frac{3+h\,n}{2}\right)}{1+hn}\end{aligned}$$ where we once again deformed the integration range, since $s\gg\sqrt{\delta}$ contributions to the integral vanish exponentially fast in the $\delta\to 0$ limit. The summation in Eq. has to be considered in the $h\to 0$ limit, where the power law contributions $\delta^{hn/2}$ become all relevant, while the $\frac{\Gamma\left(\frac{3+h\,n}{2}\right)}{1+hn}$ terms can be safely approximated as $\Gamma\left(\frac{3}{2}\right)=\sqrt{\pi}/2$ yielding $$\begin{aligned}
\label{b2_exc_den_c}
n_{\mathrm{exc}}\approx-\frac{\pi^{2}}{12}\sqrt{\delta}\lim_{h\to0}h\sum_{n=1}^{\infty}\left(\frac{\sqrt{\delta}}{6}\right)^{h\,n}=-\frac{\pi^{2}}{6}\frac{\sqrt{\delta}}{\log(\delta/6)}\propto -\sqrt{\delta}\log(\delta)^{-1}.\end{aligned}$$ As expected the case $\beta=2$ is exactly in between the pure short-range case $n_{\mathrm{exc}}\propto \sqrt{\delta}$ and the weak long range case $\beta=2-\varepsilon$, where one has $n_{\mathrm{exc}}\propto \delta^{\frac{1+\varepsilon}{2}}$ for $\varepsilon\ll 1$ and, therefore the excitation probability decays faster than in the pure short-range case. The $\varepsilon\to0$, i.e. $\beta\to 2$, limit is placed exaclty in between, with the density of excitation decaying only logarithmically faster than in the short-range case. We have numerically verified that the introduction of sub-leading $k^{2}$ terms in the expansion of $\Delta_{2}(k)$ as well as the extension of the integration range beyond the region of validity of the low momentum expansion in Eq. do not modify the scaling regime in the $\delta\to 0$ limit.
Landau-Zener Problem {#lz}
====================
As discussed in the main text, one can employ the substitution in Eq.(12) into the dynamical evolution Eq.(11) for the Bogolyubov coefficients. The resulting dynamics takes the celebrated Landau-Zener form $$\begin{aligned}
\label{lz_dyn}
i\partial_{\tau}\begin{pmatrix}u_{k}\\
v_{k}\end{pmatrix}=\begin{pmatrix}-\Omega_{k}\tau & 1\\
1 & \Omega_{k}\tau\end{pmatrix}\begin{pmatrix}u_{k}\\
v_{k}\end{pmatrix}.\end{aligned}$$ The Landau-Zener (LZ) evolution, Eq., can be solved exactly using several approaches[@Zener1932; @Wittig2005; @damski2005]. However, this exact solution is rather cumbersome, since it is obtained in terms of Weber functions. Therefore, we will rather rely on an approximate solution, which is capable to correctly reproduce the defect scaling, only sacrificing the exactness of numerical coefficients, unimportant to our scopes. The LZ Hamiltonian can be conveniently written using Pauli’s matrices $$\begin{aligned}
H_{LZ}=\Omega\tau\sigma_{z}+\sigma_{x}\end{aligned}$$ where $$\begin{aligned}
\sigma_{z}=\begin{pmatrix}
1 & 0\\
0 & -1
\end{pmatrix},\quad \sigma_{x}=\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.\end{aligned}$$ This Hamiltonian is diagonalised analogously to the Bogolyubov transformation described in the first section, one has to introduce the angle $\theta=\mathrm{arctan}\left(\frac{1}{\Omega\tau}\right)$, useful to describe the eigenstates $$\begin{aligned}
|+\rangle=\begin{pmatrix}
\cos\frac{\theta}{2}\\
\sin\frac{\theta}{2}
\end{pmatrix},\quad |-\rangle=\begin{pmatrix}
\cos\frac{\theta}{2}\\
-\sin\frac{\theta}{2}
\end{pmatrix}\end{aligned}$$ with the eigen-energies $\omega_{\pm}=\pm\sqrt{(\Omega\tau)^{2}+1}$.
The finite ramp case {#fr_corr}
====================
At the initial stage of the dynamics the system is exactly in the ground state $|\psi_{\tau=0}\rangle\equiv |-\rangle$, while at every finite time the state is given by the superposition $|\psi_{t}\rangle = \alpha_{-}(\tau)|-\rangle+\alpha_{+}(\tau)|+\rangle$, with $ \alpha_{+}(\tau)^{2}+ \alpha_{-}(\tau)^{2}=1$. According to adiabatic perturbation theory[@de_grandi2010], the excitation amplitude at first order in $\Omega$ is given by the formula $$\begin{aligned}
\alpha_{+}(\tau)\simeq - \int_{\tau_i}^{\tau}\langle+|\partial_{s} |-\rangle e^{i(\Theta_{+}(s)-\Theta_{-}(s))}\end{aligned}$$ where $$\begin{aligned}
\Theta_{\pm}(\tau)=\int_{\tau_{i}}^{\tau}\omega_{\pm}(s)ds.\end{aligned}$$ The overlap element between the adiabatic states can be computed exactly for the LZ model, $$\begin{aligned}
\langle+|\partial_{\tau} |-\rangle=\frac{\partial_{\tau}\theta}{2}=-\frac{1}{2}\frac{\Omega}{(\Omega\tau)^{2}+1}\end{aligned}$$ yielding the transition amplitude $$\begin{aligned}
\label{trans_amp}
\alpha(\tau_{f})\simeq \frac{1}{2}\int_{-\infty}^{\tau_{f}}\frac{\Omega\,d\tau}{(\Omega\tau)^{2}+1}e^{\frac{2i}{\Omega}\int_{0}^{\Omega\tau}\sqrt{s^{2}+1}ds}.\end{aligned}$$ where, without loss of generality, we imposed $\tau_{i}=-\infty$. One can explicitly integrate the phase factor $$\begin{aligned}
g(x)=2\int_{0}^{x}\sqrt{s^{2}+1}ds=\tau\sqrt{\tau^{2}+1}+\mathrm{arcsinh}(\tau).\end{aligned}$$ It is convenient to rescale the integration variable in Eq. according to $x=\Omega\tau$, $$\begin{aligned}
\alpha(\tau_{f})\simeq \frac{1}{2}\int_{-\infty}^{\Omega\tau_{f}}\frac{dx}{x^{2}+1}e^{\frac{i}{\Omega}g(x)},\end{aligned}$$ we are interested in the $\tau_{f}\gg 0$ limit of the latter we shall then separate the integral into the two contributions $$\begin{aligned}
\label{amp_2}
\alpha(\tau_{f})\simeq \frac{1}{2}\int_{-\infty}^{\infty}\frac{dx}{x^{2}+1}e^{\frac{i}{\Omega}g(x)}- \frac{1}{2}\int_{\Omega\tau_{f}}^{\infty}\frac{dx}{x^{2}+1}e^{\frac{i}{\Omega}g(x)}.\end{aligned}$$ The above expression proves that the finite ramp dynamics is always equivalent to an infinite ramp from $\tau_{i}=-\infty$ to $\tau_{f}=+\infty$ plus a correction, which is equivalent to the exctitation amplitude of a finite ramp not crossing the critical point. The phase factor $g(x)$ has no stationary points on the real line, but it possesses an inflection point at $x=0$. Therefore, the first contribution to Eq. needs to be treated separately. This computation has been already carried on in details in Ref.[@de_grandi2010], yielding $$\begin{aligned}
\label{inf_ramp}
\alpha(\infty)\simeq \frac{\pi}{3}e^{-\frac{\pi}{2\Omega}}\end{aligned}$$ where the numerical coefficient $\pi/3\approx 1.05$ is surprisingly close to the exact value $1$. The second contribution can be transformed into $$\begin{aligned}
\label{finite_ramp}
\alpha^{*}(-\tau_{f})=\frac{1}{2}\int_{-\infty}^{-\Omega\tau_{f}}\frac{dx}{x^{2}+1}e^{\frac{i}{\Omega}g(x)}.\end{aligned}$$ Since $\tau_{f}$ is positive $-\tau_{f}$ is negative and the formula describes the excitation amplitude of a ramp ending below the critical point. Therefore, the integration in Eq. does not contain the higher order stationary point $x=0$ and it can be safely integrated using the standard procedure for fast oscillating integrals[@Dingle1975] $$\begin{aligned}
\label{finite_ramp_res}
\alpha^{*}(-\tau_{f})=\frac{\Omega}{4}\frac{1}{\sqrt{1+(\Omega\tau_{f})^{2}}^{3}}.\end{aligned}$$ Coming back to the Kitaev chain problem one has $\Omega\equiv \delta/\Delta(k)^{2}$ and $\tau_{f}=(j_{\alpha}(k)-g_{f})\Delta_{\beta}(k)/\delta$. The excitation probability for a single momentum state $k$ is $$\begin{aligned}
p_{k}=\frac{\delta^{2}}{16}\frac{\Delta_{\beta}(k)^{4}}{(\Delta_{\beta}(k)^{2}+(j_{\alpha}(k)-g_{f})^{2})^{3}}.\end{aligned}$$ Therefore, the defect density for the $p$-wave superconducting Hamiltonian in Eq. after a quench starting at $g_{i}=+\infty$ and ending at $g_{f}>1$ without crossing any critical points is given by $$\begin{aligned}
n_{\mathrm{exc}}(t_{f})=\int p_{k}dk=\frac{\delta^{2}}{16}\int\frac{\Delta_{\beta}(k)^{4}dk}{(\Delta_{\beta}(k)^{2}+(j_{\alpha}(k)-g_{f})^{2})^{3}}\end{aligned}$$ where, as long as $|g_{f}|>1$ the integral remains always convergent. In the $\delta\to 0$ limit such contribution decays quadratically with $\delta$, in agreement with our expectations for adiabatic dynamics. The latter result proves that finite ramp corrections are always negligible with respect to the non analytic scaling in the defect density generated by the low momenta during the full ramp dynamics.
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abstract: |
Charged (nano)particles confined in electrodynamic traps can evolve into strongly correlated Coulomb systems, which are the subject of current investigation. Exciting physical phenomena associated to Coulomb systems have been reported such as autowave generation, phase transitions, system self-locking at the ends of the linear Paul trap, self-organization in layers, or pattern formation and scaling. The dynamics of ordered structures consisting of highly nonideal similarly charged nanoparticles, with coupling parameter of the order $\Gamma = 10^8$ is investigated. This approach enables us to study the interaction of nanoparticle structures in presence and in absence of the neutralizing plasma background, as well as to investigate various types of phenomena and physical forces these structures experience. Applications of electrodynamic levitation for mass spectrometry, including containment and study of single aerosols and nanoparticles are reviewed, with an emphasis on state of the art experiments and techniques, as well as future trends. Late experimental data suggest that inelastic scattering can be successfully applied to the detection of biological particles such as pollen, bacteria, aerosols, traces of explosives or synthetic polymers.
Brownian dynamics is used to characterize charged particle evolution in time and thus identify regions of stable trapping. An analytical model is used to explain the experimental results. Numerical simulations take into account the stochastic forces of random collisions with neutral particles, the viscosity of the gas medium, the regular forces produced by the a.c. trapping voltage, and the gravitational force. We show that microparticle dynamics is characterized by a stochastic Langevin differential equation. Laser plasma acceleration of charged particles is also approached, with an emphasize on dielectric capillaries and Paul traps employed for target micropositioning.
address:
- |
Natl. Inst. for Laser, Plasma and Radiation Physics (INFLPR),\
Atomiştilor Str. 409, 077125 Măgurele, Ilfov county, Romania
- |
Joint Institute for High Temperatures (JIHT),\
Russian Academy of Sciences, Izhorskaya 13, Bld. 2, 125412 Moscow, Russia
author:
- 'Bogdan M. Mihalcea'
- Vladimir Filinov
- Roman Syrovatka
- Leonid Vasilyak
bibliography:
- 'CCP.bib'
title: The physics and applications of strongly coupled plasmas levitated in electrodynamic traps
---
Paul trap ,ergodic dynamics ,complex plasmas ,strongly coupled plasmas ,solitary waves ,electrodynamic balance ,mass spectrometry ,aerosols ,nanoparticles ,Raman spectroscopy ,Mie theory ,laser plasma accelerated particle physics
02.20.-a ,03.65.-w ,33.15.Ta ,33.20.Fb ,37.10.Ty ,52.20.-j ,52.27.Gr ,52.27.Lw ,52.35.-g ,52.38.Kd ,52.70.-m ,78.30.-j ,78.35.+c ,78.40.-q ,82.80.-d ,89.60.-k ,92.60.H- ,92.60.Mt ,92.60.Ry ,92.60.Sz
Introduction {#Intro}
============
Until the 1950s experimental investigations performed in atomic and molecular physics were limited in sensitivity, as the spectral resolution achievable was severely limited by the thermal motion of the atoms that is responsible for the Doppler broadening of the spectral emission lines. The occurrence of Doppler-free spectroscopy techniques enabled scientists to mitigate the Doppler non-relativistic effects. Nevertheless, time-of-flight (ToF) broadening of spectral lines caused by the limited period of time that atoms spent in the interaction region with a laser beam, also imposes restrictive boundaries to the spectral resolution of resonance microwave and optical lines [@Dem10; @Dem15; @Van15; @Tome18]. In order to enhance the precision of such spectroscopic measurements ingenious methods were required to maintain the atoms within the interaction region as long as possible, under minimal perturbations produced by collisions with other atoms or molecules, or via the interaction with the containing vessel walls.
In their perpetual quest aimed at disclosing nature’s inner secrets, physicists and chemists always aspired towards the ideal of isolating and investigating single atoms in a perturbation free, pristine environment. Such an objective seemed to be quite far-fetched as E. Schr[ö]{}dinger stated in 1952: “We never experiment with just one electron or atom or (small) molecule. In thought experiments we sometimes assume that we do, this invariably entails ridiculous consequences. ... We are scrutinising records of events long after they have happened” [@Schro52]. After few years Schrodinger’s statement was invalidated by W. Paul, who invented the radiofrequency (RF) trap (also called Paul trap) [@Paul58; @Paul90; @Ghosh95] that became an invaluable instrument for modern spectroscopy, opening new pathways towards investigating atomic properties and testing quantum physics laws with unbeatable accuracy [@Major05; @Kno14].
Another ion trap pioneer is H. Dehmelt whose contributions are mainly linked to the development and use of the Penning trap [@Blaum08; @Werth08; @Werth09]. He invented ingenious methods of cooling, perturbing, trapping (one single electron was trapped for more than 10 months [@vanDyck87; @Hey11]), and probing the trapped particles, thus forcing them to reveal their intrinsic properties [@Dehm90]. In the combined electric and magnetic fields in the Penning trap, charged particles describe a complex motion [@Major05; @Quint14; @Vogel18]. Laser cooling of ions was first proposed by H. Dehmelt and D. J. Wineland since 1975 [@Wine87], and then achieved in 1978 [@Wine11; @Haro13]. Both Wolfgang Paul and Hans Dehmelt were awarded the Nobel Prize in 1989 “for the development of the ion trap technique”. Progress in the development of extremely stable laser sources allows scientists to manipulate and control the inner as well as the outer degrees of freedom of atoms and even molecules with ever increasing precision, right down to the quantum limit. Trapping and cooling techniques for electrically charged particles have expanded the frontiers of physics, leading to the emergence of new areas such as quantum engineering or quantum metrology, which explains why the Nobel Prize in Physics in 1997 was jointly awarded to S. Chu, C. Cohen-Tannoudji and W. D. Phillips “for the development of methods to cool and trap atoms with laser light” [@Haro13]. J. L. Hall and T. W. H[ä]{}nsch were awarded the Nobel Prize in Physics in 2005 “for their contributions to the development of laser-based precision spectroscopy, including the optical frequency comb technique” [@Poli13]. Finally, the Nobel Prize in Physics in 2012 was jointly awarded to Serge Haroche and David J. Wineland “for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems” [@Haro13b; @Wine13]. Thus, impressive progress was achieved in quantum metrology by enabling measurements of unprecedented accuracy on space and time [@Haro13; @Peik06; @Bush13]. Both the plasma state of matter as well as systems of charged dust particles in plasma exhibit a great interest for modern physics [@Ichi82; @Fort05; @Fort10a; @Bouf11; @Ivlev12; @Vasi15; @Usach16; @Ramaz16; @Myas17; @Ebel17]. When the potential energy associated to the Coulomb interaction between charged dust particles significantly exceeds their kinetic energy, they can evolve into ordered structures called Coulomb or plasma crystals, that represent the subject of recent investigations performed onboard the International Space Station (ISS) [@Khra16; @Pust16].
The paper is a review on strongly coupled Coulomb systems of finite dimensions, such as electrically charged particles confined in quadrupole or multipole electrodynamic traps, either under Standard Ambient Temperature and Pressure Conditions (SATP) or in vacuum [@Boll84]. An extended review of the underlying physical mechanisms is also performed. Microparticle electrodynamic ion traps (MEITs) are versatile tools than can be used to investigate properties of individual charged particles (with dimensions ranging from 10 nm to 100 $\mu$m), such as aerosols, liquid droplets, solid particles, nanoparticles, DNA sequences and even microorganisms. When the background gas is weakly ionized, the associated dynamics exhibits strong coupling regimes characterized by collective motion as damping is low. Late experimental evidence suggests that multipole trap geometries present certain advantages, among which better stability due to the existence of multiple regions of stable trapping and less sensibility to environment fluctuations. The mathematical models used and the numerical simulations performed simply create a clear and thorough picture on the phenomena that are investigated. We study particle oscillations around equilibrium positions where gravity is balanced by the trapping potential. Close to the trap centre the particles are almost frozen which means they can be considered motionless due to the very low amplitude of oscillation.
Particular examples of such systems would be electrons and excitons in quantum dots [@Boni14] or laser cooled ions levitated in Paul or Penning type traps [@Major05; @Kno14; @Haro13; @Davis02; @Bolli03]. We can also mention ultracold fermionic or bosonic atoms confined in atom traps or in the periodic potential of an optical lattice [@Boni10b]. Experimental investigations of charged particles in external potentials have largely profited from the invention of ion traps [@Paul58; @Paul90], as they have greatly influenced the future of modern physics and state of the art technology [@Ghosh95; @Major05; @Werth09; @Kno15].
Particle levitation techniques {#Sec2}
==============================
The electrodynamic (Paul) trap. The underlying physics. Electrodynamic levitation
---------------------------------------------------------------------------------
Electrically charged particles such as ions or microparticles can be levitated by using electromagnetic fields. Trapping of particles in a harmonic potential requires generating an electric restoring potential ${\vec F} = -q {\vec E}$ which increases linearly with the distance from the trap centre, such as $\vec F \propto -{\vec r}$. The corresponding trapping forces are characterized by a quadrupole potential $\Phi = \Phi_0 \left( \alpha x^2 + \beta y^2 + \gamma z^2 \right)/r_0^2$, where $\Phi_0$ denotes the electric potential applied to a quadrupole electrode configuration, $r_0$ is the trap radius, while $\alpha, \beta, \gamma$ represent weighing factors that determine the shape of the electric potential given by the solution of the Laplace equation $\Delta \Phi = 0$. For a 3D electric field we infer $\alpha = \beta = -2\gamma$. The potential is attractive in the $x$ and $y$ directions, and repulsive along the $z$ direction. According to the Earnshaw theorem a static electric field cannot achieve 3D binding [@Ghosh95; @Major05; @Kno14]. In order to overcome this problem two main types of traps have been developed: (1) the Penning trap, which employs a static electric field to achieve axial confinement and a superimposed static magnetic field that provides radial confinement, and (2) the Paul or RF trap, which relies on an oscillating, inhomogeneous electric field [@Werth08; @Kal09] that creates a dynamic pseudopotential. A sketch of a quadrupole ion trap (QIT) is shown in Fig. \[PaulWerth\]. In our review we mainly consider levitation of particles in RF or Paul traps, as the subject is extremely vast.
We return to the trapping potential to emphasize that if an oscillating electric field is applied to the trap electrodes (see Fig. \[PaulWerth\]), a saddle shaped potential is created [@Kiri16] that harmonically confines ions in the region where the field exhibits a minimum, under conditions of dynamical stability. The resulting potential is attractive in the $x$ and $y$ (radial) directions for the first half-cycle of the field, and attractive in the $z$ (axial) direction for the second half. An adequately chosen amplitude and frequency $\Omega$ of the oscillating RF field ensures trapping of charged particles of mass $m$ and charge $q$ in all three dimensions, by means of a ponderomotive force oriented towards the trap centre. The 3D Paul trap provides a confining force with respect to a single point in space called the node of the RF field, which recommends its use for single ion experiments or for the levitation of 3D crystalline ion structures [@Werth08].
Considering the electric field in a two dimensional geometry along the $x$ and $y$ axis only (linear Paul trap), we find $\alpha = -\beta, \gamma = 0$. In such case, confinement of a charged particle is accomplished only in the (radial) $x$ and $y$ direction. An additional static dc-potential applied along the $z$ direction is required to confine the particle both radially and axially [@Kal09].
The electrodynamic (Paul) trap was invented in the 1950s by Prof. Wolfgang Paul and his coworkers at the University of Bonn, Germany [@Paul58; @Paul90; @Ghosh95; @Major05]. The device was the subject of a patent awarded in 1954, for a three dimensional (3D) quadrupole ion trap (QIT) which consists of a ring electrode and two endcap electrodes [@Paul58; @Werth08; @March09]. By applying a RF potential to the ring electrode a quadrupole trapping field is produced in all three dimensions. A sketch of a QIT is shown in Fig. \[PaulWerth\]. Many other kinds of QITs exist, most notably the 2D linear ion trap with quadrupole potential in the radial plane and a d.c. field for axial trapping. The quadrupole ion trap is a remarkably versatile mass spectrometer that is capable of multiple stages of mass selection (tandem mass spectrometry), high sensitivity, and moderate mass resolution and mass range. In combination with electrospray ionization (ESI), the QIT is widely used for the study of polar molecules such as peptides and proteins [@Sny19].
Electrically charged particles and ions levitated in oscillating radiofrequency (RF) fields represent a special class of physical systems, where classical and quantum effects [@Werth09; @Haro13] compel to establish a dynamical equilibrium as they weight each other. Classical particle dynamics in a Paul trap is characterized by Mathieu equations [@Major05; @Baril74; @March97]. The stable solutions of the Mathieu equation are found by employing the Floquet theory [@Major05]. Rapid trapping field oscillations generate an effective Kapitza-Dirac potential that restrains the motion of electrically charged ions to a well defined region of space, which leads to a quasi-equilibrium distribution [@Aku14]. A picture of a three-dimensional (3D) Paul or Penning trap is given in Fig. \[PaulWerth\] [@Blaum08; @Werth09]. We do not discuss stability and solutions of the Mathieu equation, as the subject is widely covered in the literature [@Ghosh95; @Major05; @Werth08; @Werth05a; @March05].
![Basic design characteristic to three dimensional (3D) Paul and Penning traps. The inner electrode surfaces are hyperboloids. Dynamic stabilization in the Paul trap is given by supplying a RF voltage $V_0 \cos \left(\Omega t\right)$. Static stabilization in the Penning trap is achieved by means of a d.c. voltage $U = U_0$ and an axial magnetic field liable for the radial confinement. Picture reproduced from [@Werth09], by courtesy of G. Werth.[]{data-label="PaulWerth"}](PaulPenn.jpg)
The method of trapping charged particles using time-varying electric fields is not restricted to atomic or molecular ions, it equally applies to charged microparticles. The first demonstration of microparticle trapping was performed by Straubel [@Stra55], who confined charged oil drops in a simple quadrupole trap operated in air. Wuerker has obtained stable arrays of positively charged aluminum particles using a 3D Paul trap operated in vacuum [@Wuerk59]. The particles repeatedly crystallized into a regular array and then melted, owing to the dynamical equilibrium between the trapping potential and inter-particle Coulomb repulsion. A similar experiment was performed by Winter and Ortjohann, thus demonstrating storage of macroscopic dust particles (anthracene) in a 3D Paul trap operated under SATP conditions [@Wint91], as friction in air was proven to be an efficient mechanism to [*cool*]{} the microparticles. The mechanism is similar with cooling of ions in ultrahigh vacuum conditions owing to the collisions with the buffer gas molecules [@Major05]. The dynamics of a charged microparticle in a Paul trap near SATP conditions was studied by Izmailov [@Izma95] using a Mathieu differential equation with damping term and stochastic source, under conditions of combined periodic parametric and random external excitation. Further research on electrodynamic traps is presented in [@Tak04; @Kja05].
The combination between a Paul and Penning trap which uses both RF voltage and an axial magnetic field is called a combined trap [@Naka01]. Particle dynamics in nonlinear traps has also been explored. It was found that ion motion is well described by the Duffing oscillator model [@Nay08; @Kova11] with an additional nonlinear damping term [@Mih10b; @Ake10]. Regions of stable (chaos) and unstable dynamics were illustrated, as well as the occurrence of strange attractors and fractal properties for the associated dynamics. The dynamics of a parametrically driven levitated nanoparticle is investigated in [@Gies15].
Ion traps have also opened new horizons towards performing investigations on the physics of few-body phase transitions [@Fort10a; @Shuk02a; @Tsyto08; @Fort06a; @Died87; @Blu90; @Schli96] or the study of nonlinear dynamics and (quantum) chaos. Quantum engineering has opened new perpectives in quantum optics and quantum metrology [@Haro13; @Zago11]. Applications of ion traps span mass spectrometry, very high precision spectroscopy, quantum physics tests, study of non-neutral plasmas [@David01; @Dub99], quantum information processing (QIP) and quantum metrology [@Quint14; @Haro13; @Kim05; @Leibf07], use of optical transitions in highly charged ions for detection of variations in the fine structure constant [@Werth09; @Quint14] or very accurate optical atomic clocks [@Van15; @Poli13; @Rieh04; @Ved09; @March10; @Pyka14; @Lud15; @Kel15].
The ability to confine single particles or nanoparticles under conditions of minimal perturbations and then expose them to laser beams, makes it possible to perform light scattering measurements without the presence of parasitic effects owing to the interaction with other particles that may be of different sizes and morphology, and different chemical and/or optical properties. This can be achieved by trapping a charged microparticle using an electrodynamic balance (EDB) or ion trap, by optical levitation, or by acoustic levitation. The technique called electrodynamic levitation is used to isolate and analyze aerosols and microparticles, with an aim to study the mechanisms from the Earth atmosphere and supply information about the optical properties of the species of interest [@Davis10]. The EDB is an outcome of the quadrupole electric mass filter of Paul [@Paul90; @Major05], and of the bihyperboloidal electrode configuration introduced by Wuerker [*et al*]{} [@Wuerk59], which is mentioned earlier.
A hyperbolic geometry classical 3D Paul trap used at INFLPR in the experiments performed by Prof. V. Gheorghe with an aim to levitate $^{137}$Ba$^+$ ions in high vacuum $\left( 10^{-4} \div 10^{-5} \ \text{Torr} \right)$, is shown in Fig. \[PaulTrapINFLPR\] and Fig. \[PaulTrapINFLPR2\] [@Ghe76; @Giu94; @Ghe97].
![Classical Paul 3D trap with hyperbolic geometry used at INFLPR to trap $^{137}$Ba$^+$ ions in vacuum. The upper images show the 3D hyperbolic trap attached to the vacuum flange, while the lower images show the upper and lower endcap (slit) electrode, respectively. The lower endcap electrode exhibits 2 slits with dimensions $16 \times 3$ mm, corresponding to the location of two platinum belts covered with BaCl$_2$ solution [@Ghe76; @Giu94].[]{data-label="PaulTrapINFLPR"}](PaulTrapEndcapsINFLPR.pdf)
![Classical Paul 3D trap with hyperbolic geometry used at INFLPR to trap $^{137}$Ba$^+$ ions in vacuum. The left image shows the 3D hyperbolic trap where the ring electrode is pierced to allow radiation access. The right image shows the upper endcap (sieve) electrode with 256 holes of 2.4 mm diameter, equally spaced at 0.6 mm, that achieves a transparency of 56 % in the $2.5 \div 25 \ \mu$m range. This particular design of the sieve electrode facilitates collection of fluorescence radiation. The trap dimensions are $r_0 = 1.75$ cm and $z_0 = 1.24$ cm [@Ghe76; @Ghe97].[]{data-label="PaulTrapINFLPR2"}](PaulTrapINFLPRRing.pdf "fig:") ![Classical Paul 3D trap with hyperbolic geometry used at INFLPR to trap $^{137}$Ba$^+$ ions in vacuum. The left image shows the 3D hyperbolic trap where the ring electrode is pierced to allow radiation access. The right image shows the upper endcap (sieve) electrode with 256 holes of 2.4 mm diameter, equally spaced at 0.6 mm, that achieves a transparency of 56 % in the $2.5 \div 25 \ \mu$m range. This particular design of the sieve electrode facilitates collection of fluorescence radiation. The trap dimensions are $r_0 = 1.75$ cm and $z_0 = 1.24$ cm [@Ghe76; @Ghe97].[]{data-label="PaulTrapINFLPR2"}](PaulINFLPRUpperEndcap.pdf "fig:")
In the conventional operation of an EDB an alternating current (a.c.) potential is applied to the ring electrode, and a direct current (d.c.) field is generated by applying equal but opposite polarity d.c. potentials to the endcap electrodes. The a.c. field is used to trap the particle in an oscillatory mode, and the d.c. field serves the purpose of the Millikan condenser to balance the gravitational force and any other vertical forces that act upon the particle. A photodiode array is mounted on the ring electrode as shown in Fig. \[ac1\], with an aim to measure the irradiance of the scattered light as a function of angle, and a photomultiplier tube (PMT) is located at right angles to the incident laser beam to record the irradiance at a single angle as a function of time. The appropriate a.c. trapping frequency depends on the particle mass and drag force on the particle. When the vertical forces ([*e.g.*]{}, gravity) are balanced by the additional d.c. electric field applied between the upper and lower trap electrodes, the particle can be maintained at the midpoint of the EDB provided that the a.c. potential is not too large [@Sto11]. If the a.c. field is too large the particle ends up in being expelled from the balance chamber. This effect is used to isolate single particles by varying the a.c. potential and frequency to eliminate undesired particles. The principles and stability characteristics of the EDB were first analysed by Frickel [*et al.*]{} [@Kul11], while other aspects of the EDB are discussed by Davis and Schweiger [@Davis02; @Davis10; @Hart92].
Anharmonic contributions in RF linear quadrupole traps
------------------------------------------------------
As the RF linear Paul trap is widely used in physics and chemistry, an important feature is the trap capacity as there are situations that impose levitation of large ion clouds. Due to the presence of anharmonic terms in the series expansion of the trap potential [@Mih18], there is a limit on the trap capacity (the number of ions that may be confined). The effects of anharmonic terms in the trapping potential for linear chains of levitated ions are presented in [@Home11]. Two paramount effects can be distinguished, as the authors demonstrate. The first effect consists in a modification of the oscillation frequencies and amplitudes of the ions’ normal modes of vibration for multi-ion crystals, which is due to the fact that each ion experiences a different potential curvature. The second effect reported is amplitude-dependent shifts of the normal-mode frequencies, which is the outcome of elevated anharmonicity or higher excitation amplitude [@Home11]. The issue of anharmonic contributions to the electric potential generated by a linear RF trap is investigated in [@Pedre10; @Pedre18a; @Pedre18b]. Although the anharmonic part is quite comparable in the trap centre region for all the configurations investigated, the behaviour is quite different as one moves further away from the trap centre.
Particle dynamics in the trap. Equations of motion. Stability analysis
----------------------------------------------------------------------
Particle dynamics in an electrodynamic trap is described by a Mathieu equation, which is a particular case of the Hill equation [@McLac64; @Mag66]. Paul traps are able to levitate charged microparticles over a wide range of charge-to-mass ratios, by modifying the a.c. voltage amplitude and frequency [@Major05; @Davis97]. However, the spatial charge potential caused by the presence of other trapped particles changes the limits of the stability diagram depending on the type of particle ordering, particle size, and particle species. The RF and d.c. endcap voltages also influence the shape of the trapped particles. By increasing the RF voltage and the specific charge-to-mass ratio $Q/M$, the operating point of the trap can shift towards the border of the Mathieu stability diagram, case when the particle species can end up by being expelled out of the trap.
A harmonically trapped and laser-cooled ion represents a key system for quantum optics [@Orsz16], used in order to investigate pure quantum phenomena [@Wine13]. Fundamental tests on quantum mechanics have thus been made possible together with applications such as high-resolution spectroscopy, quantum metrology [@Wine11; @Sinc11], quantum computation [@Chiav05; @Bla08; @Bla12; @Har14; @Humb19], studies of quantum chaos and integrability [@Ber00; @Gra13], nonequilibrium quantum dynamics of strongly-coupled many-body systems [@San18; @Kel19], quantum sensing [@Rei18], or the realization of optical clocks with exceptional accuracy [@Van15; @Ved09; @Lud15; @Kel19]. One of the most promising systems that reduces decoherence effects [@Haro13] down to a level where quantum state engineering (state preparation) can be achieved, consists of one or more trapped ions [@Wine13; @Cler16]. In a good approximation the centre of mass (CM) of a trapped ion experiences a harmonic external trapping potential. Hence, the ion trap can be assimilated with a quantum harmonic oscillator [@Kie15].
We will shortly present the classical theoretical model proposed to characterize regular and chaotic orbits for a system of two ions (charged particles) within a 3D Paul trap, depending on the chosen control parameters [@Mih19a]. We consider the case of two ions of mass $M_1$ and $M_2$, levitated within a quadrupole (3D) Paul trap characterized by the force constants $k_1$ and $k_2$, respectively. The reduced equations of motion can be cast into [@Mih19a]:
$$\begin{cases}\label{cla1}
M_1 \ddot x = -k_1 x + a \left( x - y \right) \\
M_2 \ddot y = -k_2 y - a \left( x - y \right) \ , \\
\end{cases}$$
where $a$ represents the constant of force that characterizes the Coulomb repelling between the two trapped ions. The trap control parameters are: $$\label{}
\label{cla2}k_i = \frac{M_i q_i^2 \Omega}8 , \ q_i = 4 \frac {Q_i}{M_i} \frac{V_0}{\left( z_0 \Omega \right)^2} \ , \ i = 1, 2.$$ which implies a time average on the micromotion at frequency $\Omega$, where the higher order terms in the Mathieu equation that describes ion dynamics are discarded. $V_0$ stands for radiofrequency (RF) voltage supplied to the 3D Paul trap electrodes and $Q_i$ represents the ion electric charge. The Coulomb force constant is $ a \equiv{2Q^2}/{r^3} < 0 $, as it results from the series expansion of ${Q^2}/{r^2}$ around an average deviation of the trapped particle with respect to the trap centre $r_0 \equiv \left( x_0 - y_0 \right) < 0 $, determined by the initial conditions. The kinetic and potential energy for the system of two trapped ions are:
$$\label{cla4}T = \frac{M_1 \dot x^2}2 + \frac{M_2 \dot y^2}2 ,\ U = \frac{k_1 x^2}2 + \frac{k_2 y^2}2 + \frac 12 a \left( x - y\right)^2 \ .$$
We choose $$\label{cla6}\frac{k_1}2 = Q_1 \beta_1 , \; x = z_1, y = z_2 \ ,$$ We assume the two ions possess equal electric charges $Q_1 = Q_2$. The Lagrange function is written in the standard form $ L\left( \zeta_i,\dot \zeta_i\right) = T - U $, and the Lagrange equations are $$\label{cla7}\frac d{dt} \left( \frac{\partial L}{\partial \dot \zeta_i} \right) - \frac{\partial L}{\partial \zeta_i} = 0 , \; i = 1,2 \ ,$$ where $\zeta_i$ are the generalized coordinates, and ${\dot \zeta}_i$ represent the generalized velocities. In case of a one-dimensional system of $s$ particles (ions) or for a system with $s$ degrees of freedom, the potential energy can be expressed as: $$\label{cla11}U = \sum_{i = 1}^s \frac{k_i \zeta_i^2}2 + \frac 12 \sum_{1 \leq i < j \leq s} a_{ij} \left( \zeta_i - \zeta_j \right)^2 \ .$$
By performing a series expansion of the Coulomb potential in spherical coordinates and assuming a diluted medium, we can write the interaction potential as $$\label{cla13}V_{int}=\frac 1{4\pi \varepsilon_0} \sum_{1 \leq i < j \leq s} \frac{Q_iQ_j}{\left| \overrightarrow{r_i} - \overrightarrow{r_j} \right| } \ .$$ We denote [@Wine88] $$\label{cla14}k_1 = 2 Q_1 \beta_1 \ , \beta_1 = \frac{4Q_1 V_0^2}{ M_1 \Omega ^2 \xi^4} - \frac{2U_0}{\xi^2} \ ,$$ where $\xi^2 = r_0^2 + 2 z_0^2 $. We choose the case in which $ U_0 = 0 $ (the d.c. trapping voltage) and $r_0$ is negligible, with $r_0$ and $z_0$ the trap semiaxes. The trap control parameters are $ U_0, V_0, \xi_0 $ and $ k_i $ [@Hoff95]. We choose an electric potential $V = \frac 1{|z|} $ which we expand in series around $z_0 > 0$ $$\begin{gathered}
\label{cla18}
V\left( z \right) = \frac 1{z_0} - \frac 1{z_0^2} \left( z - z_0 \right) + \frac 1{z_0^3} \left( z - z_0 \right)^2 - \ldots \end{gathered}$$ with $z - z_0 = x - y$. Then, the expression of the potential energy in eq. \[cla4\] for a system of two ions in the trap can be cast into: $$\label{cla20}U = \frac{k_1 x_1^2}2 + \frac{k_2 x_2^2}2 + \frac 1{4\pi \varepsilon_0} \frac{Q_1 Q_2}{\left| x_1 - x_2\right| } \ ,$$ and it represents the sum of the potential energies of the two ions assimilated with two harmonic oscillators, while the third term in the relationship \[cla20\] describes the Coulomb interaction between the ions. As Hamilton’s principle requires for the potential energy to be minimum in order for the system to be stable, we search for a minimum of $U$. A system of equations results, then after some calculus we obtain $$\label{cla25}
\lambda = \sqrt[3]{\frac 1{4 \pi \varepsilon_0} \frac{k_1^2 k_2^2}{\left( k_1 + k_2 \right)^2}\, Q_1 Q_2} \ ,$$ which enables us to determine the points of minimum $x_1$ and $x_2$ of the system in an equilibrium state. We choose $$\label{cla26}z_0 = x_{1\min} - x_{2\min} = \lambda \frac{k_1 k_2}{k_1 + k_2}$$ and denote $$\label{cla27}z = x_1 - x_2 \ ,\; x_1 = x_{1\min} + x \ ,\; x_2 = x_{2\min} + y \ \Rightarrow z = z_0 + x - y \ .$$ We revert to the expansion of the trap electric potential in eq. (\[cla18\]), then express the potential energy as $$\label{cla31} U = \frac{k_1}2x_{1\min }^2 + \frac{k_2}2x_{2\min }^2 + \frac{k_1}2x^2 + \frac{k_2}2y^2 + \lambda{z_0} + \frac \lambda{z_0} \left( z - z_0 \right)^2 - \ldots \ ,$$ where $x_{1\min} = \lambda / k_1$, $x_{2\min} = \lambda / k_2$. From eqs. (\[cla4\]) we obtain $a/2 = \lambda / z_0$. According to the principle of least action the system is stable when the potential energy is minimum, which means that the Lagrange function reaches a maximum value.
### Solutions of the equations of motion
We search solutions such as $x = A \sin \omega t$ and $y = B \sin \omega t$, which we introduce in the equations of motion \[cla1\]. The resulting determinant of the system of equations is: $$\label{cla35}\left|
\begin{array}{cc}
a - k_1 + M_1 \omega^2 & -a \\
-a & a - k_2 + M_2 \omega^2
\end{array}
\right| =0 \ .$$ The stability requirement for the system demands for the determinant of this equation to be zero $$\label{cla36}\left( a - k_1 + M_1 \omega^2 \right) \left( a - k_2 + M_2 \omega^2 \right) - a^2 = 0 \ .$$ The discriminant of eq. (\[cla36\]) can be cast as $$\label{cla40} \Delta = \left[ M_1 \left( a - k_2 \right) + M_2 \left( a - k_1 \right) \right]^2 + 4 M_1 M_2 a^2$$ As it is well known a liniar homogeneous system of equations admits nonzero solutions only if the determinant associated to the system is zero. Hence, the solution of eq. (\[cla36\]) can be expressed as: $$\label{cla41}\omega_{1,2} = \frac{M_1 \left( k_2 - a \right) + M_2 \left( k_1 - a \right) \pm \sqrt{\Delta}}{2 M_1 M_2}$$ A solution for this system would be
\[subeqcla42\] $$\begin{aligned}
x = C_1 \sin \left( \omega_1 t + \varphi_1 \right) + C_2 \sin \left( \omega_2 t + \varphi_2 \right) \label{subeqcla42a} \ ,\\
y = C_3 \sin \left( \omega_1 t + \varphi_3 \right) + C_4 \sin \left( \omega_2t + \varphi_4 \right) \label{subeqcla42b} \ .
\end{aligned}$$
Such a solution represents in fact a superposition of two oscillations with secular frequencies $\omega_1$ and $\omega_2$, that stand for the eigenfrequencies of the system investigated. $\varphi_1 \div \varphi_4$ stand for the initial phases, while $C_1 \div C_4$ are a set of constants. Assuming that $a\ll k_{1,2}$ (a requirement that is frequently met in practice) in eq. (\[cla36\]), we can define the condition of strong coupling as $$\label{coup1}\left| {\dfrac{a}{k_i}} \right| \gg \left| \dfrac{M_1 - M_2}{M_2} \right| \ ,$$ where the modes of oscillation are $$\label{coup2} \omega_1^2 = \dfrac{1}{2} \left( \dfrac{k_1}{M_1} + \dfrac{k_2}{M_2} \right) , \;\ \omega_2^2 = \dfrac{1}{2} \left( \dfrac{k_1 + 2a}{M_1} + \dfrac{k_2 + 2a}{M_2} \right) \ .$$
By studying the phase relations between the solutions of eq. (\[cla1\]) we notice that the $\omega_1$ mode describes a translation of the two ions where the distance $r_0$ between them is constant, while the Coulomb repulsion between them remains unchanged, a fact that is indicated by the absence of the term $a$ in the first eq. (\[coup2\]). This mode of translation generates an axial current which can be detected. In case of the $\omega_2$ mode the distance between the ions increases and then decreases around the location of a fixed centre of mass (CM), situation when the electric current and implicitly the signal are zero. Although this mode can not be electronically detected, optical detection is possible. Therefore, for collective modes of motion the theory predicts that only a peak of the mass will be detected corresponding to the average mass of the two ions. In case of weak coupling the inequality in eq. (\[coup1\]) reverses, while from eq. (\[cla41\]) we infer $$\label{coup4} \omega_{1, 2}^2 = \left( k_{1, 2} + a \right) /M_{1, 2} \ ,$$ that is every mode of the motion corresponds to a single mass, and the resonance is shifted by the term $a$. Moreover, within the limit $m_1 = m_2$ the strong coupling requirement in eq. (\[coup1\]) is always satisfied no matter how weak the Coulomb coupling, which renders the weak coupling condition inapplicable in practice.
Fig. \[pha1\] (a) illustrates the phase portrait for a system of two ions (particles), when the ratio between the frequencies is $\omega_1 / \omega_2 = 1.5 / 2$. The theory of differential equations states that when the ratio of the eigenfrequencies is a rational number $\omega_1 / \omega_2 \in {\mathbb Q}$, the solutions of the equations of motion (\[subeqcla42\]) are periodic trajectories. Ion dynamics is then characterized as stable. If the ratio between the eigenfrequencies is an irrational number $\omega_1 / \omega_2 \notin {\mathbb Q}$, iterative rotations at irrational angles occur around a point [@Gutz90]. Such iterative irrational rotations are called [*ergodic*]{}, which is stated by the theorem of Weyl [@Buni00; @Berg03]. The angular frequencies between points located on the circles take discrete values. Fig. \[pha1\] (b) and (c) illustrate the ergodic dynamics [@Kras16] of the system under investigation. In this case ion dynamics is unstable while exhibiting traces of chaos. Fig. \[pha2\] (a) - (b) illustrate the onset of chaos or presence of chaos. According to Gutzwiller: “Ergodicity implies that the phase space $M$ cannot be decomposed into subsets with non-vanishing measure each of which remains invariant” [@Gutz90].
Dynamics of a system of two ions in a Paul trap {#ham}
-----------------------------------------------
### Hamilton function associated to the system. Hessian matrix
The relative motion of two ions of equal electric charges can be expressed as [@Blu89; @Far94a; @Far94b; @Blu95]
$$\label{ham1}
\frac{d^2}{dt^2} \begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}
+ \left[ a + 2q \cos\left( 2 t \right) \right] \begin{bmatrix}
x \\
y \\
-2 z \\
\end{bmatrix} = \frac{\mu_x^2}{|{\bf r}|^3} \begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix} \ ,$$
where $\mu_x = \sqrt{ a + \frac 12 q^2 }$ represents the radial secular characteristic frequency, while $a$ and $q$ stand for the adimensional trap parameters in the Mathieu equation, namely
$$a = \frac{8 Q U_{dc}}{M \Omega^2 \left( r_0^2 + 2z_0^2\right)} \ , \;\; q = \frac{4 Q U_{ac}}{M \Omega^2 \left( r_0^2 + 2 z_0^2\right)} \ .$$
$U_{dc}$ and $U_{ac}$ stand for the d.c and RF trap voltage respectively, $Q$ is the particle electric charge, $\Omega$ is the RF voltage frequency, while $r_0$ and $z_0$ represent the trap radial and axial dimensions. For $ a, q \ll 1$, eq. (\[ham1\]) can be investigated in the pseudopotential approximation, in which the motion is averaged over the driving terms that induce a high frequency oscillating motion (micromotion) [@Far94a]. Then, an autonomous Hamilton function results which can be expressed in scaled cylindrical coordinates $\left( \rho, \phi, z\right)$ as
$$\label{ham2} H = \frac 12 \left( {p_\rho^2} + {p_z^2} \right) + U\left( \rho, z \right) \ ,$$
where $$\label{ham3} U\left( \rho, z\right) = \frac 12 \left( \rho^2 + \lambda^2 z^2 \right) + \frac{\nu^2}{2 \rho^2} + \frac 1r$$ and $r = \sqrt{\rho^2 + z^2} \ , \; \lambda = \mu_z/\mu_x \ , \; \mu_z = \sqrt{2\left( q^2 - a\right) }$. $\nu $ is the scaled axial ($z$) component of the angular momentum $L_z$ and it represents a constant of motion, while $\mu_z$ represents the second secular frequency [@Far94b]. Both $\lambda$ and $\nu$ are positive control parameters. For arbitrary $\nu $ and $\lambda = 1/2, 1, 2$, eq. (\[ham3\]) is integrable and even separable except the case when $\lambda = 1/2$, and $ \left| \nu \right| > 0$ $\left( \nu \neq 0 \right) $. After some calculus we obtain
$$\label{ham4}\lambda^2 = 4 \frac{q^2 - a}{q^2 + 2 a}, \; \; \nu^2 = \frac{2L_z^2}{q^2 + 2a} \ ,$$
and we distinguish between three cases: (a) $\lambda = 1/2$, (b) $\lambda = 1$, and (c) $\lambda = 2$ [@Blu95; @Moore93]. By using the Morse theory [@Chang93; @Nico11], the critical points of the $U$ potential can be determined as solutions of the system of equations:
\[ham5\] $$\frac{\partial U}{\partial \rho} = \rho - \frac{\nu^2}{\rho^3} - \frac 1{r^2}\frac{\rho}r = 0 \ , \frac{\partial U}{\partial z} = \lambda^2 z - \frac 1{r^2} \frac zr = 0 \ ,$$
with $\partial r/ \partial \rho = \rho /r $ and $\partial r/ \partial z = z / r $. Further on, we find that $$\label{ham6} z \left( \lambda^2 - \frac 1{r^3} \right) = 0 \ ,$$ which obviously leads to two possible cases: $z = 0$, and $ r^3 = 1/ \lambda^2$. An extended treatment of the problem is found in [@Mih19a]. By studying the sign of the Hessian matrix eigenvalues one can analyze and classify the critical points, as well as the system stability. The degenerate critical points (characterized by $\det {\mathsf H} = 0 $) compose the bifurcation set, whose image in the control parameter space (namely the $\nu - \lambda $ plan) establishes the catastrophe set of equations which defines the separatrix:
$$\label{ham21}\nu = \sqrt{\lambda^{-8/3} - \lambda^{-2/3}}\ \mbox{ or } \ \lambda = 0$$
Fig. \[bifurc\] shows the bifurcation diagram for the system of two ions levitated in a Paul trap. The relative motion of the ions is governed by the Hamiltonian function described by eqs. (\[ham2\]) and (\[ham3\]). The diagram illustrates the stability and instability regions where the ion dynamics is integrable and non-integrable, respectively. It can be noticed that chaos prevails. Ion dynamics is integrable when $\lambda = 0.5 , \ \lambda = 1 $, and $\lambda = 2$.
![The bifurcation set for a system of two ions confined in a Paul trap. Picture reproduced from [@Mih19a].[]{data-label="bifurc"}](bif2.pdf)
We resume by emphasizing that when the electric potential is time independent or in the case of the pseudopotential approximation for RF traps, an autonomous Hamilton function can be associated to the ion system. In that case, the family of electric potentials can be classified according to the Morse theory, the bifurcation theory [@Dessup16; @Alb18], the ergodic theory and the catastrophe theory [@Gutz90; @Hilb10; @Stee14]. This classification qualitatively determines the structure of the phase space, the minimum, maximum, and critical points, the periodic orbits and the separatrices. Besides the case of axial motion, ion (particle) dynamics in a nonlinear trap is generally non-integrable [@Mih02] and chaotic orbits prevail in the phase portrait, except specific neighborhoods around the points of minimum which consist of regular orbits according to the Kolmogorov-Arnold-Moser (KAM) theory. Those points of minimum exhibit a particular interest with an aim to implement quantum logic operations for systems of trapped ions. Dynamical systems of trapped ions (particles) can be investigated by using the dynamical group theory to describe regular and chaotic dynamics of the system [@Mih18; @Mih17], depending on the control parameters determined by the RF trapping field and the trap geometry [@Mih19a].
Trap geometries. Nonlinear traps
--------------------------------
One of the major concerns for scientists was the design of ion traps with considerably reduced dimensions, required by applications such as optical clocks [@Pyka14; @Kel19; @Dele18]. Downsizing trap dimensions comes at the expense of some unwanted drawbacks, such as increased sensitivity to trap design imperfections or the occurrence of stray potentials [@Champ01].
An experiment focused on levitating single ions in a novel RF QIT with spherical shape is described in [@Nosh14]. By optimizing the spherical ion trap (SIT) geometry, nonlinearity is sensibly reduced by eliminating the electric octupole moment.
A novel miniature Paul ion trap design is reported in [@Kassa17] that exhibits an integrated optical fibre cavity. Optimal coupling of the ions to the cavity mode is achieved by moving the ion with respect to the cavity mode. The trap features a novel design that includes extra electrodes on which additional RF voltages are applied to fully control the pseudopotential minimum in three dimensions. This method lifts the need to use 3D translation stages for moving the fibre cavity with respect to the ion and achieves high integrability, mechanical rigidity and scalability. As it does not rely on modifying the capacitive load of the trap, the method leads to precise control of the pseudopotential minimum [@Kassa17].
Optical levitation and optical tweezing {#OpticLev}
---------------------------------------
It seems that the radiation pressure concept was probably first suggested by Kepler in 1619 when he advanced this hypothesis to elucidate why comet tails always point away from the sun [@Bra89]. A fundamental concept that emerged from the work of Maxwell was the idea to use electromagnetic radiation to manipulate matter using light pressure, or what is commonly known as radiation pressure. The first step is the pioneering work of Lebedev at the beginning of the 1900s, where a proof of concept is made with respect to the pressure force exerted by the electromagnetic radiation [@Lebe1901; @Bere16]. Optical trapping was first reported in 1970 when Arthur Ashkin demonstrated that micron-sized dielectric particles could be accelerated and manipulated (trapped) in stable optical potentials produced by counter-propagating laser beams [@Ash70]. Over the next 15 years Ashkin and others experimented with various optical systems that could trap microscale objects, as well as atoms and molecules. In 1986 Ashkin and his colleagues developed a single-beam optical trap [@Ash86] that was later given the name of [*optical tweezers*]{} [@Gies15]. This beam is so sharply focused that it generates an optical gradient force that balances the radiation pressure and establishes a stable optical trap in three dimensions (3D) [@Leh15]. A simplified sketch of an optical trap is given in Fig. \[OptTrap\].
![Simplified view of an optical trap. Photo reproduced from Block lab at Stanford University https://blocklab.stanford.edu/optical\_tweezers.html[]{data-label="OptTrap"}](OpticalTrap.jpg)
Until then, noncontact and noninvasive optical trapping has been mainly applied to manipulate transparent materials such as biological cells, microorganisms, or polystyrene latex spheres [@Sato96]. Many experiments have been performed in an attempt to reveal the features of optical trapping and its applicability to less transparent materials. For instance, optical trapping of micron-sized [@Sato94] and nanometre-sized [@Svobo94] metal particles has been demonstrated in the 1990s. The optical forces generated are in the femto (fN) to tenths of picoNewton (pN) range, which renders them extremely suited to investigate physics at the mesoscopics scale [@Dhola08]. Optical tweezers represent an extremely sensible nano-manipulation technique.
Optical levitation has been successfully implemented in two size ranges: (a) in the subnanometre scale where light–matter mechanical coupling enables cooling of atoms, ions and molecules down to the quantum limit, and (b) in the micron scale where the momentum transfer resulting from light scattering allows manipulation of microscopic objects such as cells. Nevertheless, it becomes problematic when one tries to employ these techniques to the intermediate — nanoscale range, that encompasses structures such as quantum dots, nanowires, nanotubes, graphene and 2D crystals. These structures are of critical importance for nanomaterials-based applications. Recently, new approaches have been developed and tested with respect to trapping of plasmonic nanoparticles, semiconductor nanowires and carbon nanostructures. A detailed review on recent advances in state-of-the-art optical trapping at the nanoscale is performed in [@Mara13]. The review focuses on some of the most remarkable advances, among which we can enumerate controlled manipulation and aggregation of single and multiple nanostructures, force measurement under conditions of femtonewton resolution, and biosensors. Thus, the ability to manipulate nanoparticles is vital in nanoscale science and technology. As object dimensions scale down to the sub-10 nm regime, it imposes a great challenge for the conventional optical tweezers. This is why a lot of effort has been invested to explore alternative manipulation methods including using nanostructures, electron beams, scanning probes, etc. [@Zheng13].
Computational modelling can be applied for example to simulate particle dynamics in optical tweezers. This approach is effective when dealing with systems that exhibit many degrees of freedom, or to perform experimental simulations. Ion dynamics simulation in optical traps is centered on modelling of the optical force, but to enhance precision it should also consider non-optical forces among which the most prominent are the Brownian motion [@Chav11] and viscous drag. A review that focuses on the theory and practical principles of simulation of optical tweezers, while also discussing the challenges and mitigation of the known issues can be found at [@Bui17].
A comprehensive review of optical trapping and manipulation techniques for single particles in air, including different experimental setups and applications can be found in the paper of Gong [*et al*]{} [@Gong18]. An experimental method that relies on employing dynamic split-lens configurations, with an aim to achieve trapping and spatial control of microparticles by means of the photophoretic force, is reported in [@Liz18].
It is known that nonspherical particles can severely influence processes that occur in the atmosphere such as radiative forcing, photochemistry, new particle formation and phase transitions [@Rama18]. Hence, in order to further investigate and characterize the properties associated with these particles and to quantify their effect on global climate, it is of utmost importance to perform measurements on single particles. Such step requires further refining of the experimental setup and a more detailed understanding of the physical processes associated with optical levitation. An investigation of optical trapping of nonspherical particles in air is performed in [@David18], where an analytical model aimed at better explaining the physical mechanisms is suggested by the authors. The 6D motion of a trapped peanut-shaped particle (3D for translation and 3D for rotation) is analyzed by means of a holographic microscope. In addition, optical forces and torques exerted by the optical trap on the peanut-shaped particle are calculated using FDTD simulations. A good agreement with the experimental results is obtained for the particle motion, while there are still some specific aspects of the particle motion that cannot be explained [@David18].
Section \[RamanFl\] refers to Raman spectroscopy and optical trapping, with respect to the state-of-the-art experimental techniques and methods. A first experimental demonstration of an optically levitated Yttrium Iron Garnet(YIG) nanoparticle in both air and vacuum is discussed in [@Seb19], as well as a scheme to achieve ground state cooling of the translational motion. The theoretical cooling scheme suggested is based on the sympathetic cooling of a ferromagnetic YIG nanosphere with a spin-polarized atomic gas. Finally, late research and results in the field of optical trapping of ions is supplied in [@Karp19].
Acoustic levitation {#AcoustLev}
-------------------
Both electromagnetic and acoustic waves exert radiation forces upon an obstacle placed in the path of the wave [@Bra89]. The forces are in a direct relationship with the mean energy density of the wave motion [@Borg53; @Live81]. Acoustic levitation uses the acoustic radiation pressure associated with plane acoustic (sound) waves to confine solids, fluids, and heavy gases. Acoustic standing waves have been widely used to trap, pattern, and manipulate particles. An overview of dust acoustic waves can be found in [@Chop14]. Nevertheless, there is still one major obstacle to cross: there is little knowledge about force conditions on particles which mainly include acoustic radiation force (ARF) and acoustic streaming (AS) [@Sepe15; @Liu17]. Recently, a numerical simulation model has been suggested with an aim to evaluate the acoustic radiation force and streaming for spherical objects located in the path of a standing wave. The model was verified by comparing the results with those obtained from the analytical solution, proposed by Doinikov [@Doi94]. Unlike analytical solutions, the proposed numerical scheme is applicable to the case of multiple spheres in a viscous fluid [@Sepe15].
Acoustic levitation uses acoustic radiation pressure forces to compensate the gravity force and levitate objects in air. Although acoustic levitation was firstly investigated almost a century ago, levitation techniques have been limited until recently to particles that are much smaller than the acoustic wavelength. Late experiments show that acoustic standing waves can be employed to steady levitate an object whose dimensions are larger with respect to the acoustic wavelength, under SATP conditions. Levitation of a heavy object (that weighs 2.3 g) is demonstrated by using a setup consisting of two 25 kHz ultrasonic Langevin transducers connected to an aluminium plate [@Andra17]. The sound wave emitted by the experimental setup generates both a vertical acoustic radiation force that compensates gravity, and a sideways restoring force that provides horizontal stability to the levitated object. An analysis of the levitation stability is achieved by means of using a Finite Element Method (FEM) to determine the acoustic radiation force exerted upon on the object. Consequently, recent advances in acoustic levitation allow not only suspending, but also rotating and translating objects in space (3D) [@Andra17; @Andra18].
In-depth informations about the acoustic radiation pressure and its applications can be found in Ref. [@Borg53; @Live81; @Andra18]. Investigation of force conditions on micrometer size polystyrene microspheres in acoustic standing wave fields is performed in [@Liu17], by using the COMSOL Mutiphysics particle tracing module. The velocity of particle movement is experimentally measured by employing particle imaging velocimetry (PIV). Use of acoustic levitation to confine microparticles in an electrodynamic trap is presented in Section \[AcoustWave\].
Instability in electrodynamic traps. Solutions
----------------------------------------------
For high values of the trapping voltage frequency, microparticles are confined in well defined regions in space and perform oscillations around points of dynamic equilibrium [@Lan12]. The particles located close to the trap centre can be practically considered as motionless. The amplitude of oscillation for trapped particles increases as they are located farther with respect to the trap centre. In addition, particle trajectories can be used to map the electric field within the trap. When the a.c. voltage frequency is reduced, particles that remain confined start to move chaotically. A structure consisting of charged particles that strongly interact via the Coulomb force is called [*quasicrystal*]{} [@Boll84]. Transition of particles from a regular motion towards a chaotic regime in case of a low frequency of the a.c. trapping voltage, is called [*melting*]{} [@Wuerk59]. A Coulomb cloud of atomic ions is obtained in [@Kja05].
Storage instabilities of electrons levitated in a Penning trap at low magnetic fields are determined in [@Paas03]. The occurrence of these instabilities is attributed to the presence of higher order static perturbations in the trapping potential. The stability properties of a linear RF ion trap with cylindrical electrodes are explored in [@Drako06]. Instabilities analogous to those experimentally reported in case of 3D ion traps are identified for an ideal linear trap. It is demonstrated they are caused by higher order contributions in the series expansion of the trapping electric potential, which deviates from an ideal one.
Experimental investigations of the stability region of a Paul trap with very strong damping demonstrate that the adimensional trapping parameter $q$ can reach values up to 25 times greater than the maximum value in absence of damping [@Nas01]. The considerable increase in the size of the stability region that is observed at high pressures can be explained by the numerical solution of the Mathieu differential equation with damping term. Nonlinear dynamics in Penning traps with hexapole and octupole terms of the electric potential is explored in [@Salas02; @Lara04].
To confine charged dust particles in a dynamical trap under SATP conditions, friction between charged particles and air must be considered. Moreover, the trap parameters must be chosen carefully because the stability region is shifted [@Major05]. One of the most difficult tasks to perform in experiments consists in charging the aerosol particles with electric charges that have the same sign [@Stern01]. In order to achieve similar charge-to-mass ratios, electrically charged oil drops have been used [@Rob99]. The drops assemble into a Coulomb structure under SATP conditions. To confine charged particles of silicon carbide (SiC) and alumina (Al$_2$O$_3$), classical quadrupole traps [@Ghe95b; @Ghe98; @Sto01] and multi-electrode traps consisting of 8, 12 or 16 cylindrical brass electrodes (with 3 or 4 mm diameter) can be used [@Mih16a; @Mih16b]. Experiments confirm that a 12 electrode trap geometry yields to higher stability and smaller amplitude of oscillation of the levitated microparticles. Confinement conditions prove to be very sensitive to the trap geometry. Quadrupole traps are used more frequently, as they provide better control of the trapping parameters. The radius of the multipole traps designed and tested [@Mih16a; @Mih16b; @Mih08] ranges between $1 \div 1.5$ cm. Negatively charged particles of Al$_2$O$_3$ with diameters between $60 \div 200$ microns (with mass of $ 4.18 \times 10^{-10}$ kg and $1.55 \times 10^{-8}$ kg, respectively) are located close to the trap axis line if their number is small. If the number of particles is larger they start to move chaotically yielding a dusty cloud. To estimate the electric charge, the particle cloud is subjected to the action of acoustic waves that induce parametric oscillations of the particles [@Sto11; @Sto08]. The value of the experimentally determined charge-to-mass ratio is around $5.4 \times 10^{-4}$ C/kg.
Ion and particle traps represent multiskilled instruments to explore nonequilibrium statistical physics, where dissipation and nonlinearity can be very well controlled. In case of ion chains dissipation is achieved by means of laser heating and cooling, while nonlinearity is the outcome of trap anharmonicity and beam shaping. Ion dynamics is governed by an equation similar to the complex Ginzburg-Landau equation. As an exotic feature, the system can be described as both oscillatory and excitable at the same time. The experiment is presented in Ref. [@Lee11], while the approach also allows controllable experiments with noise and quenched disorder.
A theoretical model used to describe the ion transient response to a dipolar a.c. excitation in a QIT is presented in [@Xu11], where high frequency ion motion components were also considered.
The electric potential is a pure quadrupole one in case of an ideal ion trap, and ion dynamics is described by a Hill (Mathieu) equation. Real traps used in experiments do not generate a quadrupole field due to geometric imperfections (electrode truncation) or misalignment of electrodes. The nonlinearity that real traps exhibit is the outcome of weak multipole fields (e.g., hexapole, octupole, decapole, and higher order fields). Ion dynamics in such a trap is governed by a nonlinear Mathieu equation that can not be solved analytically. The theoretical lines of instabilities in the first region of stability of the Paul trap for perturbations of order $n = 3$ to $n = 8$ are illustrated in Fig .\[InstabPaul\].
In Ref. [@Doro12] a technique is used to calculate the axial secular frequencies of a nonlinear ion trap with hexapole, octupole and decapole terms of the electric potential, based on the modified Lindstedt–Poincar[é]{} method. The equations of ion motion in the resulting effective potential are similar to a Duffing equation [@Kova11].
The harmonic balance method was introduced to examine the coupling effects induced by hexapole and octupole fields on ion motion in a QIT. Ion motion characteristics and buffer gas damping effects have been investigated in presence of both hexapole and octupole fields [@Wang13]. It was suggested that hexapole fields yield to higher impact on ion motion centre displacement, while octupole fields prevail on ion secular frequency shift. In addition, the nonlinear features caused by hexapole and octupole fields could add or cancel each other. The method was further used to devise an ion trap with improved performance, based on a particular combination of hexapole and octupole fields [@Wang13].
A detailed analysis based on numerical simulations of ion dynamics for a nonlinear RF (Paul) trap with terms of the electric potential up to the order 10, is performed in [@Herba14]. The Hill-Mathieu equations of motion that characterize the associated dynamics are numerically solved by employing the power series method. The stability diagram is qualitatively discussed and buffer gas cooling is implemented using a Monte Carlo method that takes dipole excitation in consideration. The method was then demonstrated in case of a real trap, and a good agreement with experiments and numerical simulations was obtained. RF ion traps encompass complex electric field shape which yields to complex ion motion. The paper of Li [@Li17] reports on SIMION simulations that show classical chaotic behaviour of ions levitated in a toroidal ion trap. Fractal-like patterns were illustrated in a series of zoomed-in regions of the stability diagram.
Late papers investigate the dynamics of a collection of charged particles (ions) in a dual-frequency Paul trap [@Saxe18; @Foot18]. In Ref. [@Saxe18] the authors infer the analytical expressions for the single particle trajectory and the plasma distribution function assuming a Tsallis statistics, by emphasizing on the difference between the trap configuration used and a conventional Paul trap. Use of a secondary RF frequency in a dual-frequency Paul trap allows one to modify the spatial extent of charged particles confinement. The plasma distribution function and temperature are periodic, provided that the ratio of the eigenfrequencies is a rational number $\omega_1 / \omega_2 \in {\mathbb Q}$, case when the solutions of the equations of motion are periodic trajectories [@Mih19a]. Then, the resulting period of plasma oscillation is given by the Least Common Multiple (LCM) of the time periods corresponding to the two driving frequencies and linear combinations of them. The plasma temperature is spatially varying and the time-averaged plasma distribution has been found to be double humped beyond a certain spatial threshold, indicating the presence of certain instabilities [@Syr19a; @Syr19b]. The effect of dual frequency in a Paul trap on the energy level shifts of the atomic orbitals is investigated, and the uncertainties in second order Doppler and Stark shift are found to be of the same order as that of a single-frequency Paul trap [@Saxe18].
In addition, it was demonstrated that the effective rotational potential can be used to describe dynamics of various diatomic particles with different centres of mass (CM) and charges, levitated in a plane quadrupole RF trap. By comparison between “the model of pseudopotential for localization of a single ion and the proposed model of the effective rotational potential for diatomic structure”, one can determine “additional positions of quasi-equilibrium for the CM of the diatomic particle and orientation angle of the molecule” [@Vasi19].
Electrodynamic (ED) levitation of one or a few charged droplets using Paul traps opens new pathways towards explaining the Rayleigh instability and the interplay between various inter-particle forces that yield to pattern formation [@Blu89; @Lee11]. Very recent papers [@Singh17; @Singh18] focus on developing a theory for two-dimensional patterns formed during levitation of two to a few droplets in an EDB. The theory is based on an extension of the classical Dehmelt approximation to interacting particle systems. Among other original contributions, the theory demonstrates solutions to the inter-particle separations, secular frequencies, stabilities of two-drop systems, and collapse of droplets onto the $X-Y$ plane. The theory is also applied to predict size and shape of closed structures that occur for few-drop systems.
Spontaneous wave-function collapse models have been proposed in an attempt to reconcile deterministic quantum mechanics with the nonlinearity and stochasticity of the measurement operation. According to these models random collapses in space of the wave function of any system occur spontaneously, leading to a progressive spatial localization free of the measurement process. Noninterferometric tests of spontaneous collapse models for a nanoparticle levitated in a Paul trap, in ultrahigh cryogenic vacuum, are reported in [@Vin19].
Chaos
-----
Quadrupole ion traps (QITs) [@Ghosh95; @Major05; @March09] have proven to be extremely versatile tools in atomic physics, high-precision spectroscopy [@Paul90], quantum metrology [@Wine11], physics of quantum information, studies of chaos and integrability for dynamical systems [@Aku14; @Ber00; @Gra13; @Blu89; @Blu95; @Blu98; @Rozh17], mass spectrometry [@March05], quantum optics [@Haro13; @Fox06], in performing fundamental tests on quantum mechanics concepts [@Orsz16; @Leibf03] and investigations of non-neutral plasmas [@Werth09; @Haro13; @Dub99; @Wine88; @Meni07]. Deterministic chaos deals with long-time evolution of a system in time. A dynamical system represents a system that evolves in time. Chaos is related to the study of the dynamical systems theory [@Gutz90; @Meiss17] or nonlinear dynamics [@Mih10b; @Blu95; @Stro15; @Schne17]. Dynamical systems can be either conservative, in case when no friction is present and the system does not give up energy in time, or dissipative when we frequently encounter a behaviour called [*limit cycle*]{}, in which the system approaches some asymptotic or limiting condition in time [@Nay08; @Gutz90; @Hilb10]. Under certain conditions, the asymptotic or limiting state is where chaos occurs. We can ascertain that chaos develops in deterministic, nonlinear, dynamical systems. Other chaos-related geometric objects, such as the boundary between periodic and chaotic motions in phase space, may also exhibit fractal properties.
The dynamics associated to a single charged particle in a Paul trap, in presence of a damping force, has been theoretically and experimentally investigated in [@Izma95; @Hase95]. Izmailov investigates damped microparticle motion in presence of combined periodic parametric and random external excitations, using the singular perturbation theory [@Izma95]. The modified stability diagrams in the parameter space have been calculated in [@Hase95], showing that the stable regions in the $a - q$ parameter plane are not only enlarged but also shifted. It was also emphasized that in certain cases the damping force may induce instability in the particle dynamics. Later, Sevugarajan and Menon developed a model relating perturbation in the ion axial secular frequency to geometric aberration, space charge, dipolar excitation, and collisional damping in nonlinear Paul trap mass spectrometers [@Sevu00]. A multipole superposition model that considers both hexapole and octupole superposition has been introduced to account for field inhomogeneities. Dipolar excitation and damping were also considered in the equation of ion motion. The perturbed secular frequency of the ion was inferred by use of a modified Lindstedt–Poincar[é]{} perturbation technique. It was also showed that the shift in ion secular frequency with the axial distance from the centre of the trap exhibits quadratic variation. Another interesting paper of Sevugarajan and Menon investigates the role of field inhomogeneities in altering stability boundaries in nonlinear Paul traps mass spectrometers, taking into account higher order terms in the equation of motion [@Sevu02]. The contribution of hexapole and octupole superposition in shifting the stable trapping region, as well as ion secular frequencies in nonlinear Paul traps have also been explored. The paper of Zhou [*et al*]{} [@Zhou10] uses the Poincar[é]{}-Lighthill-Kuo (PLK) method to infer an analytical expression on the stability boundary and ion trajectory. A multipole superposition model mainly including octupole component is proposed by the authors, with an aim to model the electric field inhomogeneities. Using the method described, both the dynamic shift and secular frequency of ions have been expanded to asymptotic series.
A nonlinear chaotic system, the parametrically kicked nonlinear oscillator, may be realized in the dynamics of a trapped, laser-cooled ion, interacting with a sequence of standing wave pulses [@Mih10b; @Ake10; @Bres97; @Adam01; @Qing17]. RF ion traps are usually associated with complex electric field shape and sensibly intricate ion motion. Classical chaotic behavior of ions in a toroidal ion trap is explored in [@Li17] by employing SIMION simulations. It is demonstrated that chaotic motion occurs due to the presence of nonlinear terms of the electric fields generated by the trap electrodes.
The quasistationary distribution of Floquet-state occupation probabilities for a parametrically driven harmonic oscillator coupled to a thermal bath is investigated in [@Dier19]. Late investigations on the stochastic dynamics of a particle levitated in a periodically driven potential are presented in [@Lan19a]. Results show that in case of atomic ions confined in RF Paul traps, noise heating and laser cooling typically respond slowly with respect to the unperturbed motion. These stochastic processes can be characterized in terms of a probability distribution defined over the action variables. In addition, Ref. [@Lan19a] presents a semiclassical theory of low-saturation laser cooling applicable from the limit of low-amplitude motion to the extent of large-amplitude motion, that fully describes the time-dependent and anharmonic trap. Thus, a detailed study of the stochastic dynamics of a single ion is achieved. Another paper investigates a single atomic ion confined in a time-dependent periodic anharmonic potential, in presence of stochastic laser cooling [@Lan19c]. It is demonstrated how the competition between the energy gain from the time-dependent drive and damping leads to the stabilization of stochastic limit cycles. Such distinct nonequilibrium behavior can be observed in experimental RF traps loaded with laser-cooled ions.
Further on, we present ion (particle) dynamics in a nonlinear quadrupole Paul trap with octupole anharmonicity. The system is better characterized as dissipative. Particle dynamics is described by a nonlinear Mathieu equation [@Brou11; @Rand16]. All perturbing contributions such as: (i) damping, (ii) multipole terms of the potential, and (iii) harmonic excitation force, are considered in this approach. In order to complicate the picture and approximate real conditions it is also considered that the particle (ion) undergoes interaction with a laser field [@Mih10b]. The resultant equation of motion is shown to be similar to a perturbed Duffing-type equation, which is a generalization of the linear differential equation that describes damped and forced harmonic motion [@Kova11].
### Dynamics of a particle confined in a nonlinear trap
We investigate the dynamics of a charged particle (ion) confined in a quadrupole nonlinear Paul trap [@Lan12], which we treat as a time-periodic differential dynamical system. Dissipation in such system is very low which leads to a number of interesting phenomena. The equation of motion along the $x$-direction for a particle of electrical charge $Q$ and mass $M$, which undergoes interaction with a laser field in a quartic potential $V\left( u \right) = \mu u^4$, $\mu > 0$ in presence of damping [@Mih10b], can be expressed as
$$\label{cha1}
\frac{d^2u}{d\tau^2} + \gamma \frac{du}{d\tau} + \left[ a - 2q\cos\left(2 \tau \right)\right] u + \mu u^3 + \alpha \sin u = F \cos \left(\omega_0 t\right) \,$$
where $u = kx$, with $k$ a constant, $\tau = \Omega t /2$, $\alpha = 2k^2\Omega_0 \cos \theta/M\Omega^2$, and $\gamma$, $\mu$ stand for the damping and anharmonicity coefficient, respectively. The adimensional parameters are
$$\label{cha2}
a = \frac{-8QU_0}{M\Omega^2 d} , \;\; q = \frac{4qV_0}{M\Omega^2 d} , \;\; d = r_0^2 + 2 z_0^2 \ ,$$
where the geometrical parameters $r_0$ and $z_0$ represent the trap semiaxes. Generally, for a typical Paul trap $a = 0.1$ and $q = 0.7$. The frequency of the applied a.c. voltage is denoted by $\Omega$, $U_0$ and $V_0$ are the static and time-varying trapping voltages, whereas $\Omega_0$ is the Rabi frequency for the ion–laser interaction, and $\cos \theta$ is the expectation value of the $x$ projection spin operator for the two-level system with respect to a Bloch coherent state [@Dodon02; @Gaze09; @Gryn10; @Combe12; @Ortiz19; @Zela19; @Cruz19]. The expression $F \cos \left( \omega_0 t\right)$ stands for the driving force, an external excitation at frequency $\omega_0$.
Equation (\[cha1\]) can also be regarded in a good approximation as Newton’s law for a particle located in a double-well potential. The force $F \cos\left( \omega t\right)$ represents an inertial force that arises from the oscillation of the coordinate system. The mathematical analysis of equation (\[cha1\]) (which is dimensionless) requires using some techniques from the global bifurcation theory [@Nay08; @Gutz90; @Alb18; @Hilb10; @Stee14; @Stro15]. We investigate equation (\[cha1\]) by means of numerical simulations, in an attempt to gain more insight on the behaviour of such kind of strongly nonlinear systems. The dynamical behaviour of the equation of motion is studied numerically by varying the damping and the driving frequency parameters, as well as the amplitude parameter. We finally discuss the possibility of observing chaos in such a nonlinear system [@Bres97; @Qing17; @Gard97; @Gard98]. Chaotic regions in the parameter space can be identified by means of Poincar[é]{} sections [@Nay08; @Gutz90; @Buni00; @Hilb10; @Stee14; @Stro15; @Cvita19], as illustrated in [@Rob18] where integrable and chaotic motion of a single ion is investigated for five-wire surface-electrode Paul traps.
### Nonlinear parametric oscillator in a Paul trap. Phase-space orbits, Poincar[é]{} sections and bifurcation diagrams. The onset of chaos. Attractors
We consider the trapped particle as a forced harmonic oscillator, described by a non-autonomous or time-dependent equation of motion. Forced oscillators exhibit many of the properties associated with nonlinear systems. Most nonlinear systems are impossible to solve analytically, which is why numerical modelling is a powerful tool to investigate the associated dynamics. The trajectory represents the solution of the differential equation starting from a set of initial conditions. A picture that illustrates all the qualitatively different trajectories of the system is called a phase portrait. The appearance of the phase portrait is controlled by fixed points. In terms of the original differential equation, fixed points represent equilibrium solutions. An equilibrium is considered as stable if all sufficiently small disturbances away from it damp out in time [@Nay08; @Gutz90; @Hilb10; @Stro15; @Cvita19].
A numerical integration of the equation of motion (\[cha1\]) is performed employing the fourth-order Runge–Kutta method [@Lynch10; @Lynch18]. In order to illustrate the dynamics of the trapped particle (ion) we represent the trajectories in the 2D phase space (phase portraits) [@Stee14; @Lynch18] and in the extended phase space as seen in Fig. \[OrbitsDuff\], with an aim to emphasize the regular and chaotic orbits.
\
\
By analyzing the associated phase portraits we observe that the cubic term in eq. (\[cha1\]) $-\mu u^3$ provides a nonlinear restoring force for large values of $x$, while the linear term pushes away from the origin. The potential for this oscillator exhibits a double-well structure. For certain initial conditions there exists an unstable equilibrium point at $x = 0$, and given some damping the particle has to fall into one side of the well or the other if it approaches the equilibrium point with just enough energy to move over it. The homogeneous problem (non-driven oscillator) contains no surprises in it. Given an initial condition there is a unique phase-space trajectory that leads to the particle winding up at the bottom of one of the two wells, after the mechanical energy is converted to heat. When the oscillator is driven by a periodic force the system can reach a limit cycle, where as much mechanical energy is lost per cycle as is dumped into the system by the crank. Chaos appears as a result of the two wells connected by the unstable equilibrium point. The phase portraits clearly reflect the existence of one (see Fig. \[OrbitsDuff\] c) or two attractors (see Fig. \[OrbitsDuff\] a, b, and d)), and of fractal basin boundaries for the trapped particle (ion) assimilated with a periodically forced double-well oscillator. For some of the parameter values presented in Fig. \[OrbitsDuff\] the system clearly exhibits two periodic attractors, corresponding to forced oscillations confined to the left or right well. Depending on the initial conditions, the system can converge rapidly to one of the two attractors after an initial transient phase. The basins of attraction generally have a complicated shape and the boundary between them is fractal [@Stro15]. As illustrated by the phase portraits, there are particular cases when the dynamics is characterized by periodicity.
The Poincar[é]{} sections are also represented in Fig. \[OrbitsDuff\]. We emphasize the appearance of what we consider to be strange attractors in the ion dynamics. A strange attractor represents the limiting set of points to which the trajectory tends (after the initial transient) every period of the driving force. Fig. \[OrbitsDuff\](f) represents a fractal set. This particular type of oscillator is also known as strange attractor, a clear indication that the system is chaotic [@Stro15; @Cvita19]. Chaos prevails too in the other cases but the system exhibits periodic orbits (see Fig. \[OrbitsDuff\] g, h and i).
From calculus we can ascertain that the frontiers of the stability diagram are shifted towards negative regions of the $a$ axis in the plan of the control parameters $\left(a, q\right)$, as already reported by Hasegawa, Sevugarajan, and Zhou [@Hase95; @Sevu00; @Sevu02; @Zhou10]. In order to better characterize the phenomena involved we have represented the phase portraits, Poincaré sections, and bifurcation diagrams for an ion (particle) levitated in the trap, both in the absence and presence of a laser field, as illustrated in Fig. \[Duff2\] and Fig.\[Duff3\]. The damped Duffing oscillator generally exhibits an aperiodic appearance as the system is chaotic [@Kova11; @Stro15]. $x\left(t\right)$ changes sign frequently which means that the particle crosses the hump repeatedly, as expected for strong forcing. The change of sign is in agreement with [@Sevu00], where the perturbed secular frequency of the ion has been obtained by using a modified Lindstedt–Poincar[é]{} perturbation technique. Sevugarajan and Menon also show that this perturbation is sign sensitive for octupole superposition and sign insensitive for hexapole superposition [@Sevu00].
A more detailed insight results from the Poincar[é]{} section, which results by plotting $\left(x\left(t\right), y\left(t\right)\right)$ whenever $t$ is an integer multiple of $2 \pi$. Practically we strobe the system at the same phase for each drive cycle. Looking at the Poincar[é]{} section we observe that the points fall on a fractal set, which we interpret as a cross section of a strange attractor for equation (\[cha1\]). The successive points $x\left(t\right), y\left(t\right)$ are found to hop erratically over the attractor while the system exhibits sensitive dependence on the initial conditions, which is the signature of chaos.
![Phase portrait, Poincar[é]{} section and bifurcation diagrams for a particle (ion) confined in a nonlinear trap. The values of the control parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 0.5$, $\mu = 1$, and $\alpha = 0$. The second bifurcation diagram corresponds to $0 < F < 5$. The first bifurcation diagram magnifies the area corresponding to $0 < F < 1$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff2"}](PhasePortDuff1.pdf "fig:"){width=".5\textwidth"} ![Phase portrait, Poincar[é]{} section and bifurcation diagrams for a particle (ion) confined in a nonlinear trap. The values of the control parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 0.5$, $\mu = 1$, and $\alpha = 0$. The second bifurcation diagram corresponds to $0 < F < 5$. The first bifurcation diagram magnifies the area corresponding to $0 < F < 1$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff2"}](PoiSectDuff1.pdf "fig:"){width=".5\textwidth"}\
![Phase portrait, Poincar[é]{} section and bifurcation diagrams for a particle (ion) confined in a nonlinear trap. The values of the control parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 0.5$, $\mu = 1$, and $\alpha = 0$. The second bifurcation diagram corresponds to $0 < F < 5$. The first bifurcation diagram magnifies the area corresponding to $0 < F < 1$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff2"}](BifurcDuff11.pdf "fig:"){width=".5\textwidth"} ![Phase portrait, Poincar[é]{} section and bifurcation diagrams for a particle (ion) confined in a nonlinear trap. The values of the control parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 0.5$, $\mu = 1$, and $\alpha = 0$. The second bifurcation diagram corresponds to $0 < F < 5$. The first bifurcation diagram magnifies the area corresponding to $0 < F < 1$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff2"}](BifurcDuff12.pdf "fig:"){width=".5\textwidth"}
In the case of a trapped ion (particle) in the presence of laser radiation, the phase portrait illustrates the existence of two attractors (as shown in Fig. \[Duff3\]) which seem to be periodic. We can discuss forced oscillations confined to the right or left well, because two basins of attraction emerge. The points on the Poincar[é]{} section fall on a fractal set which is again the signature of chaos. Thus, laser radiation renders the motion chaotic. The bifurcation diagram shows a period-doubling bifurcation for $F \approx 0.7$, and a mixture of order and chaos for $0.85 \leqslant F \leqslant 1.9$. For larger values of the kicking term ion dynamics is ordered.
![Phase portrait, Poincar[é]{} section and bifurcation diagram for an ion confined in a nonlinear Paul trap, in presence of laser radiation. The values of the parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 2$, $\mu = 1$, and $\alpha = 0.3$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff3"}](PhasePortrDuff2.pdf "fig:"){width=".5\textwidth"} ![Phase portrait, Poincar[é]{} section and bifurcation diagram for an ion confined in a nonlinear Paul trap, in presence of laser radiation. The values of the parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 2$, $\mu = 1$, and $\alpha = 0.3$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff3"}](PoiSectDuff2.pdf "fig:"){width=".5\textwidth"}\
![Phase portrait, Poincar[é]{} section and bifurcation diagram for an ion confined in a nonlinear Paul trap, in presence of laser radiation. The values of the parameters are $a = 0.1$, $q=0.7$, $\gamma = 0.3$, $\omega = 1.25$, $\Omega = 4$, $F = 2$, $\mu = 1$, and $\alpha = 0.3$. Pictures reproduced from Ref. [@Mih10b] under permission of the authors. Copyright Institute of Physics.[]{data-label="Duff3"}](BifurcDuff2.pdf "fig:"){width=".6\textwidth"}
### Nonlinear Dynamics in a Paul trap. Conclusions
We have performed a qualitative investigation of the dynamical stability of an ion confined within a nonlinear quadrupole Paul trap, with anharmonicity resulting from the presence of higher order terms in the series expansion of the electric potential [@Mih10b]. The system exhibits a strongly nonlinear character. Regular and chaotic regions of motion are emphasized within ion dynamics. System dynamics is chaotic when long-term behaviour is aperiodic. For particular initial conditions some of the solutions obtained present a certain degree of periodicity, although the dynamics is irregular. The damped parametric oscillator exhibits fractal properties and complex chaotic orbits. Chaotic (fractal) attractors were identified for particular solutions of the equation of motion. The motion on the strange attractor exhibits sensitive dependence on initial conditions. This means that two trajectories starting very close together will rapidly diverge from each other, and will exhibit utterly different behaviour thereafter. Strange attractors are often fractal sets [@Stee14; @Stro15].
Global changes in the state of a physical system are generally described by means of dynamical maps. Illustrative examples span classical nonlinear systems undergoing transitions to chaos. Ref. [@Schind13] suggests a model that extends the concept of dynamical maps to a many-body system, where both coherent and dissipative elements are taken into consideration. The stroboscopic dynamics of a complex many-body spin model is analyzed using a universal trapped ion quantum simulator. Experimental errors are discussed and experimental protocols are devised to mitigate against.
Classical motion of a single damped ion confined in a Paul trap is usually described by a damped harmonic oscillator model, such as the Duffing oscillator [@Nay08; @Kova11; @Mih10b; @Ake10]. Quantum damping motion of a trapped ion system is investigated in [@Qing17], by constructing a non-Hermitian Hamiltonian with dipole and quadrupole imaginary potential. Different real energy spectra and stable quantum states are obtained for both PT symmetry and asymmetry cases, as well as the imaginary spectrum and decaying quantum state for the PT asymmetry case. The results apply to the quantum dynamics of trapped ions. In a recent paper nonlinear dynamics of a charged particle in a RF multipole ion trap is explored using the method of direct averaging over rapid field oscillations [@Rozh17], which represents an original approach. The existence of localization regions for ion trap dynamics is identified. Illustrations of Poincar[é]{} sections demonstrate that ion dynamics is highly nonlinear [@Rozh17] as also emphasized in [@Mih10b]. Ref. [@Lan19a] predicts a regime of anharmonic motion in which laser cooling becomes diffusive, while it can also turn into effective heating. This implies that a high-energy ion could be easily lost from the trap despite being laser cooled. This loss can be counteracted using a laser detuning much larger than the Doppler detuning. It is often more meaningful to characterize systems possessing complex dynamics through certain quantities involving asymptotic time averages of trajectories. Examples of such quantities are power spectra, generalized dimensions, Lyapunov exponents and Kolmogorov entropy [@Stee14; @Stro15; @Cvita19; @Lynch18]. Under particular conditions these quantities can be calculated in terms of averages of periodic orbits.
A novel approach to characterize nonstationary oscillators by means of the so called point transformations in demonstrated in [@Zela19]. One of the main benefits of using the point transformation method lies in the fact that conserved quantities along with the structure of the inner product are preserved [@Stee14; @Stee07; @Blum10]. In addition, these transformations can be constructed to be invertible, which represents another remarkable feature. The authors demonstrate how point transformation enable one to solve the Schr[ö]{}dinger equation for a wide diversity of nonstationary oscillators. Therefore, it is expected that the method [@Zela19] can be applied to study the dynamics of particles in electromagnetic traps [@Paul90; @Major05; @Werth09; @Blaum06].
Non-neutral, complex plasmas. Coulomb systems. Examples and discussion {#Sec3}
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Complex plasmas and Coulomb systems {#ComplexCou}
-----------------------------------
Coulomb systems can be described as many-body systems consisting of identical particles that interact by means of electrostatic forces. When the potential energy associated to the Coulomb interaction is larger than the kinetic energy of the thermal (Brownian) motion [@Chav11], the system is strongly coupled as it presents a strong spatial correlation between the electrically charged particles, similar to liquid or crystalline structures [@Boll84; @Boll90a]. Strongly coupled Coulomb systems encompass various many-body systems and physical conditions, such as dusty (complex) plasmas or non-neutral and ultracold plasmas [@Werth05a; @Tsyto08; @Fort06a; @David01; @Mendo13; @Ott14].
Complex plasmas represent a unique type of low-temperature plasmas characterized by the presence of electrons, ions, neutral atoms and molecules, and highly charged nano- or microparticles, by chemical reactions, and by the interaction of plasmas with solid surfaces [@Fort05; @Boni14; @Shuk00; @Khra04; @Boni10a; @Campa14]. The particles usually bear large electrical charges and exhibit long-range Coulomb interaction which leads to the occurrence of strong coupling phenomena in the system [@Tsyto08], such as collective effects which result in the formation of plasma crystals [@Gebau03]. Complex plasmas are encountered in astrophysics as interstellar dust clouds, in comet tails or as spokes in the ring systems of giant gas planets [@Fort10a; @Shuk02a]. They are also present in the mesosphere, troposphere and magnetoshere of the Earth [@Shuk02b; @Fort11; @Pop11], near artificial satellites and space stations, or in laboratory experiments [@Boni14; @Morfi09]. Dust particle interaction occurs via shielded Coulomb forces [@Shuk00; @Shuk02b], the so-called Yukawa interaction [@Fort10a; @Donko08]. Yukawa balls are reported in case of harmonically confined dusty plasmas [@Boni10b; @Boni10a; @Trevi06; @Piel17]. Attention paid to the domain has witnessed a spectacular increase after the discovery of plasma crystals [@Schli96; @Dub99; @Piel17; @Thom94; @Tsyto07] and the detection of spokes in the rings of Saturn by the Voyager 2 mission in 1980 [@Fort10a; @Shuk02b], as illustrated in Fig. \[Saturn\].
Complex plasmas are intensively investigated in research laboratories as they are expected to shed new light on issues regarding fundamental physics such as phase transitions [@Blu90], self-organization, study of classical and quantum chaos or pattern formation and scaling [@Fort10a; @Boni10b; @Morfi09]. Present interest is focused on strongly coupled Coulomb systems of finite dimensions [@Vlad05]. Particular examples of such systems would be electrons and excitons in quantum dots [@Boni14] or laser cooled ions confined in Paul or Penning type traps [@Paul90; @Major05; @Werth09; @Vogel18; @David01].
![Cassini spacecraft images dark spokes on Saturn’s B ring. Photo reproduced from NASA https://solarsystem.nasa.gov/resources/14893/detailing-dark-spokes/. Copyright NASA.[]{data-label="Saturn"}](SaturnSpokes.pdf)
A comprehensive overview on dynamical processes that occur in complex (dusty) plasmas is written by Piel and Melzer [@Piel17; @Piel02]. Collective effects in dusty plasmas are extensively presented in [@Melz05]. It is shown that the composite structure is liable for the remarkable properties of this particular type of plasmas, among which the most prominent are the formation of liquid or solid phases under conditions of strong electrostatic coupling or using intensive nonuniform magnetic fields to levitate Coulomb clusters formed by charged diamagnetic particles [@Sav12]. Investigations begin with single particle effects, with an emphasis on levitation, harmonic and nonlinear oscillations. Few particle systems and the Yukawa interaction are also examined. Many-body systems are described by means of low-frequency electrostatic waves and instabilities that occur in complex plasmas. We also mention another paper of Piel [*et al*]{} [@Piel08] which investigates some exotic features of complex plasmas such as charge accumulation in the ion wake, ion drag (under microgravity conditions) and spherical plasma crystals [@Piel17]. The characteristic features and intrinsic properties of complex plasmas are also detailed in [@Khrap09]. Inclusive reviews on complex plasmas are written by Fortov [*et al*]{} [@Fort05] and Morfill [@Morfi09], and the reader interested in getting introduced into the domain should also refer to the books of Tsytovich, Fortov, and Bonitz [@Fort10a; @Boni14; @Tsyto08; @Boni10a].
A review on non-neutral plasmas levitated in ion traps and of their characteristic properties can be found in [@Werth05a]. Ion traps represent versatile tools to investigate many-body Coulomb systems, or dusty and non-neutral plasmas. The electrodynamic (Paul) trap uses a RF electric field which generates an oscillating saddle shaped potential that confines charged particles in a region where the electric field exhibits a minimum, under conditions of dynamical stability [@Ghosh95; @Major05; @Mih16a]. The dynamical time scales associated with trapped microparticles lie in the tens of milliseconds range, while microparticles can be individually observed using optical methods [@Davis02; @Vis13; @Vini15; @Libb18]. As the background gas is dilute, particle dynamics exhibits strong coupling regimes characterized by collective motion [@Werth05a; @Tsyto08; @Fort06a; @David01]. Dust particles may give birth to larger particles which might evolve into grain plasmas [@Tsyto08]. The mechanism of electrostatic coupling between the grains can vary widely from the weak coupling (gaseous) regime to the pseudo-crystalline one [@Werth05a; @Dub99; @Lisin13]. Complex plasmas can be described as non-Hamiltonian systems of few or even many-body particles. They are investigated in connection with issues regarding fundamental physics such as phase transitions, self-organization, study of classical and quantum chaos, pattern formation and scaling [@Fort10a; @Boni14; @Fort11].
Non-neutral plasmas. One component plasmas - OCPs
-------------------------------------------------
First experimental observations of ordered structures consisting of charged iron and aluminium microparticles confined in a Paul trap, in vacuum, were reported in 1959 [@Wuerk59]. Phase transitions occur as an outcome of the dynamical equilibrium between the trapping potential and the inter-particle Coulomb repulsion [@Died87; @Blu90]. The emergence of correlations was reported for strongly coupled ion plasmas [@Boll90a]. One year later an experiment reported the storage of macroscopic dust particles (anthracene) in a Paul trap, operating under SATP conditions [@Wint91]. Electrodynamic traps and ion trapping techniques combined with laser cooling mechanisms [@Paul90; @Ghosh95; @Werth09; @Haro13; @Kno16] allow scientists to investigate the dynamics of small quantum systems and prepare them in well-controlled quantum states [@Quint14; @Wine13; @Bush13]. Trapped ions or particles represent one-component plasmas (OCP). The OCP model is a reference one for the study of strongly coupled Coulomb systems [@David01; @Dub99; @Ott14; @Tama99].
A one-component plasma (OCP) can be regarded in a very good approximation as the simplest statistical mechanical model of a Coulomb system. This feature has drawn vivid attention on OCPs in the last 40 years. An extension of the Debye-H[ü]{}ckel (DH) theory has been employed to characterize OCPs in [@Tama99]. The approach enables one to perform analytic calculations of all the thermodynamic functions, as well as of the structure factor which is a major progress.
A non-neutral plasma can be described as a many-body collection of charged particles that does not embrace the overall charge neutrality condition. Various areas of application of non-neutral plasmas include: precision atomic clocks, trapping of antimatter plasmas and antihydrogen production, quantum computers, nonlinear vortex dynamics and fundamental transport processes in trapped nonneutral plasmas, strongly-coupled one-component plasmas and Coulomb crystals [@Boni08], coherent radiation generation in free electron devices such as free electron lasers, magnetrons and cyclotron masers, and intense charged particle beam propagation in periodic focusing accelerators and transport systems [@David01].
A classical plasma which consists of ions [*dressed*]{} with electrons exhibits a well-defined thermodynamics. In the strong-coupling regime the [*plasma*]{} is a solid while on passing to the weak-coupling regime it becomes gradually a liquid, then a non-ideal gas, and finally in the weak-coupling limit it behaves as an ideal classical gas [@Apost19].
New methods for trapping ions such as the Orbitrap, the digital ion trap (DIT), the rectilinear ion trap (RIT), and the toroidal ion trap, the development and application of the quadrupole ion trap (QIT) and the quadrupole linear ion trap (LIT), as well as introduction of high-field asymmetric waveform ion mobility spectrometry (FAIMS) are presented extensively in [@March17b].
### Electrodynamic traps as tools for confining OCPs
If only a single component charged particle species is confined in a Paul trap it represents a confined One Component Plasma (OCP). By modifying the trap parameters the plasma can perform phase transitions between gas-like, liquid-like and crystal-like states. The crystal state is the most interesting for experiments with trapped particles. The melting point of the crystal can be used to estimate a translational temperature of the trapped charged particles. The plasma state can be described by the coupling parameter [@Boll84; @Bolli03; @Werth05a; @Dub99; @Wine88; @Piel17], and it is essentially controlled by the trap parameters. Other examples of OCP systems are found in many other fields of physics, as they have been investigated theoretically over time [@Ichi82; @Ebel17; @Boll90a].
In case of quadrupole traps, the second-order Doppler effect is the result of space-charge Coulomb repulsion forces acting between trapped ions of like electrical charges. The Coulombian forces are balanced by the ponderomotive forces produced by ion motion in a highly non-uniform electric field. For large ion clouds most of the motional energy is found in the micromotion [@Major05; @Berke98; @Wang10]. Multipole ion trap geometries significantly reduce all ion number-dependent effects resulting through the second-order Doppler shift, as ions are weakly bound with confining fields that are effectively zero through the inner trap region and grow rapidly near the trap electrode walls. Owing to the specific map shape of trapping fields, charged ions (particles) spend relatively little time in the high RF electric fields area. Hence the RF heating phenomenon (micromotion) is sensibly reduced. Multipole traps have also been used as tools in analytical chemistry to confine trapped molecular species that exhibit many degrees of freedom, and thus study cold collisions and low temperature processes [@Rob18; @Trip06; @Ger03; @Ger08a; @Ger08b; @West09]. Space-charge effects are not negligible for such traps. Nevertheless, they represent extremely versatile tools to investigate the properties and dynamics of molecular ions or to simulate the properties of cold plasmas, such as astrophysical plasmas or the Earth atmosphere.
The thermodynamic properties associated to a non-ideal classical Coulomb OCP rest on a single parameter, that is the coupling parameter which we denote by $\Gamma$. In the particular case of a Yukawa OCP when the pair interaction is screened by background charges, the thermodynamic state also depends on the range of the interaction via the screening parameter. An original approach used to define and measure the coupling strength in Coulomb and Yukawa OCPs is proposed in [@Ott14], based on the radial pair distribution function (RPDF).
Stable confinement of a single ion in the radio-frequency (RF) field of a Paul trap is well known, as the Mathieu equations of motion can be analytically solved [@Ghosh95; @March05; @Rand16; @Li16]. This is no longer the case for high-order multipole fields where the equations of motion do not admit an analytical solution. In such case particle dynamics is quite complex as it is described by non-linear, coupled, non-autonomous equations of motion. The solutions for such system can only be found by performing numerical integration [@Major05; @Fort10a; @Rieh04].
Images of ordered 2D and 3D structures of microparticles levitated in ring and linear geometry microparticle electrodynamic ion traps (MEITs) are shown in Fig. \[Meit2\].
\
$(a)$
\
$(b)$
The paper of Foot [*et al*]{} [@Foot18] introduces an electrodynamic ion trap in which the electric quadrupole field oscillates at two different frequencies. By operating the trap as described, the authors report simultaneous tight confinement of ions with extremely different charge-to-mass ratios, e.g., singly ionized atomic ions together with multiply charged nanoparticles (NPs). The stability conditions for two-frequency operation are inferred from the asymptotic properties of the solutions of the Mathieu equation [@Brou11; @Rand16], which emphasizes the effect of damping on parametric resonances. The model suggested illustrates very well the effect of damping. Is is demonstrated that operation of the trap in a two-frequency mode (with frequency values as widely separated as possible) is most effective when the two species’ mass ratios and charge ratios are sufficiently large. The system can be assimilated with two [*superimposed*]{} Paul traps, each one of them operating close to a frequency optimised to tightly confine one of the species, which results in strong confinement for both particle species used. Such hybrid trap and its associated operation mode grants an advantage with respect to single-frequency Paul traps, in which the more weakly confined species develop a sheath around a central core consisting of tightly confined ions [@Foot18].
By employing of a non-neutral plasma method, a linear trapping model that accounts for large ion clouds has been demonstrated which will become the core of an atomic clock [@Pedro18].
Weakly-coupled plasmas {#weak}
----------------------
Brydges and Martin achieve an extensive review of low density Coulomb systems in Ref. [@Bryd99], which presents results on the correlations of low density classical and quantum Coulomb systems under equilibrium conditions in 3D space. The exponential decay of particle correlations in the classical Coulomb system, the Debye-H[ü]{}ckel screening, is compared and confronted with the quantum case where strong arguments are offered to account for the absence of exponential screening. The paper also examines experimental results and techniques for elaborate calculations that determine the asymptotic decay of correlations for quantum systems.
An elaborate review paper that emphasizes recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions is presented in [@Campo09]. The review is focused on a comprehensive presentation of the concept of ensemble inequivalence, as exemplified by the exact solution in the microcanonical and canonical ensembles of mean-field type models. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and convergence towards quasi-stationary states, which represents an exotic feature. Such an unusual relaxation process can be explained by introducing an appropriate kinetic theory that relies on the Vlasov equation. In addition, a statistical approach based on a variational principle established by Lynden-Bell is demonstrated to explain qualitatively and quantitatively some features of quasi-stationary states.
By introducing a modified Thirring model and then comparing the equilibrium states of the unconstrained ensemble with respect to the canonical and grand-canonical ones, it was demonstrated that systems with long-range interactions can evolve into states of thermodynamic equilibrium in the unconstrained ensemble [@Late17]. Moreover, it also results that the parameter space that determines the possible stable configurations is expanded when the system is restrained by fixing the volume and the particle number. These parameters oscillate in case of the unconstrained ensemble, along with the system energy. First-order phase transitions are also reported for the model introduced. The control parameters for the unconstrained ensemble are temperature, pressure, and chemical potential, while the replica energy represents the free energy. In contrast, macroscopic systems with short-range interactions are unable to reach equilibrium states if the control parameters are the temperature, pressure, and chemical potential.
Strongly-coupled Coulomb systems. OCPs revisited. {#strong}
-------------------------------------------------
Since the beginning of the 1980s strongly coupled and non-neutral plasma systems confined in Paul and Penning traps have been the subject of a vivid scientific interest [@Ichi82; @Boll84; @Dub99; @Wine88; @Boll90a; @Brew87; @Totsu87; @Ichi87; @Dub88; @Gil88; @Boll90b; @Dub90], due to their intriguing properties and wide area of applications.
A review on strongly coupled plasma physics and high-energy density matter is Ref. [@Muri04], which focuses on OCPs and the physical mechanisms associated to these particular type of plasmas. Examples supplied such as white dwarfs, trapped laser-cooled ions and dusty plasmas [@Piel17; @Piel08] better emphasize the wide area of applications and explain the vivid interest towards investigating strongly coupled plasmas. The kinetic theory for strongly coupled Coulomb systems based on a density-functional theory (DFT) approach, namely the Kubo-Greenwood model, is presented in [@Dufty18].
A strongly coupled non-neutral $^9$Be$^+$ ion plasma with a coupling parameter of approximately 100 or greater is reported in [@Brew87]. The ions are confined in a Penning trap, then laser cooled. Measurements were performed with respect to the plasma shape, rotation frequency, density and temperature. Among the most remarkable works on strongly-correlated non-neutral plasmas are the papers of Dubin and O’Neil [@Dub99; @Dub88; @Dub90]. We also mention the book of Fortov [*et al*]{} [@Fort06a] which explores the physics of strongly coupled plasmas. A new frontier in the study of neutral plasmas are the ultracold neutral plasmas [@Kil05; @Kil07; @Rong09; @Kil10; @Shuk10; @Lyon15; @Muri15].
Dust particles confined in an electrodynamic trap experience a large number of forces. The prevailing forces acting upon the micrometer sized particles of interest are the a.c. trapping field (whose outcome is the ponderomotive force caused by particle motion within a strongly nonlinear electric field) and gravity. The gravitational force can be expressed as [@Fort10a]
$$\mathbf{F}_g = m \mathbf{g} = \frac{4}{3} \pi a^3 \rho_d \mathbf{g} \ ,$$
where $\mathbf{g}$ stands for the gravitational acceleration, $m$ represents the dust mass, $a$ is the dust particle radius and $\rho_d$ corresponds to the dust particle density. The electric field yields a force
$$\mathbf{F}_{el} = -Q \mathbf{e} = -Ze \mathbf{E} \ ,$$
where $Q = Ze$ is the electric charge of the dust particle, and $e = 1.6 \cdot 10^{-19}$ C stands for the electric charge associated to the electron. When a temperature gradient arises within a neutral gas background, a force called thermophoretic drives the particles towards regions of lower gas temperature. The particle is also subject to the action of the ion drag force, caused by a directed ion flow [@Tsyto08; @Fort06a; @Vasi13]. To summarize, the forces that act upon a particle (or an aerosol) include the radiation pressure force, the thermophoretic force [@Piel17], the photophoretic force [@Liza18], electric forces and possibly magnetic forces, to which we add the aerodynamic drag and the force of gravity [@Tsyto08].
A one-component plasma (OCP) consists of a single species of charge submerged in the neutralizing background field [@Fort06a; @Dub99]. Single component non-neutral plasmas confined in Penning or Paul (RF) traps exhibit oscillations and instabilities owing to the occurrence of collective effects [@Werth05a]. The dimensionless coupling parameter that describes the correlation between individual particles in such plasma can be expressed as
$$\Gamma = \frac{1}{4\pi \varepsilon_0 } \frac{q^2}{a_{WS} k_B T} \ ,$$
where $\varepsilon_0$ is the electric permittivity of free space, $q$ stands for the ion charge, $a_{WS}$ is the Wigner-Seitz radius, $k_B$ denotes the Boltzmann constant, and $T$ is the particle temperature. The Wigner-Seitz radius results from the relation
$$\frac{4}{3}\pi {a}^3_{WS} = \frac{1}{n} \ ,$$
where $n$ is the ion (particle) density. Although the Wigner-Seitz radius measures the average distance between individual particles, it does not coincide with the average inter-particle distance. The $\Gamma$ parameter represents the ratio between the potential energy of the nearest neighbor ions (particles) and the ion thermal energy. It describes the thermodynamical state of an OCP. Low density OCPs can only exist at low temperatures. Any plasma characterized by a coupling factor $\Gamma > 1$ is called strongly coupled. For $\Gamma < 174$ the system exhibits a liquid-like structure, while larger values might indicate a liquid-solid phase transition into a pseudo-crystalline state (lattice) [@Werth05a; @Wint91]. OCPs are supposed to exist in dense astrophysical objects [@Horn90]. An example of a high temperature system would be a quark-gluon plasma (QGP) used to characterize the early Universe and ultracompact matter found in neutron or quark stars [@Boni10b; @Fort11].
Colloidal suspensions of macroscopic particles and complex plasmas represent other examples of strongly coupled Coulomb systems [@Kal02]. Trapped micrometer sized particles interact strongly over long distances, as they carry large electrical charges. Friction in air combined with large microparticle mass results in an efficient particle [*cooling*]{}, which leads to interesting strong coupling features. The signature of such phenomenon lies in the appearance of ordered structures, liquid or solid like, as phase transitions occur in case of such systems [@Major05; @Kno14; @Fort06a; @David01; @Dub99]. The presence of confining fields maintains the particles localized together in an OCP. Trapped and laser cooled ions offer a good, low temperature realization of a strongly coupled ultracold laboratory plasma.
The equation of motion for a particle trapped in air can be expressed in a convenient form by introducing dimensionless variables defined as $Z = z/z_0$ and $\tau = \Omega t/2$, where $z_0$ represents the trap radial dimension (a geometrical constant) and $\Omega $ stands for the frequency of the a.c. trapping voltage. The nondimensional equation of motion can be cast into [@Kul11] $$\label{extforce}
\frac{d^2 Z}{dt^2} + \gamma \frac{dZ}{dt} + 2 \beta Z \cos(2 \tau) = \sigma \ ,$$ where $\gamma$ stands for the drag parameter, $\beta$ is the a.c. field strength parameter and $\sigma$ represents a d.c. offset parameter, defined as [@Drew15] $$\gamma =\frac{6 \pi \mu d_p \kappa}{m\Omega} \ , \beta = \frac{4g}{\Omega^2}\left(\frac{V_{ac}}{V_{dc}} \right) \ , \sigma = - \frac{4g}{\Omega^2 z_0}\left(\frac{V_{dc}}{V^*_{dc}}\right) ,$$ where $ V^*_{dc} $ satisfies $$F_z - mg = q C_0\frac{V^*_{dc}}{z_0} .$$ $C_0$ and $C_1$ are two geometrical constants ($ C_0 <1 $) and $ b = z_0C_0/C_1 $. For a negative charged particle the right hand term of eq. (\[extforce\]) is a positive quantity. When the d.c. potential is adjusted to compensate the external vertical forces (gravity) $V_{dc} = V^*_{dc}$, $\sigma = 0$, and the particle experiences stable confinement. Other relevant quantities are the gas viscosity coefficient $\mu$, the particle diameter $d_p$, the particle mass $m$, while $b$ represents the geometrical constant of the electrodynamic balance (EDB) [@Hart92; @Singh18; @Davis90].
If a larger number of electrically charged particles are trapped together the Coulomb interaction will affect the individual particle motion and the spatial charge will alter the trap potentials. In case of particles with electric charges that bear the same sign, the Coulomb repulsion makes ideal trapping impossible. Coupling strongly dependens on the trap parameters. The dynamics of a strongly coupled plasma in a trap is highly nonlinear, while computer simulations using molecular or Brownian dynamics should provide accurate quantitative data. A molecular dynamics simulation for hundreds of ions confined in a Paul trap has been performed in [@Pre91]. The simulations account for the trapped particles micromotion and inter-particle Coulomb interactions. A random walk in velocity has been implemented in these calculations with an aim to bring the secular motion to a temperature that is numerically measured. When coupling is large the ions build concentric shells that undergo oscillations at the RF frequency, while the ions within a shell create a 2D hexagonal crystal-like lattice [@Pre91]. Molecular dynamics simulations with low energy ions levitated in a nanoscale Paul trap, in both vacuum and aqueous environment, are presented in [@Zhao08]. A wide collection of papers related to the domain of strongly coupled Coulomb systems is Ref. [@Kal02]. A concise introduction on the Coulomb crystals that occur when levitated ions are cooled at very low temperatures is given in [@Drew15].
As already mentioned in Section \[ComplexCou\], complex plasmas comprise electrons, ions, neutrals, and macroscopic solid dust particles of nano- or micron size [@Fort10a; @Ivlev12; @Fort11]. The dust grains [@Dra03] may exhibit a positive or negative electric charge, depending on the charging mechanisms involved. Experiments performed show that dust grains acquire a negative charge typically of the order of $10^3 \div 10^5$ elementary charges due to the electron and ion stream [@Asch12a]. As the dust particle component is strongly coupled with respect to other plasma components, occurrence of ordered structures was experimentally reported. These structures are also known as [*plasma crystals*]{} [@Thom94; @Tsyto07; @Chu94]. In Ref. [@Asch12a] micron sized dust grains are trapped in a low-temperature RF discharge, and the dynamic light scattering (DLS) technique is applied to the dust component of a complex plasma. The electric force that occurs in the plasma-wall sheath of the lower electrode compensates external forces such as gravity and ion drag. Intricate 3D crystal structures are experimentally observed. Relaxation into an equilibrium state of dust grain dynamics in plasma after interaction with a laser beam is experimentally and qualitatively investigated in [@Lisin13], where the associated physical mechanisms are also discussed.
Strong correlation effects that occur in classical and quantum plasmas are investigated in [@Ott14; @Boni08]. Coulomb (Wigner) crystallization phenomena are analyzed with an emphasis on one-component non-neutral plasmas in traps, and on macroscopic two-component neutral plasmas. The paper also explains how to achieve Coulomb crystallization in terms of critical values of the coupling parameters and the distance fluctuations and the phase diagram of Coulomb crystals. Numerical simulations of strongly coupled dust particles, that take into account the influence of the buffer gas medium (viscosity) as well as the random forces exerted upon the particles, have been carried out in [@Fili12] under SATP conditions. The simulations help in identifying optimum operating values for the electric field amplitude and frequency, that are required in order to achieve stable capture of dust particles in a linear Paul trap. The possibility to levitate Coulomb clusters, that consist of charged diamagnetic particles in a nonuniform magnetic field, is investigated both theoretically and experimentally in [@Sav12]. The numerical simulations performed illustrate the appearance of standing waves of the dust particle density, caused by the dynamic effects induced by the periodic external low frequency electric field. The neutral gas pressure effect over the crystallization of 3D cylindrical complex plasmas under laboratory conditions, is investigated in [@Stein17].
Calculation results of the inner pressure and energy of strongly coupled Coulomb systems levitated in a Paul trap, have been carried out by employing the statistical theory applied to the liquid state. Pair correlation functions of the Coulomb systems have been inferred. The system total energy, the internal pressure, and the Coulomb coupling parameter $\Gamma$ have been computed using the inter-particle Coulomb potential [@Lapi18a].
Mesoscopic systems. Phase Transitions
-------------------------------------
Experimental investigations of charged particles levitated in external potentials have recorded significant progress that coincided with the advent of ion traps. The paper of Wuerker [*et al*]{} reports ordered structures obtained with aluminium microparticles that are cooled as an outcome of collisions with a background buffer gas [@Wuerk59]. Experiments describe crystallization into a regular array, followed by melting when both the RF trap potential and cooling force change. Other experiments also report similar phenomena [@Blu89]. The transition from an ion cloud to a crystal-like structure corresponds to a chaos-order transition. These intriguing phenomena are a convincing proof that a Paul trap represents a powerful tool to investigate the physics of few-body phase transitions and thus gain new insight on such mesoscopic systems [@Schli96; @Walth95].
The dynamics of a single stranded DNA (ssDNA) molecule can also be investigated by using nanoscale, linear 2D traps operating in vacuum [@Jose10]. By employing molecular dynamics simulations, it is demonstrated that a line charge can be efficiently trapped for a well defined range of stability parameters. A 40 nm long ssDNA does not fold or curl in the Paul trap, but could experience rotations around the CM. Rotations could be prevented by applying a stretching field in the axial direction,which leads to enhanced confinement stability.
A novel versatile method to achieve one-by-one coupling of single nano- and microparticles is demonstrated in Ref. [@Kuhl15]. The setup relies on a segmented linear Paul trap that levitates particles in combination with an optical microscope, as this particular geometry enables fast particle characterization and alignment-free assembly of particles. Using fluorescent quantum dot clusters and dye-doped polystyrene beads electromagnetic coupling is achieved by attaching them to spherical silica microresonators. Thus, coupled systems of levitated particles can be extensively investigated. The particles can also be deposited on the facets of optical fibers which is also demonstrated experimentally. An “optical fiber microfluidic control device based on photothermal effect induced convection” is demonstrated in [@Zhan19], “which can achieve manipulation and particle sorting in range of hundreds of microns”. The setup design is simple as it uses three single-mode optical fibers regularly arranged, reducing the cost associated with optical tweezers.
Phase transitions represent an intriguing feature of complex plasmas, and an important research focus in many-body physics over the last decades. The ordered structures collapse (sometimes the process is called melting) due to a variation of the discharge pressure or discharge power. The phenomenon is the outcome of the high thermal energies that dust particles acquire. Phase transitions of a 3D complex plasma can be investigated by means of the DLS technique [@Asch12b]. Anisotropy in the plasma-wall sheath plays an important role in this case [@Asch12a]. Physical mechanisms such as ion flow in the sheath yield to an ion two-stream instability and a phonon instability, both liable for the phase transitions and for the remarkably high kinetic energy of the dust component in the disordered phase state [@Schwei98]. Measurements on phase transitions phenomena are very delicate and very few experiments have been performed up to now, especially on systems with very few crystal layers [@Tsyto08].
An area of vivid interest regarding nonneutral plasmas is connected to phase transitions to liquid and crystal states, when the coupling parameter [@Kul11; @Ott14; @Drew15]
$$\label{coup}
\Gamma = \frac {e^2}{a k_B T}$$
is sufficiently large, a condition that is satisfied for low temperatures. As seen in eq. \[coup\], the coupling parameter represents the ratio between the nearest-neighbour Coulomb energy $\left(e^2/a \right)$ and the thermal energy of the particle (ion), where $ a = \frac 34 \pi \bar{n}$ stands for the Wigner-Seitz radius and $\bar{n}$ is the average particle density.
Structural phase transitions for trapped ions are investigated in [@Marci12], where molecular dynamics simulations are employed to investigate the structural transition from a double ring to a single ring of ions. Qualitative investigations on the process of two-ion crystals splitting in segmented Paul traps, as well as the challenges raised by the precise control of this process are presented in [@Kauf14]. Due to strong Coulomb repulsion cold ions represent a remarkable tool to explore phase transitions under stable confinement conditions. A recent paper reports on ordered stuctures and phase transitions of up to sixteen laser-cooled $^{40}$Ca$^+$ ion crystals in a custom-made surface-electrode trap (SET) [@Yan16]. Experimental results are shown to exhibit very good agreement with the numerical simulations. Preparation and coherent control of the angular momentum state of a two-ion crystal is demonstrated in [@Urban19].
Spontaneous symmetry breaking is a fundamental concept in many areas of physics, including cosmology, particle physics and condensed matter. An example would be the breaking of spatial translational symmetry which underlies the formation of crystals and the phase transition from liquid to solid state. Using the analogy of crystals in space, the breaking of translational symmetry in time and the emergence of a [*time crystal*]{} was recently proposed, but was later shown to be forbidden in thermal equilibrium. However, non-equilibrium Floquet systems which are subject to a periodic drive can exhibit persistent time correlations at an emergent subharmonic frequency. This new phase of matter has been dubbed a [*discrete time crystal*]{}. The experimental observation of a discrete time crystal in an interacting spin chain of trapped atomic ions is reported in [@Zhang18]. A periodic Hamiltonian is applied to the system under many-body localization conditions, and a subharmonic temporal response that is robust to external perturbations is monitored. Observation of such a time crystal opens the door to the study of systems with long-range spatio-temporal correlations and novel phases of matter that emerge under intrinsically non-equilibrium conditions [@Zhang18].
Strongly coupled plasmas confined in quadrupole and multipole electrodynamic traps {#Sec4}
==================================================================================
Stable 2D and 3D structures. Phase transitions
----------------------------------------------
As a general rule the particle size and inter-particle distance allow one to use simple diagnostic means, e.g. optical measurements in the visible range. Results of such investigations can be used to interpret properties of crystals, liquids and colloidal suspensions [@Ivlev12]. So far, plasma crystals have been obtained in plasmas of glow and high-frequency discharges at low pressures $\left( 1 \div 100 \textrm{Pa} \right)$ or using a beam generated by a charged particle accelerator [@Fort05; @Fort10a]. Investigations have been performed on crystal-liquid-gas phase transitions, on the temperature influence on dusty structures (starting from SATP conditions down to cryogenic ones) [@Fort02], or about oscillations and waves that propagate in dusty plasma structures [@Fort05; @Fort10a]. The response of dusty plasma structures with respect to external actions such as pulsed electric [@Pust09; @Vasi07; @Vasi08] and magnetic fields, laser radiation, thermophoretic force [@Vasi03a], electron beam [@Vasi03b], etc., has been extensively investigated. 3D structures consisting of a large number of particles have been explored in experiments performed aboard the orbital space station “MIR” and the International Space Station (ISS) [@Fort10a; @Myas17; @Pust16]. Some technological applications of dusty plasmas are considered in [@Bouf11].
As Coulomb crystals consisting of dust particles with electric charges of identical sign are unstable and particles are pushed apart by the Coulomb repulsion, a stable crystal can occur only in the presence of a trap potential [@Died87; @Blu89; @Marci12; @Kauf14; @Yan16]. These traps result for low pressure electrical discharges in regions of strong non-uniform electrical field. The trap can develop in the near-electrode layer of a capacitive RF discharge, or in a striation of the glow discharge positive column. The gravity force acting upon a charged particle is balanced by a longitudinal electrical field aligned in an axial direction, in case of a discharge that exhibits cylindrical symmetry. The radial electric field emerges due to the ambipolar diffusion of electrons and ions to the walls, which results in the confinement of charged particles in the radial direction. The electric charge of the dust particles can rise up to values that are several thousand times larger with respect to the electric charge of the electron, as it is determined by the electron temperature and the particle size [@Fort05; @Fort10a]. The dimension of levitated dust particles generally lies in the $1 \div 50 \ \mu$m range, but not necessarily [@Wint91; @Ghe98; @Sto01; @Sto08; @Vis13]. The size range is determined by the electric charge of the particle and the electrical discharge fields that accomplish levitation. The parameter fields for ordered dusty structures in low pressure electrical discharges are determined by plasma parameters that limit the investigated properties of plasma crystals.
An additional restriction concerning the occurrence of Coulomb crystals in gas-discharge plasmas results from the fact that the electrical charges of the dust particles depend on the electron temperature, which in turn depends on the reduced electrical field intensity. If the electrical charges of dust particles and the trap that confines them are created independently, ordered structures of charged dust particles can also be produced for various conditions. A method to generate dusty plasma crystals uses a proton beam with an average energy of about 2 MeV. The proton beam ionizes the gas and additional trap electric fields are generated by electrodes supplied with d.c. voltages [@Fort06b]. Once the pressure increases the crystal is destroyed. Then, levitation of charged particles no longer exists and particles are expelled out of the discharge as the electric fields that create the trap vanish. For example, the phenomenon occurs when the glow d.c. discharge striations fade, and the longitudinal electrical field gradient that provides stability of levitating particles (compensating the gravitational force) also vanishes. When the pressure increases, the radial electrical force drops off. Hence, in presence of an ion drag force and of the thermophoretic forces that develop due to a temperature gradient, confinement of particles in the radial direction becomes impossible [@Fort05; @Fort10a].
### Microparticle electrodynamic traps-MEITs
Further on we will shortly present MEITs and the physics associated to them [@Libb18]. MEITs have demonstrated to be versatile instruments, that are suited to characterize the physico-chemical properties of single charged particles with dimensions ranging between $100$ nm $\div 100 \ \mu$m, such as aerosols [@Kul11; @Vogel13a; @Kurt07], liquid droplets [@Singh17; @Singh18; @Lamb96], solid particles [@Wuerk59; @Wint91; @Ghe95b; @Sto16], nanoparticles [@Sande14], and even microorganisms [@Peng04] or DNA segments [@Jose10]. Stable 2D and 3D structures consisting of $26 \ \mu$m diameter particles levitated under SATP conditions by ring-type microparticle electrodynamic ion traps (MEITs), are shown in Fig. \[Meit\]. The trapping voltage supplied to the planar ring electrode is 6 kV a.c. at 60 Hz frequency. The particles are negatively charged monodispersed Lycopodium club-moss spores, illuminated with laser light [@Libb18].
![Stable 2D and 3D structures consisting of $26 \ \mu$m-diameter particles trapped under SATP conditions by ring-type microparticle electrodynamic ion traps (MEITs). The trapping voltage supplied to the planar ring electrode is 6 kV a.c. at 60 Hz frequency. The particles are negatively charged monodispersed Lycopodium club-moss spores, illuminated with laser light. Source: images reproduced from [@Libb18] by courtesy of K. Libbrecht.[]{data-label="Meit"}](Meit1.jpg "fig:") ![Stable 2D and 3D structures consisting of $26 \ \mu$m-diameter particles trapped under SATP conditions by ring-type microparticle electrodynamic ion traps (MEITs). The trapping voltage supplied to the planar ring electrode is 6 kV a.c. at 60 Hz frequency. The particles are negatively charged monodispersed Lycopodium club-moss spores, illuminated with laser light. Source: images reproduced from [@Libb18] by courtesy of K. Libbrecht.[]{data-label="Meit"}](Meit3.jpg "fig:")
### Coulomb structure in a linear Paul trap affected by electrical pulses
Fig. \[setpuls\] shows the schematic diagram of the power supply for an electrodynamic trap. Oscillations are excited by rectangular electric pulses applied to the additional electrodes located at the trap ends, separated by a distance of 5 cm. The pulses amplitude ranges between $10 \div 320$ V and the frequency lies between $1 \div 20$ Hz. The relative pulse duration can be adjusted in the interval $0.01 \div 0.99$. The electrodynamic trap presented in [@Syr18] consists of four cylindrical parallel, longitudinal electrodes, placed at the vertices of a square with a side of 2 cm. The electrodes are 10 cm long with a 4 mm diameter. A sinusoidal voltage is applied, of opposite phase between adjacent electrodes. The a.c. voltage amplitude is around 4.5 kV at a frequency of 50 Hz.
![Scheme of the linear electrodynamic trap. Notations: trap electrodes -– 1, dust particles -– 2, generator of rectangular pulses –- 3. Source: picture reproduced from [@Syr18] with permission of the authors.[]{data-label="setpuls"}](setpuls.png)
Polydisperse $Al_2O_3$ particles are used in the experiments described in [@Syr18]. Electrical charging of the particles is achieved by employing the induction method. The particles are placed on a flat electrode that is gradually shifted towards the electrodynamic trap. Then, a positive electric potential is applied to the electrode. When the potential is higher than 5 kV the particles are attracted and absorbed into the trap. The motion of the particles is recorded using a CCD camera HiSpec1, at a maximum resolution of $1240 \times 1024$ pixels. The frame rate can reach up to 524 frames per second at maximum resolution. Illumination of the particles is implemented by a flat laser beam with a diameter of 1 mm. The laser wavelength is 532 nm and the output power ranges up to 150 mW.
![Coulomb structure in a linear electrodynamic trap at moments of highest compression (right column) and stretching (left column): a), b) pulse frequency 1 Hz; c), d) pulse frequency 5 Hz, e), f) pulse frequency 9 Hz. Source: images reproduced from [@Syr18] with permission of the authors.[]{data-label="pulse"}](pulse.png)
Fig. \[pulse\] illustrates oscillations of the Coulomb structure in a linear electrodynamic trap, for different values of the pulse frequency. The pulse amplitude is 320 V. The relative pulse duration is 0.5, while pulses are in phase. By increasing the frequency, the amplitude of the oscillations drops to almost undetectable at 20 Hz; the oscillations cease to affect the middle region of the structure. At a frequency of 1 Hz, the structure at the moment of maximum compression corresponds to the structure in case of a constant potential supplied at the additional electrodes.
Confinement regions in the voltage–frequency plane
---------------------------------------------------
Let us consider the confinement region in the $U_{\omega} - f$ plane, shown in Fig. \[fig:5\]. For each value of the frequency, the confinement region is bounded by lower and upper values of the a.c. trapping voltage. Besides this region the trap cannot confine dust particles. On the other hand, if the a.c. voltage is low then the total restoring force that holds the particle inside the trap will be smaller then the gravity force, and particles will fall down out of the trap. Otherwise, if the a.c. voltage value is too high then the trapping field might be large enough to push the particle out of the trap during a half-period of oscillation.
![Boundaries for dust particle confinement in the $U_{\omega} - f$ plane. Experimental parameters: $f = 50 \div 500$ Hz , $U_{end} = 4000$ V, $\rho_{particle} = 0.38 \cdot 10^4$ kg/m$^3$, $r_{particle} =7 \ \mu$m, $Q_{particle} = 50000 e$, $\eta = 17 \cdot 10^{-6}$ Pa $\cdot$ s - dynamic viscosity, T = 300K. Source: picture reproduced from [@Lapi13] with permission of the authors.[]{data-label="fig:5"}](Uw_k_f_i.pdf)
Mass dependent particle separation in electrodynamic traps
----------------------------------------------------------
Fig. \[fig:6\] shows the results of simulations performed for dust particle confinement in a four wire (electrode) trap. We can also observe levitation of trapped particles for three different values of the charge-to-mass ratio. Inside the trap there are 30 particles of type A (black circles) with ${Q/M}_A = 6.9 \cdot 10^{16}e$/kg and $m_{pA} =1.3 \cdot 10^{-12}$ kg, 5 particles of type B (grey squares) with ${Q/M}_B = 6.9 \cdot 10^{16} e$/kg and $m_{pB} = 2.6 \cdot 10^{-12}$ kg, and finally 5 particles of type C (grey diamonds) with ${Q/M}_C = 3.4 \cdot 10^{16}e$/kg and $m_{pC} = 2.6 \cdot 10^{-12}$ kg.
![Typical dust particle levitation in the linear Paul trap. Left panel – side view, right panel – end view. Experimental parameters: $f = 50$ Hz , $U_{end} = 900$ V, $U_{\omega} = 4400$ V, $r_{p}=7\ \mu$m, $\eta = 17 \cdot 10^{-6}$ Pa$\cdot$s – dynamic viscosity, $T = 300$ K. Symbols: black circles – particles of type A: $\rho_{pA}=0.38 \cdot 10^4$ kg/m$^3$, $Q_{pA} = 9 \cdot 10^5 e$; grey squares – particles of type B: $\rho_{pB}=0.76 \cdot 10^4$ kg/m$^3$, $Q_{pB} = 1.8 \cdot 10^6 e$; grey rhombus – particles of type C: $\rho_{pC}=0.76 \cdot 10^4$ kg/m$^3$, $Q_{pC} = 9 \cdot 10^5 e$.[]{data-label="fig:6"}](DiffParticles_i.png)
Fig. \[fig:6\] illustrates how an electrodynamic trap can be used to perform dust particle separation. Type C particles are located below type A particles, as they have identical electric charge but are two times more heavier. Particles of type B and those of type A possess the same charge-to-mass ratio, but type B ones are two times more heavier and carry a double electrical charge in comparison with type A particles. Due to their double electric charge, type B particles are more strongly attracted and thus levitate closer to the trap axis. As illustrated in Fig. \[fig:6\], type A particles establish an oscillating cylinder under the trap axis. The inter-particle interaction also contributes to the spatial separation and to the ordering of dust particles along the trap axis [@Lapi17].
Confinement of a maximum number of particles
--------------------------------------------
Fig. \[capas\] presents the regions of confinement for a maximum number of particles (corresponding to the maximum trap capacity), for a given value of the trapping voltage frequency and different values of the charge-to-mass ratio $Q/M$, under SATP conditions. The region that confines a maximum number of particles lies within the curve corresponding to a specific value of the trapping voltage frequency. The range of $Q/M$ values modifies for a certain value of the maximum number of trapped particles, and it is determined by the boundary points of an intersection of the straight line drawn through a point corresponding to a specific number of particles with the curve obtained in the simulation. As the particle number in a dusty structure rises, the confinement region shrinks. Such a phenomenon is illustrated by the solid lines in Fig. \[capas\].
![Dependence of the maximum number of trapped particles on the frequency of the a.c. voltage and on the charge-to-mass ratio $Q/M$, under SATP conditions. The values of the a.c. voltage frequency are : 1–30 Hz, 2–50 Hz, 3–80 Hz, 5–150 Hz, 6–200 Hz. Source: picture reproduced from [@Vasi13] with permission of the authors.[]{data-label="capas"}](capas.pdf)
In order to establish the maximum trap capacity, a large number of particles are injected into the trap. As the frequency rises, the region that confines a fixed number of particles extends. The maximum number of particles corresponds to a maximum of the curve in Fig. \[capas\]. The highest number of particles confined in the trap is reached only for a single value of the charge-to-mass ratio $Q/M$.
### Experimental setup
The experimental setup that uses a linear quadrupole trap to capture and confine electrically charged dust particles is presented in Fig. \[setup\]. The electrodes of the dynamic trap and the endcap electrodes are made of 12 cm long copper rods with a diameter of 3 mm. The distance between the electrode axes is 1.3 cm. The endcap electrodes are placed along the trap axis, at a distance of 65 mm apart. The cylindrical trap electrodes achieve radial confinement of the particles, while the endcap electrodes maintain the particles contained within the axial region of the trap. The a.c. voltage applied across the four cylindrical electrodes ranges from $0 \div 2000$ V, its frequency can be modified by means of a generator. A d.c. voltage of 900 V is applied between the endcap electrodes. To mitigate against the action of external electric fields, the trap is placed inside a grounded metal shield (chassis). The high voltage supply wires are placed at a safe distance, in order to prevent the electric field they generate from perturbing the trapping field. The wires are cabled along the metal surface of the shield. Particles are detected by means of a high speed video camera HiSpec, at a maximum resolution of $1280 \times 1024$ pixels. To better observe the particles with dimensions ranging between $10 \div 150$ microns, a laser diode ($\lambda = 550$ nm, power $= 10 \div 100$ mW) is used or a laser sheet with a diameter of 150 microns. The trap is placed inside a transparent box to severely limit the influence of air flows on the particle dynamics [@Lapi18b].
![Scheme of the experimental setup used to study the behaviour of charged dust particles in a quadrupole dynamic trap. Legend: 1, 2, 3, 4 are the trap dynamic electrodes, 5 denotes the endcap electrodes, 6 is the laser diode, 7 stands for the high speed camera, $S_{ac}$ is a source of a.c. voltage, $S_{dc}$ is a source of d.c. voltage. Source: picture reproduced from [@Vasi13] with permission of the authors.[]{data-label="setup"}](setup.png)
### Experimental details and characteristic parameters
Polydisperse $Al_2O_3$ particles with dimensions ranging between $10 \div 80$ microns, and hollow borosilicate glass spheres with dimensions that lie within the $30 \div 100$ micron range have been used in the experiments. As the numerical simulations illustrate, the quadrupole trap exhibits selectivity with respect to the mass and the electric charge of the particles. Therefore, the trap will capture only those particles that specifically meet the required confinement conditions. In order to levitate the microparticles and build up a stable ordered structure, a special source of electrically charged particles has been devised and tested. Electrical charging of the microparticles is achieved by employing a streamer discharge in the electric field of a plane capacitor. The source allows us to positively charge particles up to $10^5 \div 10^6 e$, while in the same time it delivers them an initial velocity of $0.5 \div 2$ m/s. Selection of particles with the desired charge-to-mass ratio is achieved in the electric field of a plane capacitor, towards which a flow of charged particles is directed. The capacitor plates are arranged in a horizontal plane and the distance between them is around 10 cm. In order to achieve stable confinement, a positive potential generated by a d.c. voltage source is applied to the lower capacitor plate. The voltage value is modified such as to balance the gravity force acting upon the charged particles. The particles that satisfy this condition move along the capacitor plates, and they finally end up confined in the trap. Moreover, in order to provide stable confinement of the particles a mandatory condition lies in reducing the particle velocity. The neutral drag force efficiently brakes the articles by air friction (SATP conditions) when the capacitor electric field vanishes. The particles move across the quadrupole trap where low-velocity grains that possess an adequate charge-to-mass ratio are captured. We will describe the method used to determine the particle electric charge. The trap supply voltage is initially switched off. As an outcome of gravity particles fall and precipitate on a substrate. The dimensions of the precipitated particles are estimated using an optical microscope. The particle weight is evaluated by assuming it exhibits a spherical shape. The electric charge of the particle is determined from the condition that the gravity force in the plane capacitor gap is balanced by the electric field in the area. The principle of the method in described in Section \[specharge\].
Ordered Coulomb particle structures
------------------------------------
Examples of experimentally observed ordered Coulomb structures consisting of aluminum oxide particles with dimensions ranging between $10 \div 80 \ \mu$m at a frequency $f = 80$ Hz, are shown in Fig. \[fig5\]. The ordered structure usually comprises 50 up to 100 particles, as their number depends on the frequency and amplitude of the a.c. trapping voltage.
![a) Photo of the trap and ordered structure of aluminum oxide particles with $10 \div 15 \ \mu$m diameter. $U_{ac} = 2000$ V, $f = 80$ Hz, $U_{dc} = 900$ V; b) Magnified image of the ordered Coulomb structure. The area of the view is $17 \times 9.5$ mm$^2$. The inter-particle distance between grains of similar size ranges between $400 \div 800 \ \mu$m, the distance between a big particle and a smaller one is around $1800 \div 2000 \ \mu$m. Source: images reproduced from [@Vasi13] with permission of the authors.[]{data-label="fig5"}](fig5.png)
![Snapshot of a segment of the ordered Coulomb structure for hollow glass spheres with a diameter ranging between $30 \div 50 \ \mu$m. $U_{ac} = 2000$ V, $f = 80$ Hz, $U_{dc} = 900$ V. The area of the view field is $11\times 7.5$ mm$^2$. The inter-particle distance spans values from $1300 \div 1800 \ \mu$m. Source: image reproduced from [@Vasi13] with permission of the authors.[]{data-label="fig6"}](fig6.png)
Fig. \[fig6\] shows the structure established by hollow glass microspheres of diameter $30 \div 100 \ \mu$m. The maximum number of particles that compose the ordered structure does not exceed 50. Owing to a large friction force, the chaotic movement of the particles is negligible as most of them oscillate at small amplitudes, in synchronism with the a.c. electric field. The particles that are closest to the trap axis exhibit low displacements, while only several particles located far from the trap axis oscillate with large amplitude. It seems likely that these particles are positioned at the border of the stability boundary.
The ordered Coulomb structure was observed for about 12 hours. We show the Coulomb interaction is strong and the mean inter-particle distance stays about the same. The electrical charge of the trapped particles is estimated using the method described above, and we obtain values that span an interval $10^5 \div 10^6 \ e$. We emphasize that in contrast to dusty structures in a plasma, the trapped microparticles settle into a Coulomb system in absence of the neutralizing plasma background. Since the trap confines particles with equal charge-to-mass ratios, the ordered structures can incorporate particles of different sizes. An example of such a structure is shown in (Fig. \[fig5\] b), where in the lower part of the dusty structure one can distinguish a large particle. The inter-particle distance between equally sized grains varies between $400 \div 800 \ \mu$m, and the distance between the large particle and smaller ones is around $1800 \div 2000 \ \mu$m. By considering that the trap confining force is proportional to a space shift and the repulsive force between the particles is Coulombian, it follows that that the electric charge of the large particle is $6 \div 9$ times greater than the electric charge of smaller sized particles.
![Photo of the trap and ordered Coulomb structure made of aluminum oxide particles with dimensions of $10 \div 15$ microns. $U_{ac}= 2000$ V, $f = 80$ Hz, a) $U_{dc} = 900$ V; b) $U_{dc} = 750$ V. Source: images reproduced from [@Vasi13] with permission of the authors.[]{data-label="fig7"}](fig7.png)
By further lowering the value of the endcap voltage, particles reach the trap extremities and leave it by moving along the endcap electrodes. When the number of particles in the trap drops down to 10 or even less, the microparticles align along the trap axis and set up a linear chain. Even if the length of the trap electrodes is equal to 12 cm, numerical simulations were performed for a length of 15 cm of the cylindrical electrodes. Simulations show that a change in the trap electrodes length could result into a slight change in the electric field at a peripheral region near the electrodes end. As illustrated in Fig. \[fig5\] and Fig. \[fig7\] particles are not present in this region. The same remark is valid with respect to the small difference between the $L_h$ values used in experiment and numerical simulation, respectively.
Ordered Coulomb structures that encompass a large number of particles are of interest for applications such as nuclear batteries [@Filip05], where the stable structures of radioactive particles have to levitate in a high density gas media. These structures can be obtained using electrodynamic traps. Fig. \[large\_stable\] shows a fragment of a large stable Coulomb structure that encloses around 2000 particles.
![Fragment of the large stable Coulomb structure containing about 2000 particles[]{data-label="large_stable"}](large_stable.pdf)
### Comparison of experimental and simulated results
The left panel in Fig. \[fig:Experiment\_Simulation\] presents the experiment of dust particle confinement in a slim Paul trap. Aluminum oxide Al$_{2}$O$_{3}$ ($\rho_p = 0.38 \times 10^4$ kg/m$^3$) dust particles with radius $r_p =1 0 \div 15 \ \mu$m are injected inside a two wire trap (2WT). The trap electrodes are supplied at an a.c. voltage $U_{\omega}\sin({\omega}t)$, where $U_{\omega} = 4400$ V. The estimated value of the electrical charge of trapped microparticles is $Q_{p} \sim 10^5 \div 5\times 10^5 \ e$. The experimental trap parameters are: wire (electrode) length – $L_{m} = 15$ cm, distance between wires – $L_{b} = 1.3$ cm, endcap electrode spacing – $L_{h} = 6$ cm, endcap electrode length – 4.5 cm. The d.c. voltage supplied to the endcap electrodes is $900$ V.
A numerical simulation of this experiment has been performed. The values of the trap parameters used in the simulation are: $r_p = 10 \ \mu$m, $\rho_p = 0.38 \times 10^4$ kg/m$^3$, and $Q_p =4.5 \times 10^5 e$. The simulation results on dust particle confinement are presented in the right panel of Fig. \[fig:Experiment\_Simulation\]. The experimental and numerical simulation results are in good agreement, as they validate each other.
![Comparison between experimental results and numerical simulations with respect to dust particle confinement in a linear Paul trap. Left panel – the typical experimental observation of the particle confinement, right panel – numerical simulation of experiment. Experimental parameters: $f = 50$ Hz , $U_{end} = 900$ V, $U_{\omega} = 4400$ V, $r_p = 10 \div 15 \ \mu$m, $\rho_p = 0.38 \times 10^4$ kg/m$^3$ , $\eta = 17 \times 10^{-6}$ Pa$\cdot$s, $T =300$ K; Numerial simulation parameters: $r_{p}=10 \ \mu$m, $\rho_p = 0.38 \times 10^4$ kg/$m^3$, $Q_p = 4.5 \times 10^5 e$. Source: pictures reproduced from [@Lapi13] with permission of the authors.[]{data-label="fig:Experiment_Simulation"}](ExpReal_i.png "fig:") ![Comparison between experimental results and numerical simulations with respect to dust particle confinement in a linear Paul trap. Left panel – the typical experimental observation of the particle confinement, right panel – numerical simulation of experiment. Experimental parameters: $f = 50$ Hz , $U_{end} = 900$ V, $U_{\omega} = 4400$ V, $r_p = 10 \div 15 \ \mu$m, $\rho_p = 0.38 \times 10^4$ kg/m$^3$ , $\eta = 17 \times 10^{-6}$ Pa$\cdot$s, $T =300$ K; Numerial simulation parameters: $r_{p}=10 \ \mu$m, $\rho_p = 0.38 \times 10^4$ kg/$m^3$, $Q_p = 4.5 \times 10^5 e$. Source: pictures reproduced from [@Lapi13] with permission of the authors.[]{data-label="fig:Experiment_Simulation"}](Experiment_i.pdf "fig:")
Hamiltonians for systems of $N$ particles. Semi-classical dynamics in combined 3D traps. Ordered structures
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Collective dynamics for systems of ions confined in quadrupole 3D traps with cylindrical symmetry is characterized in [@Major05; @Ghe00]. We consider a system consisting of $N$ ions (particles) in a space with $d$ dimensions, ${\mathbb R}^d$. The coordinates in the space (manifold) of configurations ${\mathbb R}^d$ are denoted by $x_{\alpha j}\ , \; \alpha = 1, \ldots , N\ ,\; j=1, \ldots , d$. In case of linear, planar or space models, we choose $d = 1$, $d = 2$ or $d = 3$, respectively. We introduce the kinetic energy $T$, the linear potential energy $U_1$, the quadrupole trap potential energy $U$, and the anharmonic trap potential $V$ [@Pedre10; @Pedre18a; @Pedre18b]:
\[stq2\] $$\label{stq2a}T = \sum_{\alpha = 1}^N \sum_{j=1}^d \frac 1{2m_\alpha }\ p_{\alpha
j}^2 \ , \ \ U_1 = \frac 12 \sum_{\alpha =1}^N \sum_{j=1}^d \delta_j x_{\alpha j}\ ,\;$$ $$\label{stq2b}
U = \frac 12 \sum_{\alpha = 1}^N \sum_{i, j=1}^d \kappa _{ij} x_{\alpha j}^2 \ , \ \ V = \sum_{\alpha = 1}^N v\left( {\mathbf x}_\alpha , t \right)$$
where $m_\alpha $ is the mass of the ion denoted by $\alpha $, ${\mathbf x}_\alpha = \left(
x_{\alpha 1}, \ldots , x_{\alpha d} \right) $, and $\delta_j$ and $\kappa_{ij} $ might eventually be functions that depend on time. The Hamilton function associated to the system of $N$ ions, denoted by $H$, can be cast as $$H = T + U_1 + U + V + W \ ,$$ where $W$ represents the interaction potential between the ions.
If the ions (particles) possess equal masses, we introduce $d$ coordinates $x_j$ of the CM of the system $$\label{mp9}x_j = \frac 1N \sum_{\alpha =1}^N x_{\alpha j} \ , \;$$ and $d\left( N - 1\right) $ coordinates $y_{\alpha j}$ with respect to the relative motion of the ions
$$\label{mp9b}\ y_{\alpha j} = x_{\alpha j} - x_j \ , \;\ \sum_{\alpha = 1}^N y_{\alpha j}=0 \ .$$
Further on we introduce $d$ collective coordinates $s_j$ and the collective coordinate $s$, defined as $$\label{mp10b}s_j = \sum_{\alpha =1}^N y_{\alpha j}^2 \ , \; s = \sum_{\alpha = 1}^N \sum_{j=1}^d y_{\alpha j}^2\ .$$ By calculus, we obtain $$\label{mp10}\sum_{\alpha =1}^N x_{\alpha j}^2 = N x_j^2 + \sum_{\alpha = 1}^N y_{\alpha j}^2 \ .$$ $$\label{mp11}\ s_j = \frac 1{2N} \sum_{\alpha, \beta = 1}^N \left( x_{\alpha j} - x_{\beta j}\right)^2 \ , \; \ s = \frac 1{2N} \sum_{\alpha , \beta = 1}^N \sum_{j=1}^d \left( x_{\alpha j} - x_{\beta j}\right)^2 \ .$$
From eq. (\[mp11\]) it results that $s$ represents the square of the distance measured from the origin (fixed in the CM) to the point that corresponds to the system of $N$ ions from the space (manifold) of configurations. The relation $s = s_0$ with $s_0 > 0$ constant determines a sphere of radius $\sqrt{s_0}$, with its centre located in the origin of the space (manifold) of configurations. In case of ordered structures of $N$ ions, the trajectory is restricted within a neighborhood $\left \|s-s_0\right\| < \varepsilon $ of this sphere, with $\varepsilon $ sufficiently small. On the other hand the collective variable $s$ can be also interpreted as a dispersion:
$$\label{mp11b}s = \sum_{\alpha =1}^N \sum_{j=1}^d \left( x_{\alpha j}^2 - x_j^2 \right) \ .$$
We further introduce the moments $p_{\alpha j}$ associated to the coordinates $x_{\alpha j}$. We also introduce $d$ impulses $p_j$ of the CM, and $d \left( N - 1 \right) $ moments $\xi _{\alpha j}$ of the relative motion characterized as
$$\label{mp11c}p_j = \frac 1N \sum_{\alpha =1}^N p_{\alpha j}\ ,\; \ \xi _{\alpha j}=p_{\alpha j} - p_j \ , \; \ \sum_{\alpha = 1}^N \xi _{\alpha j} = 0 \ ,$$
with $p_{\alpha j} = - i\hbar \left( \partial / \partial x_{\alpha j} \right)$. We also introduce $$D_j = \frac 1N \sum_{\alpha = 1}^N \frac{\partial}{\partial x_{\alpha j}} \ , \;\; D_{\alpha j} = \frac{\partial}{\partial x_{\alpha j}} - D_j \ , \;\; \sum_{\alpha = 1}^N D_{\alpha j} = 0$$ In addition $$\sum_{\alpha = 1}^N \frac{\partial^2}{\partial x_j^2} = ND_j^2 + \sum_{\alpha = 1}^N D_{\alpha j}^2$$
When $d=3$ we denote by $L_{\alpha 3}$ the projection of the angular momentum of the $\alpha $ particle on the axis $3$. Then, the projections of the total angular momentum and of the angular momentum of the relative motion on the axis $3$, denoted by $L_3$ and $L_3^{\prime }$ respectively, satisfy
\[mp12\] $$\label{mp12a}\sum_{\alpha = 1}^N L_{\alpha 3} = L_3 + L_3^{\prime }\; , \; \; L_{\alpha 3} = x_{\alpha 1} p_{\alpha 2} - x_{\alpha 2} p_{\alpha 1} \ ,$$ $$\label{mp12b}L_3 = p_1 D_2 - p_2 D_1 \ ,\; L_3^{\prime } = \sum_{\alpha = 1}^N \left( y_{\alpha 1} \xi_{\alpha 2} - y_{\alpha 2} \xi_{\alpha 1} \right) \ .$$
In case of a quadrupole combined (Paul and Penning) trap that exhibits cylindrical symmetry, for a constant axial magnetic field $B_0$, the Hamilton function for the system of $N$ ions of mass $M$ and equal electric charge $Q$ can be cast into [@Major05; @Mih18; @Mih11]
$$\label{mp13}
H = \sum_{\alpha = 1}^N \left[ \frac 1{2M}\sum_{j=1}^3 p_{\alpha j}^2 + \frac{K_r}2 \left( x_{\alpha 1}^2 + x_{\alpha 2}^2 \right) + \frac{K_a}2 x_{\alpha 3}^2 - \frac{\omega_c}2 L_{\alpha 3}\right] + W \ ,$$
with $$K_r = \frac{M\omega_c^2}4 - 2 Q c_2 A \left( t\right) \ ,\; K_a = 4 Q c_2 A\left( t\right) \ ,\;\omega_c = \frac{Q B_0}M\ ,$$ where $\omega_c$ is the cyclotronic frequency in the Penning trap, $c_2$ depends on the trap geometry, and $A\left( t\right) $ represents a function that is periodic in time [@Mih18]. Then $H$ can be expressed as the sum between the Hamiltonian of the CM of the system $H_{CM}$, and the Hamiltonian $H^{\prime }$ which characterizes the relative motion of the ions:
\[mp14\] $$\label{mp14a}
H = H_{CM} + H^{\prime }\ ,$$ $$\label{mp14b}
H_{CM} = \frac 1{2NM} \sum_{j=1}^3 p_j^2 + \frac{NK_r}2 \left( x_1^2 + x_2^2 \right) + \frac{NK_a}2 x_3^2 - \frac{\omega_c}2 L_3\ ,$$ $$\label{mp14c}
H^{\prime} = \sum_{\alpha = 1}^N \left[ -\frac{\hbar ^2}{2M} \sum_{j=1}^3\xi_{\alpha j}^2 + \frac{K_r}2 \left( y_{\alpha 1}^2 + y_{\alpha 2}^2\right) +\frac{K_a}2y_{\alpha 3}^2\right] -\frac{\omega_c}2L_3^{\prime} + W \ .$$
We assume $W$ as an interaction potential that is invariant to translations (it only depends of $y_{\alpha j}$). The ordering of ions within the trap can be illustrated by standard numerical programming, using the Hamilton function given below
$$\begin{gathered}
\label{mp15}
H_{sim} = \sum_{i=1}^n \frac 1{2M} \vv{p_{i}}^{2} + \sum_{i=1}^n \frac M2\left( \omega_1^2 x_i^2 + \omega_2^2 y_i^2 + \omega _3^2z_i^2\right) + \\
\sum_{1 \leq i < j \leq n}\frac{Q^2}{4 \pi \varepsilon_0}\frac 1{\left| \vec r_i - \vec r_j \right| }\ ,\end{gathered}$$
where the second term describes the effective electric potential of the electromagnetic trap and the third term accounts for the Coulomb repulsion between the ions (particles).
Applications: The electrodynamic trap with corona discharge
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Multiple experiments have been performed under SATP conditions, in an attempt to generate structures of electrically charged dust particles in the thermal plasma of a gas burner or in a corona discharge. In case of the thermal plasma Coulomb crystals do not occur [@Fort05], due to strong gas flows and high temperature gradients that result in fast destruction of dust particles in the high temperature gas. Since many years the corona discharge was used to remove dust in electrostatic precipitators, to separate particles, and to achieve colouring of powder. Therefore, the mechanisms used to electrically charge dust particles as well as their subsequent dynamics in the corona discharge are quite well known [@Cha95]. Nevertheless, stable dusty structures in the corona discharge have not been obtained yet.
The possibility to use a corona discharge in a nuclear excited dusty plasma, with an aim to achieve better stability of the plasma-dust structures and to accomplish a more efficient conversion of nuclear energy into radiation, has been investigated in [@Fort10b]. The electric field distribution and other characteristics of the corona discharge in a nuclear excited dusty plasma have been obtained in the special case of a cylindrical geometry, at pressures ranging between $1 \div 100$ atm. A mathematical model was developed to describe the behaviour of dust particles in a nuclear excited plasma. The model can characterize the following basic physical processes: (1) screening of Coulomb forces, (2) energy exchange and the stochastic nature of the interaction of dust particles with a buffer gas and the surrounding plasma, and (3) the strong spatial inhomogeneity of a nuclear excited plasma. Calculations for potential and nonpotential forces that act on dust particles have been carried out. In case of potential forces, stationary crystal-like dust structures are observed. For nonpotential forces, ordered vortices rotating towards each other are reported [@Ryk02a]. The production of well ordered plasma-dust structures out of a fissionable material is also shown to be possible under conditions of a corona discharge, in a nuclear excited plasma. Experimental studies of an argon-xenon gas mixture [@Ryk02a] demonstrate that micron sized dust particles can evolve into levitating stable and rotating ordered structures, and even the occurrence of dust crystals in an adequate trap is demonstrated.
A study on the possibility and the conditions required to confine dust particles in a linear Paul trap operating under SATP conditions, in a corona discharge plasma or in a nuclear-excited plasma is performed in [@Lapi13]. The behaviour of dust particles is simulated by means of Brownian dynamics [@Chav11]. Numerical simulations are carried out for an adequate choice of the dust particle parameters and of the a.c. trapping voltages. Optimum values for the dust particle parameters and for the trapping voltage are established, with an aim to achieve stable confinement. The simulations performed allow one to identify the regions of stable confinement, as well as the influence of the particle mass and electric charge, or of the trapping voltage and frequency. The results of numerical simulations are in good agreement with the experimental results obtained in a linear Paul trap [@Lapi13].
In the framework of the statistical theory of liquids, the thermodynamic parameters of a strongly nonideal Coulomb microparticle structure confined in a linear Paul trap (operated under SATP conditions) are calculated using the Brownian dynamics method. The Coulomb potential of inter-particle interaction and the calculated pair correlation functions of the Coulomb structure are used. The average inter-particle interaction parameter (the coupling parameter $\Gamma$), the internal energy of the Coulomb structure, and the pressure it imposes on the trap are inferred. It has been found that these parameters decrease with increasing size and charge of particles, caused by an increase in the average equilibrium inter-particle distance in the electrodynamic trap. As the system approaches a steady state, the energy and pressure also drop off due to an increase in the average inter-particle distance caused by a partial ordering of the Coulomb system of particles [@Lapi19a].
The electrodynamic trap that creates a corona discharge is investigated in [@Vlad18]. Neutral dust particles are injected into the trap, where they are electrically charged in the plasma of the corona discharge. As a result the trap confines charged particles in presence of a plasma cloud. The electrodynamic trap consists of four horizontally positioned electrodes made of wire. Each electrode is 10 cm long and its diameter is 300 $\mu$m. The distance between the electrodes is 10 mm. The Corona discharge between the electrodes is ignited using a voltage of 5 kV. The ac voltage frequency is 50 Hz. The experiments use polydisperse Al$_2$O$_3$ powder [@Vlad18]. To observe the associated dynamics particles are illuminated using a laser that operates at a wavelength of 532 nm, at a maximum power of 150 mW. Particles are recorded by means of a HiSpec 1 digital camera, at a maximum resolution of $1280 \times 1024$ pixels. The camera allows recording at a maximum frame rate of up to 506 frames/s.
The electrically neutral particles are injected inside the trap from the upper side. The particles acquire electric charge in the field of a corona discharge during their downwards motion. Then some of the particles are captured inside the trap. For particles with a diameter larger than 1 $\mu$m the main charging mechanism is field charging. In this case the charge of the particle can be estimated using the expression [@Pau32]: $Q = 3 \varepsilon_0 E d^2 \left( 1 + 2\frac{\varepsilon - 1} {\varepsilon + 2}\right) $, where $E$ is the electric field intensity, $\varepsilon_0$ represents the (absolute) vacuum permittivity, $d$ is the size of the particle, and $\varepsilon$ stands for the relative permittivity. The electric field intensity essentially depends on the particle position inside the trap. For example, a 10 $\mu$m diameter particle located near an electrode can accumulate up to $4\times 10^4$ units of electron charge $e$. The captured particles create a stable structure near the trap axis. The presence of the electric wind caused by the direct movement of ions complicates the particle confinement mechanism in a trap with corona discharge, with respect to the case of a classical electrodynamic trap. The photo of a stable structure of aluminium oxide particles in a linear electrodynamic trap with corona discharge is shown in Fig. \[coroneld\]. The amplitude of the a.c. voltage is 7 kV and the electric current value is around 80 $\mu$A. The trapped particles size is estimated to lie between $10 \div 40 \ \mu$m [@Vlad18]. The dimensions of the structure are $1.4 \times 0.2 \times 0.15$ cm, and it consists of approximately 55 particles. The average inter-particle distance is 0.8 mm.
![Stable structure of aluminium oxide particles in a linear electrodynamic trap with corona electrodes. Source: image reproduced from [@Vlad18] with permission of the authors.[]{data-label="coroneld"}](coroneld.png)
Applications: Charged microparticle confinement by oscillating electric fields in a gas flow {#gasflow}
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Dust particles (microparticles) are often present in the atmosphere around nuclear power plants or in energy devices, such as etching and fusion installations. Removal of microparticles out of these devices is an important issue. One of the ways to improve the filtration efficiency lies in perturbing the microparticles by various physical factors, with an aim to alter their physical properties. For example, in case of electrostatic filters the microparticles build up electric charge in the corona discharge, and then accumulate at the electrodes [@Peuk01]. Under certain conditions a corona discharge sensibly enhances the efficiency of electrostatic filters, thus providing a high degree of cleaning [@Von67; @Fink89]. However, electrostatic filters are not efficient in capturing microparticles with dimensions ranging between $0.6 \div 1.6 \ \mu$m. Unfortunately, the issues of selective particles removal cannot be solved by corona discharge precipitators.
Confinement of electrically charged microparticles in an electrodynamic trap has been investigated under static conditions in gas media, for SATP conditions [@Vasi13; @Lapi13]. The regions of particle capture have been studied for a wide range of parameters such as: the microparticle electric charge, mass and radius, the oscillating electric field strength and frequency. Different versions of improved linear electrodynamic trap geometries that achieve a more effective capture of microparticles have been suggested in [@Ghe98; @Sto01; @Mih08; @Lapi13; @Ghe96a].
![Regions of confinement versus the microparticle diameter $d_p$. The regions are located between the upper and lower bounds presented by the corresponding lines for different electric field frequencies: $1$ – $f$ = 30 Hz, $2$ – $f$ = 50 Hz, $3$ – $f$ = 100 Hz, $4$ – $f$ = 200 Hz. Line $5$ that describes $\Phi_m$ and the region denoted as 6 are related to the trap and particle parameters used in experiments performed in [@Lapi15c]. Source: picture reproduced (modified) from [@Lapi15c] under permission of the authors.[]{data-label="Hz"}](HzGray.pdf)
Ref. [@Lapi15c] presents theoretical and experimental studies of charged microparticle capture in a linear Paul trap, in gas flow. The regions of microparticle and trap parameters required to achieve particle confinement are investigated under normal SATP conditions. The model used shows that the interaction force between the microparticle and the trap electrodes (2) depends on: $$\label{intforce}
\Phi_p = \frac{U_{ac}Q_p}{2 \ln(R2/R1)} \ .$$ The numerical simulation performed in [@Lapi15c] considers a microparticle characterized by a density $\rho_p = 3.99$ g/cm$^3$. We mention that $U_{ac}$ stands for the trapping voltage, while $R_2$ and $R_1$ represent the radii of the grounded cylindrical shell surrounding the trap and trap electrode respectively.
Fig. \[Hz\] presents the confinement regions for charged microparticles (areas $1$, $2$, $3$, $4$), in case of a gas flow velocity value around $5$ cm/sec. The lower and upper bounds of particle confinement regions are denoted by the corresponding lines. Above the upper bound the electric field is strong enough to push the microparticle out of the trap. Below the lower bound the particle cannot be captured, as the trap field cannot compensate gravity forces. The black line (denoted as 5) corresponds to
$$\Phi_m = \frac{U_{ac}Q_m}{2 \ln(R2/R1)}, \ Q_m = 3 \pi \varepsilon_0 d^2 E \left( 1 + 2\ \frac{\varepsilon - 1} {\varepsilon + 2}\right) \ ,$$
where $Q_m$ is the maximum electric charge that a particle can accumulate in the corona discharge [@Pau32]. Within the simulations performed the magnitude of the electric field strength in the corona discharge was chosen $E = 20$ kV/cm. The shaded region (denoted as 6) characterizes microparticles used in the experiments discussed below.
From Fig. \[Hz\] one can draw the following conclusions:
- the regions of confinement for a particle with a diameter less than 10 $\mu$m are located above curve $5$, so these particles cannot be trapped,
- confinement regions for frequencies ranging between $f \sim 30 \div 50$ Hz (areas $1$ and $2$) start for particle dimensions around $d \sim 0.6 \mu$m; in order to capture a particle of smaller dimensions, the electric field frequency has to reach a value of about $\sim 100$ Hz (areas $3$ and $4$),
- as the electric field frequency rises, the voltage amplitude value required to achieve particle capture also increases.
Fig. \[Hz2\] shows the analogous regions of confinement versus microparticle diameter at frequency $f = 100$ Hz, for gas flow velocities equal to $5$, $10$, and $20$ cm/sec respectively. Fig. \[Hz2\] also illustrates that the confinement region becomes narrower by increasing the gas flow velocity, owing to the shift of the lower bound (border) upwards for virtually an unmodified position of the upper bound.
![Regions of confinement (between corresponding lines) versus microparticle diameter $d_p$ for different gas flow velocities: $1$ – $v_f = 5$ cm/s, $2$ – $v_f = 10$ cm/s, $3$ – $v_f = 20$ cm/s. Line $4$ corresponds to $\Phi_m$. Source: picture reproduced from [@Lapi15c] under permission of the authors.[]{data-label="Hz2"}](Hz2Gray.pdf)
Fig. \[Hz3\] shows the upper bound of the particle capture region as a function of the gas flow velocity versus the electric field frequency (black line). The physical reason for the black line rise lies in the fact that all dynamical time scales associated to particle motion in the trap should be reduced by augmenting the gas flow velocity.
![The upper bound of the particle capture region. $\Phi_p = 5 \times 10^{-13} \div 5.3 \times 10^{-9}$ J ($q_p = 4 \times 10^3 \div 4.2 \times 10^7 \ e$, $U_\omega = 8$ kV, $d_p = 2 \ \mu$m). Source: picture reproduced from [@Lapi15c] under permission of the authors.[]{data-label="Hz3"}](Hz3.pdf)
### Experimental setup for the study of particle capture in gas flows
To study the physical mechanisms responsible for particle capture, the charged particles have to be drifted by the gas flow across the trap. An image of the experimental setup is presented in Fig. \[ExSheme\]. The setup consists of three separate modules located in a gas channel: $1$ represents the corona discharge module used to charge microparticles, $2$ is the trap module, and $3$ stands for the air-exhauster module.
![The sketch of the experimental setup used for charged microparticle capture within a linear trap, in a gas flow. The gas channel consists of 3 modules: $1$ represents the corona discharge module, $2$ is the trap module, $3$ denotes for the air-exhaust module. Source: image reproduced (modified) from [@Lapi15c] under permission of the authors.[]{data-label="ExSheme"}](ExScheme.pdf)
The corona discharge module consists of one row of discharge electrodes, and two rows of grounded electrodes located above and below the discharge electrodes, at a distance of 12 mm apart. The discharge electrodes are made of wires with a diameter of $70 \ \mu$m, arranged at a distance of 1 cm apart with respect to each other. A d.c. voltage $U_c$ with a peak value up to $15$ kV is applied to the discharge electrodes. The grounded electrodes are made of metal rods with $d = 3$ mm diameter. This particular design generates an ion breeze (wind) in two opposite directions, thus compensating its influence on the column of air within the channel.
Diagnostics and observation of microparticles is achieved using a laser light sheet with a 1 cm spot. The sheet height allows microparticle observation both within the trap volume and outside of it. The laser light sheet parameters are: wavelength $532$ nm and power up to $230$ mW. Microparticles are recorded using a HiSpec 1 Fastec Imaging camera (with a characteristic resolution of $1280 \times 1024$ pixels), located along the laser light sheet (Fig. \[ExSheme\]).
The air-exhaust module is located at the end of the gas channel. The exhaust fan blows out (generates) a gas flow with velocities that reach up to $v_f = 50 \pm 11$ cm/s. Polydisperse aluminium oxide Al$_2$O$_3$ powder was used in the experiments [@Lapi15c]. The microparticle density is $\rho_p = 3.99$ g/cm$^3$ for a typical size ranging between $4 \div 80 \ \mu$m.
The experiment starts by turning off the exhaust fan and by supplying a d.c. voltage $U_c = 15$ kV to the discharge electrodes. Then, particles are injected in the corona module. By traveling through the corona discharge area the particles acquire a positive charge $q$ and fall inside the trap module, where they are captured between the electrodes. Fig. \[StrNoStr\](a) shows an example of the stable Coulomb structure of charged microparticles captured in the trap. The captured microparticles oscillate around equilibrium positions at a frequency of 50 Hz. Most of the particles are captured inside the trap below the central axis, while others are trapped above it.
![The structure of charged microparticles captured by the electrodynamic trap ($a$) in a static gas media ($v_f = 0$ cm/s) and ($b$) in the gas flow ($v_f = 50$ cm/s). Source: image reproduced from [@Lapi15c] under permission of the authors.[]{data-label="StrNoStr"}](StrNoStr.pdf)
To study the effect of the gas flow on an ensemble of captured microparticles, the exhaust fan is turned on. Most of the microparticles are blown out of the trap and only a few particles remain trapped inside it (see Fig. \[StrNoStr\](b)). The particle structure shifts below the central axis of the trap if the inter-particle distances increase.
We emphasize that the regions of particle trapping observed in the experiments turn out to be larger than those obtained from numerical simulations. Thus, from Fig. \[Hz3\] it follows that particle capture at a frequency $f = 50$ Hz is possible if the gas flow velocity is lower than $v_f$ = 10 cm/s, while in experiments the capture is observed for $v_f$ = 50 cm/s. To explain this deviation let us note that in simulations the drag force of gas flow is estimated using the Stokes formula for spherical particles, while in the experiments performed the particle shape varies from spherical to slab. An alternative estimation of the drag force can be obtained from Newton’s regime [@Soo67] for particles of different shapes. In that case the drag force is characterized by the expression $F_{\mathrm{drag}} = \rho_f {v_f}^2 S_p C_x/2$, where $\rho_f$ is the mass density of the gas, $v_f$ is the velocity of a microparticle relative to the velocity of the gas flow, $S_p$ is the reference area of the microparticle, and $C_x$ is the drag coefficient that lies between $0.09 \div 1.15$. The dependence of the Stokes force and of the drag force $F_{\mathrm{drag}}$ versus the microparticle diameter is illustrated in Fig. \[Stok\], for a value of the gas velocity $v_f = 50$ cm/s. In Fig. \[Stok\] the characteristic Stokes force is at least ten times larger than the drag force $F_{\mathrm{drag}}$, under identical conditions. Therefore, to achieve microparticle capture the gas velocity value could be higher, while the value of $\Phi_p$ might be lower. The capture regions in Fig. \[Hz\] and Fig. \[Hz2\] are shifted downwards, while the capture region in Fig. \[Hz3\] is shifted upwards. The numerical estimations of gas velocity for dust particle capture validate the experimental results.
![Dependence of Stokes and drag forces on microparticle diameter $d_p$. Source: picture reproduced (modified) from [@Lapi15c] under permission of the authors.[]{data-label="Stok"}](Stokes.pdf)
To measure the dimensions of microparticles captured in the electrodynamic trap we employ two methods that are explained in Fig. \[GetP\] (a) and (b). The first method (see Fig. \[GetP\] (a)) uses a hole in the wall of the gas channel between the trap electrodes. During the microparticle confinement experiment the hole is closed. When microparticles are captured the exhaust fan is turned off, the hole is opened, and the clean ebonite wand that is initially negatively charged due to friction is inserted through the hole inside the gas channel. Positively charged microparticles are then captured by the ebonite wand that is extracted from the gas channel afterwards. Then, the ebonite wand is discharged and microparticles are removed by shaking it and precipitate on the subject glass. The second method (see Fig. \[GetP\] (b)) uses a slit in the wall of the gas channel under the trap. The slit is closed during the experiment. After the microparticles are captured, the exhaust fan is turned off and the slit is opened. The clean subject glass is inserted within the gas channel. The electrodynamic trap is then turned off and microparticles fall down on the subject glass. For both methods used the microparticle dimensions are measured by means of a microscope.
![Picture of the mechanism of captured microparticles collection: ($a$) by using an electrically charged ebonite wand, and ($b$) by means of a subject glass slide. Source: picture reproduced (modified) from [@Lapi15c] under permission of the authors.[]{data-label="GetP"}](GetP.pdf)
In contrast to the simulations performed where spherical shaped particles were considered, the experiments use powder of polydisperse microparticles of complex shape. To define the effective particle size for each microparticle, the smallest particle size $d_{\mathrm{min}}$ and the largest one $d_{\mathrm{max}}$ have been considered. The effective microparticle diameter is defined as $d_p = \frac{d_{\mathrm{min}} + d_{\mathrm{max}}}{2}$. The accuracy of the microparticle diameter measurement is around 4 $\mu$m. Fig. \[Histo\] shows three distributions of microparticles versus the effective particle sizes. The first one represents the initial distribution $P_i\left(d_p\right)$ of particles introduced in the flow. The second distribution refers to microparticle dimensions obtained by using the ebonite wand $P_e\left(d_p\right)$, while the third distribution is the one obtained by using the subject glass slide $P_g\left(d_p\right)$.
![The initial and captured microparticle distributions: $1$ represents the distribution of particles on the subject glass; $2$ stands for the distribution of particles entrained by the ebonite wand; $3$ is the initial distribution. The average diameter of the microparticles in the experiment (curves $1$ and $2$) is $d_p = 32 \ \mu$m with rms deviations $\sigma = 0.02$ and $\sigma = 0.01$, respectively. Source: picture reproduced (modified) from [@Lapi15c] under permission of the authors.[]{data-label="Histo"}](HistoGray1.pdf)
![Relative distribution of particles versus the effective sizes. Source: picture reproduced (modified) from [@Lapi15c] under permission of the authors.[]{data-label="Ratio"}](Ratioi.pdf)
Fig. \[Ratio\] presents the function $P\left(d_p\right)$ described by the expression $$P\left(d_p\right) = \frac{P_e\left(d_p\right) + P_g\left(d_p\right)}{2P_i\left(d_p\right)}\ .$$ The function $P\left(d_p\right)$ represents in fact the relative distribution of particles depending on their effective dimensions. In Fig. \[Ratio\] the maximum of the captured microparticles distribution is shifted towards small sized particles, with respect to the initial powder. The result indicates that the trap is selective as the capture process is more efficient for smaller sized particles.
Applications: Cleaning dusty surfaces using electrodynamic traps
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In Ref. [@Deput19] a method is proposed to achieve cleaning of dusty surfaces, as an application of linear electrodynamic traps. A linear quadrupole electrodynamic trap (shown in Fig. \[trap\]) consisting of four parallel cylindrical steel electrodes, each one with a diameter of 3 mm and a length of 30 cm, has been designed and tested. The electrodes are located on the vertices of a square that has a side of 2 cm. The a.c. voltage $U_a\sin(\omega t)$ at frequency $f = \omega /2\pi = 50$ Hz is applied to the trap electrodes (where opposite electrodes are in phase), while the electric potential phase shift between neighboring electrodes is equal to $\pi$. The electrodynamic trap is placed inside an optically transparent plastic box to guard against instabilities induced by air flows [@Lapi18b]. To forbid particle escape at the left end of the trap (see Fig. \[trap\]) an additional electrode is mounted, supplied with a d.c. electrical potential $U = 1$ kV. We remark that trapped particles are restricted to escape even when there is no endcap electrode present, as a result of the repulsive potential that occurs, created by charged particles that accumulate at the left wall of the plastic box. Electrically neutral polydisperse Al$_2$O$_3$ particles are placed on a glass substrate located below the trap. Video recording of the particles that are illuminated by means of a laser beam is achieved using a HiSpec 1 video camera. All experiments are carried out under SATP conditions.
![Trap design, front and side views. Notations: 1$ \div$ 4 – trap electrodes, 5 – endcap electrode, 6 – levitated particles, 7 – glass substrate with dust particles, 8 – optically transparent plastic box. Source: picture reproduced (modified) from [@Deput19] under authors permission.[]{data-label="trap"}](trap.pdf)
#### Experimental results. Capture and retention of particles
As the amplitude $U_a$ of the a.c. voltage increases to a critical value (for example 2.5 kV, in case of a 0.5 cm distance between the substrate and the trap electrodes), the electric field begins to attract and then capture dust particles. Fig. \[fig9.2\] shows images of the Al$_2$O$_3$ particles captured within the trap. These particles are attracted from the glass substrate located at a distance of 0.5 cm apart with respect to the trap electrodes. By observing the space delimited by the substrate (3) and the electrode (1), one can notice how particles are dragged from the substrate towards the trap. Particles captured by the trap that slowly move towards the free end are also visible. When the $U_a$ voltage value rises from 3.5 kV (Fig. \[fig9.2\]$a$) to 5 kV (Fig. \[fig9.2\] $b$), the number of confined particles significantly increases. Fig. \[fig9.2\] $c$ shows the open end of the trap. We notice how trapped particles shift towards the open end, then leave the trap and fall down. This effect can be used to clean very dusty surfaces, and then collect the trapped particles into a special container located at the open end.
![Images of Al$_2$O$_3$ particles dragged from the glass substrate located at a distance of 0.5 cm with respect to the trap electrodes, for different trapping voltage amplitudes: $a)$ $U_a = 3.5$ kV, $b)$ and $c)$ $U_a = 5$ kV; 1 and 2 indicate the trap electrodes, 3 denotes the glass substrate that holds the particles. Source: images reproduced (modified) from [@Deput19] under authors permission.[]{data-label="fig9.2"}](figure2.pdf)
Fig. \[fig9.3\] shows images of a glass substrate located at 1 cm distance (\[fig9.3\]a), 0.5 cm distance (\[fig9.3\]b), and 0.15 cm distance (\[fig9.3\]c) with respect to the trap electrodes supplied at $U_a = 5$ kV. Dark areas in the images indicate substrate regions that are cleaned of particles. These areas are located near the projection of electrodes (indicated by solid lines) on the substrate. When the distance between the substrate and the electrodes decreases, the surface area rendered free of particles increases. Fig. \[fig9.3\]d shows an image of the substrate obtained after it was shifted normal to the axis of the electrodes, parallel to its own surface. The wide dust-free areas are clearly visible. Dashed lines denote the initial projections of the trap electrodes on the substrate, while solid lines mark their final position.
![Images of a cleaned substrate placed at different distances $l$ apart with respect to the electrodes of the electrodynamic trap for $U_a = 5$ kV: $a) l = 1$ cm, $b)$ and $d) l = 0.5$ cm, $c) l = 0.15$ cm. Projections of the electrodes are represented by solid lines. In image d) the initial and final projections of the electrodes are recorded by dashed and solid lines, respectively. Source: images reproduced (modified) from [@Deput19] under authors permission[]{data-label="fig9.3"}](figure3.pdf)
Fig. \[fig9.4\] shows an image of the particle trajectories (observed from the end of the trap) for an exposure time that is equal to one period $T = 2\pi/\omega$ of the a.c. voltage. As illustrated some of the particles are dragged inside the trap. A certain number of the particles exhibit curved trajectories near the electrodes, an indication of the phenomenon of particle reflection followed by a shift (movement) oriented downwards. Fig. \[fig9.4\]b shows an enlarged image of a single particle trajectory near the bottom right electrode (the electrode is marked by a black circle, while the white spot represents a glare). The particle is reflected when the electrode changes its polarity. An analysis of Fig. \[fig9.4\] allows us to advance an interesting assumption about the mechanism of particle drag into the trap. Electrical charging of dust particles might be caused by polarization and triboelectric physical mechanisms. A neutral dust particle located on a substrate is polarized by the electric field. Then, the particle begins to move and accumulates electric charge as an outcome of friction with the substrate and with other particles [@Low80].
![The trajectory of particles in an electrodynamic trap (observed from the end of the trap). Voltage amplitudes: a) $U_a = 4$ kV, b) $U_a = 3$ kV. Source: image reproduced (modified) from [@Deput19] under permission of the authors[]{data-label="fig9.4"}](figure4.pdf)
The charged particles whose velocity is directed into the trap (for an appropriate polarity of the electrode) can be captured. However, even in this case the trap not will hold all particles. Fig. \[fig9.5\] shows several successive images of the trajectory of a particle whose velocity is directed towards the trap volume, after reflection from the electrode (Fig. \[fig9.5\]a). Since the voltage amplitude value $U_a = 2.5$ kV is too low for confinement, the particle falls down (Fig. \[fig9.5\]b, \[fig9.5\]c).
![The trajectory of the particle (marked by an arrow) for an a.c. voltage amplitude $U_a = 2.5$ kV, a value that is insufficient to achieve trapping. The time interval between two successive images is $3T$, while the exposure time is equal to $T$ (the period of the a.c. voltage). Source: images reproduced (modified) from [@Deput19] under permission of the authors.[]{data-label="fig9.5"}](figure5.pdf)
When the a.c. voltage amplitude increases, the conditions for particle uplift and capture are enhanced. Fig. \[fig9.6\] shows three images of the trajectory of a particle reflected from the electrode towards the trap volume, for a voltage amplitude value of 3.5 kV. In that case the confining forces generated by the oscillating electric field intensify and the particle climbs towards the trap centre. Other captured particles behave in a similar manner.
![The trajectory of the trapped particle (marked by an arrow), for an a.c. voltage amplitude $U_a = 3.5$ kV. The particle shifts towards the trap centre. The time interval between successive images is $3T$, while the exposure time is equal to $T$. Source: images reproduced (modified) from [@Deput19] under permission of the authors.[]{data-label="fig9.6"}](figure6.pdf)
Numerical simulation of dust particles behaviour in electrodynamic traps {#s:model}
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The experimental observations discussed above agree with the results of the numerical simulations performed with respect to charged particle confinement. Two independent numerical approaches have been used, namely the methods of Brownian and molecular dynamics. By using the methods specific to Brownian dynamics [@Vasi13; @Syr16a], we have performed numerical simulations of the motion of charged particles for different linear trap setups. The trapping conditions for both a single particle and an ensemble of particles have been obtained.
The idea behind the simulations was to find out the conditions which trigger the uplift of charged particles from the substrate, followed by their capture inside the trap. It is quite natural that for low trapping voltages the electric field is not able to lift the charged particle. Experimental results show that this issue occurs for trapping voltage amplitudes that are below 2.5 kV. As we increase to higher voltage amplitudes $U_a$, if the particle velocity at an initial moment of time is oriented towards the trap, the capture probability is high. Fig. \[fig9.7\] shows the simulated particle trajectory for $U_a = 3.7$ kV, using the molecular dynamics method. At an initial moment, the particle with a radius of 20 $\mu$m and an electric charge of around $10^5e$ is placed below the bottom of the trap. The particle velocity value is about 0.5 m/s, directed upwards. The value of the voltage amplitude under simulation was close to the experiment value of 3.5 kV, in Fig. \[fig9.6\]. The particle is captured inside the trap and its trajectory is similar to the one observed experimentally, as shown in in Fig. \[fig9.6\].
![The simulated trajectory of a particle whose velocity is directed upwards. The horizontal and vertical coordinates of the particle relative to the trap axis are plotted. Source: picture reproduced (modified) from [@Deput19] under permission of the authors.[]{data-label="fig9.7"}](figure7.pdf)
Using numerical simulations we have also tried to establish the conditions under which the charged particle is lifted from the surface of the substrate, enters the trap and then leaves it, as experimentally observed. Fig. \[fig9.8\] shows the simulation of the trajectory for a charged particle initially located under the trap (asterisk), for $U_a = 8$ kV; the particle radius is $r = 15 \ \mu$m and the electric charge value $q = 2.3 \times 10^6 e$. Numerical simulations have been performed using the Brownian dynamics method. The particle is lifted inside the trap where it follows a complex motion between the electrodes, before quickly falling down in the end as an outcome of the agreement in the direction of gravity and electric forces. Simulations performed allow us to identify the conditions that enable lifting and capture of charged particles inside the trap, for sufficiently large values of the electric charge and of the trapping voltage. Both the experiment and simulations performed demonstrate that in order to achieve particle confinement the electric charge should be about $3 \times 10^5 e$, while the voltage amplitude range would lie between $3 \div 5$ kV. For higher values of these parameters ($10^6e$, 8 kV) the efficiency of particle capture deteriorates significantly. This conclusion is important in practice, especially when using linear traps to clean dusty surfaces, since it indicates that an increase of the oscillating voltage amplitude does not always result into enhanced efficiency for particle capture mechanisms.
![Numerical simulation of the dynamics of a particle with a radius of $15 \ \mu$m and an electric charge of $2.3 \times 10^6 e$ captured inside the trap, for an a.c. voltage amplitude of 8 kV. The small dot close to the $X$ axis denotes the initial particle position. The horizontal and vertical coordinates of the particle relative to the trap axes are plotted. Source: picture reproduced (modified) from [@Deput19] under permission of the authors.[]{data-label="fig9.8"}](figure8.pdf)
In order to establish the regions of stable particle confinement, mathematical (numerical) simulations on the dynamics of a single particle or of an ensemble of charged particles confined in a quadrupole Paul trap have been carried out in [@Vasi13], for different values of the air pressure. The sketch of the quadrupole trap geometry used is shown in Fig. \[fig:trap\]. The trap consists of four cylindrical electrodes with diameter $d = 3$ mm and length $L_m = 15$ cm, supplied by means of an a.c. voltage, and two 4.5 cm long endcap electrodes supplied with a d.c. voltage. The endcap electrodes are located at each extremity of the geometrical trap axis, separated at a distance of $L_h = 6$ cm apart. The distance between the axes of dynamical electrodes is $L_b = 1.3$ cm.
![Sketch of the linear quadrupole trap. 1, 2, 3, 4 denote the trap dynamic electrodes, and 5 stands for the endcap electrodes. Source: picture reproduced (modified) from [@Vasi13] with permission of the authors.[]{data-label="fig:trap"}](Goodtrap.png)
In [@Lapi13] the motion of dust particles is described by means of the Brownian dynamics, taking into account the viscosity of the buffer gas and the stochastic forces that act on dust particles, owing to the neutral and plasma particles. The inter-particle interaction and dust particle interaction is taken into account by considering the Coulomb forces. The dynamics of dust particles is assumed to be described by a Langevin equation:
$$\label{Lang}
m_d\frac{d^2 r_i}{dt^2} = F_{tr}(r_i) + F_{int}\left( r_i \right) - 6 \pi \eta R_d \left( \frac{d r_i}{d t} - v_f \right) + F_{Br}\left( r_i \right) + F_{mg} \ ,$$
where $m_d$ is the dust particle mass, $r_j$ is a radius-vector for the j-th particle; j = 1, 2, ..., $N$; $N$ represents the number of dust particles; $F_{tr}(r_j) = -\nabla_i \overline{U}$ is an external potential force of the trap; $F_{int}(r_i) = -\nabla_i U$ is the force that describes pair interaction between particles, while $U$ stands for the potential energy of dusty particle interaction; $F_{mg}$ is the gravity force; $F_{Br}$ is the Langevin delta correlated source of forces which models random forces acting on a dust particle owing to the buffer gas particles; $R_d$ is the radius of a dust particle, $\eta \approx 0.02$ mPa $\cdot$ s represents the dynamical viscosity of air at SATP conditions, and $v_f$ is velocity of the air flow. The dust particle dynamics is simulated by the numerical integrator of the Langevin equation.
The physical factors that affect dust particle dynamics can be estimated by means of the typical values of the forces that act upon the particles. The typical physical parameters values used in the numerical simulations are: inter-particle distance $r \sim 10^{-4}$ m; typical electric charge of the particle $Q \sim 10^5e$; trapping voltage $U \sim 10^3$ V. Thus, we obtain the following estimations for the forces in the Langevin equation, in case of particles with a diameter of several microns:
$$\begin{aligned}
F_{int}\left( r_{s},r_{i} \right) = \frac{k \cdot q_{s} \cdot q_{i}}{r^2} \sim 10^{-10} N, \
F_{tr}\left( r_{i} \right) = U q_i/r \sim 10^{-8} N, \
\\ F_{mg}^{i} = m_{i}g \sim 10^{-10} \div 10^{-13} N. \end{aligned}$$
As it approaches a neutral molecule the ion polarizes it, and hence interacts with the induced dipole. The force that describes that interaction is $$F_{Br}\left( r_{i}\right) = \frac{2 \tilde{\alpha} q^2}{r^5} \sim 10^{-10} ,$$ where $\tilde{\alpha}$ stands for the polarizability of the molecule, and $\tilde{\alpha} \sim r_{neutral}^2 / 8$ [@Rai91]. The Stokes force $F_{Stokes} = 6 \pi \eta R_{d} dr_{i}/dt$ takes values ranging between $ 10^{-11} \div 10^{-13} N$, for particle velocities $dr_{i}/{dt} \approx 10^{-2} \div 10^{-4}$ m/s.
### Electromagnetic field of the trap
To simulate the electromagnetic field within the trap we assume the wires are surrounded by a grounded cylindrical electrode that creates a cylindrical capacitor, with a diameter much larger than the distance between the wires $D >> L_b$. Under this assumption each electrode and the surrounding cylinder can be considered as a cylindrical capacitor. By assigning a voltage difference between an electrode and an external cylinder, one can calculate the charge accumulated on the internal electrode. To allow for finite length of electrodes and calculate the forces acting on a dust particle, each electrode is mathematically divided into small sections and it is assumed that each of these sections can be regarded as a point charge. The resulting Coulomb force acting on a charged dust particle is estimated as a sum of the contribution of each point charge located at the electrodes.
By dividing the wires on large number $\tilde{N}$ of uniformly charged points, the total force acting upon any particle from a wire can be calculated as a sum of Coulombian forces: $$F_{int}(r_{i}) = \sum_{k}\frac{LUQ_{i}}{2\tilde{N} \ln \left(\frac{R_2}{R_1}\right)\left(r_i - r_k\right)^2} \ ,$$ where $L$ is the electrode length, $U$ represents the amplitude of the applied a.c. voltage, $\omega =2\pi f$ is the a.c. voltage frequency, $Q_i$ stands for the electric charge of particle i, $r_i$ and $r_k$ are the radius – vectors of the dust particle and of the point on the wire respectively, $R_2$ is the radius of a grounded cylinder, and $R_1$ denotes the radius of a wire.
![Contour plots of the time evolution of the electric field. Two diagonally connected electrodes are supplied at $\tilde{U} = U_{\omega}cos({\omega}t)$, while the other two electrodes are supplied at $U = U_{\omega}sin({\omega}t)$. The darker regions in the figure correspond to potential wells that attract the particles, while the lighter ones describe potential hills that repel the grains. In the left picture the inner region of the trap corresponds to a potential hill, while in the right picture it corresponds to a potential well. The potential fields are located at the trap centre, normal to the trap axis, so that the contribution of the electrostatic field generated by the endcap electrodes can be disregarded. Source: image reproduced from [@Lapi13] with permission of the authors.[]{data-label="fig:surface"}](fig2_i.pdf)
A uniform charge distribution along the trap wires can occur in case of more or less uniform dust particle distribution along the trap axis. In case of endcap electrodes with fixed potential such a situation does not appear. To estimate the electric field generated by the endcap electrode, it is necessary to solve a cylindrical Laplace’s equation in orthogonal elliptic - hyperbolic coordinates ($\sigma - \tau$).
$$\Delta U = \frac{1}{\alpha^2(\sigma^2 - \tau^2)} \left[ \frac{\partial}{\partial\sigma}
\left[(\sigma^2 - 1)\frac{\partial U}{\partial \sigma}\right]
+ \frac{\partial}{\partial\tau}\left[(1 - \tau^2) \frac{\partial U}{\partial \tau}\right] \right] \ ,
$$
where $2 \alpha = L_h$ is the distance between the endcap electrodes. Thus, for a dust particle the electrostatic intensity at a point in space of coordinates ($\sigma$, $\tau$) can be expressed as:
$$\begin{aligned}
E\left(\tau, \sigma\right) \propto \exp \left(- \frac{1}{1 - \alpha^2}\ln \left(\tau^2 + \frac{\alpha^2 \sigma^2 - 1}{1 - \alpha^2}\right) + \frac{1}{1 - \alpha^2}\ln \left(\frac{1}{\alpha^2 - 1}\right) -1 \right) \\
\nonumber
\tau = \sqrt{\left(z + \alpha \right)^2 + x^2 + y^2} - \sqrt{\left( z - \alpha \right)^2 + x^2 + y^2} \\
\nonumber
\sigma = \sqrt{\left(z + \alpha \right)^2 + x^2 + y^2} + \sqrt{\left( z - \alpha \right)^2 + x^2 + y^2}\end{aligned}$$
The a.c. voltage $U_{\omega}$ is applied to the trap electrodes as follows: two diagonal electrodes are supplied with an a.c. voltage $U = U_{\omega}\cos({\omega}t)$, while the other two electrodes are supplied at $\tilde{U} = U_{\omega}\sin({\omega}t)$. The trapping voltage frequency is ${\omega}= 2 \cdot \pi f$ (as shown in Fig. \[fig:trap\]). The setup generates an oscillating electric field inside the trap. The equipotential surfaces of the electric field for three different instances of time : $t = \frac 14 \pi$, $\frac 34 \pi$, $\frac 54 \pi$ are illustrated in Fig. \[fig:surface\]. In order to prevent dust particle escape from the trap, a d.c. voltage $U_{end}$ is supplied between the endcap electrodes that repels particles towards the inner volume of the trap.
### Regions of single particle confinement
We revert to the dynamic equation (\[Lang\]). The trapping force exerted upon a dust particle depends on the product between the dust particle charge and the a.c. trapping voltage. Thus, for the lower boundary of the dust particle confinement region an inverse dependence between the a.c. voltage and the dust particle charge is expected. The left panel in Fig. \[fig:3\] confirms this dependence that is predicted by calculations.
![The lower boundary of the regions of dust particle confinement. Left panel: $U_\omega$ versus particle charge; Right panel – $Q$ versus particle mass for two different a.c. voltages. The area above the line corresponds to particle confinement. Left panel parameters: $Q_{particle} = 10 e$, $U_{\omega} = 135$ V, $f = 50$ Hz, $U_{end} = 135$ V, $\rho_{particle} = 1,5 \cdot 10^4$ kg/m$^3$ , $r_{particle} = 1 \ \mu$m, $\eta = 17 \cdot 10^{-6}$ Pa $\cdot$ s - dynamic viscosity, $T = 300$ K. Right panel parameters: $f = 100$ Hz, $U_{end} = 700$ V, $U_{\omega} = 2,2 \div 22$ kV, $\rho_{particle} = 0.38 \cdot 10^4 $ kg/m$^3$, $r_{particle} = 1 \ \mu$m, $\eta = 17 \cdot 10^{-6}$ Pa $\cdot$ s - dynamic viscosity, T = 300 K. Source: picture reproduced from [@Lapi13] with permission of the authors.[]{data-label="fig:3"}](U_k_Q_i.pdf "fig:") ![The lower boundary of the regions of dust particle confinement. Left panel: $U_\omega$ versus particle charge; Right panel – $Q$ versus particle mass for two different a.c. voltages. The area above the line corresponds to particle confinement. Left panel parameters: $Q_{particle} = 10 e$, $U_{\omega} = 135$ V, $f = 50$ Hz, $U_{end} = 135$ V, $\rho_{particle} = 1,5 \cdot 10^4$ kg/m$^3$ , $r_{particle} = 1 \ \mu$m, $\eta = 17 \cdot 10^{-6}$ Pa $\cdot$ s - dynamic viscosity, $T = 300$ K. Right panel parameters: $f = 100$ Hz, $U_{end} = 700$ V, $U_{\omega} = 2,2 \div 22$ kV, $\rho_{particle} = 0.38 \cdot 10^4 $ kg/m$^3$, $r_{particle} = 1 \ \mu$m, $\eta = 17 \cdot 10^{-6}$ Pa $\cdot$ s - dynamic viscosity, T = 300 K. Source: picture reproduced from [@Lapi13] with permission of the authors.[]{data-label="fig:3"}](Q_k_m_Pri_diff_Uw_i.pdf "fig:")
The right panel in Fig. \[fig:3\] also shows the analogous lower boundary of dust particle confinement for two different a.c. voltages ($U_{\omega} = 2200$ V and $U_{\omega} = 22000$ V). The lower boundary is presented by the dependence of the dust charge versus its mass. These results support the prominent role of the charge-to-mass ratio $\frac{Q}{m}$, as it results from equation \[Lang\] and from the above estimations of the r.h.s. forces in eq. \[Lang\], for the experimental conditions considered. The applied a.c. voltages are lower than the breakdown voltage for air.
![Regions of stable confinement of a single dust particle in dependence of frequency f and charge-to-mass ratio $Q/m$, for different dynamical viscosities $\eta =1.7 \ \mu$Pa $\cdot$ s (shaded region 1) and $\eta =17 \ \mu$Pa $\cdot$ s (unshaded region 2). The particle and trap parameters are: $Q = 20500 \div 685000 e$, $U = 4400$ V, d.c. voltage = $900$ V, particle density $\rho = 0.76 \times 10^{-4}$ kg/m$^{3}$, particle radius $9 \ \mu$m. Source: picture reproduced from [@Vasi13] with permission of the authors.[]{data-label="sumnjp"}](Sumnjp.png)
Let us consider regions of stable confinement for a single particle and investigate the dependence on the a.c. voltage frequency. For each value of the frequency, the confinement region is limited by upper and lower values of the charge-to-mass ratio $Q/m$ (Fig. \[sumnjp\]). In a similar manner, for each value of the charge-to-mass ratio $Q/m$ the confinement region is limited by lower and upper frequencies. Beyond that region the trap cannot confine particles. By increasing the dynamical viscosity of the medium, the confinement region becomes wider. As an example, Fig. \[sumnjp\] shows the results of simulations for two different values of the viscosity: $\eta = 1.7 \ \mu$Pa $\cdot$ s (squares) and $\eta = 17 \ \mu$Pa $\cdot$ s (circles). Dissipation of dust particles kinetic energy in case of a low value dynamical viscosity of the medium is lower, and therefore velocity and kinetic energy of a particle are larger due to the electric field. For higher particle velocities, the boundary frequency should be higher in order to prevent particle escape out of the trap at half–cycle of the a.c. voltage. The results of calculations presented in Fig. \[sumnjp\] demonstrate that by using quadrupole dynamic traps it is possible to confine particles with larger mass and dimensions, with respect to the situation of low pressure plasmas of RF or d.c. glow discharges.
Waves in Plasmas {#waves}
================
Solitary waves in long charged particle structures confined in the linear Paul trap {#SolWave}
-----------------------------------------------------------------------------------
Charged dust particles embedded into plasmas do not only change the electron – ion composition and thus affect conventional wave modes (e.g., ion – acoustic waves), but they also: (a) introduce new low-frequency modes associated with the microparticle motion, (b) alter the dissipation rates and, (c) give rise to instabilities [@Piel17], etc. Investigations on dust acoustic solitary waves in plasmas were carried out over more than several decades [@Pecs13]. The particle electric charges vary in time and space which leads to important qualitative differences between dusty plasmas [@Piel17] and usual multicomponent plasmas [@Fort05; @Fort10a; @Boni14; @Chop14; @Khra04; @Vlad05; @Shuk92; @Chat12; @Ghai17; @He18; @Set18], because in addition to positive ions and electrons other components are present such as negative ions or dust particles. Dust acoustic waves in a charged dust component were investigated in a complex plasma at low-pressure, a model for which the analytical and numerical fluid models and kinetic approaches are treated extensively in [@Fort05; @Fort10a; @Khra04]. The main difficulty in solving this issue lies in the fact that a complex plasma represents a nonlinear medium, where the waves of finite amplitude cannot be considered independently. Nonlinear phenomena in complex plasmas are very diverse, due to a large number of different wave modes that can be sustained. The wave amplitude can reach a nonlinear level because of different physical processes and mechanisms. This is not necessarily an external forcing or an outcome of the wave instabilities – it can also be a regular collective process of nonlinear wave steepening. In absence of dissipation (or for low dissipation) nonlinear steepening can be balanced by wave dispersion, which in turn can result in the formation of solitons [@Lan13]. When dissipation in the system is large it can overcome the role of dispersion, case when the balance between nonlinearity and dissipation can generate shock waves [@Fort05; @Fort10a; @Khra04]. A very effective method for studying these phenomena is the particle-in-cell (PIC) simulation [@Ludwig12; @Zhang14; @Gao16; @Med18a].
Propagation of nonlinear waves in dusty plasmas is investigated in [@Pak09], where the Kadomtsev–Petviashivili (KP) equation is inferred in an unmagnetized dusty plasma with variable dust charge and two temperature ions. Moreover, the system is better characterized by means of a Sagdeev potential, which enables one to explore the stability conditions and the existence of solitonic solutions. A travelling rarefaction soliton propagates in most cases. The amplitude of solitary waves of the KP equation diverges at critical values of the plasma parameters. Solitonic solutions of the modified KP equation with finite amplitude are derived for such case [@Pak09]. A study of the characteristics of freak waves in dusty plasma containing two temperatures ions is presented in [@Set18], by modulating the KP equation to infer the nonlinear Schrödinger equation (NLSE). Nonlinear propagation of dust-acoustic (DA) solitary waves [@Ghai17; @Denra18] in a three-component unmagnetized dusty plasma consisting of Maxwellian electrons, vortex-like (trapped) ions and arbitrarily charged cold mobile dust rain, is investigated in [@Rah14]. It is demonstrated that the dynamics of small but finite amplitude DA waves is regulated by a nonlinear equation of modified Korteweg-de Vries (mK-dV) type. The basic characteristics and propagation of dust-acoustic (DA) shock waves (DASHWs) [@Chop14] in self-gravitating dusty plasmas containing massive dust of opposite polarity, trapped ions, and Boltzmann electrons are explored in [@Ema18; @Sumi19]. The reductive perturbation technique is applied to infer the standard modified Burgers equation (mBE). Collision of ion-acoustic solitary waves in a collisionless plasma [@Vlad11] with cold ions and Boltzmann electrons is studied using numerical simulations in [@Med18a; @Med18b]. All these nonlinear phenomena have important applications in space and astrophysical environments.
In contrast to the studies mentioned above, we firstly discuss the experimental generation of density waves in a strongly coupled one component Coulomb system consisting of micron sized particles (interacting via a non screened Coulomb potential) under SATP conditions (in air). Microparticles are levitated in a long, linear quadrupole electrodynamic Paul trap [@Syr19a]. In particular, the possibility of generating density waves in the form of individual solitary humps is also demonstrated. The physical possibility of generating waves that are equivalent to the solitary density waves is discussed by Arnold, as it is caused by the nonuniform velocity distribution of moving particles [@Arno92].
The analysis of the experimentally observed solitary density waves [@Syr19a] is based on some statements of the catastrophe theory [@Arno92], that allow one to identify these waves as [*caustics*]{}. The conclusion is based on the fact that the density profile of the experimentally obtained solitary waves can be described under a good accuracy, by the theoretical dependence predicted by the caustic theory [@Arno92] ($const/\sqrt{\epsilon}$, where $\epsilon$ is the distance with respect to the singularity of the caustic). Strictly speaking, this classification can be applied to collisionless system of particles, when small deviations from constancy in the initial velocity distribution lead to particle accumulations for sufficiently long periods of time. According to Arnold’s remarks [@Arno92] [*’this conclusion still holds when one goes from a one dimensional medium to a medium filling a space of any dimension, and when one allows for the effects on the motion of particles of an external force field or a field originating from the medium, and also when the effects of relativity and the expansion of the universe are accounted for …. Ya. B. Zel’dovich called such caustics pancakes (’bliny’ in Russian ; at first pancakes were interpreted as galaxies, later as clusters of galaxies) … The predictions of the theory of singularities for the caustics geometry, wave fronts and their metamorphoses, have been completely confirmed in experiments …’*]{}.
As demonstrated in [@Syr19a; @Syr19b] solitary caustics can be considered as new experimental support of the general versatility of the caustic theory in describing different physical phenomena, not only in collisionless systems of particles but also when the inter-particle interaction and the interaction with external fields in viscous media are strong. Further on we show that under SATP conditions, the generation of solitary density waves in strongly coupled one component Coulomb systems of particles is possible, when the energy losses caused by air viscosity can be compensated by the energy contribution of the a.c. electric field in the trap.
Multiparticle trapping in linear Paul trap
------------------------------------------
Experimental investigations on the excitation and development of dust particle density disturbances in charged particle structures were performed using a linear quadrupole electrodynamic trap (Fig. \[Trap\]) [@Syr19a]. The trap consists of four parallel steel cylindrical electrodes, with a diameter of 3 mm and a length of 30 cm. An a.c. electric potential $U_a\sin (2\pi ft)$ with frequency $f = 50$ Hz is applied between the four trap electrodes, placed at the corners of a square with a side of 2 cm. The phase shift of the electric potential between adjacent electrodes is equal to $\pi$. At the left end of the trap (see Fig. \[Trap\]) an additional electrode is mounted, supplied at a constant electric potential $U = 1$ kV, that prevents the particles from escaping along the trap axis. The right end of the trap is left open, but the end electrode effects [@Lapi15b] also prevent particle escape.
The electrodynamic trap is placed in a optically transparent plastic box to mitigate against air flows [@Lapi18b]. Polydisperse Al$_2$O$_3$ particles are trapped, with diameter ranging between $10 \div 40 \ \mu$m. The particles are positively charged on the surface of a flat electrode (labeled as 8) supplied at 10 kV, by using an induction method (see Fig. \[Trap\]). The electrode is brought inside the trap from below, through a hole drilled in the plastic box. The particles are accelerated by an electric field produced between the charging electrode and the trap electrodes, and then attracted in the trap. After this the flat electrode is removed. Video recording of the particles that are illuminated using a laser beam is performed by means of a HiSpec 1 video camera. The trap captures particles for a certain range of charge-to-mass ratios, as presented in [@Vasi13].
Fig. \[Stable\_structure\] shows the stable structures obtained for charged dust particles, for an a.c. voltage $U_a = 3.6$ kV. Although the observed structure slightly oscillates with the frequency of the a.c. supply voltage, the inter-particle distances remain approximately constant. In order to characterize particle correlations, the electric charge of the particles and the average inter-particle distance have been estimated. The electric charge of the particles was not measured in [@Vasi13]. The average electric charge value was estimated using some of the results previously obtained in [@Syr16b; @Vasi18], and the average value found was about $5 \times 10^4 e$. The average inter-particle distance is around 1 mm, estimated from typical photos of particle configurations using a specially written computer code. Thus, the Coulomb interaction at the average inter-particle distance is approximately equal to $10^5$ in units of atmospheric gas temperature at $300$ K.
Pair correlation functions
--------------------------
The average pair correlation function was obtained for structures analogous to those presented in Fig. \[Stable\_structure\]. By video recording the particle motion at a frame rate of two hundred frames per second, the pair correlation function is estimated using a special computer code for each frame. The average pair correlation function is shown in Fig. \[cor\]. To obtain the correct correlation function, the diameter of the laser beam should not exceed the mean inter-particle distance. Two cylindrical lenses are used to create a flat laser beam with a diameter of roughly 0.25 mm. The lenses are arranged so that the constriction region is achieved at the centre of the Coulomb structure.
The pair correlation function presented in Fig. \[cor\] exhibits two peaks, a feature that can be explained due to various dimensions and electric charge values of the particles that compose the Coulomb structure. The main peak occurs for an inter-particle distance of roughly 1 mm. The suggested pair correlation function makes it possible to state the existence of short – range order structures and the absence of long – range ones, which is typical for liquid – like particle ordering. Hence, the Coulomb structure consists of dust particles that are strongly correlated for short distances, while the pair correlation function points out to the liquid – like ordering, with a correlation radius of roughly two average inter-particle distances. From the physical point of view, such behaviour of the pair correlation functions can be explained by chaotic overlapping of the long – range Coulomb [*tails*]{} of the inter-particle interaction.
Occurrence and evolution of solitary density waves in a linear trap
-------------------------------------------------------------------
Density waves can arise in such type of Coulomb structures after the injection of additional particles (Fig. \[injection\]). Additional particles are injected into the trap originating from the surface of the flat electrode (Fig. \[injection\]$(a)$), after which a hump density wave occurs in the Coulomb structure that propagates towards the right edge of the trap at a velocity of around 4 cm/s (Fig. \[injection\]$(b)$).
\
$(a)$
\
$(b)$
Density waves can also occur in a stable Coulomb structure when the operating parameters of the electrodynamic trap are modified, for example when the a.c. trapping voltage rises up to a value of 5.1 kV. In this case, dust density waves appear in the long range dust particle structure (see Fig. \[Stable\_structure\]). During propagation the wave shape changes very little. Propagation of a density wave in the vicinity of an end electrode is illustrated in Fig. \[reflection\]. The wave propagates towards the end electrode with a velocity of 5.9 cm/s (Fig. \[reflection\]$(a)$). After approaching the end electrode within a minimum distance of 1 cm (Fig. \[reflection\]$(b)$), the wave stops for approximately 0.5 s then starts propagating in the reverse (backward) direction with a velocity of 4.1 cm/s (Fig. \[reflection\]$(c)$). We emphasize that the characteristic time of all dust particle dynamic processes in the trap is related to a frequency value that is typical of the order of several Hz, or tens of Hz. An analogous reflection occurs when the hump wave approaches the right end of the trap. The reflection is caused by the longitudinal electric field that is present at the boundary of the trap, as an outcome of the curvature of the electric field lines [@Lapi15b].
\
$(a)$
\
$(b)$
\
$(c)$
Fig. \[separation\] illustrates the time evolution of the density hump wave that appears in the central region of the trap. One second later after its ocurrence, the hump wave divides into two separate humps (Fig. \[separation\]$(b)$) that propagate in opposite directions with a relative speed of 7.3 cm/s.
\
$(a)$
\
$(b)$
\
$(c)$
More interesting experimental observations are shown in Fig. \[five\_caustics\], which illustrates the simultaneous existence of five hump density waves for $U_a = 5$ kV and $f = 50$ Hz. For these particular values of the trap parameters density waves chaotically propagate along the trap in different directions, while they can also merge and split.
\
One-dimensional particle-in-cell simulations
--------------------------------------------
In order to describe the dynamics of a system of electrically charged particles levitated in a Paul trap, a fairly complex theoretical model is required that takes into account both the time-varying electric field created by the trap electrodes and the space-charge field created by the particles. Up to now, there exists no adequate theoretical model in literature to describe such phenomenon. That is the main reason why the theoretical description of such phenomenon starts from considering a one component plasma (OCP) fluid model, an approach that can at least qualitatively clarify certain physical features of the system [@Syr19a].
Simple simulations of density waves are performed using the one-dimensional kinetic approximation, under which a system of identical charged particles is considered. Each particle has an associated mass $m$ and an electric charge $q$. It is assumed that the particles are [*cold*]{} and their dynamics can be described in a collisionless approximation. The corresponding system of Vlasov equations can be expressed as:
$$\label{eq1}
\frac{\partial f}{\partial t} + v \frac{\partial f}{\partial x} - \frac{\partial \varphi}{\partial x} \frac{\partial f}{\partial v} = 0 , \quad
\frac{\partial^2\varphi}{\partial x^2} = - n \,, \quad
n = \int\limits_{-\infty}^{\infty}\!f(x,v,t)\,dv\,,$$
where $f$ is the particle distribution function, $n$ denotes the particle density, $\varphi$ represents the electric field potential, while $ x $, $ t $ and $ v $ represent the coordinate, time, and velocity, respectively. The equations are given in normalized form with
------------------- ---------------------------- --------------------------------
$n_{0}$, $L$, $[m/4\pi q^2n_{0}]^{1/2}$, $L [4\pi q^2n_{0}/m]^{1/2}$,
$4\pi qn_{0}L^2$, $4\pi qn_{0}L$, $(mn_0)^{(1/2)}/\sqrt{4\pi}qL$
------------------- ---------------------------- --------------------------------
$$\label{eq2}$$
as units of density, length, time, velocity, potential, electric field and particle distribution function respectively. $n_0$ stands for the maximum value of the particle density at $t = 0$, while $L$ is the half-length of the particle structure under consideration.
### Initial and boundary conditions
The system of equations (\[eq1\]) should be built up using additional initial and boundary conditions that characterize a particular problem. Several possible initial particle distributions have been taken into account. However, it should be noted that the experimental particle density is nonuniform, for example due to the end effects that crop up in the endcap electrodes area [@Syr19a]. Hence, to study the occurrence of density waves and perform numerical simulations it is necessary to use a nonuniform density distribution as an initial condition. We emphasize on the fact that density waves start to develop precisely in the end regions, but only for a.c. supply voltages higher than 5.1 kV. As an example we present the evolution of a system of identical positively charged particles, characterized by the following density distribution at an initial moment of time $t = 0$:
$$n =
\left \{
\begin{tabular}{ccc}
0 & \mbox{at} & $-1.1 \leq x \leq -1.0$ \\
$\cos[\pi (x + 0.5)]$ & \mbox{at} & $-1.0 \leq x \leq 0.0$ \\
$\cos[\pi (x - 0.5)]$ & \mbox{at} & $1.0 \leq x \leq 1.1$ \\
0 & \mbox{at} & $1.0 \leq x \leq 1.1$
\end{tabular}
\right \}$$ $$\label{eq3}$$ We assume the particles are [*cold*]{} and their flow velocity is zero at an initial moment of time $t = 0$. The constant value of the electric field $E = E_0$ is defined on the left boundary of the region ($x = -1.1$), while $ E = - E_0 $ defines the right boundary ($ x = 1.1 $). Calculations are performed by choosing a value $ E_0 = 0.1 $. Thus, it is assumed that the electrodes are located at the boundaries of the system, and the electric fields in the area prevent the electrically charged particles from escaping out of the region $ [- 1.1, 1.1] $.
The time evolution of such a system of charged particles is studied using the particle-in-cell (PIC) method, developed and widely used to investigate the dynamics of a collisionless plasma [@Ludwig12; @Zhang14; @Gao16; @Med18a]. By using the PIC method the material medium is represented by a set of particles that move under the influence of an electromagnetic field, which in turn is determined by the particle distributions and their associated electric currents.
Generally, the particle motion obeys certain boundary region conditions. In this case, for particles located at the boundaries $ x = \pm 1.1 $ there are no additional conditions other than the influence of the given electric field. Sufficiently fast particles can leave the system at one of the boundaries. The motion of each charged particle depends on its initial position. It turns out that for a convenient representation of the solutions obtained, it makes sense to distinguish particles located at $t = 0$ in the region $x < 0$ (particles 1), from particles whose initial coordinates lie in the region $x > 0 $ (particles 2). In calculus we consider an identical value $8 \cdot 10^5$ for the number of particles in each ’sort’. To represent the dynamics in the phase plane (space) only the data for each 200th particle have been used.
Caustic solitary density waves in collisionless approximation
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We consider the results of the numerical simulation of the problem under investigation. Fig. \[figC\] shows the evolution of the particle system state at different moments of time. For each instant of time, the phase plane and the density distribution are shown in a double figure. It can be seen that for $t = 1.9$ (Fig. \[figC\](a)), the initial cosine distribution of the density for each [*sort*]{} of particles (shown by the dotted curves) changes into a homogeneous one.
At the left and right boundaries of the region of uniformly distributed particles, as well as in the point of contact ($ x = 0 $) of particles 1 and particles 2, a sharp change in the particle velocities is reported. Basically, it is exactly at these points that the derivatives of the particle velocities with respect to the coordinate tend to infinity, and the flow begins to acquire a two-stream nature. More clearly, these phenomena are observed later at $t = 2,7$ (Fig. \[figC\](b)). At the points at which the two streams merge, density peaks appear for each [*sort*]{} of particles. The peak density of particles is several times larger than the initial density amplitude. The location of particle stream mergers can be defined as a caustic by analogy with an optical caustic, that represents the place where the rays merge [@Arno92]. At $t = 2.7$ the formation of four caustics is clearly visible. Two caustics develop near the boundaries of the region, due to particle reflection at the electrodes. In the centre caustics arise because the initial non-uniform particle distribution evolves with the emergence of particle flows that move towards each other. The phase curve $v\left(x\right)$ of all particles (namely, particles 1 and particles 2) is three-valued, corresponding to a three-stream flow.
At subsequent moments of time corresponding to $ t = 4 $ (Fig. \[figC\](c)) and $ t = 7 $ (Fig. \[figC\](d)), the peak values of the density in caustics drop while the phase space curves illustrate well the fact that density peaks are located at the points where the streams merge. We emphasize that the two extreme caustics developed near the electrodes practically remain still for a long time. Simultaneously, the two central caustics diverge from the centre and give rise to caustic waves. We point out that a caustic motion is observed in the study of the nonlinear dynamics of one-dimensional flows, for a self-gravitating cold nondissipative substance [@Gure93]. In this case particles are attracted to each other and the particle countermotion arises naturally. The development of caustics in a system consisting of several kind of charged particles is also possible. Such a phenomenon was observed in a simulation of the expansion of a plasma with negative ions in vacuum [@Med10].
At subsequent instants of time $ t \ge 9$, the central caustics goes across almost immobile extreme caustics and vanishes at the electrodes. In that case the particles leave the region located between the electrodes. It is interesting to note that only particles 2 (initially located in the region $ x > 0 $) exit (leave) to the left, while particles 1 (initially located in the region $ x < 0 $) exit to the right. In this situation with two near-electrode caustics, the escape of particles continues till $ t = 13 $. Then, particles cease to cross the boundaries and begin to stem near-electrode caustics, which now consists of particles of another [*sort*]{} (Fig. \[figC\](e), $ t = 13.5 $). These new caustics slowly approach the previously developed near-electrode caustics and together with them new particle streams arise.
Stream interaction leads to the formation of ring-like structures in the phase plane (space), around which there are large enough [*humps*]{} of the particle density (Fig. \[figC\](f), $t = 19$). Gradually, these [*humps*]{} of density contract and then turn into peaks with increasing height. The maximum density is achieved at $t = 22.1$ (Fig. \[figC\](g)). Simultaneously, closed ring-shaped structures in the phase plane begin to gradually rotate and remodel (Fig. \[figC\](h), $t = 24$). Moreover, the number of streams increases and new caustics are generated both in the centre and at the peripheral region. In time, the number of caustics will increase and the distance between them will shrink.
Thus, a system of particles with identical mass and electric charge confined between the reflecting electrodes evolves such as to create two near-electrode caustics and two central caustics, which eventually diverge from the centre and reach the end electrodes. Experimentally obtained central caustics are ilustrated in Fig. \[separation\]. Particles can leave the interelectrode region. As time goes by, development of new caustics is possible both near the electrode area and in the central region. New central caustics also diverge from the centre. Eventually, the flow will acquire a multi-caustic and multi-stream character.
Caustic solitary density waves in the strongly coupled Coulomb charges in Paul trap
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The solitary density waves appear in the long range structure of charged dust particles (see Fig. \[Stable\_structure\] (a) and \[Stable\_structure1\] (a)), when the a.c. voltage supplied to the trap electrodes rises up to 5.1 kV [@Syr19a]. In such case the density waves chaotically propagate along the trap axis in different directions.
\
$(a)$\
\
$(b)$
Fig. \[Stable\_structure1\] (b) presents the upper part of the axillary symmetric solitary density hump (wave), with its external upper bound approximated by the smooth line. Similar accumulations of particles have been considered in [@Arno92]. Following the basic ideas revealed in [@Arno92] we will consider the physical reason which leads to the occurrence of density waves in experiments and simulations [@Syr19a]. To achieve such step we will briefly remind the results of density wave simulation. As mentioned before, at $t = 1.9$ (Fig. \[figC\](a)) the initial cosine distribution of the particle density of each [*sort*]{} of particles changes into a space homogeneous one, but nonuniform in the velocity distribution. The field $v\left( x \right)$ at time $t = 1.9$ can be considered as the initial velocity distribution of the particles that are uniformly distributed along the trap axis.
\
When the particles begin to move, starting from a certain moment in time faster particles begin to leave the slower ones behind. Thus, the initially uniform space distribution (density) of particles changes with time. So, deviations from stability in the initial velocity distribution lead to the accumulation of particles as illustrated in Fig. \[figC\] (b) and (c) (density peaks are analogues of traffic congestion). If the density profile in this accumulations can be described by a dependence proportional to $\epsilon^{-1/2}$ ($\epsilon$ is the distance from the top of the density hump), we can identify the observed density humps as the [*caustics*]{} [@Arno92].
The bottom panel in Fig. \[abc\] presents the density profiles of the experimental and simulated particle accumulations, as well as their dependence $const/\sqrt{\epsilon}$ versus the distance $\epsilon$, with fit constants ($const_1$ and $const_2$) at different moments of time. Experimental and simulated density profiles of the wave humps are in a good agreement with each other, and depend on $const/\sqrt{\epsilon}$. Thus, we can identify particle accumulations as caustic density waves.
Energy income and energy loss for density waves in a Paul trap
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Let us consider the energy that is characteristic to particle motion in the classical linear Paul trap [@Paul90]. If the $z$ axis is directed along the trap symmetry axis, the Mathieu equation that describes the dynamics of a single particle can be expressed as
$$\begin{aligned}
\label{dimensional}
\frac{d^2x}{d\tau^2} + (a_x + 2 q_x \cos{2 \tau})x + b\frac{dx}{d\tau} = 0,\\
\frac{d^2y}{d\tau^2} + (a_y + 2q_y \cos{2 \tau})y + b\frac{dy}{d\tau} = 0,
\end{aligned}$$
where $$a_x = -a_y = \frac{4QU}{mr_0^2\omega^2}, \ q_x = - q_y = \frac{2QV}{mr_0^2\omega^2} \ ,$$ are the adimensional characteristic constants for the trap geometry, $V$ is the amplitude of the a.c. voltage, $U$ stands for the d.c. voltage ($V = 0$ in our case), $\tau = {\Omega t}/2$ represents the dimensionless time, $b = {2\gamma}/{m\omega}$ is the damping parameter, $Q$ denotes the particle charge, $\gamma = 6\pi \eta R$ stands for the linear damping coefficient, $\eta$ is the dynamic air viscosity under SATP conditions, while $R$ represents the radius of a spherical particle.
The stable behaviour of the dust particle structure or the occurrence of density waves depend on the energy income and energy loss for the dust particle levitated in a Paul trap. We consider the mechanical work per particle of the electrical field in the trap $W_{E}$, and the mechanical work of the viscosity forces $W_{fr}$. The work of the electric field can either increase or decrease the charged particle energy, while particle energy losses are defined by the the work peformed by the viscosity forces. The corresponding mechanical works are described by the scalar product between forces and particle displacements:
$$\begin{aligned}
\label{Wtr}
W_{E}(t)= \int_0^t d\tau \left( \mathbf{F}^{E}(\tau) | \frac{d\mathbf{r}}{d\tau}\right), \\
W_{fr}(t)= \int_0^t d\tau \left( \mathbf{F}^{fr}(\tau) | \frac{d\mathbf{r}}{d\tau}\right),\end{aligned}$$
where brackets denote the scalar product of vectors $\mathbf{F}^{E}(\tau)$ and $\frac{d\mathbf{r}}{d\tau}$. Particle motion within the trap can occur along the classical trajectories $\mathbf{r}(\tau)$ with velocity ${d\mathbf{r}}/{d\tau}$, where $\mathbf{F}^{E}(\tau)$ is the electrical trap force acting upon a particle, and $\mathbf{F}^{fr}(\tau) = -6 \pi \eta R \frac{d\mathbf{r}}{d\tau}$ represents the damping force. Fig. \[work\] illustrates the time evolution of the energy income $W_{E}(t)$ and energy loss $W_{fr}(t)$, for a single particle confined in the trap.
![(Left panel) Energy income and energy loss for one particle confined in the trap. Stable particle structure (Fig. \[Trap\] (b)): lines 1 - $W_{E}$, 2 -$ W_{fr}$, 3 - ($W_{E}-W_{fr}$). Density wave generation (Fig. \[Stable\_structure1\] (a)) : lines 4 - $W_{E}$, 5 -$ W_{fr}$, 6 - ($W_{E}-W_{fr})$. Mechanical work $W$ is given in conditional units. (Right panel) Stability diagram of a single particle in a quadrupole Paul trap in the ($a_x - q_x$) plane. Above the stability lines $1$ and $2$ the particle is ejected from the trap (more details can be found in [@Nas01; @Hase95; @Vini15]). For line 1 the air viscosity is zero; in case of line 2 air viscosity is considered for SATP (normal) conditions. Points presenting experimental results: 3 – stable particle structure (Fig. \[Trap\] (b)), 4 – dust wave generation (Fig. \[Stable\_structure1\] (a)).[]{data-label="work"}](logwork.pdf "fig:") ![(Left panel) Energy income and energy loss for one particle confined in the trap. Stable particle structure (Fig. \[Trap\] (b)): lines 1 - $W_{E}$, 2 -$ W_{fr}$, 3 - ($W_{E}-W_{fr}$). Density wave generation (Fig. \[Stable\_structure1\] (a)) : lines 4 - $W_{E}$, 5 -$ W_{fr}$, 6 - ($W_{E}-W_{fr})$. Mechanical work $W$ is given in conditional units. (Right panel) Stability diagram of a single particle in a quadrupole Paul trap in the ($a_x - q_x$) plane. Above the stability lines $1$ and $2$ the particle is ejected from the trap (more details can be found in [@Nas01; @Hase95; @Vini15]). For line 1 the air viscosity is zero; in case of line 2 air viscosity is considered for SATP (normal) conditions. Points presenting experimental results: 3 – stable particle structure (Fig. \[Trap\] (b)), 4 – dust wave generation (Fig. \[Stable\_structure1\] (a)).[]{data-label="work"}](diagrstab.pdf "fig:")
Shifts in oscillations of lines observed in Fig. \[work\] illustrate that changes in particle velocity are lagging with respect to changes in the electrical trap forces, that are proportional to the particle acceleration. That is the physical reason behind the energy exchange between particles and the electric field within the trap. The decay of the difference ($W_{E}-W_{fr}$) indicated by line $3$ shows the transition to a stable regime in Fig. \[Trap\] (b), while line $6$ indicates that higher a.c. voltages supplied to the trap electrodes result in an increase of the energy income which represents the physical reason for density waves that develop in many particle structures (Fig. \[Stable\_structure\] (a)).
The right panel in Fig. \[work\] shows the stability diagram for $30 \ \mu$m particles (which is the most probable particle size according to [@Syr16b]) in the $a_x - q_x$ plane, when $\gamma = 0$ (line 1) and $\gamma = 6 \pi \eta R$ (line 2). Point 3 in Fig. \[work\] corresponds to the experimental parameters of the stable Coulomb structure in Fig. \[Trap\] (b). Point 4 located near the boundary of the stability region corresponds to density wave generation (Fig. \[Stable\_structure1\] (a)). These points are obtained as numerical solutions of equations (\[dimensional\]). Further on, we are going to obtain the experimental diagram of exciting density waves in strongly coupled Coulomb systems of particles confined in air, under SATP (normal) conditions.
The physical mechanism responsible for particle motion along the trap axis in a viscous gas is not yet completely explained. The mechanisms responsible for the energy transfer from transverse oscillations of the trap electric field into the longitudinal direction are still unknown. This change is likely possible due to the strong inter-particle interaction that results in fast [*thermalization*]{} in case of strongly coupled nonequilibrium systems of particles. Other possibilities are shown in [@Bark10] where dielectric nanoparticles levitated in a dynamic trap are considered, and both degrees of freedom are coupled to the cavity field. The longitudinal electric field can also occur as a result of the accumulating volume charge, that appears after the build up of charge density accumulations. Contrary to the case of the glow gas-discharge plasma, such physical mechanisms as the energy transfer in all particle degrees of freedom related to the dependence of the electric charge on the space position or the random fluctuations [@Bel99; @Vaul99; @Fili05] do not occur in our experiments, due to the electric charge conservation for particles confined in the trap, for periods of time of up to several hours [@Syr16b]. A detailed treatment of the problem of energy redistribution in the particle motion along the trap axis represents an objective of further investigations.
### Discussion
The paper [@Syr19a] represents the first pioneering work on density wave propagation in a linear structure of strongly coupled charged particles. The solitary density wave is demonstrated to occur as a single hump for a quadrupole electrodynamic trap operating under SATP (normal) conditions, when energy losses due to air viscosity can be compensated by the energy increase owing to the a.c. trapping fields. Generation of density waves is demonstrated by adding charged particles to the electrodynamic trap or by performing a smooth increase of the a.c. voltage amplitude. The physical possibility of analogous hump density waves (caustics) is discussed by V. I. Arnold [@Arno92], as it is caused by the nonuniform velocity distribution of dust particles.
The density waves that crop up can be identified as caustics according to Arnold’s definition [@Arno92], as the experimental wave density profile can be described by theoretical dependencies predicted by the caustic theory ($const/\sqrt{\epsilon}$, where $\epsilon$ is the distance from the caustic singularity). The results of experimental investigations can be considered as new proofs that confirm the versatility of the caustic theory in describing different physical phenomena, not only for collisionless systems of particles but also when inter-particle interaction and interaction with external fields in viscous media are strong. Generation of a density wave in a quadrupole electrodynamic trap has to be demonstrated, when energy losses due to the drag force can be compensated by the energy contribution of the a.c. trapping fields. Therefore, these waves can be regarded as the outcome of some sort of self-sustaining waves in a non-equilibrium environment and may be identified as autowaves. The mathematical apparatus required to describe the wave phenomenon has to be further developed. In opposition, the energy of solitons has to be preserved as they are required to satisfy a certain set of additional physical criteria [@Lan13].
Studies on the interaction between an acoustic wave and levitated microparticles in a linear Paul trap {#AcoustWave}
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We study the effect that a low frequency acoustic wave inflicts upon a cloud of particles levitated in a linear Paul trap, under SATP conditions. An acoustic wave represents a type of longitudinal wave that consists of a sequence of pressure pulses or elastic displacements of the material, whether gas, liquid, or solid, in which the wave propagates. The speed of an acoustic wave is determined by the temperature, pressure, and elastic properties of the medium it crosses. In case of an electrodynamic trap, application of an acoustic wave creates an additional force field that superposes over the trapping field. The combined effect of the two force fields may result in thermalization and stabilization of the particle. Thus, improved manipulation of the levitated particles is achieved by implementing better control over the position in space, as well as a more precise selection of the specific charge ratio $Q/M$. In case of a 3D Paul trap the particle dynamics strongly depends on the specific charge ratio. As the effect of an acoustic wave is essentially mechanic, it is possible to decouple the mass $M$ and the electric charge $Q$ respectively, in the equation of motion. The parameters that characterize an acoustic wave are the intensity and the associated frequency. Both parameters span a wide range of values, so that the mechanical effect of the acoustic wave can be determined in a precise manner. The experiments are presented in [@Sto11; @Sto08].
Generally, a two-dimensional (2D) quadrupole field is generated by applying a trapping RF-voltage $V_0 \cos \Omega t$ between diagonally connected pairs of cylinders mounted parallel and equidistantly spaced around the trap central axis, where $\Omega$ stands for the RF frequency. Particle dynamics in the $x - y$ (radial) plane that is normal to the trap longitudinal axis $z$, is governed by the same Mathieu type of equation as is the case for a 3D cylindrical Paul trap [@Major05; @Stock16]. The setup used in [@Sto11] consists of 4 equidistant, radially spaced electrodes (E1 $ \div$ E4) and two endcap electrodes (E5 and E6), as illustrated in Fig. \[ac1\]. The a.c trapping voltage $V_{ac} = V_0 \cos \left( \Omega t\right)$ is applied between the E2 and E3 electrodes, where $\Omega = 2 \pi \nu_0$, $\nu_0 = 50$ Hz. A d.c. voltage $U_x$ is applied to the lower electrode E4 with an aim to perform particle diagnosis and compensate gravity, while the upper trap electrode E1 is connected to the ground potential. An axial confinement voltage denoted by $U_0$ is applied between the endcap electrodes E5 and E6 in order to prevent particle escape along the trap axis. Under the assumption that the trap length $L$ is sensibly larger than the trap radius $R$, the electric potential near the axis region (afar from the ends) is [@Major05]
$$\label{acpot}
\Phi \left(x, y, t \right) = \frac {x^2 - y^2}{2R^2} \left( U_0 + V_0 \cos \Omega t \right) \ , \ \ x, y \ll R$$
![Experimental setup for acoustic wave excitation. Picture reproduced from [@Sto11] with kind permission from O. Stoican[]{data-label="ac1"}](acoustic1.jpg)
The trap geometry used in [@Sto11] is variable ($L = 3 \div 7$ cm), a common feature that is shared with other previously designed linear traps [@Ghe98; @Sto01]. During the experiments, the value of $L$ was kept constant at 3.5 cm. According to [@Sto11] the equations of motion in the $x - y$ (radial) plane for a particle levitated in a Paul trap and located in the region close to the trap axis, that undergoes action of an acoustic wave, can be expressed as
$$\begin{aligned}
\label{Acou1}
M \frac{d^2 x}{dt^2} = - \frac{Q}{R^2}\left( U_0 + V_0 \cos \Omega t \right) x - k \frac{dx}{dt} + F_{ac x} \left( t \right) \ , \\
M \frac{d^2 y}{dt^2} = \frac{Q}{R^2}\left( U_0 + V_0 \cos \Omega t \right) y - k \frac{dy}{dt} + F_{ac y} \left( t \right) \ ,\end{aligned}$$
where $F_{ac}$ is the force exerted upon the particle by the acoustic wave, while the second term in each equation describes the force of drag experienced by an object that moves through a fluid, either liquid or air. The drag force is contrary to the direction of the incoming flow velocity. For example, the drag force acting upon an object that moves in a fluid is
$$D = \frac 12 C \rho A v^2 \ ,$$
where $C$ stands for the drag coefficient that varies with the speed of the body in motion, $\rho$ is the density of the fluid, $A$ represents the cross-section area of the body that is normal to the flow direction, and $v$ denotes the speed of the object with respect to the fluid. Considering spherical particles, the force of viscosity (or the drag force) on a small sphere moving through a viscous fluid is given by the Stokes law
$$F_d = 6 \pi \eta R_p v \ ,$$
where $\eta $ is the dynamic viscosity (in case of air, under SATP conditions $\eta = 18.5 \ \mu$Pa$\cdot $s), $R_p$ stands for the particle radius, and $v$ denotes the flow velocity relative to the particle. The microparticle weight is considered negligible. Further on we introduce the adimensional time denoted as $\xi = \Omega t/2$ [@Major05], which changes the system of equations (\[Acou1\]) into
$$\label{Acou2}
\frac{d^2u}{d\xi^2} + \delta \frac{du}{d\xi} + \left( a_u + 2 q_u \cos \left( 2\xi \right)\right) - f_{ac\ u} = 0 ,\ u = x, y \ ,$$
with $$\label{Acou3}
a_x = - a_y = \frac{4QU_0}{M\Omega^2 R^2} , q_x = -q_y = \frac{2QV_0}{M\Omega^2 R^2} , \ \delta = \frac {12 \pi \eta R_p v}{M\Omega} , \ f_{ac\ u}\left( t\right) = \frac{4 F_{ac\ u}}{M\Omega^2} \ .$$
The values of the adimensional parameters determine the frontiers of the stability domains for the solutions of the Mathieu equation [@Ghosh95; @Major05; @March05]. The first stability domain of the Mathieu equation is illustrated in Fig. \[PaulStab\]. If the adiabatic approximation is valid ( $|a_x|, |a_y| \ll 1$ and $|a_x|, |a_y| \ll 1$ ), the solutions to the system of eqs. (\[Acou2\]) are bound and may be expressed as [@Major05] $$\begin{aligned}
x\left( t\right) = x_0 \cos\left(\omega_x t + \varphi_x\right) \left( 1 + \frac{q_x}{2}\cos\Omega t \right) , \\
y\left( t\right) = y_0 \cos\left(\omega_y t + \varphi_y\right) \left( 1 + \frac{q_y}{2}\cos\Omega t \right) \ ,\end{aligned}$$ where $x_0, y_0, \varphi_x, \varphi_y$ are determined from the initial conditions, and $$\omega_x = \frac{\Omega}{2}\sqrt{\frac{q_x^2}{2} + a_x} , \ \omega_y = \frac{\Omega}{2}\sqrt{\frac{q_y^2}{2} + a_y} \ .$$
Thus, the dynamics of a single particle (ion) in the radial ($x-y$) plane is a harmonic oscillation at two characteristic frequencies $\omega_x$ and $\omega_y$ (called [*secular*]{} motion), which is amplitude modulated at the drive frequency $\Omega$ ([*micromotion*]{}) [@Libb18; @Wang10]. If one disregards the micromotion, the ion dynamics in the radial plane $x-y$ can be assimilated with the motion of a particle in a harmonic potential.
### Experimental setup
The experimental setup (shown in Fig. \[ac1\]) relies on the method described in [@Schle01]. The beam generated by a cw laser diode (650 nm, 5 mW) is directed along the longitudinal axis of the trap. The electric potential along the trap $z$ axis is well characterized by eq. (\[acpot\]). A photodetector (PD) placed outside the trap, normal to the beam direction, is used to collect a fraction of the radiation scattered by the trapped microparticles. Background light effects are mitigated using a dedicated electronic circuit. Acoustic excitation of trapped microparticles is performed by means of a loudspeaker, that generates a monochromatic acoustic wave with frequency $\nu_A$. The scattered radiation intensity is amplitude modulated by the particle motion. The spectrum of frequency components or more precisely the frequency-domain representation of the signal is visualized by means of a spectrum analyzer. The experimental setup (shown in Fig. \[ac2\]) is an enhanced version of the previous setup used in [@Sto08].
![Experimental setup for acoustic wave excitation. Measurement chain. Picture reproduced from [@Sto11] with kind permission from O. Stoican.[]{data-label="ac2"}](acoustic2.pdf)
Polydisperse Al$_2$O$_3$ microparticles with dimensions ranging between $60 \div 200 \ \mu$m have been used. When the operating point in the stability diagram of the Mathieu equation reaches the boundary ([*springpoint*]{}, according to [@Davis90]), large amplitude oscillations of the microparticle occur which end in particle loss. Experimental data taken in absence and in presence of the acoustic excitation are reproduced in Fig. \[ac3\] and Fig. \[ac4\].
![Spectra of the photodetector output voltage in absence of acoustic excitation. Experimental parameters: $\nu_0 = 80 $Hz, $V_0 = 3.3$ kV, $U_x = 0$ V. Picture reproduced from [@Sto11] with kind permission from O. Stoican.[]{data-label="ac3"}](acoust31.pdf)
![Spectrum of the photodetector output voltage in case of acoustic wave excitation microparticles. Experimental parameters: $\nu_0 = 80$ Hz, $V_0 = 3.3$ kV, $U_x = 0$ V, $U_0 = 920$ V, sound level $\approx$ 85 dB. $f_A$ in the picture corresponds to $\nu_A$ in our case. Picture reproduced from [@Sto11] with kind permission from O. Stoican.[]{data-label="ac4"}](acoust32.pdf)
Fig. \[ac4\] clearly illustrates how an acoustic wave induces additional lines in the motional (dynamic) spectrum of the trapped microparticles. If the value of the acoustic wave frequency $\nu_A$ is close to the value of the a.c. voltage frequency $\nu_0$, an amplitude modulation of the photodetector voltage is reported. The phenomenon is identical with the beat frequency between two wavelengths. A numerical analysis is also performed in [@Sto11], while the results show a good agreement with the experiment. Based on the experimental results it is supposed that the operating point in the stability diagram of the linear trap under test corresponds to $q_x \approx 0.908$ (namely $Q/M \approx 5.4 \times 10^{-4}$ C/kg, according to eq. \[Acou3\]). If the specific charge ratio $Q/M$ is known as well as the microparticle mass, then the value of the electric charge $Q$ can be estimated.
Multipole linear electrodynamic traps. Discussion and area of applications {#Sec6}
==========================================================================
We review some of the most important milestones that describe experiments with microparticles in quadrupole and multipole traps, in an attempt to describe state of the art in the field. Among the first papers aimed at investigating confinement of ion clouds in multipole traps we can mention Refs. [@Walz93] and [@Mih99]. Particle (ion) dynamics has been studied by means of the numerical integration of the equations of motion, while ion stability was shown to be phase and position dependent. The trajectories envelope and the phase-dependence of ion location in the quadrupole and hexapole RF traps was emphasized.
An electrodynamic trap, used for investigating charging processes of a single grain under controlled laboratory conditions, is proposed in [@Bera10]. A linear cylindrical quadrupole trap was used, where every electrode was split in half in order to achieve a harmonic trapping potential and thus perform precise measurements on the specific charge-to-mass ratio, by determining the secular frequency of the grain. The approach relies on a method previously applied in case of microparticles [@Sto08]. In Ref. [@Vasi13] mathematical simulations are used to investigate a dust particle’s behaviour in a Microparticle Electrodynamic Ion Trap (MEIT) with quadrupole geometry. Regions of stable confinement of a single particle are reported in dependence of frequency and charge-to-mass ratio. An increase of the medium’s dynamical viscosity results in extending the confinement region for charged particles. Ordered Coulomb structures of charged dust particles obtained in a quadrupole trap operated under SATP conditions are also reported by the group from the JIHT Institute in Moscow, RAS [@Lapi16b]. Very recent papers focus on measuring the charge of a single dust particle [@Deput15a; @Lapi16a], on the effective forces that act on a microparticle levitated in a Paul trap [@Lapi15b; @Lapi16c], or confinement of microparticles in a gas flow [@Lapi15a; @Deput15b]. Other papers report on the dynamics of micrometer sized particles levitated in a linear electrodynamic trap that operates under SATP conditions, where time variations of the light intensity scattered by the microparticles are recorded and analyzed in the frequency domain [@Sto11; @Vis13]. The paper of Marmillod [*et al*]{} presents a RF/high voltage pulse generator that provides suitable waveforms required to operate a planar multipole ion trap/time-of-flight mass spectrometer [@Marmi13].
A late paper focused on MEITs and the physics associated to them [@Libb18] investigates ring electrode geometry and quadrupole linear trap geometry centimeter sized traps, used as tools for physics teaching labs and lecture demonstrations. A viscous damping force is introduced to characterize the motion of particles confined in MEITs operating under SATP conditions [@Vini15]. With respect to the microparticle species used, it is demonstrated that Stokes damping describes well the mechanism of particle damping in air. The secular force in the one-dimensional case is inferred. Nonlinear dynamics is also observed [@Nay08], as such mesoscopic systems [@Porr08] are excellent tools to perform integrability studies and investigate quantum chaos [@Mih10b].
Investigations on higher pole traps intended to be used for ion trapped based frequency standards were first reported in Ref. [@Prest99]. A 12-pole trap was tested and ions have been shuttled into it originating from a linear quadrupole trap. The outcome of the experiments was a clear demonstration that a multipole trap holds larger ion clouds with respect to a quadrupole trap. The paper also emphasizes on the issue of space charge interactions that are non-negligible in a multipole trap. The Boltzmann equation that characterizes large ion clouds in the general multipole trap of arbitrary order was solved. The authors state that fluctuations in the numbers of trapped ions influence the clock frequency much less severely than in the quadrupole case, which motivates present and future quest towards developing and testing multipole traps for high-precision frequency standards [@Kno14; @March10]. An interesting paper [@Burt06] reports on the JPL multipole linear ion trap standard (LITS) which has demonstrated excellent frequency stability and improved immunity from two of its remaining systematic effects: the second-order Doppler shift and the second-order Zeeman shift. The authors report developments that reduce the residual systematic effects to less than $ 6 \times 10^{-17} $, and the highest ratio of atomic transition frequency to frequency width (atomic line quality factor) ever demonstrated in a microwave atomic standard operated at room temperature.
An Electron Spectrometer MultiPole Trap (ES-MPT) setup was devised by Jusko and co-workers [@Jusko12]. A radio-frequency (RF) ion trap and an electron spectrometer were used in the experiment, with an aim to estimate the energy distribution of electrons produced within the trap. Results of simulations and first experimental tests with monoenergetic electrons from laser photodetachment of $O^{-}$ are presented. Other papers of the group from the Charles Univ. of Prague approach the study of associative photodetachment of $H^{-} + H$ using a 22-pole trap combined with an electron energy filter [@Rou09], study of capture and cooling of $OH^{-}$ ions in multipole traps [@Trip06; @Otto09] or $H^{-}$ ions in RF octupole traps with superimposed magnetic field [@Rou10].
Numerical simulations of the kinetic temperature of ions levitated in a cryogenic linear multipole RF trap, that are subject to elastic collisions with a buffer gas are presented in [@Asva09]. The ion temperature dependence on the trapping parameters is analyzed in detail. A disadvantageous ion-to-neutral mass ratio combined with high trapping voltages might result in ion heating, at temperatures which are much above the trap temperature. Transversal probability distributions are obtained for a 22-pole ion trap and various trapping voltages. As the trapping voltage increases, the ions are pushed closer to the RF electrodes. A new design of the 22-pole cryogenic electrodynamic trap is discussed in [@Asva10], where buffer gas-cooled H$_2$D$^+$ ions are confined at a nominal trap temperature of 14 K. One of the experimental issues in case of cryogenic traps lies in the realization of a RF resonator that should dissipate very little power [@Gando12]. A linear octupole ion trap that is suitable for collisional cryogenic cooling and spectroscopy of large ions is investigated in [@Boya14]. The strong radial confinement achieved by the trap is superior to the performance of a 22-pole trap as it enhances the dissociation yield of the stored ions to 30 %, which recommends it for laser spectroscopy studies [@Dem15]. A 22 pole ion trapping setup intended for spectroscopy has been tested in [@Asva14], using CH$_5^+$ and H$_3$O$^+$ ions that are buffer gas cooled by using He at a temperature of 3.8 K.
Multipole ion traps designs based on a set of planar, annular, concentric electrodes are presented in [@Clark13]. Such millimeter (mm) scale traps are shown to exhibit trap depths as high as tens of electron volts. Several example traps were investigated, as well as scaling of the intrinsic trap characteristics with voltage, frequency, and trap size. Stability and dynamics of ion rings in linear multipole traps as a function of the number of poles is studied in [@Carta13]. Multipole traps present a flatter potential in their centre and therefore a modified density distribution compared to quadrupole traps. Crystallization processes in multipole traps are investigated in [@Champ13], where the dynamics and thermodynamics of large ion clouds in traps of different geometry is explored. Applications of QIT and multipole ion traps span areas such as Quantum Information Processing (QIP), metrology of frequencies and fundamental constants [@Quint14; @Kno15; @Orsz16], production of cold molecules and the study of chemical dynamics at ultralow temperatures (cold ion–atom collisions) [@Gian19]. A most recent paper reports on a microfabricated planar ion trap featuring 21 d.c. electrodes, intended to study interactions of atomic ions with ultracold neutral atoms. The design also integrates a compact mirror magneto-optical chip trap (mMOT) for cooling and confining neutral $^{87}$Rb atoms, that are transferred into an integrated chip-based Ioffe-Pritchard trap potential [@Bahra19].
Improved stability in multipole ion traps. Optimizing the signal-to-noise ratio
-------------------------------------------------------------------------------
In case of quadrupole traps the second-order Doppler effect is the result of space-charge Coulomb repulsion forces acting between trapped ions of like electrical charges. The Coulombian forces are balanced by the ponderomotive forces produced by ion motion in a highly non-uniform electric field. For large ion clouds most of the motional energy is found in the micromotion. Multipole ion trap geometries significantly reduce all ion number-dependent effects resulting through the second-order Doppler shift, as ions are weakly bound with confining fields that are effectively zero through the inner trap region and grow rapidly close to the trap electrode walls. Because of the specific map shape of trapping fields, charged particles spend relatively little time in the high RF electric fields area. Hence the RF heating phenomenon (micromotion) is sensibly reduced. Multipole traps have also been used as tools in analytical chemistry to confine trapped molecular species that exhibit many degrees of freedom [@Trip06; @Ger08a; @Ger08b; @West09]. Space-charge effects are not negligible for such traps. Nevertheless, they represent extremely versatile tools to investigate the properties and dynamics of molecular ions or to simulate the properties of cold plasmas, such as astrophysical plasmas or the Earth atmosphere. Stable confinement of a single ion in the radio-frequency (RF) field of a Paul trap is well known, as the Mathieu equations of motion can be analytically solved [@Ghosh95; @Major05; @Baril74; @March05].
New Structures for Lossless lon Manipulations (SLIM) module were recently developed to explore ion trapping at a pressure of 4 Torr [@Zhang15]. Based on such setup ions can be trapped and accumulated with up to 100 % efficiency, contained for minimum 5 hours with insignificant losses and then rapidly ejected outside the SLIM trap. Such features open new pathways for enhanced precision ion manipulation.
Regions of stability in multipole traps. Discussion
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A linear quadrupole Paul trap uses a combination of time varying and static electric potentials, to create a trapping configuration that confines charged particles such as ions, electrons, positrons, micro or nanoparticles (NPs) [@Kno15; @Orsz16; @Hart17]. A radiofrequency (RF) or a.c. voltage is used in order to generate an oscillating quadrupole potential in the $x-y$ plane, which achieves radial confinement of the trapped particles. Axial confinement of positively charged ions (particles) is achieved by means of a static potential applied between two endcap electrodes located at the trap ends, along the trap axis ($z$ plane). The RF or a.c. field induces an effective potential which harmonically confines particles in a region where the field exhibits a minimum, under conditions of dynamic stability [@Ghosh95; @Major05; @March05; @Vini15; @Libb18]. As it is almost impossible to achieve quadratic a.c. and d.c. endcap potentials for the whole trap volume, it can be assumed that near the trap axis the electric potential can be regarded as harmonic, which is a sufficiently accurate approximation.
Stable confinement of a single ion in the radio-frequency (RF) field of a Paul trap is well known, as the Mathieu equations of motion can be analytically solved [@Ghosh95; @Major05; @March05]. This is no longer the case for high-order multipole fields where the equations of motion do not admit an analytical solution. In that case particle dynamics is quite complex, as it is described by non-linear, coupled, non-autonomous equations of motion. The solutions for such system can only be found by performing a numerical integration [@Major05; @Fort10a; @March05; @Rieh04]. The assumption of an effective trapping potential still holds and ion dynamics can be separated into a slow drift (the [*secular motion*]{}) and the rapid oscillating micromotion [@Otto09; @Champ09]. The effective potential in case of an ideal cylindrical multipole trap can be expressed as
$$V^{*}(r) = \frac{1}{4} \frac{n^2(qV_{ac})^2}{m \Omega^2 {r_0}^2}{\left(\frac{r}{r_0}\right)}^{(2n - 2)} \ ,$$
where $V_{ac}$ is the amplitude of the RF field and $n$ is the number of poles (for example $n=2$ for a quadrupole trap, $n = 4$ for an octupole trap and $n = 6$ for a 12-pole trap). The larger the value of $n$, the flatter the potential created within the trap centre and the steeper the potential close to the trap electrodes.
Recent experiments show that in case of linear multipole RF (Paul) traps, single ring structures occur as an outcome of the radial potential geometry. By investigating the stability of these tubular structures as a function of the number of trap poles, it turns out that the ions arrange themselves as a triangular lattice folded onto a cylinder [@Carta13]. Numerical simulations performed for different lattice constants supply the normal mode spectrum for these ion structures.
Nonlinear dynamics of a charged particle in a RF multipole ion is investigated in [@Rozh17] and the expression of the 2D electrical potential of the trap is supplied. The existence of localization regions for ion dynamics in the trap is demonstrated. The associated Poincar[é]{} sections illustrate ion dynamics to be highly nonlinear. Another paper proposes a new method to analyze the potential field in multipole ion traps [@Rud19], which shows a good agreement with both the numerical calculations and experimental data. Results show that for multipole traps the associated dynamics exhibits multiple stable quasi-equillibrium points. The high-order Laplace fields that appear in multielectrode trap systems yield quasi-periodic or chaotic nonlinear ion motion, which is demonstrated as an Ince-Strutt-like stability diagram. A numerical demonstration is performed for a 22-pole trap [@Rud19].
### Multipole trap geometries. Experimental setups
We report three setups of Multipole Microparticle Electrodynamic Ion Traps (MMEITs). The first geometry consists of eight brass electrodes of 6 mm diameter equidistantly spaced around a 20 mm radius, and two endcap electrodes located at the trap ends that are 65 mm apart. The second trap geometry consists of twelve equidistantly spaced brass electrodes (a$_1 \div$ a$_{12}$) and two endcap electrodes ($b_1$ and $b_2$), as illustrated in Fig. \[12pole\]. The electrode diameter is 4 mm, the trap radius is 20 mm, while the electrode length does not exceed 85 mm.
A sketch of the 12-electrode trap geometry designed and tested is shown in Fig. \[12pole\]. Both setups are intended to study the appearance of stable and ordered patterns for different microparticle species. Alumina microparticles (with dimensions ranging between $60 \div 200$ microns) were used in order to illustrate the trapping phenomenon, but other species can be considered. Specific charge measurements over the trapped microparticle species can also be performed by refining the setup. The 12-electrode trap tested is characterized by a variable geometry [@Sto11; @Ghe98; @Sto01], as its length can vary between $10 \div 75$ mm. The endcap electrode denoted as $b_2$ is located on a piston which slides along the $z$ axis of the trap. An electronic supply system deliveres an a.c. voltage $V_{ac}$ with an amplitude of $0 \div 4$ kV and a variable frequency in the $40 \div 800$ Hz range, required in order to achieve radial trapping of charged particles. A high voltage step-up transformer driven by a low frequency oscillator (main oscillator) $O_1$ delivers the $V_{ac}$ voltage, as shown in Fig. \[supply\]. An auxiliary oscillator $O_2$ is used to modulate the amplitude of the $V_{ac}$ voltage. The modulation ratio can reach a peak value of $100\%$, while the modulation frequency lies in the $10 \div 30$ Hz range.
![A sketch of the 12-electrode linear Paul trap geometry. a) cross-section; b) longitudinal section; c) electrode wiring. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole"}](12eltrap.pdf)
Photos of both 8-electrode and 12-electrode traps are presented in Fig. \[traps\]. The electronic system also supplies a variable d.c voltage $U_x$ (also called diagnose voltage) applied between the upper and lower multipole trap electrodes, used to compensate the gravitational field and shift the particle position along the $x$ axis. Ions confined in linear Paul traps under ultrahigh vacuum conditions do arrange themselves along the longitudinal $z$ axis and within a large region around it, where the trapping potential is very weak. The situation is sensibly different in case of electrically charged microparticles, which we explain in Section \[model\]. An extra d.c. variable voltage $U_z$ is applied between the trap endcap electrodes, in order to achieve axial confinement and prevent particle loss at the trap ends. The $U_x$ and $U_z$ voltages range between $100 \div 700$ V, and their polarity is reversible. Both the $U_x$ and $U_z$ voltages are obtained using a common d.c. double power supply that delivers a voltage of $\pm \ 700$ V at the output. Because the absorbed d.c. current is very low, the d.c. voltages are supplied to the electrodes by means of potentiometer voltage dividers as shown in Fig. \[supply\].
![Block scheme of the electronic supply unit. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="supply"}](ElectrSupply.pdf)
The supply system is a single unit which delivers all the necessary trapping voltages, as illustrated in Fig. \[ACDCsupply\]. A microcontroller based measurement electronic circuit allows separate monitoring and displaying of the supply voltage amplitudes and frequencies (for the a.c. voltage and modulation voltage). In order to visualize and diagnose the trapped particles, the experimental setups are equipped with two different illumination systems. The first system is based on a halogen lamp whose beam is directed normal to the trap axis. The second system consists of a laser diode solidary attached to one of the endcap electrodes, whose output beam is directed parallel to the trap axis.
![Photos of the 8-electrode and 12-electrode linear Paul trap geometries designed and tested in INFLPR. Images reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="traps"}](8poletrap.pdf "fig:") ![Photos of the 8-electrode and 12-electrode linear Paul trap geometries designed and tested in INFLPR. Images reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="traps"}](12poletrap.pdf "fig:")
If the a.c. voltage is lower than 3 kV stable particle oscillations are observed, until the d.c. potential is adjusted to balance vertical forces such as gravity. When such condition is satisfied the oscillation amplitude becomes vanishingly small and the particle experiences stable trapping [@Kul11]. The traps that we tested are fitted to study complex Coulomb systems (microplasmas) confined in multipole dynamic traps operating at SATP conditions. The experimental results clearly demonstrate that the stability region for multipole traps is sensibly larger in comparison to the case of a quadrupole trap (electrodynamic balance configuration) [@Hart92; @Libb18; @Vasi13; @Prest99].
The electrolytic tank method was used to map the radiofrequency field potential within the inner volume of the 12-electrode and 16-electrode traps. A precision mechanical setup has been designed and realized in order to chart the electric field within the traps. Dedurized water was used as an electrolyte solution. Two a.c. supply voltage values were used: 1 V and 1.5 V, respectively. We emphasize that these are the amplitude values measured. Measurements were performed for an a.c. frequency value of $\Omega = 2\pi \times 10$ Hz. Figures \[12pole1V\] and \[12pole15V\] show the experimentally obtained contour and 3D maps of the trap potential (rms values).
![Contour and 3D plots for the 12-electrode Paul trap potential when $V_{ac} = 1 V$. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole1V"}](Contour1V.pdf "fig:") ![Contour and 3D plots for the 12-electrode Paul trap potential when $V_{ac} = 1 V$. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole1V"}](meshc1V.pdf "fig:") ![Contour and 3D plots for the 12-electrode Paul trap potential when $V_{ac} = 1 V$. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole1V"}](Surfc1V.pdf "fig:")
![Contour and 3D plots for the 12-electrode Paul trap potential when $V_{ac} = 1.5 V$. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole15V"}](Contour15V.pdf "fig:") ![Contour and 3D plots for the 12-electrode Paul trap potential when $V_{ac} = 1.5 V$. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole15V"}](meshc15V.pdf "fig:") ![Contour and 3D plots for the 12-electrode Paul trap potential when $V_{ac} = 1.5 V$. Picture reproduced from [@Mih16a] with kind permission of the authors. Copyright AIP.[]{data-label="12pole15V"}](Surfc15V.pdf "fig:")
### Results
Different linear multipole electrodynamic trap geometries operating under SATP conditions have been investigated. Work was focused on an 8-electrode, a 12-electrode, and a 16-electrode trap geometry respectively, with an aim to study conditions for stable microparticle confinement and to illustrate the emergence of planar and volume structures for these microplasmas. Microparticles are radially confined due to the a.c. trapping voltage $V_{ac}$, which is set at a value of 2.5 kV. Levitated microparticles can be shifted both axially and vertically using two d.c. voltages: $U_z$ and $U_x$. The geometry of the 8-electrode trap has proven to be very critical, different radii have been tested and the trap has gone through intensive tests with an aim to optimize it. It presents a sensibly higher degree of instability compared to the 12-electrode and 16-electrode geometries. A 16 electrode trap designed and tested in INFLPR is shown in Fig. \[16pole\]. The trap geometry consists of 16 brass electrodes of 60 mm length and 4 mm diameter, equidistantly spaced on a 46 mm diameter. If the AC potential is not too large $\left(V_{ac} < 3.5 \ \text{kV}\right)$, stable oscillations will occur until the d.c. potential is adjusted to balance vertical forces such as gravity. When this condition is fulfilled the oscillation amplitude becomes vanishingly small and the particle is stable trapped [@Sein16].
![Image of a 16 electrode trap with variable geometry, designed and tested in INFLPR, for levitating microparticles under SATP conditions. The trap radius is 23 mm. Image reproduced from [@Mih16b] with kind permission of the publisher.[]{data-label="16pole"}](16poletrap.jpg)
The 12-electrode (pole) trap has been studied more intensively as particle dynamics is far more stable. The traps are loaded with microparticles by means of a miniature isolated screwdriver. The peak of the screwdriver is inserted into the alumina powder. When touching the screwdriver to one of the trap electrodes, the particles are instantly charged and a small part of them are confined, depending on their energy and phase of the a.c. trapping field. Thus a trapped particle microplasma results (very similar to a dusty plasma, which is of great interest for astrophysics), consisting of tens up to hundreds of particles. Such a setup is suited in order to study and illustrate particle dynamics in electromagnetic fields, as well as the appearance of ordered structures, crystal like formations [@Ghe98; @Sto01; @Mih16a; @Vini15; @Libb18].
The a.c. potential within the 12 and 16 electrode trap was mapped using an electrolytic tank filled with dedurized water. Further on, we will describe the method used for the 16-pole trap. The trap was immersed within the water around 38 mm of its total length. A needle electrode was used to measure the trap voltage, immersed at 16 mm below water level. The needle electrode can be displaced 36 mm both horizontally and vertically, using a precision mechanism. The trap potential was mapped using a transversal (radial) section located at 22 mm with respect to the immersed end, for 2 mm steps on both vertical and horizontal position. The experimental setup including the trap and the electrolytic tank is shown in Fig. \[tank\]. The Paul trap was supplied with a sine wave delivered by a function generator at 46.1 Hz frequency. An oscilloscope was used to monitor the sine wave. The rms value of the a.c. voltage was measured using a precision voltmeter. Maps of the trap potential were obtained for a 0.5 V amplitude sine wave supplied to the trap electrodes. The sine wave was applied between even and respectively uneven electrodes, connected together. In Fig. \[mapchart15V\] we show maps of the trap potential for a supply voltage of 1.5 V amplitude (1.037 V rms value) sine wave.
![Use of electrolytic tank method to chart the map of the electric field within the 16 electrode trap. Image reproduced from [@Mih16b] with kind permission of the publisher[]{data-label="tank"}](ElctrTank.jpg)
Practically, stable confinement is achieved especially in the 12-electrode (pole) and 16-electrode traps, as we have observed thread-like formations (strings) and especially 2D (some of them zig-zag) and 3D structures of microplasmas. The stable structures observed have the tendency of aligning with respect to the $x$ component of the radial field. The ordered formations are not located along the trap axis, but rather in the proximity of the electrodes. The laser diode has been shifted away from the initial position (along the trap axis), towards outer regions of the trap. We report stable confinement for hours and even days. Laminary air flows that cross the trap volume break an equilibrium which might be described as somehow fragile and some of the particles can neutralize at the electrodes. Nevertheless, we have also tested the trap under conditions of intense air flows and we have observed that most of the particles remain trapped, even if they rearrange themselves after being subject to intense and repeated perturbations. When a transparent plastic box was used in order to shield the trap, a sensible increase in the dynamical stability of the particle motion has been achieved, as well as the emergence of larger clouds. Experimental evidence presented demonstrates that a 16 electrode trap design is characterized by an extended region of lower field with respect to the 8-electrode and 12-electrode geometries we have tested. Moreover, the 16-electrode trap design is suitable for various applications where larger signal-to-noise ratios are required.
We illustrate below maps of the trap potential for a supply voltage of 1.5 V amplitude (1.037 V rms value) sine wave.
![Maps of the 16-electrode trap potential: contour plot, pseudocolor plot, surface plot and ribbon plot for an input sine wave of 1.5 V amplitude (1.037 V rms). Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="mapchart15V"}](ContPlot.pdf "fig:"){width="0.45\linewidth"} ![Maps of the 16-electrode trap potential: contour plot, pseudocolor plot, surface plot and ribbon plot for an input sine wave of 1.5 V amplitude (1.037 V rms). Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="mapchart15V"}](PseudColorPlot.pdf "fig:"){width="50.00000%"} ![Maps of the 16-electrode trap potential: contour plot, pseudocolor plot, surface plot and ribbon plot for an input sine wave of 1.5 V amplitude (1.037 V rms). Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="mapchart15V"}](SurfPlot.pdf "fig:"){width="48.00000%"} ![Maps of the 16-electrode trap potential: contour plot, pseudocolor plot, surface plot and ribbon plot for an input sine wave of 1.5 V amplitude (1.037 V rms). Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="mapchart15V"}](RibbonPlot.pdf "fig:"){width="48.00000%"}
In Fig. \[microplasmas\] we present some images that illustrate the stable structures we have been able to observe and photograph in the 12-pole trap. All photos were taken with a high sensitivity digital camera, using the halogen lamp in order to illuminate the trap. Pictures taken using the laser diode were less clear due to reflection of light on the trap electrodes. We report filiform structures consisting of large number of microparticles far from the trap centre, where the trapping potential is extremely weak. This leads to the conclusion that microparticle weight is not balanced by the trapping field in the region located near the trap centre. Moreover, most of the ordered structures observed were generally located very close to the trap electrodes, at distances about $2 \div 10$ mm apart from them. The a.c. frequency range was swept between $\Omega = 2\pi \times 50 \div 2 \pi \times 100$ Hz.
![Photos of the ordered structures observed in the 12-electrode Paul trap. Images reproduced from [@Mih16a] with kind permission of the authors. Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="microplasmas"}](figure5b.pdf "fig:") ![Photos of the ordered structures observed in the 12-electrode Paul trap. Images reproduced from [@Mih16a] with kind permission of the authors. Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="microplasmas"}](figure5c.pdf "fig:") ![Photos of the ordered structures observed in the 12-electrode Paul trap. Images reproduced from [@Mih16a] with kind permission of the authors. Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="microplasmas"}](figure5d.pdf "fig:") ![Photos of the ordered structures observed in the 12-electrode Paul trap. Images reproduced from [@Mih16a] with kind permission of the authors. Picture reproduced from [@Mih16b] with kind permission of the authors. Copyright AIP.[]{data-label="microplasmas"}](figure5g.pdf "fig:")
The trap geometries investigated are characterized by multiple regions of stable trapping, some of them located near the trap electrodes. Most of the photos taken illustrate such phenomenon along with the numerical simulations performed. At least three or four regions of stable trapping were identified, but due to camera limitations focusing was achieved for a limited region of space. When looking with bare eyes and for an adequate angle of observation, spatial structures were observed in different regions within the trap volume. We can ascertain that the multipolar trap geometries investigated, and especially the linear 12- and 16-electrode Paul traps exhibit an extended region where the trapping field almost vanishes. The amplitude of the field rises abruptly when approaching the trap electrodes, which leads to stable trapping along with the occurrence of planar and volume structures in a layer of about a few millimeters thick, which practically spans the inner electrode space. We also report particle oscillations around equilibrium positions, where gravity is balanced by the trapping potential. Close to the trap centre particles are almost [*frozen*]{}, which means they can be considered motionless due to the very low amplitude of their oscillation. The oscillation amplitude increases as one moves away from the trap centre. We also report regions of dynamical stability for trapped charged microparticles located far away from the centre of the trap, as illustrated in Fig. \[microplasmas\]. The values of the specific charge ratios for the alumina microparticles range between $5.4 \times 10^{-4} \div 0.13 \times 10^{-3}$ C/kg [@Sto01; @Sto08].
The experimental results reported deal with trapping of microparticles in multipole linear ion (Paul) traps (MLIT) operating in air, under SATP conditions. We suggest such traps can be used to levitate and study different microscopic particles, aerosols and other constituents or polluting agents which might exist in the atmosphere. The research performed is based on previous results and experience [@Ghe98; @Mih08; @Sto08; @Vis13; @Fili12; @Lapi15b]. Numerical simulations were run in order to characterize microparticle dynamics.
Analytical and numerical modelling. Instabilities. Mitigation {#model}
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In addition to experimental work, numerical simulation of charged particle dynamics was carried out under conditions close to the experiment. Brownian dynamics was used to study charged microparticle motion and thus identify regions of stable trapping. Numerical simulations take into account stochastic forces of random collisions with neutral particles, viscosity of the gas medium, regular forces produced by the a.c. trapping voltage and gravity. Thus, microparticle dynamics is characterized by a stochastic Langevin differential equation [@Vasi13; @Fili12]:
$$\label{eq.1}
m_p \frac{d^2 r}{dt^2} = F_t(r)-6 \pi \eta r_p \frac{dr}{dt} + F_b + F_g$$
where $m$ and $r_p$ represent the microparticle mass and radius vector, $\eta$ is the dynamic viscosity of the gas medium with $\eta = 18.2 \:\mu$Pa$\cdot$s, and $F_t(r)$ is the ponderomotive force. The $F_b$ term stands for the stochastic delta-correlated forces accounting for stochastic collisions with neutral particles, while $F_g$ is the gravitational force. We consider a microparticle mass density value $\rho_p = 3700$ kg/m$^3$ [@Sto08; @Vis13], valid for all simulations performed [@Mih16a; @Lapi16d]. In order to solve the stochastic differential equation (\[eq.1\]), the numerical method developed in [@Skeel02] was used. The average Coulomb force that acts on a microparticle (as an outcome of the contribution of each trap electrode) can be expressed as the vector sum of forces of point-like charges uniformly distributed along the electrodes, as demonstrated in [@Vasi13; @Lapi15b]:
$$\label{eq.2}
|F_t(r)| = \sum \limits_s \frac{L U q}{2 N \ln{\left(\frac{R_2}{R_1}\right)}(r_s - r)^2},$$
where $L$ is the length of the trap electrodes, $U$ is the trapping voltage: $V_{ac} \sin(\Omega t)$ or $V_{ac} \sin(\Omega t+\pi)$, $q$ is the microparticle charge, $N$ is the number of point-like charges for each trap electrode, $R_2$ and $R_1$ represent the radii of the grounded cylindrical shell surrounding the trap and trap electrode respectively, while $r$ and $r_s$ denote the vectors for microparticle and point-like charge positions respectively. Numerical simulations were run considering the following trap parameters: length of electrodes $L = 6.5$ cm, $V_{ac} = 2$ kV, $R_2 = 25$ cm, $R_1 = 3$ mm, and a trap radius value $r_t = 2$ cm. For such model the results of the computations depend on $\Phi_p$, defined in Sec. \[gasflow\] by eq. \[intforce\].
To investigate the influence of the number of trap electrodes on the stability of alumina (dust) particles, we study the average amplitude of particle oscillations. We consider the dynamics of a number of 20 microparticles confined within the trap and average the amplitudes of particle oscillations. To achieve that we choose a period of time long enough (around ten periods of the a.c. trapping voltage), so as to obtain stable particle oscillations. Trajectories for 20 particles confined in 8-electrode, 12-electrode, and 16-electrode traps respectively, are shown in Fig. \[tracks8-16\]. As the number of trap electrodes increases, particle trajectories are shifted downwards because of a smaller gradient of the a.c. electric field, as illustrated in Fig \[tracks8-16\].
![End views of the microparticle tracks in (a) 8-electrode, (b) 12-electrode, and (c) 16 electrode traps at $f = 60$ Hz. Big black dots correspond to trap electrodes (not in scale). The microparticle electric charge value was chosen $q = 8 \cdot 10^4 e$. The d.c. voltage was disregarded in the simulations. Images reproduced from [@Mih16a] with kind permission of the authors under AIP Copyright, and from [@Mih16b] with kind permission from the publisher.[]{data-label="tracks8-16"}](8el.pdf "fig:") ![End views of the microparticle tracks in (a) 8-electrode, (b) 12-electrode, and (c) 16 electrode traps at $f = 60$ Hz. Big black dots correspond to trap electrodes (not in scale). The microparticle electric charge value was chosen $q = 8 \cdot 10^4 e$. The d.c. voltage was disregarded in the simulations. Images reproduced from [@Mih16a] with kind permission of the authors under AIP Copyright, and from [@Mih16b] with kind permission from the publisher.[]{data-label="tracks8-16"}](12el.pdf "fig:") ![End views of the microparticle tracks in (a) 8-electrode, (b) 12-electrode, and (c) 16 electrode traps at $f = 60$ Hz. Big black dots correspond to trap electrodes (not in scale). The microparticle electric charge value was chosen $q = 8 \cdot 10^4 e$. The d.c. voltage was disregarded in the simulations. Images reproduced from [@Mih16a] with kind permission of the authors under AIP Copyright, and from [@Mih16b] with kind permission from the publisher.[]{data-label="tracks8-16"}](16el.pdf "fig:")
In order to achieve dust particle confinement, besides the a.c. trapping voltage there are several other parameters that have to be considered: the a.c. voltage frequency, the influence of the trap geometry (by means of the trap geometrical parameters), the particle dimension and its geometrical shape, or the inter-electrode distance value. Ref. [@Lapi15a; @Lapi16d] demonstrate the connection between stable dust particle confinement and the a.c. field frequency or the specific charge ratio $Q/M$, for the two trap geometries considered and two different values of the viscosity. The first trap geometry consists of 4 wires (see Fig. \[fig:trap\]), while the second geometry comprises only 2 wires, supplied at an a.c. voltage $U_{\omega}\sin({\omega}t)$.
As it follows from Fig. \[fig:Regions\_geometries\], for each frequency of the a.c. voltage the regions of confinement are bounded by the lower and upper values of the specific charge ratio. Beyond these regions the trap cannot confine dust particles. In the left panel of Fig. \[fig:Regions\_geometries\] the confinement region characteristic to the two wire trap (2WT) is shifted, while in the same time it is wider with respect to the confinement region in case of a 4 wire trap (4WT). This result is an outcome of the smaller boundary gradients for the potential field inside the 2WT trap, with respect to case of the 4WT. Thus, in order to maintain particles levitated inside the 2WT trap a higher value of the charge-to-mass ratio is required.
![Regions of dust particle confinement in the $f$ – $Q/M$ plane, for two trap geometries and two different values of the viscosity. The parameter values are: $f = 30 \div 200$ Hz , $U_{\mathrm{end}} = 900$ V, $U_{\omega} = 4400$ V, $\rho_{p}=0.38 \times 10^4$ kg/m$^3$, $r_{p}=7 \ \mu$m, $Q_{p} = 2 \times 10^4 e \div 6.8 \times 10^5 e$, $\eta=17 \times 10^{-6}$ Pa$\cdot$s – dynamic viscosity, $T = 300$K. Source: picture reproduced (modified) from [@Lapi15a] with permission of the authors.[]{data-label="fig:Regions_geometries"}](NormalN.pdf "fig:") ![Regions of dust particle confinement in the $f$ – $Q/M$ plane, for two trap geometries and two different values of the viscosity. The parameter values are: $f = 30 \div 200$ Hz , $U_{\mathrm{end}} = 900$ V, $U_{\omega} = 4400$ V, $\rho_{p}=0.38 \times 10^4$ kg/m$^3$, $r_{p}=7 \ \mu$m, $Q_{p} = 2 \times 10^4 e \div 6.8 \times 10^5 e$, $\eta=17 \times 10^{-6}$ Pa$\cdot$s – dynamic viscosity, $T = 300$K. Source: picture reproduced (modified) from [@Lapi15a] with permission of the authors.[]{data-label="fig:Regions_geometries"}](NotNormalN.pdf "fig:") ![Regions of dust particle confinement in the $f$ – $Q/M$ plane, for two trap geometries and two different values of the viscosity. The parameter values are: $f = 30 \div 200$ Hz , $U_{\mathrm{end}} = 900$ V, $U_{\omega} = 4400$ V, $\rho_{p}=0.38 \times 10^4$ kg/m$^3$, $r_{p}=7 \ \mu$m, $Q_{p} = 2 \times 10^4 e \div 6.8 \times 10^5 e$, $\eta=17 \times 10^{-6}$ Pa$\cdot$s – dynamic viscosity, $T = 300$K. Source: picture reproduced (modified) from [@Lapi15a] with permission of the authors.[]{data-label="fig:Regions_geometries"}](normalNotNormal.pdf "fig:")
A comparison between the left and central panels in Fig. \[fig:Regions\_geometries\] shows that the confinement region becomes wider for both trap geometries as the viscosity increases. Simulations were carried out for two different values of the dynamic viscosity: $\eta = 1.7 \times 10^{-6}$ Pa$\cdot$s, and $\eta = 17 \times 10^{-6}$ Pa$\cdot$s. The confinement regions also depend on the complicated interplay between the trap confining forces and viscosity, suppressing dust particle oscillation and possible resonance. A lower viscosity value results in smaller dissipation of the dust particle energy, thus increasing the velocity and implicitly the energy that a particle acquires from the trap field. Experimental results suggest that the 4WT geometry is better as it implies a lower value of the specific charge ratio $Q/M$ or of the a.c. voltage $U_{\omega}$ required to confine particles.
In order to solve the stochastic differential equation (\[eq.1\]), the numerical method developed in [@Skeel02] is used. Numerical simulation is performed considering the following trap parameter values: electrode length $L = 6.5$ cm, $R_2 = 25$ cm, $R_1 = 3$ mm, trap radius $r_t = 4$ cm, and a.c. voltage amplitude $U_{\omega} = 2$ kV.
In Fig. \[468\] hills correspond to potential barriers and pits correspond to potential wells that attract microparticles. White holes inside the hills correspond to trap electrodes. Every half-cycle period of the a.c. voltage barriers and wells swap positions, and each particle oscillates between these positions. Particle oscillations result in dynamic confinement [@Vasi13]. Fig. \[468Stab\] presents the confinement regions for electrically charged microparticles confined in 8, 12 and 16 electrode trap geometries, that are investigated in [@Mih16a; @Mih16b]. The confinement regions are identified as follows: for an 8 electrode trap the trapping region is located between solid gray lines, for a 12 electrode trap the area delimited by dark gray dash lines, and for a 16 electrode trap the zone delimited by black dash-dot-dot lines. Beyond these regions multipole traps cannot confine particles. In case of low values of the electric charge, the a.c. field can no longer compensate the gravity force and particles flow across the trap. When the particles reach the right-hand area of the trapping region, the electric field pushes them out of the trap during one half-cycle of the oscillation.
The dependence between the particle oscillation amplitude and the number of electrodes is illustrated in Fig. \[468ampl\]. In Fig. \[468ampl\](a), for a rather low value of the electric charge, the dependency of the averaged amplitude of motion on the frequency $f$ exists only for 12 and 16 electrode traps, because the regions of particle confinement for these traps are wider than the one corresponding to an 8 pole trap. The dependence of the oscillation amplitude on the number of trap electrodes is complex and it depends on the electric charge and inter-particle interaction, as mentioned before. The higher the frequency of the a.c. field, the lower the oscillation amplitude. For an 8 electrode trap, a resonance effect at $60$ Hz is identified for particles with an electric charge value of $q = 8 \times 10^4 e$.
![3D plots for a 8 (a), 12 (b) and 16 (c) pole traps. Source: pictures reproduced from [@Mih16a; @Lapi16d] with kind permission from the authors. Copyright AIP.[]{data-label="468"}](8.pdf "fig:") ![3D plots for a 8 (a), 12 (b) and 16 (c) pole traps. Source: pictures reproduced from [@Mih16a; @Lapi16d] with kind permission from the authors. Copyright AIP.[]{data-label="468"}](12.pdf "fig:") ![3D plots for a 8 (a), 12 (b) and 16 (c) pole traps. Source: pictures reproduced from [@Mih16a; @Lapi16d] with kind permission from the authors. Copyright AIP.[]{data-label="468"}](16.pdf "fig:")
![Regions of single particle confinement depending on the frequency $f$ of the a.c. trapping voltage and on the particle charge $q_p$. Calculations are performed for the following parameter values of the microparticle species used: radius $r_p = 5 \ \mu$m, electric charge $q_p = 3 \times 10^4 e \div 5 \times 10^{11} e$. Vertical lines 1 – 4 correspond to electric charge values of $q_p = 4, 6, 8, 10 \times 10^4 e$, considered in order to estimate the oscillation amplitude in the trap. Source: graphic reproduced from [@Mih16b; @Lapi16d] with kind permission of the authors. Copyright AIP.[]{data-label="468Stab"}](468Stab.png)
![Dependence of the average oscillation amplitude on the number of trap electrodes, the particle charge $q_p$ and the a.c. voltage frequency $f$. Calculations are performed using the following parameters: particle radius $r_p = 5 \ \mu$m, electric charge $q_p = 3 \times 10^4 e \div 1.2 \times 10^5 e$. Source: picture reproduced from [@Lapi16d] with permission of the authors.[]{data-label="468ampl"}](468ampl.png)
Experimental investigations are performed in quadrupole and octupole vertically oriented traps. Fig. \[quadro\] and Fig. \[octu\] show photographs of a horizontal cut of dusty structures confined for various frequencies of the a.c. trapping voltage. The voltage amplitude is 4.25 kV. The bright spot in the centre of the photos represents the end electrode that provides particle levitation.
![Dust structure in a quadrupole trap: $f = 50$ Hz (a), $f = 110$ Hz (b).[]{data-label="quadro"}](quadro.png)
![Dust structure in an octupole trap: $f = 50$ Hz (a), $f = 110$ Hz (b).[]{data-label="octu"}](octu.png)
In case of the quadrupole configuration for a frequency $f = 50$ Hz, the particles are located near the symmetry axis of the trap. By increasing the frequency the amplitudes of particle oscillations decrease, while the inter-particle distances expand. When the frequency increases from 50 Hz to 130 Hz the oscillation amplitude drops off by 65%, from $0.65$ mm down to $0.23$ mm. In case of the octupole trap the particles are approximately arranged in a circle, whose diameter increases with the frequency (from $16$ mm to $27$ mm, when the frequency rises from $50$ Hz to $130$ Hz). The oscillation amplitude and the inter-particle distance depend on the frequency, in a similar manner to the situation in a quadrupole trap. However, the dependence is less significant: by incrementing the frequency from 50 Hz to 130 Hz the oscillation amplitude drops off only by 40%, from $0.62$ mm down to $0.37$ mm.
Perspectives and new technological approches
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A most recent paper introduces a multipole plasma trap that represents a novel approach based on using RF electric multipole fields to levitate charged particles of a plasma within a 3D volume. A new concept is illustrated in [@Hicks19] whereupon one of the trap electrodes is replaced with the aperture of a linear multipole stage, which acts as a pipe by channeling plasma out of a source region into the trap volume. This linear multipole plasma transport can be also used to investigate 2D multipole confinement, whether an axial magnetic field is present or not.
Mass spectrometry using electrodynamic traps {#Sec7}
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Electrodynamic traps have proven to be versatile tools to perform chemical characterization of microparticles [@Huang97; @Jones97; @Song06]. A 3D Paul trap can be used to perform both microparticle or even molecule levitation and mass analysis [@Jons97]. Chemical characterization is then achieved by employing optical methods such as absorption, fluorescence, and Raman spectroscopy [@Dem15]. After investigation by means of optical methods the microparticle is analyzed using mass spectrometry. Current methods for ion generation include laser ablation, desorption or photoionization.
Accurate control and manipulation of trapped ion species plays an important role in advancing many technologies, including ion-based quantum information processing (QIP), mass spectrometry (MS), quantum metrology and quantum engineering. In the field of life sciences and chemical analysis, MS using ion and charged particle traps is a versatile and powerful analytical tool to investigate chemical and biological species. Specimens with diameters ranging between $10$ nm $\div 20 \ \mu$m, i.e. DNA sequences, proteins, virus, cells, bacteria, and generally NPs can be diagnosed with high resolution and accuracy. In an attempt to further increase the resolution of MS, integration of advanced optical detection is considered a key approach [@Jiang11]. Similarly, optical spectroscopy of nanometre sized particles depends on ion trap capacity. Gas phase nanoparticles ranging from 3 nm to 15 nm in diameters have been confined and levitated, using a Trapped Reactive Atmospheric Particle Spectrometer (TRAPS) with a maximum trap capacity of $5 \times 108$ particles and a residence time of 12 sec, for X-ray and optical spectroscopy [@Mein10]. Recently, a surface-electrode ion trap was developed with integrated fiber optics capable of efficiently collecting fluorescence light from the ion [@VanDev10]. It is obvious that ion manipulation finds use in various applications such as miniaturized chemical detectors, vacuum technology, ultra-precise atomic clocks and secure communication schemes. Therefore, new approaches for particle (ion) control and manipulation are a mandatory step towards implementing the next generation of advanced technologies [@Kolo07; @Smith10].
A linear ion (Paul) trap (LIT) uses a superposition of time varying, strongly inhomogeneous (a.c.) and d.c. electric potentials, to achieve a trapping field that dynamically confines ions and other electrically charged particles [@Major05; @March97; @March05; @Vini15; @March95a; @March17a]. When the a.c. trapping voltage frequency lies in the few Hz up to MHz or even GHz range, electrons, molecular ions or electrically charged micro and nanoparticles with masses of more than 10 [*u*]{} (atomic mass units) are confined [@Smith10]. Ion dynamics in a Paul trap is described by a system of linear, uncoupled equations of motion (Hill equations [@Mag66]), that can be solved analytically [@McLac64; @Rand16; @Vasi13; @Lebe12]. The linear trap geometry can be used as a selective mass filter or as an actual trap, by creating a potential well for the ions along the $z$ axis of the electrodes [@Sto01; @Otto09]. In addition the linear design results in increased ion storage capacity, faster scan times, and simplicity of construction. A Paul trap runs in the mass-selective axial instability mode by scanning the frequency of the applied a.c. field [@Staff84; @Nie08; @Smith08]. Pictures of a linear and of an annular trap geometry intended for microparticle levitation under SATP conditions are presented in Fig. \[LinTrap\] [@Ghe98; @Sto01].
Recent sampling and ionization methods extend mass spectrometry (MS) to many applications, such as investigation of biomolecules, aerosols, explosives, petroleum, and even microorganisms [@Peng04]. Recent advances towards the miniaturization of mass spectrometers are presented in [@Ouya09; @Peng11; @Guo18]. The approaches used consist either in minimizing the RF electronics and the vacuum system, or in using new materials such as polymers or ceramics. One of the challenges in chemical physics lies in the ability to control ion temperatures. To achieve this goal low-temperature photoelectron spectroscopy instruments have been developed, aimed at investigating complex anions in the gas phase, including multiply charged ones and molecules [@Wang08]. Such an experimental setup consists of an electrospray ionization (ESI) source, a 3D Paul trap where ions are laser cooled, a ToF mass spectrometer (MS), and a magnetic-bottle photoelectron analyzer. The device allows a good control of the ion temperatures in the $ 10 \div 350 \ K$ range.
The quadrupole Paul trap as a mass spectrometer
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Since its early days the Paul trap (3D QIT) [@Paul58; @Paul90; @Major05; @Blaum98; @Doug09] has proven to be a versatile device that uses path stability as a means of separating ions according to their specific charge ratio. Mass analysis is mainly performed using either one of two techniques: (a) mass selective resonance detection, when presence of ions is detected by means of an external electronic circuit connected between the endcap electrodes, or (b) mass selective storage, in which the positive ions are expelled through holes pierced in the endcap electrodes onto the first dynode of an electron multiplier or into a [*channeltron*]{}. Mass selective storage is based on external detection, which lifts many of the issues associated with mass-selective detection [@March97; @March05; @March95a; @Kais91; @March95b; @Xu09; @Gross17].
The resonance detection technique represents the original method to perform mass analysis using an ion trap [@Paul58; @Blaum98]. It employs supplying a small a.c. voltage of frequency $\omega_{ac}$ between the endcap electrodes, in addition to the d.c. and RF ion trapping voltages $U_0$ and $V\cos\left(\omega t\right)$, applied between the endcap and ring electrodes. Any trapped ions that exhibit a mass to charge ratio $m/z$ [@Kais91] in such a way that their fundamental frequency of motion in the axial direction $\omega_z$ is equal to the perturbing field frequency value $\omega_{ac}$, will resonate. Ions get neutralized at the endcap electrodes, while the resulting current is detected [@Staff84; @Fisch59; @Schwa91; @Guan93]. Broad-band nondestructive ion detection can be achieved in a QITMS by excitation of a cloud of trapped ions with different masses, followed by recording of ion image currents induced on a small detector electrode embedded in the surrounding endcap electrode [@Soni96; @Nappi98].
The mass selective storage technique for mass analysis of ions was devised by Dawson and Whetten [@Gross17; @Daws69]. Mass spectrometers based on this technique have been experimentally tested and investigated [@March09; @March97; @March05; @Schwa91; @Daws69]. Operation of a QIT based on this technique requires that the applied d.c and RF trapping voltages are chosen such that the range of $m/z$ values trapped in the device is made so narrow, that only ions of a particular $m/z$ ratio can be confined. Ejection and detection of the trapped ions is usually accomplished by applying a voltage pulse between the endcaps. The ions expelled reach an electron multiplier and the corresponding electric current is measured. To achieve a mass scan over a wide range of ion masses, an experiment must be performed for every possible $m/z$ value within the chosen range [@Staff84; @Schwa91; @Guna09].
If a scan line is selected that crosses the stable region of a Paul trap near its upper corner, only a narrow range of masses will achieve stable confinement within the trap volume (see Fig. \[MassScan\]). The resolving power of the trap is enhanced as the scan line approaches the crest of the stable area.
![ Scan line with constant ratio $a/q$ for operation of a Paul trap as a mass spectrometer. Picture reproduced from [@Werth09] by courtesy of G. Werth.[]{data-label="MassScan"}](MassScan.jpg)
The Quadrupole Ion Trap Mass Spectrometry (QITMS) is a promising technique to perform mass analysis of micron-sized particles such as biological cells, aerosols, and synthetic polymers [@Dale91; @Batey14]. The trap can be operated in the mass-selective axial instability mode [@Nie08; @Smith08]. Microparticle diagnosis can be achieved by operating the QIT as an EDB, for low frequency values of the a.c. field applied to the trap electrodes (typically less than 1 kHz) [@March10; @Doug09; @March95c]. Because of this low frequency, in order to achieve high mass measurement accuracy (better than 1 ppm) only one particle is analyzed at a time, over a time period ranging from seconds up to minutes. A specific charge $m/z$ can be isolated in the ion trap by ejecting all other $m/z$ particles, when scanning various resonant frequencies. Moreover, an ion trap can be coupled to an Aerosol Mass Spectrometer (AMS) to investigate atmospheric aerosol (nano)particles [@Kul11; @Kwa11; @Ula11; @Signo11; @Cana07].
Advantages of ion trap mass spectrometry
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Due to intrinsic features such as high sensitivity, compact size, somewhat large operating pressure and the unique capacity to achieve multistage tandem mass analysis (MS$^n$), QITs have been largely used as mass analyzers [@March97; @March10; @March17b; @March17a; @Nie08; @Doug05]. First experimental setups were based on classical 3D Paul traps with hyperbolic-shaped electrodes [@March05; @Blaum06; @March95a; @Blaum98; @Doug09; @Schwa91]. Practical realization of such geometries is frequently associated with mechanical imperfections and electrode misalignments [@Tak07], that degrade the strength of the quadrupole trapping field and lead to the occurrence of higher order terms in the series expansion of the electric potential [@March05; @Beat87; @Aust10; @Zhang11; @Wang14]. Ion dynamics in a nonlinear resonance ion trap used to perform MS is characterized by a nonlinear Mathieu differential equation [@Wang93]. The performances of commercial quadrupole mass spectrometers (QMS) characterized by imperfections have been investigated with respect to an ideal hyperbolic 3D Paul trap geometry, using the computer simulation program SIMION 3D [@Blaum98]. In addition, the case of a QMS in which a static magnetic field is applied axially in the $z$-direction along the length of the mass filter is presented in [@Syed10]. A technique that relies on the variation of the electric field within a cylindrical ion trap (CIT) mass spectrometer while it is in operation, is presented in [@Sona13]. By employing this technique, the electrodes of the CIT are split into a number of mini-electrodes, supplied at different voltages in order to achieve the desired field. The fundamental principles of quadrupole mass spectrometers (QMS) are discussed in [@Batey14].
An issue of high interest for MS experiments is related to the effects induced by supplying various d.c. magnitudes and polarities to only one of the endcaps of a 3D QIT. A monopolar d.c. field is obtained by supplying a d.c. potential to the exit endcap electrode, while the entrance endcap electrode is kept at ground potential. Control over the monopolar d.c. magnitude and polarity during time periods associated with ion accumulation, mass analysis, ion isolation, etc., leads to increased ion capture efficiency, increased ion ejection efficiency during mass analysis, and effective isolation of ions using lower a.c. resonance ejection amplitudes [@Koizu09; @Prent11]. As an outcome of these remarkable experimental improvements the performance of a 3D ion trap used for MS experiments is greatly enhanced.
Late experiments show that decent performance can also be achieved using simplified trap geometries such as hybrid [@Arkin99], spherical [@Nosh14], linear [@Suda12], planar [@Clark13; @Bahra19; @Song06; @Aust10; @Zhang11; @Alda16], and cylindrical (toroidal) setups [@Li17; @Taylor12; @Higgs16; @Kot16; @Kot17; @Higgs18] that exhibit the advantage of enhanced compactness, while being easier to design and machine [@Taylor12]. Reduced power operation of a mass analyzer under conditions of minimum loss of spectral resolution and mass range is beneficial when discussing in terms of portability. Given that the RF amplitude required to perform mass analysis scales with the square of the analyzer dimensions, minimization of QITs has represented a primary concern. The performance of a miniature, stainless steel, rectilinear ion trap (RIT) is investigated in [@Hend11]. Portable mass spectrometers usually employ cylindrical ion traps (CIT). A LIT exhibits sensibly improved trapping capacity and trapping efficiency with respect to a 3D QIT [@Major05; @March09; @March05]. A rectilinear ion trap geometry used in a miniaturized MS system was able to achieve a resolution of $\Delta m/z = 0.6$ Th (FWHM -full width at half-maximum) and a mass range of up to $m/z \sim 900$ [@Li14a]. Electrode misalignments in linear ion traps have been investigated in [@Wu15], by employing SIMION to model the case of a two-plate linear ion trap. Geometric misalignments in six degrees of freedom were analyzed with respect to the resolving power and ion detection efficiency. A new fabrication method, numerical simulations performed with SIMION, and experimental results for micromachined CIT (m-CIT) arrays suggested for use in miniaturized mass spectrometers are presented in [@Chaud14].
Intensive efforts of the scientific community have been performed towards miniaturization of ion traps for portable MS applications. The most demanding technological challenge lies in limiting the energy consumption of particular components of a mass spectrometer, such as the vacuum system, the ion source, and the RF amplifier [@Jones97; @Math06; @Jau11]. The development of a multichannel arbitrary waveform generator intended for QIP using ITs is described in [@Baig13]. A technique that achieves active stabilization of the harmonic oscillation frequency of a laser-cooled atomic ion confined in a RF Paul trap is demonstrated in [@Johns16]. The solution relies on sampling and rectifying the high voltage RF applied to the trap electrodes. Thus, the 1 MHz atomic oscillation frequency is stabilized to a value that is below 10 Hz. Use of this technique is expected to enhance the sensitivity of ion trap (IT) based MS, and the fidelity of quantum operations in IT quantum information processing (QIP). A low power RF amplifier circuit for ion trap applications is presented and tested in [@Nori16]. A different paper describes the design and operation of a multichannel, low-drift, low-noise d.c. voltage source, particularly designed for supplying the electrodes of a segmented LIT [@Beev17].
Digital QMS(s) exhibit certain unique features and render more simple various ion handling operations, as they provide an enhanced control over the frequency, duty cycle, and amplitude of the trapping potentials. The work of Brabeck [*et al*]{} [@Brabe16] demonstrates how matrix solutions of the Hill differential equation can be used to explore the influence of the additional degrees of freedom on ion stability. In order to illustrate these effects, the paper provides the stability diagrams corresponding to a digital mass filter that employs asymmetric driving potentials.
Chemical physics relies strongly on QIT as versatile investigation tools. ITs are currently used to investigate both external and internal dynamical systems. An interesting discussion focused on linear and nonlinear resonances in QITs is carried out in [@Sny17a], with an emphasis on the effect of quadrupole field nonlinearity.
The mass/charge range of a MS that operates either at the boundary of the first stability region of the Mathieu equation or in the resonance ejection mode [@Koizu09; @Remes15], is usually limited by the highest RF voltage that can be supplied to the trap electrodes. High voltages are dangerous for miniature instruments as they might induce discharges onto the electrodes which are separated by small distances. To overcome this technological challenge an alternative approach to mass range extension has been used, based on a method of scanning the resonance ejection frequency nonlinearly in the form of an inverse Mathieu $q$ scan [@Sny17b; @Sny17c]. The method has been tested experimentally in case of a benchtop LTQ linear ion trap and for the Mini 12 miniature LITMS. In both situations, an increase in mass range of up to 3.5 times is reported [@Sny17c].
Trap operation in the mass-selective axial instability mode
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Operation of an ion trap in the mass-selective axial instability mode was reported for the first time in 1984 [@Staff84]. It is based on the fact that if the working point $\left(a_z, q_z\right)$ moves along a [*scan line*]{} which crosses either the $\beta_z = 0$ or $\beta_z = 1$ boundary, then the trapped ion will rapidly develop axial instability and get expelled onto an appropriately positioned detector. Experimentally, the simplest way to achieve axial instability lies in supplying the ring electrode with RF power only, which renders the scan line coincident with the $q_z$ axis [@March05; @Staff84]. The method begins by creating ions using electron bombardment or photoionization, and maintaining the RF trapping voltage constant. Then, the RF voltage is ramped linearly in such a way that the $a_z, q_z$ values of the ions shift to the $\beta_z=1$ boundary, at which point ions are expelled out of the trap whilst the $m/z$ value increases, as illustrated in Fig. \[axinstab\].
![Stability diagram for the quadrupole ion trap plotted in ($a_z, q_z$) space. The points marked $m_1$, $m_2$, and $m_3 \left(m_1 < m_2 < m_3\right)$ refer to the coordinates of three ions: $m_1$ has already been ejected, $m_2$ is on the point of ejection, while the species $m_3$ is still trapped. Reproduced from R. E. March and J. F. J. Todd (Eds.), Practical Aspects of Ion Trap Mass Spectrometry, Vol. III, page 11, Figure 1.7. CRC Press, Boca Raton, FL, 1995. Copyright 1995 by CRC LLC. Reproduced with permission of CRC Press LLC in the format Journal/magazine via Copyright Clearance Center.[]{data-label="axinstab"}](QITAxiInstab.pdf)
Charge detection QITMS has demonstrated to be a remarkable technique suited for high-speed mass analysis of micron-sized particles, such as biological cells and aerosols [@Smith08]. The paper of Nie [@Nie08] describes the technique, and explains how the trap can operate in the mass-selective axial instability mode by scanning the frequency of the applied a.c. field. A calibration method is presented, aimed at finding the points of ejection in the stability diagram $\left(q_{eject}\right)$ of the individual particles that are investigated.
Specific charge measurement techniques based on ion trap mass spectrometry {#specharge}
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We basically describe the method employed to measure the specific charge for different microparticle species levitated in different Paul trap geometries tested at INFLPR, operated under SATP conditions. The equation of motion for a particle of mass $M$ and charge $Q$ confined within the trap is [@Ghe98; @Sto01]:
$${\bf F} = Q {\bf E} - K{\bf r} + m{\bf g} + Q\bf{E_{dc}} \ ,$$
where ${\bf r} = \left( x, y, z\right)$ is the particle vector position, and $K \left(K > 0\right)$ is the coefficient describing the aerodynamical drag force. ${\bf E}$ represents the trap electric field, $M{\bf g}$ stands for the gravitational force, and ${\bf E_{dc}} = V_2/\left(2 r_0\right)$ is the electric field produced by applying a static potential difference between the upper and lower rods (electrodes) of a quadrupole LIT separated at $2 r_0$, in order to shift the particles towards the trap centre.
We introduce the $\tau = \Omega t/2$ and $\Lambda = K / \left( M \Omega \right)$ parameters, where $\Omega$ is the RF drive frequency. We also denote $X = e^{\Lambda \tau}x, Y = e^{\Lambda \tau}y, Z = e^{\Lambda \tau}z$. Then, the Mathieu equations that characterize the trapping process are homogeneous along the $X$ and $Y$ axes, with well known solutions and stability domains. The Z-axis dynamics is described by an inhomogeneous Mathieu equation. In this case stability is obtained for (a) $i\mu $ real; (b) $\mu$ real and $|\mu| < \Lambda$; (c) $\mu -i$ real and $|\mu - i| < \Lambda$. Hence, the stability regions corresponding to the solutions of the inhomogeneous equations of motion in presence of drag forces include not only the stability regions of the homogeneous equations but also a part of the instability regions, which means they are extended.
Typically, parametric excitation of the trapped microparticle motion is achieved by applying a low amplitude and variable frequency additional a.c. voltage in series with the $V_{ac}$ trapping voltage. When the supplementary a.c. field frequency is twice that of the secular motion, microparticles (located at the limit of the first Mathieu stability domain) resonantly absorb energy from the field while their secular motion amplitude exponentially increases [@Ghe96b]. If the trap contains a number of noninteracting particles, besides normal resonance at the secular motion frequency and parametrical resonance at the double secular frequency, other weak resonances are observed [@Ved90] as a consequence of the presence of coupled terms for various combinations of the motion frequencies.
When the coupling parameter between particles (the ratio between Coulombian and thermal energy) is greater than 175, solid effects occur in the levitated particles system [@Major05; @Werth09; @Boll84; @Werth05a; @Dub99; @Wine88]. As the inter-particle distance is larger for trapped microscopic particles than for trapped ions, it is much easier to observe such behaviour. In Fig. \[interpart\] we supply a picture of the inter-particle distance $d$ for a variable linear trap geometry designed and tested in INFLPR [@Ghe98; @Sto01] and presented in Fig. \[LinTrap\], for SiC microparticles with homogeneous, well-defined dimensions ranging between $ 50 \div 1000 \ \mu$m. $N$ represents the number of microparticles, while $L$ is the trap length.
![Picture of inter-particle distance $d$ for a variable linear trap geometry designed and tested in INFLPR [@Ghe98; @Sto01] presented in Fig. \[LinTrap\], for SiC microparticles with dimensions ranging between $ 50 \div 1000 \ \mu$m. $N$ represents the number of microparticles, while $L$ is the trap length. Reproduced from [@Sto01] with kind permission of the publisher.)[]{data-label="interpart"}](Interpart.jpg)
One up to thousands of microparticles can be trapped along the trap axis. Ordered structures (planar, zig-zag, and volume structures) are reported within the LIT confinement volume. The $Q/M$ specific charge ratio for SiC microparticles has been estimated from the $Q / M = 2 g \ {\Delta z_0} /\Delta V_2$ relation as about $10^{-4}$ C/kg, for a static trapping voltage $V_0 = 500$ V [@Sto01], where $\Delta z_0$ stands for the axial shift of the microparticle motion corresponding to a variation of the $V_2$ voltage denoted by $\Delta V_2$. The absolute electric charge is evaluated from the electrostatic equilibrium conditions for microparticles confined along the trap axis, considering the $V_2/2$ generated electrical forces and the inter-particle Coulombian repulsion. For a value $V_2 = 500$ V and $L = 75$ mm, an electric charge value $Q$ of around $10^{-11}$ C is inferred. By knowing the specific charge value, the microparticle mass can be instantly evaluated. As the SiC density value is known, the microparticle radius [@Lapi18c] can be inferred.
The specific charge has been estimated for different trapped microparticle species, and the values are obtained are displayed in Tabel \[specific\] [@Sto01].
**Microparticle species** **SiC** **Anthracene** **Alumina** **Hydroxyl appatite**
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$\bar{Q}/\bar{M} \times \ 10^{-4} \left[ \text{C/kg}\right]$ $3.482$ $2.974$ $4.147$ $4.873$
: Averaged values of the specific charge ration for different microparticle species levitated in the linear Paul trap presented in Fig. \[LinTrap\][]{data-label="specific"}
A different diagnosis method employs parametrical excitation of the microparticle motion. The parametrical excitation voltage frequency is $\omega/2\pi$. When the value of $\omega$ is twice the secular motion frequency, parametrical resonance occurs. Experimental measurements performed for $100 \ \mu$m diameter SiC microparticles, $\Omega/2 \pi = 100$ Hz and $V_{ac} = 2.5$ kV, reveal parametrical resonance at about 25 Hz. Hence, the specific charge of the SiC microparticles was evaluated as $5 \times 10^{-4} $ C/kg [@Ghe98].
Measurements of charge density of tapered optical fibers using charged polystyrene particles confined in a linear Paul trap operating under SATP conditions are reported in [@Kami16].
Aerosols and nanoparticles {#Sec8}
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According to [@Kul11] aerosols can be described as two-phase systems consisting of the suspended solid or liquid phase, levitated (immersed) in the surrounding gas phase. Despite their small size aerosols yield a major impact on global climate and health, but the underlying mechanisms and subsequent effects are still far from being explained. Typically, ambient aerosol particles are characterized by dimensions ranging from a few nm to a few hundred micrometers, while they consist of a wide variety of materials. About 90 % of the aerosols found in the atmosphere are generated by natural sources. Volcanoes eject huge amounts of ash and volcanic debris in the troposphere, as well as sulfur dioxide and other gases. Sea spray is another important source of natural aerosols [@Sult17], or the wind carrying desert sand particles or soil dust (including specific minerals) to either sea or land areas [@Kond06]. Both species of previously described aerosols are larger particles with respect to their human-made counterparts. Phytoplankton is also responsible for the emission of gases such as dimethyl sulfate.
Solid and liquid aerosol particles that consist of a significant fraction of biological materials are called bioaerosols. Their dimensions vary between $0.1 \div 250 \ \mu$m. Airborne bacteria, viruses, toxins and allergens, or fungal spores, represent important types of bioaerosols [@Heo17]. Bioaerosols have a tremendous economic impact in the transmission of diseases of humans (e.g., flu, severe acute respiratory syndrome (SARS)), other animals, agricultural crops, and other plants, while also causing asthma or allergies [@Srik08; @Kim18]. Hence, detection and characterization of bioaerosols is an issue of vivid interest. Moreover, as the world is confronted with an increasing wave of terrorist attacks, detection of bioaerosols or other aerosols associated with traces of explosives or chemical weapons is a matter of utmost importance. Hence, there is a need for improved methods and instrumentation for rapidly characterizing harmful bioaerosols [@Kim08; @Islam18]. Such instruments could also be used in studying and monitoring disease dissemination [@Ghosh15; @Fuji17].
We also mention the contribution of cosmic aerosols that is negligible with respect to the aerosol sources described above. Space weather is a relatively new and important field of research, as it represents a branch of space physics and aeronomy. It strongly influences radio communications, space missions, diagnostics of ionospheric and space plasmas, detection of pollutants and re-entry objects, weather prediction, and the phenomenon of global warming. Recent scientific progress and results clearly show that nano- and micrometre-sized electrically charged particles coming from the interplanetary space and present in the Earth’s atmosphere, can alter both the local properties and diagnostics of the interplanetary, magnetospheric, ionospheric and terrestrial complex plasmas. The various sources of charged dust particles and their effects on the near-Earth space weather are extensively presented in Ref. [@Pop11]. The remaining 10 % of aerosols are considered anthropogenic or of human origin, as they are produced by a variety of sources. Even if less abundant than their natural counterparts, anthropogenic aerosols can prevail the air downwind of urban and industrial areas [@Kul11; @Sande14; @Tom17; @Mao14; @Marin18; @Marin19].
Aerosols play a prominent role in a wide range of scientific areas. We emphasize that aerosols are also intensively used in the pharmaceutical industry, while technological applications include spray drying or delivery of fules for combustion.
Atmospheric aerosols. Sources. Classification. Physical and chemical properties {#atmaero}
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We can distinguish between [*primary*]{} and [*secondary*]{} aerosol sources. The primary sources of aerosols are mostly of natural origin. Among them we can enumerate oceans (sea salt aerosols), desert and semiarid regions, biological material, smoke resulted from burning of biological material, direct anthropogenic particle emissions such as soot, road dust, suspended particulate matter, or interplanetary dust particulates. Secondary (indirect) aerosol sources which represent the major source of particles below 1 $\mu$m in radius, are generated by conversion of available natural and man made atmospheric trace gases into solid and liquid particles. Extraterrestrial particles are generated by meteor showers or comet debris that disintegrate on collision, get trapped in the Van Allen radiation belts and occur mostly over the Earth poles [@Rama18]. Atmospheric aerosols represent a mixture of solid or liquid particles suspended in air [@Mull08]. Similar to astrophysical dusty plasmas, atmospheric aerosols are made out of micro- and nanoparticles (NPs) [@Rama18; @Sein16; @Agran10].
Human industrial activity that leads to the release of anthropogenic aerosols in the atmosphere is responsible for adverse health effects, producing haze in urban areas or even deposition of acids [@Kim08; @Islam18; @Via13; @Kelly15]. Anthropogenic aerosols alter the radiation balance of the Earth (albedo) [@Pand95]. Although NPs represent the largest portion of ambient aerosol concentration, in fact they rarely account for a significant fraction of the total mass [@Sein16]. NPs can emerge from primary or secondary processes [@Via16]. Primary particles are generated directly at source, while secondary ones occur from the condensation of gas-phase species. The main aerosol species that exist in the atmosphere include sea salt, mineral dust, sulfate, nitrate, and carbonaceous aerosols (black carbon - BC [@Liu18] and organic carbon) [@Rama18].
Different micron sized organisms immersed in air are called airborne particles or bioaerosols, a category which mainly includes pathogenic and non-pathogenic, live or dead fungi and bacteria, viruses, allergens, pollen, etc., most of them held responsible for the increasing incidence of health issues of human beings and other living animals [@Heo17; @Srik08; @Kim18; @Nowo16]. Unfortunately, pathogenic bioaerosols are also used as biological weapons, such as the bio-terrorist attacks that took place in 2001 using [*Bacillus antracis*]{} spores. Together with the pandemic outbreak of flue due to influenza AH1N1 virus in 2009, these events represented strong arguments that emphasize the significance of bioaerosol research [@Kim08; @Ghosh15]. It is a matter of utmost importance, as it is considered that bioaerosols are considered to represent future weapons of mass destruction (WMD). Currently, there is a vivid interest to investigate and explain the mechanisms responsible for the occurrence and transport of bioaerosol, with an aim to identify the associated health risks for population and limit the exposure. In-depth details on indoor bioaerosol sampling and dedicated analytical techniques are found in [@Ghosh15; @Fuji17], where the authors show that apart from bioaerosol recognition further development of their control mechanisms represents a primary concern [@Mull08; @Nowo16].
The complex picture associated to an aerosol, which is the outcome of the heterogeneity in particle size, chemical composition, phase, and mixing state, is further complicated by the inherently rapid coupling with the surrounding gas phase [@Wills09].
Global climate and environment protection. Aerosols and quality of life {#globcli}
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The direct relationship between the presence of aerosols in the atmosphere and factors such as the quality of life, raises scientific challenges and makes chemical analysis of aerosol particles an issue of great interest [@Kul11; @Scha12]. Ocean-derived microbes in sea spray aerosol (SSA) possess the ability to affect global climate and weather, by acting as ice nucleating particles in clouds [@Sult17; @Smirn17]. As the atmosphere is characterized by a large variety of organic compounds with complex structure and in many cases with intrinsic fragility, study of atmospheric chemistry and global climate evolution uses different tools and methods in order to achieve such goal [@Rama18; @Tom17; @Bou15; @Box15]. Many questions are still left unanswered, as well as the accompanying mechanisms that account for these effects. For example, the processes responsible for the formation of secondary organic aerosol (SOA) components are not yet explained. Different types of particles can clump together to form hybrids that are difficult to discriminate. Changes in humidity or temperature may result in drastic changes in how certain aerosol species behave and interact with cloud droplets. Hence, there is a vivid interest towards developing new sensitive mass spectrometric methods with soft ionization techniques that can help in identifying such components among atmospheric particles, while also characterizing the underlying mechanisms. The main objective lies in building new tools for laboratories and ground-based aerosol monitoring stations, able to supply relevant data that are missing or are rendered incomplete using methods other than ion trap MS.
Late studies demonstrate that presence of atmospheric aerosol particles in the atmosphere has significant direct and indirect effects on air quality, global climate and hydrological processes [@Kim08; @Tom17; @Smirn17; @Kelly16], things related to the phenomenon of global warming and the quality of life [@Kul11; @Nowo16; @Bou15; @Box15]. The impact produced by aerosol particles is the outcome of their physical and chemical properties [@Davis02; @Davis97; @Kond06; @Mao14]. Aerosols are presumed to have a larger impact on climate compared to greenhouse gases [@Smirn17], according to the Intergovernmental Panel on Climate Change (IPCC) Working Group (WG) I 4-th and 5-th Assessment Report (AR4 & AR5), released in 2007 and 2013 respectively [@IPCC6]. Nevertheless there still is a high uncertainty about it, owing to the aerosol complex composition and a still incomplete picture needed to characterize the interactions between aerosols and global climate [@Kul11; @Sein16; @Kim08; @Mao14; @Kelly16]. We can distinguish between two types of interaction: direct and indirect interactions. By indirect effect hydrophilic aerosols act as cloud condensation nuclei (CCN), affecting cloud cover and implicitly the radiation balance. Direct interactions account for the light scattering mechanism on aerosols, resulting in cooling effects. On the other hand, aerosols containing black carbon (BC) [@Liu18; @Liou11; @Liu19] or other substances absorb incoming light, thus heating the atmosphere [@Kul11; @Rama18]. According to the measurements performed, the direct radiative effect of BC would be the second-most important contributor to global warming, after absorption by carbon dioxide (CO$_2$) [@Liu19]. In addition to scattering or absorbing radiation, aerosols can alter the reflectivity or albedo of the planet. The impact of aerosols and the associated interactions are shown in Fig. \[Brook\].
![Natural and anthropogenic aerosol sources. Aerosol particles alter global climate, both individually and by performing as cloud condensation nuclei (CCN), thus influencing the energetic balance of the Earth (or albedo). Thus, gathering of accurate data to help understand the physical and chemical mechanisms characteristic to aerosols is a mandatory step to explain their effects on Earth’s climate. Picture reproduced from [@Kelly16].[]{data-label="Brook"}](BrookAerosol.jpg)
In addition, nonspherical particles have a major impact on the Earth atmosphere as they influence processes such as radiative forcing [@Rama18; @Liou11; @Lee16], photochemistry, new particle formation, and phase transitions [@David18]. The scientific community is highly interested in explaining the physical properties and chemical composition of submicron aerosol particles, in an effort to better characterize radiative forcing [@Liou11; @Lee16] and air quality [@Kul11; @Sein16; @Cana07; @Kim08; @Tom17; @Kulma11]. Organic Aerosol (OA) represents the prominent fraction of non-refractory submicron particle mass (between 18 and 70 %), while in tropical forest regions OA stands for almost 90 % of the total fine aerosol mass. Thus, OA is the most abundant but least characterized fraction of atmospheric aerosol particles, as an outcome of its highly complex chemical composition [@Vogel13a].
There is a large interest towards minimizing the uncertainties associated with data collection when evaluating the impact of aerosols on global climate [@Lebe12; @Islam18; @Agran10; @Lee16; @Kirch08; @David08]. Different approaches and methods have been developed for analyzing particles ranging from 10 nm to $10 \mu$m in diameter size, which consist of salts, soot, crustal matter, metals, and organic molecules, often mixed together [@Nash06; @Wang06]. Late investigations show that the largest uncertainty in the radiative forcing of climate stems from the interaction of aerosols with clouds [@Rama18; @Nowo16; @Kelly16]. Thus, accurate characterization of aerosols and of their intrinsic properties is a mandatory step towards explaining many important processes in the atmosphere while also characterizing the energy balance of the Earth [@Rama18; @Sein16; @Tom17; @Bou15]. The incoming solar radiation is modified when it passes through the atmosphere by two fundamental processes: light scattering and light absorption [@Liou11]. One can frequently find the term attenuation or light extinction, which includes both processes. The attenuation of the light by these processes results in important climatic consequences. Aerosol particles both absorb and scatter the light, with the efficiency of the processes being highly dependent on their size and shape, chemical composition, morphology, and the wavelength of the incident radiation. Aerosol dimensions vary between $0.001 \div 100 \ \mu$m, but the maximum in scattering efficiency is found for aerosol particles with dimensions ranging from $0.1 \div 1 \ \mu$m [@Rama18].
The European Space Agency (ESA) considers that Earth observation will result in “a reliable assessment of the global impact of human activity and the likely future extent of climate change” [@Earth]. Radiative forcing can not be estimated with precision at current time, so reliable assessment depends on the development of global models for aerosols and clouds that are well supported by observations [@Kim08; @Islam18; @Lee16]. The ESA Living Planet Programme includes the EarthCARE (Earth Clouds, Aerosol and Radiation Explorer) mission, which investigates the aerosols’ interaction with clouds. The primary target of the EarthCARE mission scheduled for launch in 2021, is to advance knowledge “of the role that clouds and aerosols play in reflecting incident solar radiation back into space and trapping infrared radiation emitted from Earth’s surface” [@Earth]. An enhanced understanding and better modeling of the relationship between clouds, aerosols, and radiation is considered to be one of the highest priorities in climate research, weather prediction and quality of life assurance. The mission will employ state of the art high-performance lidar and radar technology to perform global observations of the vertical structure of clouds and aerosols [@Islam18], while in the same time performing measurements of radiation [@Kim08; @Mull08; @Come17; @Hal18; @Ans05; @Come13].
The field application of an aerosol concentrator, in parallel with the use of an atmospheric pressure chemical ionization ion trap mass spectrometer (APCI-IT-MS), represents a state of the art technique that is used to characterize aerosol particles, with an emphasize on OA [@Vogel13a]. Field measurements showing that atmospheric secondary organic aerosol (SOA) particles can be present in a highly viscous, glassy state have triggered intensive investigations that approach the issue of low diffusivities of water in glassy aerosols [@Bast17]. Moreover, water diffusion experiments on highly viscous single aerosol particles levitated in an EDB demonstrate a typical shift behaviour of the Mie scattering resonances [@Bast18]. This represents a clear indicator on the changing radial structure of the particle, which provides an experimental method and approach aimed at monitoring the diffusion process inside the particle.
Bioaerosols largely influence the Earth system, especially as an outcome of the interplay between the atmosphere, biosphere, climate, and public health. As shown in Section \[atmaero\], airborne bacteria, fungal spores, pollen, and other bioparticles can induce or aggravate human, animal, and plant diseases [@Nowo16]. In addition, bioaerosols may serve as nuclei for cloud droplets, ice crystals, and precipitations, thus strongly influencing climate and rainfalls [@Tom17].
Hazardous effects of aerosols and nanoparticles on humans. Fine particulate matter and coarse particle pollution {#hazard}
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Late studies aimed towards the assessment of the impact of airborne NP exposure on human health has resulted in extensive research activity directed towards better characterizing these particles and understanding which properties are the most prominent in the context of health effects [@Kelly15]. Therefore, comprehension of the sources, chemical composition, physical structure and ambient concentrations of nanoparticles has enhanced significantly in the last decade [@Sande14; @Islam18; @Via13; @Via16].
Airborne particles with dimensions $< 100$ nm are called ultrafine particles or NPs. Ultrafine aerosol concentrations in urban areas strongly depend on time and physical location [@Bzdek12]. Aerosol particles with diameter less than 10 $\mu$m enter the pulmonary bronchi, while those whose diameter is lower than 2.5 $\mu$m reach the pulmonary alveoli, a region where gas exchange takes place, and from here they are carried into blood [@Srik08]. Aerosols and NPs are known to inflict harmful effects on humans. There is a strong effort towards limiting the maximum concentration and enforce safety levels for the atmospheric aerosol micro and nanoparticles (including dust) most hazardous to human health [@Heo17; @Kim18; @Fuji17]. We distinguish between two main categories: (i) fine particles with a diameter less than 2.5 microns (also called fine particulate matter or PM$_{2.5}$), which are the most dangerous, and (ii) larger particles with a diameter less than 10 microns but larger than 2.5 microns, namely the particulate matter PM$_{10}$ (also called coarse particles). Important steps are being taken worldwide to limit pollution levels caused by PM$_{2.5}$ and PM$_{10}$ NPs, in order to minimise their harmful effects on humans and biological tissue. In the EU and USA directives are enforced which regulate the PM$_{2.5}$ and PM$_{10}$ NP levels. Member States must set up [*sampling points*]{} in urban and also in rural areas. In some fields of engineering, PM2.5 and PM10 particles are called NPs [@Via13; @Via16]. Besides particulate matter, these sampling points must perform measurements on the concentration of sulphur dioxide, nitrogen dioxide and oxides of nitrogen, lead, benzene and carbon monoxide. Measurements are performed to assess the impact of street characteristics and traffic factors, in order to evaluate the exposure of people to BC [@Dons13; @Willi16].
Strong evidence indicates that breathing in PM$_{2.5}$ over the course of hours to days (short-term exposure) and months to years (long-term exposure) can cause serious public health effects that include premature death, adverse cardiovascular and respiratory effects, or even harmful developmental and reproductive effects [@EPA19]. Lung cancer seems to be associated with the emission of NPs produced by diesel engines [@Kirch08]. Scientific data also indicates that breathing in larger sizes of particulate matter (coarse particles or PM$_{10}$) may also have public health consequences [@Kelly15]. In addition, particle pollution degrades public welfare by producing haze in cities or constantly increasing the rate of allergies for population living in urban areas. People with obesity or diabetes are more vulnerable to increased risk of PM - related health effects. This is why constant monitoring of the various polluting agents is a major concern for health services and life quality assurance in Europe, USA, and in other countries throughout the world [@EPA19; @IRSST]. Aerosol characteristics exhibit large fluctuations in time as the anthropogenic part is quite substantial in urban areas, causing incidence of associated health effects to be sensibly higher. This is an issue of utmost interest, as the largest fraction of the population in Europe is located around urban areas and it is therefore severely exposed to degraded and degrading air quality effects. Current ambient air quality monitoring network is solely based on fixed monitoring sites, which does not not always reflect the exposure impact and the associated effects on humans [@Srik08; @Kim18; @Tom17; @Nowo16; @Bou15].
Investigations demonstrate that breathable aerosols with dimensions ranging between $1 \div 10 \ \mu$m, are mostly dangerous for human health as they are responsible for respiratory (allergic asthma and rhinitis, airway inflammation) and infectious human diseases [@Ghosh15]. Regular or ordinary human activities generate bio-aerosols. Airborne bio-aerosols are carried by air flows, which means they can be inhaled or attached to human bodies [@Heo17; @Kim18]. A mandatory requirement to enforce occupational safety and public health purposes lies in monitoring and controlling bio-aerosol concentration in the air.
Recent research indicates that NPs are also associated with toxic effects on humans [@IRSST], as they are widely used by the cosmetics industry. Many sunscreens contain NPs of zinc oxide or titanium dioxide. There are manufacturers that have added C60 fullerenes into anti-aging creams, because these particles can act as antioxidants. Strong evidence suggests that normally inert materials can become toxic and damaging when they are nano-sized. Evidence collected indicates that NP effects on human (living) tissue are extremely dangerous [@IRSST]. As many metals are toxic, the metallic content of the nanoparticulate burden represents an important factor to consider when attempting to assess the impact of NP exposure on human health [@Sande14]. Special care must be taken for people occupied in this area or exposed to nanomaterials, which strongly motivates the need to further analyze and characterize nanomaterial systems [@Wang06]. NPs can be classified into different classes based on their properties, shapes or sizes. The main categories include fullerenes, metal NPs, ceramic NPs, and polymeric NPs. NPs possess unique physical and chemical properties due to their high surface area and nanoscale size [@Khan17].
Investigation methods. Quantitative uncertainties. Mitigation {#InvestMeth}
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Laboratory techniques aimed at investigating aerosol properties and associated physico-chemical processes usually involve the study of single particles, particle ensembles, or they collect aerosol properties out of bulk phase studies [@Kwa11; @Pin11]. Techniques aimed at investigating single aerosol particles help in better characterizing the sources, chemistry, and evolution of atmospheric aerosols. Single-particle techniques provide for: (1) measurement on the composition of single particles of roughly one picogram of mass, even if they occur in very low concentration, (2) rapid measurement of temporal and spatial fluctuations of specific aerosol species, especially those that exhibit low concentration, and (3) information on the morphology and internal structure of single particles [@Pin11]. The techniques that are used to characterize single particles in atmospheric aerosols are: (a) laser-induced-fluorescence (LIF), (b) aerosol laser-ablation mass spectroscopy, and (c) two-dimensional (2D) angular elastic scattering.
On the other hand, quantitative uncertainties in the amount of aerosols, and especially aerosol properties need to be mitigated. It is obvious that only more accurate measurements combined with complex computer modelling will provide the critical information that scientists need, with an aim to fully integrate aerosol impact into climate models and thus minimize uncertainty about how the climate will evolve in the future. Aerosol MS has proven to be a very promising and versatile technique to perform diagnosis of such aerosol components [@Davis10; @Kul11; @Sein16]. Aerosol mass spectrometers generally use time-of-flight mass spectrometers or linear quadrupole mass filters. Recent investigations demonstrated that an ion trap is a very powerful tool to perform chemical analysis of aerosol particles.
LIDAR techniques provide important information regarding the particle effective radius for anthropogenic aerosols, in case of forest fire smoke or for mixtures of anthropogenic aerosols and biomass burning generated aerosols [@Hal18; @Ans05; @Come13]. Moreover, multiwavelength Raman lidar observations performed in Europe and worldwide supply important data regarding natural sources of pollution and human produced pollution in the free troposphere [@Mull08].
The effect of aerosols is considered to be the major cause of uncertainty in the global climate radiation balance calculations. Aerosol extinction rests on the wavelength, size, concentration, composition, and to a lesser extent, the aerosol shape. Consequently, new methods and techniques are required to characterize and model these quantities. The size distribution of larger aerosols can be monitored by means of a multistatic lidar, at least in the spherical approximation. Aerosols that are small compared to the incident wavelength exhibit a Rayleigh-like scattering dependence, as their size cannot be estimated using multistatic lidar techniques. Raman lidar measurements, Mie models of extinction and backscatter Angstrom ratios demonstrate that small aerosols bring a relevant contribution to optical scattering, while also indicating that size information can be extracted from lidar gathered data.
Measurements performed on particle ensembles, that represent selected narrow size ranges of accumulation mode particles ($1 \ \mu$m diameter), generally produce a rapid assessment of a statistically averaged property. This approach allows scientists to explore the aerosol optical properties, chemical aging, and phase behaviour [@Kul11; @Wills09; @Bou15; @Krieg12; @Guan11].
The Earth atmosphere consists of absorbing aerosol particles ranging from high-absorbing BC to less-absorbing mineral dust and brown carbon [@Liu18; @Liu19; @Scha12]. The absorption of aerosol particles [@Liou11] has a major impact on climate, while it also poses a high risk to public health [@Tom17; @Bou15; @Box15]. This explains the primary concern towards monitoring and characterizing the regional and global distribution of these aerosol particulates. Characterization of the temporal and spatial distribution of absorbing aerosol particles demands a complex and varied approach. Remote sensing using laser and radar ground stations, as well as satellite monitoring, is a mandatory approach to achieve extended spatial and temporal coverage. In situ airborne and surface measurements bring important supplementary data. To these, we add single-particle measurements that provide critical insights that would be lost when performing ensemble measurements [@Kul11; @Tom17; @Wills09; @Bou15; @Walt19].
Cloud physics research has been long time performed in the laboratory, using a wide range of techniques. Experimental studies of cloud particles (haze droplets, cloud droplets, ice crystals, etc.) may be classified in two main categories: (a) those that require large populations of particles such as cloud chambers [@Hagen89; @Song94b], and methods that isolate individual particles [@Davis02; @Davis97; @Davis90; @Shaw00] such as EDB [@Hart92; @Lamb96], optical tweezers [@Gies15; @Wills09], acoustic levitation [@Sto11; @Andra18], etc. Single-particle studies avoid population effects, by containing the particles in a very well defined region of space under conditions of minimal perturbation [@Stra55; @Wuerk59; @Wint91; @Ghe95b; @Ghe98; @Shaw00].
In Fig. \[SegmTrapINFLPR\] we supply an image of a segmented quadrupole linear Paul trap designed and tested in INFLPR, used to levitate anthropogenic aerosols, in our case humic acid or ammonium sulphate (NH$_4$)$_2$SO$_4$. Experiments are in progress and different trap geometries are currently under test, as well as a new microsystem based programmable voltage source that delivers three d.c. voltages ($0 \div 1000$ V), and an a.c. voltage source that delivers a 3.5 kV peak voltage at a variable frequency.
![Levitation of solid aerosol particles of humic acid in a segmented Paul trap geometry under test in the spring of 2019 at INFLPR. The trap is operated under SATP conditions. The three independent sections of the trap can be observed in the images. Above the central section of the trap lies the collimation lens, located at one of the ends of an optical fiber that picks up the fluorescence signal and delivers it to a spectrometer. Work in progress[]{data-label="SegmTrapINFLPR"}](SegmPaul1.jpg)
Experiments performed demonstrate that a time series of optical resonance spectra of an evaporating, non-spherical, irregular aerosol particle levitated in an EDB, exhibits patterns which are associated to its evaporation kinetics [@Zardi09]. Simulated spectra of an evaporating, model aerosol particle show comparable features. These patterns can be used to characterize the particle size variation with time. The complex physical and chemical mechanisms that govern the size, composition, phase and morphology of aerosol particles in the atmosphere, represent a major challenge for scientists which try to characterize and model them. Measurements performed on single aerosol particles ($2 \div 100 \mu$m in diameter) levitated in electrodynamic, optical and acoustic traps, allow one to investigate the individual processes under minimal perturbation and well controlled laboratory conditions [@Krieg12]. In addition, particle size measurements can now be prepared with unprecedented accuracy (sub-nanometre) and over a wide range of timescales. Hence, one can precisely identify the physical state of a particle, while its chemical composition and phase can be determined under conditions of high spatial resolution.
Femtosecond spectroscopy opens new perspectives in bioaerosol sensing. On one side, the induced nonlinear light emission from the particles is remarkably peaked in the backward direction which is favorable for remote detection, and on the other side quantum control schemes allow discrimination from major interference such as soot and other organic non-biological particles [@Wolf10]. Optical trapping represents another versatile tool intended for the investigation of aerosol droplets, where the measurements techniques and methodologies include the observation of elastically-scattered light, the measurement of a Raman or fluorescence fingerprint, and the application of conventional brightfield microscopy. An in-depth treatment of optical trapping and the associated techniques can be found in [@Wills09].
Use of the light scattering mechanism to explore small airborne particles and reveal their intrinsic features represents a remarkable analytical tool in aerosol science. The scientific community makes intensive efforts to characterize the scattering and absorption cross-sections of aerosol particles, as there is a large uncertainty in explaining the physico-chemical mechanisms responsible for the complex interaction between atmospheric particles and solar radiation in Earth’s atmosphere [@Kul11; @Tom17; @Via16; @Bou15; @Box15]. The optical properties associated to an individual aerosol particle determine the amount of scattering and absorption that occurs during its interaction with electromagnetic radiation, and depend upon the size, shape, morphology, real and imaginary parts of the particle refractive index and of the incident radiation wavelength. Scattering of electromagnetic radiation by particles can be divided into three main categories: (a) Rayleigh scattering, (b) Mie scattering, and (c) optical scattering. We can distinguish among the three regimes as the wavelength of the radiation is: (a) much larger, (b) of the same order, or (c) much smaller, with respect to the particle size. The Rayleigh and optical scattering can be regarded as approximate solutions to the scattering problem, while the Lorenz-Mie theory supplies an analytical solution to investigate scattering of an electromagnetic plane wave by an isotropic, homogeneous sphere [@Bain18; @Mie1908; @Lock09].
The Lorenz-Mie theory is valid for all three scattering regimes, but its application is often restrained due to its high demands on computational time and the friendly features of the approximate models that are used to investigate the Rayleigh or geometric regimes [@Wrie09]. A Lorenz-Mie calculation is based on the evaluation of the scattering coefficients. Ref. [@Bain18] presents an optical trapping setup intended for accurate studies of the physico-chemical processes, which enables performing single particle measurements. The paper demonstrates that Mie resonances should be used to determine the size and refractive index of optically trapped particles. State of the art results are focused on methods to infer the refractive index of aerosol particles from measured optical properties, using refractive index retrievals (also known as inverse Mie methods). As stated in [@Rad18], retrievals of the aerosol refractive index are based on two fundamental methods: (i) measurements of the extinction, absorption and/or scattering cross-sections or efficiencies of size- (and mass-) selected particles for mass-mobility refractive index retrievals (MM-RIR), and (ii) measurements of aerosol size distributions and a combination of the extinction, absorption and/or scattering coefficients for full distribution refractive index retrievals (FD-RIR). The method is demonstrated for the study of pure and mixtures of ammonium sulfate and nigrosin aerosol [@Rad18].
To conclude, we emphasize that the wide variety of atmospheric aerosol chemical composition and physico-chemical properties requires single-particle analysis, in order to to fully characterize the chemical evolution of particles and their effects on the environment, clouds, global climate, and health. Late progress and advances with respect to state-of-the-art techniques for individual particle analysis are presented in a comprehensive manner in [@Sulli18]. The approach focuses on “online single-particle mass spectrometry, experiments performed with levitated isolated particles using an EDB or aerosol optical tweezers, and the use of electron, X-ray, and Raman spectromicroscopies for detailed analysis and chemical mapping of particles collected on substrates.”
High precision mass measurements for aerosols and nanoparticles
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Mass spectrometry (MS) provides high sensitivity coupled with a fast response time to probe chemically complex particles [@March10; @March17a; @March95c]. Off-line MS techniques require sample collection on filters, but can provide detailed molecular speciation. In particular, off-line MS techniques using tandem MS experiments and high resolution mass analyzers provide improved insight into secondary organic aerosol formation and heterogeneous reaction pathways [@Prat11]. Mass spectrometry techniques are used to perform real‐time measurements and thus acquire information about the aerosol properties. The Aerosol Mass Spectrometer (AMS) uses aerodynamic lens inlet technology together with thermal vaporization and electron‐impact (EI) MS to measure the real‐time non‐refractory (NR) chemical speciation and mass loading as a function of the size of fine aerosol particles (with aerodynamic diameters ranging between $\sim 50$ and 1,000 nm). The original AMS is based on a quadrupole mass spectrometer (QMS) with EI ionization. A review of aerosol mass spectrometry is given in Ref. [@Nash06]. More recent versions employ ToF mass spectrometers and produce full mass spectral data for single particles [@Cana07]. AMS are used to perform [*in situ*]{} measurements on gas vehicle emission and aerosol phase composition [@Chiri11].
A nanoaerosol mass spectrometer (NAMS) employed for real-time characterization of individual airborne NPs is described in Ref. [@Wang06]. The NAMS includes an aerodynamic inlet, a quadrupole ion guide, a quadrupole ion trap, and a ToF mass analyzer. Charged particles in the aerosol are drawn through the aerodynamic inlet, focused through the ion guide, and then captured in the ion trap. Trapped particles are irradiated with a high-energy laser pulse to reach the “complete ionization limit” where each particle is thought to be completely disintegrated into atomic ions. Within this limit, the relative signal intensities of the atomic ions give the atomic composition. The method is first demonstrated with sucrose particles produced with an electrospray generator. A method to deconvolute overlapping multiply charged ions (e.g. C3+ and O4+) is presented in [@Wang06].
A novel Ion Trap Aerosol Mass Spectrometer (IT-AMS) for atmospheric particles has been developed and characterized in [@Kurt07]. Using this instrument the chemical composition of the non-refractory component of aerosol particles can be measured quantitatively. The setup makes use of the well-characterized inlet and vaporization/ionization system of the Aerodyne Aerosol Mass Spectrometer (AMS). While the AMS uses either a linear quadrupole mass filter (Q-AMS) or a time-of-flight mass spectrometer (ToF-AMS) as the mass analyzer, the IT-AMS utilizes a 3D QIT. The main advantages of an ion trap are the possibility of performing MS$^n$-experiments as well as ion/molecule reaction studies. The IT-AMS is operated under full PC control and can be used as a field instrument due to its compact size. A detailed description of the setup is presented. Experiments show that a mass resolving power larger than 1500 can be reached. This value is high enough to separate different organic species at $ m/z \sim 43$. Calibrations with laboratory-generated aerosol particles indicate a linear relationship between signal response and aerosol mass concentration. These studies, together with estimates of the detection limits for particulate sulfate (0.65 $\mu$g/m$^3$) and nitrate (0.16 $\mu$g/m$^3$) demonstrate the suitability of the IT-AMS to measure atmospheric aerosol particles. An inter-comparison between the IT-AMS and a Q-AMS for nitrate in urban air yields good agreement [@Kurt07].
Aerosol mass spectrometry (AMS) investigations for organic and inorganic aerosols
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Particulate matter (PM) emissions from various sources represent a major source of concern as they contribute to the degradation of air quality, especially in urban environments but not restricted to. Adverse effects of PM$_{10}$ and PM$_{2.5}$ particles on human health are discussed in Section \[hazard\]. One of the methods used to evaluate their impact and determine the exposure level is based on using Optical Remote Sensing (ORS) techniques. The principle of the method relies on using path-integrated multi-spectral light extinction measurements in a vertical plane by ORS instruments downwind of a passing PM source. The light extinction measurements performed assist in extracting path-averaged PM$_{2.5}$ and PM$_{10}$ mass concentrations, using inversion of the Mie extinction efficiency matrix for a wide range of size parameters [@Kim08]. Light extinction can be computed using the Lorenz-Mie theory [@Herg12], provided one knows the optical properties and the size distribution of PM. If the PM is non-absorbing, then light extinction is the outcome of scattering alone. Hence the refractive index will exhibit a non-zero real part and a zero imaginary part. The extinction efficiency depends on the particle diameter $d$, and electromagnetic radiation wavelength $\lambda$. In case of a broad spectral band the extinction efficiency must be calculated for each combination of particle size and corresponding wavelength. Mass aerosol concentrations can be determined by means of the ORS method [@Kim08].
Generally, properties of boundary-layer aerosols associated with local and regional emissions of particles and gases are very different with respect to those of free-tropospheric particles. Climate models show that particles persist in the boundary layer over the continents and they are precipitated within 2-4 days, around a radius of up to 2000 km with respect to their origin. Instead, free-tropospheric particles are often carried over large distances across different continents, where the transport times may are usually longer than 1 week. These particles play a prominent role in the cloud formation processes, a feature that makes them important agents with respect to the aerosol indirect effect [@Kim08]. Multiwavelength Raman lidars have been used in the recent years to characterize geometrical, optical, and microphysical properties of free-tropospheric pollution. Different aerosol types have been monitored such as anthropogenic pollution, forest-fire smoke particles from North America and Siberia, arctic area pollution, and the drift of dust particles coming from the Sahara desert [@Kim08].For a quite comprehensive overview of light detection and ranging (lidar) techniques aimed at characterizing desert dust, the reader should refer to [@Mona12]. Another review paper that approaches fundamentals, technology, methodologies and state-of-the art of the lidar systems used to retrieve aerosol information is the paper of Comer[ó]{}n [*et al*]{} [@Come17]. Late results and directions of research can be also found in the paper of Hallen [@Hal18]. Recent advances and “state of the art science in techniques for individual particle analysis” are found in [@Sulli18].
Analytical and numerical modeling of the problems of interest with respect to aerosol dynamics and elaboration of prediction models
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Detailed information about the extent and distribution of stratospheric aerosols is necessary to perform climate modeling, as well as to validate aerosol microphysics models and investigate geoengineering. In addition, stratospheric aerosol loading has to be determined with sufficient precision, so as to enhance the retrieval accuracy of key trace gases (e.g. ozone or water vapour) when explaining remote sensing measurements of the scattered solar radiation. The most frequently used markers to characterize stratospheric aerosols are the aerosol extinction coefficient and the [Å]{}ngstr[ö]{}m coefficient. It is considered that a best approach is based on the use of particle size distribution parameters together with the aerosol number density. The paper of Malinovska [*et al*]{} [@Mali18] suggests a new retrieval algorithm to infer the particle size distribution of stratospheric aerosol from space-borne observations of the scattered solar radiation in the limb-viewing geometry.
Need for a complete system covering both non-refractory and refractory particles found in the atmosphere
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Single particle mass spectrometers (SPMSs) represent versatile instruments employed to infer the chemical composition and mixing state of aerosol particles in the atmosphere under ever changing environmental conditions. Moreover, SMPSs can deliver high temporal resolution without prior sample preparation and produce [*in situ*]{} single particle structure data. SMPSs have demonstrated excellent abilities in detecting microbes, and late experiments show that aerosol ToF mass spectrometers are able to identify aerosolized microbes in ambient sea spray aerosol [@Sult17].
A very recent paper demonstrates how mass spectrometry can be achieved with arrays of 20 multiplexed nanomechanical resonators, where each resonator is designed with a distinct resonance frequency which becomes its individual address [@Sage18]. The paper also reports on mass spectra of metallic aggregates in the MDa range characterized by more than one order of magnitude enhancement in the analysis time with respect to individual resonators. A 20 NEMS array is probed in $150$ ms, under conditions of the same mass limit of detection as a single resonator [@Sage18]. Excellent agreement is reported with a conventional ToF spectrometer operated in the same system.
A laser ablation aerosol particle ToF mass spectrometer (LAAPTOF) is used in [@Shen19] to investigate single aerosol particles. During the experiments seven major particle classes have been identified. As an outcome of the precise particle identification and well-characterized overall detection efficiency (ODE) characteristic to this instrument, particle similitude can be conveyed into corrected number and mass fractions without requiring a reference instrument. An aerosol mass spectrometer (AMS) was also used, where the two MS exhibit good correlation regarding total mass for more than 85 % of the measurement time, which represents a trace that non-refractory species measured by AMS may originate from particles consisting of internally mixed non-refractory and refractory components. The paper reports on finding specific relationships between LAAPTOF ion intensities and AMS mass concentrations for non-refractory compounds, for particular measurement periods [@Shen19]. The approach used enables the non-refractory compounds measured by AMS to be assigned to different particle classes.
Single particles confined in electrodynamic traps. Methods of investigation {#Sec9}
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Light scattering mechanisms
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Investigation of light scattering mechanisms by small random particles has raised an enormous interest for science since the 1950s. Among the most prominent works in the domain we mention the books of van der Hulst [@Hulst81] and Bohren [@Bohr98], to which we add [@Kul11; @Rama18; @Sein16; @Tom17; @Agran10; @Bou15; @Box15; @Kokha08]. We distinguish between (a) elastic scattering where the photon energy does not change, and (b) inelastic scattering when the photon energy modifies as well as the inner energy of the scattering particle. We can further classify elastic scattering as:
1. Rayleigh scattering $\longmapsto$ photon scattering from small, uncharged but polarizable particles (such as atoms or molecules), whose dimensions are considerable smaller than the radiation wavelength. Polarization of scattering particles makes the scattered radiation to be emitted uniformly in all directions. Rayleigh scattering exhibits wavelength dependence where shorter wavelengths are scattered more than higher ones,
2. Mie or Debye scattering $\longmapsto$ photon scattering from relatively large particles or molecules with dimensions comparable or larger than the incident radiation wavelength or larger. The resulting radiation is non-uniformly scattered and shows little dependence on the wavelength. Forward Mie scattering is stronger than backward scattering, because the relative phase differences of contributions from different scattering locations on the particles dwindle,
3. Thomson scattering $\longmapsto$ photon scattering on charged particles.
Inelastic scattering can be also classified as:
- Brillouin scattering $\longmapsto$ a mechanism which typically occurs in photon scattering from solid materials. It involves acoustical phonons with characteristic frequencies in the GHz region. The incident radiation wavelength is altered by the energy levels of sound waves or phonons in the solid material, but these shifts are quite small,
- Raman scattering $\longmapsto$ a mechanism where the frequency of the scattered radiation changes. It is used to perform diagnostic analysis. Raman scattering is typically weak and considerably less intense than the Rayleigh scattered light, which explains the special care in the design and operation of Raman spectrometers. For Raman scattering at gas molecules, the vibration and rotation states of the molecules change; as a general rule the molecules possess a higher energy after scattering, which suggests a correspondingly lower photon energy of the scattered light (Stokes components). When molecules are previously excited, anti-Stokes components with increased optical frequency are reported. Analogously, Raman scattering can arise in solids, involving phonons with frequencies in the THz region, also called optical phonons.
We can also remind Compton scattering which is basically an inelastic form of Thompson scattering that occurs when the energy of the incident radiation starts to become comparable to the rest energy of the charged particle.
Single particles levitated in particle traps
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As already shown, an electrodynamic balance (EDB) represents a device used to levitate charged particles/droplets or ions, either in vacuum or under SATP conditions. Particular trap geometries such as ring shaped [@Nosh14] and hyperboloidal shaped electrodes [@Major05; @March05; @Singh17; @Ghe94; @Davis11], segmented [@Pyka14] and planar geometries [@Clark13; @Bahra19; @Song06; @Aust10; @Zhang11; @Alda16], or even toroidal shaped electrodes [@Li17; @Taylor12; @Higgs16; @Kot16; @Kot17; @Higgs18] are employed to generate a quadrupole electric field. Use of EDBs has pushed forward the domain of mass spectrometry, while providing chemists and physicists an invaluable tool to explore microparticles and NPs and thus investigate their physico-chemical properties [@Kwa11; @Pin11].
In the last two decades a considerable effort was invested to devise new methods and techniques to investigate chemical reactions and NPs at the single molecule level. Single molecule manipulation techniques include scanning probe microscopy (SPM) such as the Atomic Force Microscope (AFM) and the Scanning Tunneling Microscope (STM) [@Hey11], near-field scanning optical microscopy (NSOM) and variations of. However, these techniques raise technological challenges with respect to the highly complex tip-sample and sample-surface interactions. One of the first attempts to circumvent the issues related to SPM based techniques consists in developing a new single molecule control method, which relies on a trap setup that is largely modified with respect to a conventional 3D QIT [@Seo03]. The method enables [*in situ*]{} investigation of single molecules and NPs by enabling ultra-sensitive detection, and the device is called single nanoparticle ion trap (SNIT).
New storage devices based on inhomogeneous, time-dependent electric fields, characterized by trap geometries such as quadrupole, linear octupoles, higher-order multipoles or series of ring electrodes are described in [@Ger03]. Different species of charged particles can be levitated, starting from electrons via molecular ions and NPs.
Manipulation of microparticles and NPs (including molecules) in vacuum represents one of the challenges of modern science, as it enables investigations on proteins, DNA segments, viruses and bacteria [@Jose10; @Peng04], dusty plasmas [@Fort05; @Morfi09], or detection of tiny forces [@Li18; @Haya18]. Measurements on single aerosol particles (ranging from $2$ to $100 \ \mu$m in diameter) levitated in electrodynamic, optical, and acoustic traps or deposited on a surface, allow individual processes and physical mechanisms to be studied in isolation under controlled laboratory conditions [@Krieg12].
Non-destructive, optically detected mass measurements on single microparticles and NPs particles using electrodynamic (Paul) traps have been performed in several groups around the world. A technical limitation of these setups comes from the fact that they use different particle injection methods. The major drawback when using such methods is caused by an imprecise control over the particle mass and range, which strongly affects the reproducibility and control of the experiment. Nanoparticle Mass Spectrometry (NPMS) is a very powerful and versatile tool to perform NP diagnosis. Very recent experiments suggest a novel technique called electrospray ionization (ESI) NP source [@Howd14]. The method is based on using a pair of radiofrequency (RF) ion guides, that provide collisional cooling and focusing of the NP beam generated by the source, while also acting as prefilters (in fact, mass filters) and sorting species as a function of the specific mass-to-charge ($Q/M$) ratio.
Trapping of single particles or colloidal molecules in a planar aqueous Paul trap (PAPT) setup is described in [@Guan11]. Optical trapping of an ion is described in [@Schne10]. Trapping is demonstrated in regions that are located in closest vicinity of the electrodes, while showing that hybrid setups combining both optical and RF potentials can efficiently cover the full spectrum of composite confinements - from RF to optical. Experiments demonstrate the perturbing influence of the static electric potential, that can easily restrain successful optical trapping. Different analytic models have been tested, suggesting that recoil heating stands for the most prominent heating effect, as the experimental results indicate [@Schne12]. The ion lifetime in an optical dipole trap is constrained by photon scattering. State-of-the-art experimental techniques such as optical repumping may lead to a remarkable lifetime increase by three orders of magnitude [@Lamb17]. By employing these techniques optical trapping and isolation of ions can be performed on a level that is comparable to neutral atoms. Such almost decoherence-free techniques enable scientists to accomplish isolation from the environment. This achievement represents a major breakthrough that opens new pathways towards novel regimes of ultracold interactions of ions and atoms at collision energies considered previously inaccessible. This approach enables performing a novel class of experimental quantum simulations with ions and atoms in a variety of versatile optical trapping geometries, such as bichromatic traps or higher-dimensional optical lattices [@Lamb17].
A late experiment reports on single micron-sized melamine-formaldehyde particles levitated in the sheath of an RF-plasma and exposed to an intense laser beam, while being trapped in optical tweezers [@Wieb19]. A reversible change in the particles’ properties is observed, such as a gain in particle charge where the initial charge restores within minutes. Another state-of-the-art experiment reports on a cold-damping scheme used to cool one mode of the CM motion of an optically levitated NP in ultrahigh vacuum ($10^{-8}$ mbar), whose temperature drops off from room temperature down to an extremely low value of 100 $\mu$K [@Tebb19].
The outcome of laser phase noise heating on resolved sideband cooling, in the context of cooling the CM motion of a levitated NP in a high-finesse cavity is analyzed in Ref. [@Meyer19]. As phase noise heating does not represent a fundamental physical constraint, the paper explores the regime where it becomes the main limitation in Levitodynamics. The interest in investigating this regime is motivated given that it represents the main restriction in reaching the motional ground state of levitated mesoscopic objects under conditions of resolved sideband cooling.
The dynamical backaction effect and the possibility of cooling a levitated nanosphere in the absence of a cavity, or when the nanosphere is trapped outside a cavity and excites a continuum of electromagnetic modes is explored in [@Abba19], where both Stokes and anti-Stokes processes are considered.
Another field of large scientific interest is the study of micro and nanomechanical oscillators [@Gies15] and especially of their associated quantum mechanical motion, as the results are expected to provide novel insights into the boundary between the quantum and classical worlds. Cooling of a dielectric nanoscale particle trapped in an optical cavity is investigated in [@Bark10]. An experimental spectrometer setup based on two separate cryogenic ion traps that enable formation and characterization of solvated ionic clusters, such as water, methanol, and acetone around a protonated glycylglycine peptide is described in [@Marsh15]. A novel technique demonstrates that [*levitated electromechanics*]{} can be successfully employed for electronic detection, cooling and precision sensing of (single) microparticles and NPs [@Gold19]. The technique uses charged particles levitated in an ion trap, coupled to an RLC circuit. Sub-Kelvin temperatures are achievable with room temperature circuitry, and it is estimated that trap operation in a cryogenic environment will enable ground-state cooling for micron-sized particles. It is also suggested that hybrid levitated opto-electromechanical setups are supposed to open new pathways towards deep cooling into the quantum regime, enabling scientists to engineer highly nonlinear states such as squeezed states [@Gold19]. Levitated nano-mechanical resonators, and especially experimental aspects related to the precise control of the nonlinear and stochastic bistable dynamics of a levitated nanoparticle in high vacuum are investigated in [@Ricci17].
Investigation of chemical and optical properties of trapped particles or microdroplets using optical techniques. General considerations
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Charge detection quadrupole mass spectrometry (CD-ITMS) driven by rectangular and triangular waveforms (rect-CD-ITMS and tri-CD-ITMS) has been developed in an attempt to achieve a better characterization of microparticles. Experimental results indicate that the rect-CD-ITMS and tri-CD-ITMS can operate well and perform mass measurement of microparticles by using the frequency scan. By increasing the applied voltage and signal-to-noise ratio (S/N) of the charge detector, the mass resolution can be further improved. In addition, the rect-CD-ITMS and tri-CD-ITMS can be used to characterize red blood cells (RBCs) [@Xio12].
On the other hand, handling and control of single cells and large biomolecules [@Sato96] is a prerequisite for various medical and biological techniques, starting with in vitro fertilization to genetic engineering. To achieve this different techniques and devices for micromanipulation have been devised and tested: (a) optical tweezers [@Gies15; @Ash86; @Sato94; @Svobo94; @Mara13; @Zheng13] that provide single particle manipulation but limited trapping capacity as an outcome of the strong focusing requirement; and (b) dielectrophoresis that enables massive manipulation under conditions of insufficient spatial resolution. A promising approach is demonstrated in [@Flor15], that associates advantages characteristic to both methods and allows optical trapping and manipulation of microparticles suspended in water, due to laser-induced convection currents. Experimental results show that for low laser power (0.8 mW) particles are trapped at the centre of the beam, while at higher powers $\left( \sim 3 \ \text{mW} \right)$ particles arrange themselves in a ring that encircles the beam. This behaviour is the result of the action of two competing forces: the Stokes and the thermo-photophoretic forces. Numerical simulations establish that thermal gradients achieve trapping. A comprehensive guide that approaches subjects such as the optical properties of aerosol particles, multiple light scattering and Fourier optics of aerosol media, or optical remote sensing techniques, is Ref. [@Kokha08].
Laser trapping of NPs is demonstrated in [@Leh15], with an emphasize on colloidal metal NPs that exhibit remarkable features due to their particular interaction with electromagnetic radiation, caused by surface plasmon resonance effects. An experimental method that relies on the use of dynamic split-lens configurations with an aim to achieve trapping and spatial control of microparticles through the photophoretic force, is presented in [@Liza18]. Optical trapping allows high precision investigations of many microphysical and chemical processes, as it enables measurements on the single-particle level as demonstrated in [@Bain18], where the size and refractive index of single aerosol particles are determined using angular light scattering and Mie resonances.
Light scattering measurements in particle traps
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As a rule when studying light scattering, photoemission from a particle surface, processes of evaporation and condensation of a drop, or chemical reactions with a participation of solid particles [@Major05; @Davis97; @Ghe95b; @Shaw00], one preliminary charged particle has to be placed in the trap. Particle confinement is achieved at low gas pressures within the trap. The dynamics of charged dust particles and their capability to build up Coulomb structures in the Paul trap at low pressures are examined in [@Wuerk59]. The single trapped particle moves along Lissajous curves depending on the parameters of the applied trapping voltage. As mentioned in [@Wuerk59], such a trap with a single particle could serve as an analog computer for solving the Mathieu equation. Large number of particles can be confined in a trap, but in this case particles move chaotically depending on the frequency and amplitude of the trapping voltage applied to the electrodes.
Experiments performed have demonstrated that alumina or SiO$_2$ spheres (grains) with diameter ranging between $0.1 \div 1 \ \mu$m can be simply detected by means of light scattering. Relevant information about the $q/m$ ratio (also called specific charge) can be inferred by performing the Fourier analysis of the recorded signal, which is modulated by the particle secular motion [@Ger03]. Recent techniques and late results with respect to the measurement of elastic scattering and absorption of aerosols are presented in [@Ula11]. Both spectroscopy and angular measurement of intensity can be employed to infer the size of aerosol particles. Moreover, additional properties can be retrieved from the angular dependence of scattering.
The paper of Gouesbet reviews laser based optical techniques used to characterize discrete particles embedded in [*flows*]{} [@Goue15]. Measuring the absorption of a single aerosol particle is a difficult task, due to the complexity associated with the problem. Among the few techniques available, there is none suitable for measuring the single-particle absorption of coarse-mode nonspherical aerosols. Nevertheless, analysis of two-dimensional angular optical scattering (TAOS) patterns provide a possible solution to perform such task. TAOS patterns of single aggregate particles made out of spheres are reported in [@Zardi09], where optical resonance spectroscopy is employed to size evaporating solid, non-spherical particles. Single bi-sphere particles levitated in an EDB have been used to record 2D angular scattering patterns at different angles of the coordinate system of the aggregate, with respect to the incident laser beam [@Krieg11]. Numerical simulations were also used to perform a qualitative analysis of the experimental results, based on using the Mackowski code. The Brownian motion allows one to sort out high symmetry patterns for analysis. In another experiment, by employing a Multiple-Sphere T-Matrix (MSTM) code, captured TAOS patterns with geometries similar to a previously designed instrument have been simulated [@Walt19]. By discriminating the speckle size and the integrated irradiance of these simulated TAOS patterns, one can distinguish between high-absorbing, weak-absorbing, and non-absorbing particles over the size range of $2 \ \mu$m to $10 \ \mu$m. The size of the particle can be estimated due to the presence of the speckle in the scattering pattern. The next step lies in characterizing particle absorption using the integrated irradiance.
Elastic scattering. Lorenz-Mie theory
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Important aspects on the Lorenz-Mie scattering theory [@Herg12] have been presented in Section \[InvestMeth\]. Many key technologies heavily rely on NPs and the area of applications is exponentially growing, starting from pygments used in paints and cosmetics, drug delivery, chemical and biological sensing, gas sensing, CO$_2$ capture, etc. [@Khan17]. Beyond their important applications related to multiple high technology areas (see Section \[hazard\]), NPs also present a vivid interest for fundamental research, especially for the areas of complex plasmas and astrophysics.
[*In situ*]{} analysis of NPs can be performed by employing the method of multiple-wavelength Rayleigh–Mie scattering ellipsometry. In Ref. [@Gebau03] the method is applied to characterize NPs levitated in low-pressure plasmas. The experiments demonstrate that the size distribution and the complex refractive index can be determined with high precision. Moreover, the Rayleigh–Mie scattering ellipsometry also applies to achieve [*in situ*]{} analysis of NPs under high gas pressures and in liquids.
Interpretation of elastic scattering data based on Mie theory [@Herg12] enables accurate measurements on the size and refractive index of microspheres. These measurements can be subsequently employed to analyze droplet evaporation and condensation processes for pure components and multicomponent droplets [@Davis10]. Moreover, the Rayleigh limit of charge on a droplet can be investigated using elastic scattering, in order to determine the size and charge corresponding to droplet explosions. Phase function measurements (intensity versus scattering angle) and data gathered from morphology-dependent resonances (MDRs) provide alternative methods to determine the size and refractive index of droplets and microspheres [@David18; @Smith08; @Krieg12; @Krieg11], and thus characterize coated spheres [@Davis10; @Trevi09].
A QIT calibrated for microparticle MS that is employed to levitate dye-labeled polystyrene microspheres and study the associated fluorescence spectra, is presented in [@Trevi07]. The absolute mass and electric charge of the particle are resolved by measuring its secular oscillation frequencies. The microsphere radius is calculated by employing the Lorenz-Mie theory [@Mie1908; @Lock09; @Herg12; @Mish09] to interpret the fluorescence emission spectrum, that is dominated by optical cavity resonances [@Trevi07]. The laser-induced coalescence of two conjoined polystyrene spheres levitated in a QIT, is investigated by monitoring optical morphology dependent resonances (MDRs) that show up in the fluorescence emission spectrum [@Trevi09]. Surface tension drives the heated bisphere into an individual sphere. At the end of the structural transformation phase, the particle dimensions are inferred by analyzing the frequency shifts of the non-degenerate azimuthal MDRs. The relaxation time of the viscous sphere is then used to infer the polystyrene viscosity and temperature. An overview of light scattering experiments aimed at measuring one or more elements of the scattering matrix as functions of the scattering angle, for ensembles of randomly oriented small irregular particles in air is given in [@Munoz11].
Elastic light scattering shows sensitivity to aerosol particle size, shape, complex refractive index, and the molecular density distribution within the particle. As it provides high finesse, the technique is frequently used to achieve real-time information and perform [*in situ*]{} classification of the aerosol type. Elastic light scattering is especially recommmended in discriminating hazardous bioaerosols among normal atmospheric background constituents [@Kul11; @Tom17; @Bou15; @Box15]. Remarkable progress recorded due to improved theoretical models and enhancement of the computing power has enabled performing numerical simulations that are able to infer light scattering patterns and mechanisms out of very complex and highly irregular systems. A new method that provides simultaneous measurements of the back-scattering patterns and of images of single laser-trapped airborne aerosol particles is presented in [@Fu17].
An algorithm used to calculate electric and magnetic fields inside and around a multilayered sphere is developed in [@Ladu17]. The algorithm includes explicit expressions for the Mie expansion coefficients inside the sphere.
An EDB can be used to levitate highly viscous single aerosol particles for water diffusion experiments, as demonstrated in [@Bast18]. The particle growth and reduction of the shell refractive index are experimentally observed via redshift and blueshift behaviour of the Mie resonances, respectively. The particle radius as well as a core-shell radius ratio are inferred from the measured shift pattern and Mie scattering calculations.
A system used to characterize individual aerosol particles using stable and repeatable measurement of elastic light scattering is described in [@Lane18]. The technique relies on using a linear electrodynamic quadrupole (LEQ) particle trap that levitates charged particles injected using ESI. The particles are confined along the stability line which coincides with the trap centre. Optical interrogation is employed to investigate the particles, the scattered light is collected and intensities are calculated based upon the Lorenz-Mie scattering theory. Correlation of scattered light measurements for different wavelengths enables one to distinguish and categorize inhomogeneous particles [@Lane18].
Measurements performed towards determining the size and absorption of single nonspherical aerosol particles from angularly-resolved elastic light scattering are presented in [@Walt19]. The paper describes how the analysis of two-dimensional angular optical scattering (TAOS) patterns provides a technique that enables measuring the absorption of a single aerosol particle.
Investigations with respect to the optical absorption cross section for homogeneous spheres that interact with electromagnetic radiation in the visible range, calculated with the Mie equations, are performed in [@Soren19]. Four regimes can be distinguished: the Rayleigh Regime, the Geometric Regime, the Reflection Regime, and a Crossover Regime.
Inelastic scattering. Raman and fluorescence spectroscopy {#RamanFl}
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Inelastic scattering (Raman and fluorescence scattering) is employed to determine the chemical composition of microparticles and to study gas/particle and gas/droplet chemical reactions. In addition, inelastic scattering was also used to monitor polymerization of monomer microparticles. It is demonsrated that the temperature of a microparticle can be inferred by measuring the ratio of the anti-Stokes to Stokes scattering intensities. Moreover, inelastic scattering can be used to detect of biological particles such as pollen and bacteria [@Davis10; @Davis11].
Characterization of ions produced as a result of the interaction between a high energy laser pulse and NPs is indispensable to perform quantitative determinations of the composition and size of NPs, by means of single particle mass spectrometry (SPMS). The paper of Zhou [*et al*]{} [@Zhou07] introduces a one-dimensional hydrodynamic model to explain the interaction mechanisms, showing that the laser field is coupled to the non-equilibrium time-dependent plasma hydrodynamics of the heated aluminum particles. The properties of ions generated during the interaction with a strong laser pulse (532 nm wavelength, 100 mJ/pulse) are investigated, for nanosecond pulses and particle dimensions ranging between $20 \div 400$ nm, that are most relevant for SPMS.
Raman spectroscopy of single particles levitated in an EDB is described in [@Signo11]. The most prominent applications include the study of atmospheric aerosols. The EDBs provide the advantage of enabling investigations of various atmospheric processes and phenomena that involve kinetic effects. Both static EDBs and the recently developed scanning EDBs (SEDBs) techniques are extremely suited to perform hygroscopic measurements and investigate phase transformations of levitated aerosol particles. Moreover, homogeneous reactions of Organic Aerosol Particles (OAP) can be investigated using this approach [@Signo11].
The synergy between optical trapping and Raman spectroscopy [@Signo11] opens new pathways to explore, characterize, and identify biological micro-particles. More precisely, optical trapping lifts the limitation imposed by the relative inefficacy associated to the Raman scattering process. As a result of applying this technique, Raman spectroscopy can be employed to study individual biological particles in air and in liquid, providing the potential for particle identification with high specificity, or to characterize the heterogeneity of individual particles in a population [@Wang15]. An introduction about the techniques used to integrate Raman spectroscopy together with optical trapping, and thus achieve qualitative studies on single biological particles levitated in liquid and air can be found in [@Red15].
The time evolution of fluorescence and Raman spectra of single solid particles optically trapped in air is observed in [@Gong17]. Initially the spectra exhibit strong fluorescence with weak Raman peaks, then the fluorescence is bleached within seconds and only clean Raman peaks are reported. The experiment uses an optical trap that is assembled by using two counter-propagating hollow beams. Both absorbing and non-absorbing particles in the atmosphere are strongly trapped. The setup provides a novel method to investigate dynamic changes in the fluorescence and Raman spectra collected from a single optically trapped particle, under SATP conditions.
A custom-geometry LIT designed for fluorescence spectroscopy of gas-phase ions at ambient to cryogenic temperatures is described in [@Raja18]. Laser-induced fluorescence emitted by the trapped ions is collected from between the trapping rods, normal to the excitation laser beam directed along the axis of the LIT. The original design allows an enhanced optical access to the ion cloud with respect to similar designs. As a general rule, studies of ions in the gas phase are associated with fluorescence spectroscopy due to the possibility to precisely control the solvation state of a chromophore. There are certain technical challenges to be solved when combining mass spectrometry with fluorescence spectroscopy, as ion density is low and scattering elements are persistent along the excitation laser beam path [@Raja18].
Characterization of the physico-chemical properties of molecules implies studying their temporal reactions within a micro-sized particle in its natural phase. A late experiment reports on measurements performed of temporal Raman spectra in different submicron positions of a laser-trapped droplet composed of diethyl phthalate and glycerol [@Kalu18]. Thus, laser-trapping Raman spectroscopy (LTRS) allows one to characterize single airborne particles, single cells, spores, etc., by supplying information on the particle’s size, shape, surface structure, extinction coefficient, refractive index, elastic scattering pattern, fluorescence, and Raman shifts, etc.
Particle detection and investigation. Laser induced breakdown spectroscopy (LIBS).
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Laser Induced Breakdown Spectroscopy (LIBS) is a non-destructive atomic emission spectroscopy technique that employs a high energy laser pulse as the excitation source [@Dem15]. The laser atomizes and excites a small amount of material at the sample’s surface (in the nanogram to picogram weight range), developing a plasma plume that emits light at frequencies characteristic to the chemical elements present in the sample. As LIBS is basically a surface interrogation technique, it may not identify species present in the inner side of a sample unless the sample is homogenized prior to analysis. The sensitivity of LIBS lies in the ppm range, the analysis is performed in real-time, and no preliminary sample preparation is typically required (unless homogenation is desired). LIBS yields atomic emission and plasma emission spectra that can be used to determine elemental composition of single particles [@Pin11].
The influence of water content, droplet displacement, and laser fluence on the laser-induced breakdown spectroscopy (LIBS) signal of precisely controlled single droplets is investigated in [@Jarvi16]. NP analysis is extremely valuable for many applications. One of the disadvantages associated to optical excitation techniques stems from the bounded resolution which is the outcome of the diffraction limit. Single particle nanoanalysis was previously unattainable by means of LIBS, but recently attogram-scale absolute limits of detection have been achieved by employing optical trapping and levitation to isolate single particles [@Puro17]. The multielemental capabilities of this approach are demonstrated by subjecting two different types of nanometric ferrite particles to LIBS analysis [@Puro19].
Linear Paul traps as tools in laser plasma accelerated particle physics {#Sec10}
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Strongly nonlinear longitudinal waves in plasma are introduced in [@Kriv92], in a incessant quest to identify physical mechanisms for plasma accelerators of charged particles that allow one to achieve high acceleration rates and energy.
One of the major issues in physics was to find a method to confine charged particles. As a combination of static fields cannot achieve that, a solution was devised in the 1950s when the concept of [*alternating gradient (AG) focusing*]{} was implemented [@Coura52; @Coura58]. The basic idea lies in creating an array of electrostatic lenses arranged in an AG configuration. The particles can be accelerated or decelerated by applying an appropriate high-voltage switching sequence to the lenses. By periodically switching the orientation of the two transverse fields, a net focusing force is generated, oriented along both transverse coordinates. Thus, confinement of a charged particle beam in both transverse dimensions is achieved by combining two magnetic quadrupoles of opposite polarity into a doublet [@Kjae05; @Dorf06; @Fuku14; @Fuku16]. The method is applied to confine ions in quadrupole mass filters, in Paul traps [@Paul58; @Paul90; @Stra55], and in all type of particle accelerators [@Scha12]. Generally, the time-varying sinusoidal voltage applied to a quadrupole trap can exhibit an arbitrary waveform, provided that a stable solution of the Hill (Mathieu) equation [@Mag66] that describes particle dynamics results. Thus, a square time-varying trapping voltage can be used. Two decades ago it was suggested to make use of the analogy between the dynamics of charged particles in the non-homogenous magnetic quadrupole field of accelerators or storage rings, and that of an electric quadrupole trap excited by square voltage pulses [@David00; @Kjae01]. Instead of using typical sinusoidal voltages, controllable, pulsed RF voltages are applied to the trap electrodes with an aim to investigate collective beam dynamics in various lattice structures [@Tak04; @Kelli15]. Moreover, the collective dynamics of a one-component plasma (OCP) in a RF quadrupole trap is physically equivalent to that of a charged-particle beam that propagates throughout a periodic magnetic lattice. Experiments focused on using linear Paul traps to study space-charged-dominated beams have been reported in the last two decades [@Tak04; @Kjae05; @Dorf06; @Gils04; @Chung07; @Gils07]. Moreover, Ref. [@Kjae02a; @Kjae02b] focus on emulations of beams in the ultra strongly coupled regime of Coulomb crystallization.
In addition to the significant contributions of the group at the Aarhus University, the group from the Hiroshima University has designed two different classes of trap systems, one of which uses a RF electric field (Paul trap) while the other uses an axial magnetic field (Penning trap) to achieve transverse plasma confinement [@Fuku14; @Oht10; @Oka14]. These systems are called S-POD (Simulator of Particle Orbit Dynamics) [@Ito08]. Their intrinsic feature is the capability to approximately reproduce the collective motion of a charged-particle beam propagating through long alternating-gradient (AG) quadrupole focusing channels using the Paul trap, and long continuous focusing channels using the Penning trap. Such an interesting feature allows one to investigate various beam-dynamics issues using compact setups, without the need for large-scale accelerators. Linear and nonlinear resonant instabilities of charged-particle beams traveling in periodic quadrupole focusing channels have been experimentally investigated using a compact non-neutral plasma trap [@Oht10]. The linear Paul trap system named S-POD was applied to investigate a collection of space-charge-induced phenomena. To emulate lattice-dependent effects periodic perturbations are applied to the quadrupole electrodes, which yields to additional resonance stop bands that shift depending on the plasma density. The loss rate of trapped particles was measured as a function of bare betatron tune, with an aim to identify resonance bands in which the plasma becomes unstable. When an imbalance is created between the horizontal and vertical focusing, the instability bands split [@Fuku14]. Experimental results suggest that the instability band is somewhat insensitive to the phase of the quadrupole focusing element location within the doublet configuration over a significant range of parameters. The Penning trap with multi-ring electrode geometry has been employed to study beam halo formation driven by initial distribution perturbations [@Oka14]. A similar apparatus to the S-POD, the Intense Beam Experiment (IBEX), has been designed at the Rutherford Appleton Lab (RAL)-UK. To use either of these experimental setups to investigate beam dynamics under conditions of more intricate lattice configurations, novel diagnostic techniques have to be devised and tested for Paul traps. Such a technique is demonstrated in [@Marti18], where a new method is described to measure the beta function and the emittance at a given time in a Paul trap. A novel technique is also demonstrated in [@Ito19], where direct measurement of low-order coherent oscillation modes in a LPT is performed by detecting the image currents induced on the electrodes’ surfaces. The technique is based on picking up weak signals from the dipole and quadrupole oscillations of a plasma bunch, which are then subjected to Fourier analysis. Late experiments report on Paul-trap systems that allow positioning of fully isolated micrometer-scale particles with micrometer precision as targets in high-intensity laser-plasma interactions [@Ost19].
Besides quadrupole traps, multipole geometries have been also investigated. The configuration that has been experimentally tested includes additional electrodes that enable one to control the strengths and time profiles of the low-order nonlinear fields that occur in the trap, independently of the linear focusing potential [@Fuku15]. Thin metallic plates are inserted in between the cylindrical electrodes (rods) of a quadrupole Paul trap. The size and arrangement of the extra electrodes are optimized by using a Poisson solver. Simple scaling laws are inferred, to perform a quick estimate of the sextupole and octupole field strengths as a function of the plate dimensions. Particle tracking simulations that were performed clearly demonstrate the possibility to achieve controlled excitation of nonlinear resonances in the multipole Paul trap.
Trapping of nanoparticles using dielectric containers and capillary tubes
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A transportable setup intended for trapping an assembly of solid nanoparticles-NPs (with dimensions in the micrometer or nanometer range), within a limited and well-defined region of space and without mechanical contact with the walls of an enclosure or any other supporting element, is described in [@Sto14]. The NPs are levitated within a dielectric container of random shape, fully closed or partially open, placed between the electrodes of a linear quadrupole electrodynamic (Paul) trap, as shown in Fig. \[dieltrp\]. There are several methods to achieve electrical charging of the NPs as follows: (i) as an outcome of the physical contact due to the interaction with the dielectric wall of the container, (ii) by providing additional electrodes where high voltage pulses are applied, (iii) by sudden variation of the electric voltage supplied to the trap electrodes, or (iv) as a consequence of an ionizing radiation beam (generated by an external source) which acts upon the particles. The container, identified as (8) in Fig. \[dieltrp\], is made of a dielectric material. The dielectric container can be fully transparent, half-transparent, or opaque, as it can be also filled with gas, with a mixture of gases, or it might be evacuated. Even capillary tubes can be used as containers. The system devised is able to confine conductive powders (either semiconductor, dielectric, or even radioactive), different types of germs, bacteria, viruses, seeds, or other microbiological material. The system is also adequate for use in setups and equipments intended for investigation, manipulation, measurement and determination of the physico-chemical characteristics and properties of trapped microparticles or NPs. In addition, the system provides a large optical access for the trapped microparticle species, and it can also be employed to investigate the effect of fields of force or of external radiation sources upon micron sized particles levitated in electrodynamic (Paul) traps.
![Longitudinal and two transversal (corresponding to the A-A anb B-B plans) cross-sections of the micro- and nanoparticle trapping setup in capillary dielectric tubes, based on a quadrupole linear Paul trap. Legend: 1a $\div$ 1d – trap electrodes; 2a and 2b – cylindrical endcap electrodes; 3 – electrode cross section tangent to the circle of radius $r_0$; 4 – quadrupole trap $z$ axis; 5 – laser beam generated by the laser diode denoted as $6$; 8 – dielectric container and electric charging system; 9a and 9b – spacer rings; 10 – trapped nanoparticles. Picture reproduced from [@Sto14] with kind permission from O. Stoican.[]{data-label="dieltrp"}](capillary.pdf)
The NPs located inside the container are levitated in a well defined gaseous environment, whose pressure and chemical composition are controlled. Moreover, the particles can be also trapped in vacuum. Such a system enables manipulation and investigation of toxic, radioactive or biological material, thus excluding the possibility of contaminating human personnel or the surrounding environment. The recipient containing the solid particles that are to be trapped can be prepared at a different location, then safely carried (in absence of contamination) to the location where the setup or analysis and measurement system is placed.
We will further explain the operation of the experimental setup described in [@Sto14], used to confine NPs in dielectric containers and capillary tubes. The system relies on a quadrupole electrodynamic trap that consists of four cylindrical, equidistantly spaced electrodes. The electrode cross section is a circular surface, tangent to a circle of radius $r_0$, denoted as (3) in Fig. \[dieltrp\]. Moreover, two additional endcap electrodes are placed in a plane that is normal to the one holding the trap electrodes, at equal distance with respect to the trap ends. The trap geometry can exhibit a larger number of electrodes, provided that this number is even. Fig. \[dieltrp\] shows a longitudinal section and two cross sections of the experimental setup tested. Two cylindrical endcap electrodes are located coaxially with respect to the trap axis (4). One of or both endcap electrodes are perforated along the longitudinal axis, with an aim to allow illumination of the trapped NPs (denoted as 10) by means of a laser beam (5) generated by a cw laser diode (6). The laser diode is not required if the confined particles are observed directly, or indirectly by means of radioactive or X-ray emission.
Solid particles to be confined are located within a container made of a dielectric material, be it transparent, semi-transparent, or opaque. The container is located within the space delimited by the trap cylindrical electrodes 1a $\div$ 1d, and by the trap endcap electrodes, 2a and 2b. The container vessel (8) may exhibit different shapes and geometries, its cross section might change along the longitudinal axis, it can either exhibit or not cylindrical symmetry, it can be open at both ends or it can be open at one end and closed in the opposite end, it can be fully closed or somewhat closed at both ends. The setup in Fig. \[dieltrp\] shows a dielectric container that has a cylindrical shape, while being fully closed at both ends. Either one of them or both endcap electrodes can change position and shift along the trap axis, which modifies the extent of the inner space delimited by the trap cylindrical electrodes. Fig. \[dieltrp\] also illustrates the possibility of shifting the (2b) electrode along the trap axis (4), as its new position is denoted by a dashed line. Thus, containing recipients of various lengths can be introduced in the inner inter-electrode space. The setup also allows for the possibility to detach one or more trap electrodes, in an aim to remove or introduce the container that holds the particles to be confined [@Sto14]. In addition, Fig. \[dieltrp\] also illustrates the possibility to detach the (1a) electrode in an attempt to insert the dielectric container (8) in the inner inter-electrode space, while its new position is represented by a dashed line. The maximum dimensions of the dielectric container cross-section must be chosen so as to enable its location inside the inner volume delimited by the cylindrical electrodes (1a) $\div$ (1d), (2a) and (2b). Thus, even capillary tubes can be used as particle containers. Mechanically, the container (8) can be positioned by means of two spacer rings (9a) and (9b) (see Fig. \[dieltrap2\]), located at both ends. The radiation (12) emitted by the trapped particles (10) is observed in a plan that is normal to the trap axis (4), and it can either be scattered light, radioactive emission, or X-radiation. The emitted radiation crosses the walls of the container (8) and is observed by means of an instrument (denoted as 13), such as the objective of a microscope or of another optical instrument, a CCD camera, a photomultiplier, an array of photoelectric cells, a Geiger counter, etc.
![View of the experimental setup for trapping nanoparticles based on a dielectric container. Legend: 1a and 1b – trap electrodes; 2a and 2b – cylindrical endcap electrodes; 4 – quadrupole trap $z$ axis; 5 – laser beam generated by the laser diode denoted as $6$; 8 – dielectric container; 9a and 9b – spacer rings; 10 – levitated nanoparticles; 22- conducting wire; 23 – electronic module that delivers high voltage pulses of amplitude $U_p$. Picture reproduced from [@Sto14] with kind permission from O. Stoican.[]{data-label="dieltrap2"}](capillary2.pdf)
Analogous to other traps [@Sto01; @Mih16a; @Mih16b], the power supply delivers a high a.c. voltage $U_{ac}$ and two d.c. voltages, namely an $U_z$ voltage applied at the endcap electrodes to achieve axial confinement, and a diagnosis voltage $U_x$ applied between the upper (1a) and lower (1b) trap electrodes, whose electrical polarity can be reversed by means of an electronic switch. All supply voltages are measured with respect to the (1a) electrode that is connected to the ground potential, taken as reference. The diagnosis voltage $U_x$ compensates gravity and enables the experimentator to shift the particle position in the radial plane.
The experimental parameters that allow stable trapping for particles with a particular value of the specific charge $Q/M$ depend on the NP dimensions, on the $U_x$ and $U_z$ voltages, as well as on the trapping voltage $U_{ac}$ amplitude ($V_0$) and frequency $f_0$. Generally, for a linear electrodynamic trap with $2n$ cylindrical electrodes, where $n \in N$ is a natural number ($n = 2, 3, \ldots $), when both the $U_x$ and $U_z$ voltages are zero and the polar coordinate $r$ is much smaller than the distance between the trap axis and the electrode surface $r_0$, the analytical expression of the multipole field generated in a point close to the trap axis is [@Asva09; @Ger92]
$$\label{diel1}
\Phi \left(r, \phi, t\right) = V_0 \left({\frac r{r_o}}\right)^n \cos \left(n\phi \right) \sin\left(\omega_0 t\right) \ ,$$
where $r$ and $\phi$ are the polar coordinates of the specific point, and $\omega_0 = 2 \pi f_0$. The experimental setup suggested in [@Sto14] can exhibit either a quadrupole or a multipole geometry with an even number of electrodes, and the NPs are confined within a dielectric container. According to literature [@March97], trapping of particles is performed in optimum conditions if
$$\label{diel2}
\frac QM < \frac{1.816 \ \pi^2 r_0^2\ f_0^2}{V_0}$$
In order to find the optimum trapping point, depending on the dimensions and specific charge $Q/M$ of the naniparticle species used, the a.c. voltage $U_{ac}$ amplitude $\left(V_0\right)$ and frequency $f_0$ can be modified, along with the values of the trapping voltages $U_x$ and $U_z$. Such an operation practically shifts the operating point in the Mathieu equation stability diagram, with an aim to achieve stable trapping. Electrical charging of the NPs can be achieved:
1. as an outcome of the physical contact between particles confined in the dielectric container (8), without the need of extra electrodes or of another ionizing system,
2. as a result of the interaction between the NPs confined in the dielectric container (8) and its dielectric surface,
3. by applying a high voltage electric pulse $U_p$ across an electrode located within the dielectric container (8). According to Fig. \[dieltrap2\] the electrode can consist of a thin conducting wire (22) that crosses the container (8) parallel to the trap axis (4), located close to the inner surface in order to severely reduce perturbations of the electric field distribution near the trap axis. Thus, the trapping electric field can be considered as approximately harmonic. The conducting wire (22) is connected to an electronic module (23) that delivers high voltage pulses of amplitude $U_p$ at the output,
4. an overvoltage can be applied for a short period of time, superimposed over the $U_x$ diagnosis voltage. To achieve that, the secondary winding of a step-up transformer is connected to the (1b) electrode. The output of a pulse generator is fed into the primary winding of the step-up transfomer,
5. by switching the electrical polarity of the d.c. voltage $U_x$, applied between the electrodes denoted by (1a) and (1b). It can be done by either acting upon a mechanical switch, or by means of an electronic switch,
6. the NPs located within the container (8) can be acted upon for a specific period time by an ionizing radiation radiation beam, such as ultraviolet (UV) radiation, X radiation or radiation emitted by a radioactive source. The radiation interacts with the NPs that are to be trapped, by ionizing and implicitly charging them.
Use of linear Paul traps for target positioning {#Target}
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Theoretical and experimental investigations with respect to laser plasma acceleration of particles also use isolated mass-limited targets, that result in increased ion energies and peak spectral distributions of interest for a large number of applications [@Bul08; @Klu10]. Different approaches used to realize mass-limited targets use micrometer sized water droplets, gas-cluster targets with dimensions in the 100 nm range, and objects carried by thin structures with dimensions ranging between 100 nm up to a few microns [@Ost16]. For all cases presented the range of target dimensions is restricted. In order to lift all target restrictions as well as unintended interactions, Paul traps have been used in combination with high-power laser systems in an attempt to position isolated micron-sized targets in the laser focus with high accuracy. The Paul trap employed in the experiment described allows sensibly higher laser energies and a significantly larger range in what concerns the diameter size of the target [@Ost16; @Ost18]. The trap represents a major progress with respect to recent Paul trap geometries used in similar experiments [@Sto14; @Sok10]. The most notable difference lies in the fact that instead of friction between the target particles with the surrounding gas, the Paul trap used in [@Ost16; @Ost18] employs an ion gun to achieve charging. Thus, specific charge ratios $Q/M$ that are a few orders of magnitude higher are obtained, which leads to a minimization of the residual motion of trapped targets at comparable kinetic energies. The particle motion and implicitly its kinetic energy is not damped by friction with air, but it is produced by an opto-electronic feedback loop that runs continuously, which allows trap operation at pressure values that go below $10^{-5}$ mbar [@Ost18]. Thus, particle diagnostics is not influenced by the background gas environment. The setup allows target positioning with an accuracy value that lies within the micron range, due to large access angles that the Paul trap system exhibits.
Similar to the setup described in [@Sto14], the transportable setup used in [@Ost18] uses a quadrupole Paul trap with cylindrical rods and provides a large optical access to the particle species of interest. There is a vivid scientific interest towards using micron-scale particles as targets for high-intensity laser-plasma acceleration experiments. A tightly focused laser pulse interacts with the strongly coupled plasma that consists of levitated particles, confined within a well defined region in space located in the trap centre. Both the focusing optics and laser diagnosis packages should be provided for the experiment. The resulting ultra-short and intense X-ray, ion, and electron bursts are expected to have important applications in physics and medical research. In comparison with metallic foils or bulk foils targets [@Groza19], well isolated targets could exhibit major advantages, especially in what regards the maximum energy of energetic ions, as well as their energy distribution and source size distribution. The setup can provide targets of any solid material with diameters currently ranging from 500 nm $\div$ up to $50 \ \mu$m. The system is able to confine both cluster and droplet targets and allows scans across a large diameter-range. An almost interaction free environment is created as any particle interaction with nearby structures or background gas is sensibly reduced, while the positioning accuracy is controlled down to micron level precision [@Ost18].
The trap uses a cylindrical electrode geometry which yields to the occurrence of higher order terms of the electric potential, such as the those corresponding to the 12-pole and 20-pole order. As the trapping field is far from being a pure quadrupole one (in fact it is anharmonic), the equations of motion in the radial plane ($x$ and $y$ directions) are coupled. Higher order terms lead to unstable trajectories and may even result in chaotic motion, as an outcome of the phenomenon of resonant heating [@Blau98]. The trap is operated with $q < 0.4$, namely in a region of the stability diagram where only very high order instabilities exist. On the other hand, due to the fact that the target levitates close to the trap centre higher order instabilities are reduced to the minimum. Axial confinement along the $z$ axis is achieved by supplying additional d.c. voltages on the endcap electrodes of 3 mm diameter (located $10 \div 20$ mm apart), which leads to an additional motion along the trap axis. Anharmonic fields vanish in the trap centre and they are discarded in the discussion. The particle trajectory is measured electro-optically and filtered electronically. An electrical feed-back field is used with a proper phase-shift to turn the harmonic macromotion into the dynamics of a damped harmonic oscillator, which efficiently damps the macromotion [@Ost18].
The Paul trap based setup developed for laser-plasma experiments is shown in Fig. \[PaulTarg1\]. A vacuum chamber allows reaching pressures down to the $10^{-6}$ mbar. Generally, laser plasma accelerators operate at pressure values around $10^{-5}$ mbar to minimize the effects of the gaseous environment on the laser beam, and to provide appropriate detection of laser-plasma accelerated particles. The Paul trap consists of four cylindrical copper rods of 5 mm diameter equidistantly spaced around the trap centre. The trap radius is $r_0 = 8.1$ mm. Due to the large optical access provided by the geometry used, large-numerical-aperture optics can be employed, characteristic for experiments with ultrahigh-intensity lasers. All a.c. and endcap electrodes are positioned in precision manufactured ceramic seats, which electronically isolate them with respect to each other. Moreover, each a.c. electrode and endcap electrode is connected to its individual voltage supply. The a.c. trapping voltage can reach a maximum value of 3 kV with $\Omega = 2 \pi \cdot 5$ kHz, while the d.c voltage applied between the endcap electrodes can rise up to 450 V.
![Paul trap setup used for taget positioning. The setup comprises a vacuum chamber, the trap and connected power supplies, an ion gun to charge the target particles, a laser to illuminate target particles, and the optical measurement and feedback setup. Picture reproduced from [@Ost18] by courtesy of T. Ostermayr.[]{data-label="PaulTarg1"}](PaulTrapVer1.pdf)
The setup includes a particle reservoir located above the trap centre that is filled with a few milligrams commercial sample of monodisperse spherical particles, used as target. By means of a mechanical loading system, particles fall into the trap through a 100 mm hole. An ion beam – generated by an ion gun – is fed into the trap and charges the particles. Thus, a few particles are confined within the trap. The endcap voltage is dropped off to a value of a few volts before trapping, which yields a strong Coulomb repulsion that expels most of the particles out of the trap leaving just a single one. Then, the ion gun is shut off and the pressure is reduced to a lower value. Simultaneously, the a.c. and d.c. trapping voltages are increased to their maximal values to ensure tight confinement in the radial plan. To visualize the particles a 660 nm laser diode (delivering 50 mW of output power) is used. Two thirds of the output power are coupled into a single-mode fiber. An adjustable collimator lens performs laser beam focusing to a spot size between 0.25 and 1 mm FWHM (Full-Width at Half Maximum) in the trap centre. Three independent imaging systems collect the stray light from target. A position-sensing diode (PSD) is employed to record the CM motion of the particle’s image. The frequency of the particle motion is inferred by means of a Fast Fourier Transform (FFT) algorithm. The coordinate signal is phase-shifted and delivered as an additional voltage to the trap electrodes, which leads to an effective damping of the particle motion in absence of a particular gas.
The key parts of the concept used are the optical setup and the optoelectronic damping that are responsible for achieving experiments with well positioned targets at high-vacuum conditions, which eliminates the need for buffer gas that would affect the high-power laser and diagnostics for the laser plasma interaction. The setup represents a remarkable improvement as it can be customized to fit any interaction chamber geometry. Another critical issue lies in the fact that the system should be able to collect the small amount of stray light created by microscopic target particles. Since only the CM of the particle’s image is tracked by the PSD, a large aperture lenses is used with comparably small depth of focus ($\sim 1 \div 2 \ \mu$m) to provide enhaced collection accuracy. The fiber-coupled laser is fed into the vacuum chamber by means of a vacuum feedthrough, to reduced the influence of stray light to a minimum.
Particle acceleration using PW class lasers and strongly coupled plasmas levitated in electrodynamic traps {#PWTargets}
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Plasma acceleration is a technique to accelerate charged particles, such as electrons, positrons and ions, using an electric field associated with electron plasma wave or other high-gradient plasma structures (like shock and sheath fields). The plasma acceleration structures are created using ultra-short laser pulses or energetic particle beams, that are matched to the plasma parameters. These techniques offer a way to build high performance particle accelerators of much smaller size than conventional devices. Electron acceleration by laser-plasma interaction is a mechanism devised and first proposed by Tajima and Dawson in 1979 [@Taji79]. Similar to a beam-driven wake, a laser pulse can be used to excite the plasma wake. As the pulse propagates across the plasma volume, the electric field separates the electrons and nucleons. If the fields are sufficiently strong, all of the ionized plasma electrons can be removed from the centre of the wake: this is known as the [*electron bubble*]{} or [*blowout regime*]{}, first reported in 2002 [@Joshi06]. The electron bubble regime is a 3D nonlinear phenomenon in laser-plasma interactions. Relativistic plasma waves driven by pulsed femtosecond high power lasers allow electron acceleration, due to the very intense longitudinal electric fields that occur in the plasma [@Down14a].
When an intense laser pulse penetrates a low-density plasma, a bucket will occur as all the ionized plasma electrons that interact with the focused laser pulse are expelled by the radiation pressure force, while the fully stripped ions are assumed to be stationary at the time-scales of plasma electron response to the exciting fields. The charge separation field attracts bulk electrons near the axis and an electron bubble (cavity) occurs, which trails the driving laser at relativistic speed. The wakefield is a plasma wave with a relativistic phase velocity. Electrons accelerated in such regime are self-injected from the ambient plasma into a single bucket [@Li14b]. In the linear regime plasma electrons aren’t completely removed from the centre of the wake [@Hook13]. In such case the linear plasma wave equation [@Piel17] is valid. However, the wake appears very similar to the blowout regime and the physics of acceleration is the same [@Down14b].
It was suggested for the first time in 2009 to use compact Laser Plasma Accelerators (LPAs), with an aim to produce broadband radiation similar to the outer space and use it for radiation hardness testing of electronic parts [@Rosen11; @Gan15]. LPAs offer cost-effective radiation generation, which makes them extremely suited for testing radiation resistivity of electronic components [@Gan19]. Besides that, LPAs have the potential to become versatile tools for reproducing certain aspects of space radiation, under far better reproducibility than state-of-the-art technology [@Hook14], among which we can enumerate existing linear particle accelerator, cyclotron (MeV energy protons), synchrotron (energetic protons in the GeV range), or electron synchrotron facilities [@Hook13]. Ground-based conventional particle accelerators cannot reproduce the exponential electron beam distribution spectrum that occurs in space, and a typical approach to ground-based radiation hardening assessment (RHA) involves investigating the impact of radiation at multiple monochromatic energies, and then extrapolating the results to other energies based on modeling and simulation. The primary deficiency in such ground-based approaches is that the naturally occurring space radiation environment energy spectrum is dramatically different from the output of a conventional Earth-bound accelerator. On Earth, accelerators based on classical technology do not generate particle energy spectra that follow the exponential distribution found to characterize particle energy spectra in space, but deliver instead nearly monoenergetic distributions.
Petawatt (PW) laser technology and the mechanism of plasma acceleration of charged particles (especially electrons and protons) finds extremely important areas of applications for space technology. The advantage of plasma acceleration lies in the fact that the acceleration field is much stronger than that of conventional RF accelerators. In case of RF accelerators the field exhibits an upper limit, determined by the threshold for dielectric breakdown of the acceleration tube. Such a phenomenon limits the acceleration gain over any given area, requiring large dimension accelerators in order to achieve high energies. Present interest is focused on using ultrashort high intensity laser pulses that are matched to the plasma parameters. Using such mechanisms it is possible to create very large electric field gradients and very compact structures, with respect to classical accelerators. The basic idea lies in using LPA techniques with an aim to reproduce the cosmic radiation environment, in terms of particle species (electrons and protons) [@Groza19; @Gan19]. This is an issue of large interest for the European Space Agency, in support of the Jupiter Icy Moon Explorer (JUICE) mission aimed at exploring Jupiter and three of its moons: Ganymede, Europa and Callisto [@juice]. The basic idea is to demonstrate the concept of laser plasma acceleration of electrons and protons [@Groza19] up to energies of tens of MeV, or even larger than 100 MeV. The outcome lies in achieving an exponential energy distribution of the generated electrons, that would be similar to the distribution met in the Jovian system. Nevertheless, while space radiation can be characterized as continuous, the LPA technique generates a discontinuous flux of charged particles [@Gan15].
As shown in Section \[Target\], custom designed Paul trap systems allow accurate positioning of micrometer-sized particles with micron precision, which makes them extremely useful as targets in high intensity laser-plasma interactions. A setup intended for a laser plasma experiment is proposed in [@Ost18]. The sketch of the setup is reproduced in Fig. \[PaulTarg3\]. The high power laser beam is delivered via the beam-line system (BLS) and then focused on the trap using an off-axis parabolic (OAP) mirror. The laser focus is monitored via an optical system consisting of a microscope objective and a mirror. The black dotted line labeled as focus diagnostics denotes the optical path of the microscope. Electron, ion, and X-ray detectors can be put in the plane of laser propagation, marked as green.
![Paul trap setup used in laser plasma acceleration experiment. Picture reproduced from [@Ost18] by courtesy of T. Ostermayr.[]{data-label="PaulTarg3"}](PaulTrap3.pdf)
Fast mechanical shutters are used to reduce the influence of stray light. Laser beam powers of up to 150 J have been used. The secular frequency of the levitated microparticles is measured by real-time tracking of the PSD signal. As the trap parameters ($r_0$, $\Omega$, and $V$) are known, by measuring the secular frequency one can determine the charge-to-mass ratio $Q/M$ and the strength of the confinement. Thus, the particle position can be established with remarkable accuracy. Repetitive measurements performed for 10 $\mu$m diameter polystyrene particles yield a value of 0.29 C/kg for the charge-to-mass ratio, and an electric charge value $Q = 9.9 \times 10^5 e$ (for a secular frequency value $\omega_sec/2 \pi = 125$ Hz). Thus, measurements performed on single levitating particles demonstrate reproducibility, stability, and high accuracy in determining the particle location [@Ost18]. The most important parameter that reflects the quality of damping is the residual motion of a levitated particle, which can be estimated using the [*in situ*]{} focus diagnostics. The method used is based on determining the CM and then plot the 2D histogram of the occurrence of a specific particle position [@Ost18]. The very low values of the Full-Width at Half Maximum (FWHM) of the motion demonstrate that strong confinement is achieved, which leads to high accuracy in pinpointing the particle position and long trapping times. Thus, CM tracking is shown to be the most accurate method to determine the residual motion.
An experimental demonstration of the acceleration of a clean proton bunch is performed in [@Hilz18]. To achieve that, a microscopic and three-dimensionally (3D) confined near critical density plasma is created that evolves from a $1 \ \mu$m diameter plastic sphere, which is levitated and positioned with micrometer precision in the focus of a Petawatt laser pulse. The emitted proton bunch is reproducibly observed with central energies between 20 and 40 MeV and narrow energy spread (down to 25%), showing almost no low-energetic background. Using 3D particle-in-cell (PIC) simulations the acceleration process is fully characterized, emphasizing the transition from organized acceleration to Coulomb repulsion. This reveals limitations of current high power lasers and viable paths to optimize laser-driven ion sources.
Finally, theoretical and experimental aspects of relativistic intense laser-microplasma interactions and potential applications are presented by T. Ostermayr in [@Ost19]. The book describes a Paul-trap based target system, developed and tested so as to provide fully isolated, well defined and well positioned micro-sphere-targets, for experiments with well focused PW laser pulses. The interaction between the PW laser pulses and the micro-spheres converts these targets into microplasmas [@Dub99; @Dub90; @Boll90b], able to emit proton beams with kinetic energies higher than 10 MeV [@Ost19; @Hilz18]. The spatial distribution and the kinetic energy spectrum of the proton beam can be modified by changing the acceleration mechanism. The results span from broadly distributed spectra in cold plasma expansions, to spectra with relative energy spread that is reduced down to 25 % in spherical multi-species Coulomb explosions and in directed acceleration processes. Both analytical modeling and the numerical simulations show good agreement with the experimental results. The 3D PIC simulations performed characterize the complete acceleration process and clearly emphasize the transition from regular acceleration to Coulomb repulsion [@Hilz18]. This is a strong argument towards employing microplasmas to engineer laser-driven proton sources.
A sketch of the experimental setup used in [@Hilz18] is shown in Fig. \[PaulTarg4\]. The targets are extracted out of a reservoir by mechanical vibration. As particles fall freely inside the operating trap, electrical charging is achieved by means of an ion gun. The trapped particle amplitude is reduced using a two phase mechanism: buffer gas cooling at $10^{-4}$ mbar pressure, followed by electro-optical damping for a vacuum pressure lower than $10^{-5}$ mbar. Similar to the experiment described in [@Ost18] stray light is collected onto a PSD, and a small fraction of it is imaged onto a CCD camera. The hollow spheres are created by covering polystyrene spheres with a PMMA layer. Then, the polystyrene cores are subsequently dissolved away.
![Paul trap setup used in laser plasma proton acceleration experiment: (a) Incoming PHELIX laser beam, (b) Paul trap power supply, (c) 660 nm illumination laser, (d) electrooptical diagnostics for target damping and positioning, (e) Paul trap electrodes, (f) trapped target scatter screen for transmitted light with (g) and without (h) target, (i) magnetic spectrometer, (j) IP raw proton/ion data (without degraders). Picture reproduced from [@Hilz18] by courtesy of T. Ostermayr.[]{data-label="PaulTarg4"}](PaulTrap4.pdf)
In a different experiment, tungsten micro-needle-targets were used as targets for a well focused PW laser, with an aim to generate few-keV X-rays and 10-MeV-level proton beams simultaneously. Measurements performed show that the beams exhibit only few-$\mu$m effective source-size. Thus, the technique can be used to [*demonstrate single-shot simultaneous radiographic imaging with X-rays and protons of biological and technological samples*]{} [@Ost19; @Hilz18]. Other issues are also approached in Ref. [@Ost19], such as future perspectives and directions of action for the study of laser-microplasma interactions using non-spherical targets.
Conclusions and perspectives {#Concl}
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The paper is a review on the physics and applications of strongly correlated Coulomb systems levitated in electrodynamic (Paul) traps. After a short introduction into the physics of particle traps at the beginning of Section \[Sec2\], the issue of the anharmonic contributions to the electric potential generated by a linear RF trap is discussed, with respect to recent investigations [@Pedre18b]. Particle dynamics is examined, for a classical system of two particles levitated in a 3D Paul trap. A theoretical model is introduced, with an aim to characterize regular and chaotic orbits for the system described, depending on the chosen control parameters [@Mih19a]. The phase portraits illustrate ordered and chaotic orbits, while particle dynamics is shown to be ergodic. By studying the sign of the eigenvalues of the Hessian matrix one can analyze and classify the critical points, as well as the system stability. Recent advances in the development of nonlinear traps are mentioned, before introducing optical levitation and acoustic levitation techniques. The problem of instabilities in electrodynamic traps is also approached, with respect to the most relevant and recent contributions to the domain. Chaos is reviewed in Sec. \[chaos\], where the dynamics of a charged particle confined in a quadrupole nonlinear Paul trap is investigated. Phase portraits, Poincar[é]{} sections and bifurcation diagrams illustrate the onset of chaos, as well as the existence of attractors in the associated dynamics, which can be characterized as strongly nonlinear.
Non-neutral, complex plasmas and Coulomb systems are investigated in Section \[Sec3\]. Strongly coupled Coulomb systems encompass various many-body systems and physical conditions, such as dusty (complex) plasmas or non-neutral and ultracold plasmas [@Werth08; @Werth05a; @Tsyto08; @Fort06a; @Morfi09]. We emphasize that ion traps represent versatile tools to investigate many-body Coulomb systems, or dusty and non-neutral plasmas. The dynamical time scales associated with trapped microparticles lie in the tens of milliseconds range, while microparticles can be individually observed using optical methods. As the background gas is dilute, particle dynamics exhibits strong coupling regimes characterized by collective motion [@Major05; @Piel17; @Melz05; @Sav12]. Weakly-coupled plasmas and strongly-coupled Coulomb systems are presented in Sections \[weak\] and \[strong\], respectively. Section \[Sec3\] illustrates how a Paul trap represents a versatile tool to investigate the physics of few-body phase transitions and thus gain new insight on such mesoscopic systems [@Schli96; @Walth95].
Strongly coupled plasmas confined in quadrupole electrodynamic traps are reviewed in Section \[Sec4\]. Stable 2D and 3D structures are discussed, including Coulomb structures [@Vasi13; @Syr18; @Lapi19a; @Vlad18; @Lapi15c; @Vasi18], as well as Microparticle electrodynamic ion traps (MEITs) [@Libb18], with an emphasize on latest techniques, experimental results, and applications. Hamiltonians for systems of $N$ particles are used to describe ordered structures and collective dynamics for systems of charged particles (ions) levitated in quadrupole 3D traps with cylindrical symmetry. The Hamilton function for the system of $N$ particles of mass $M$ and equal electric charge $Q$, confined in a combined quadrupole 3D trap is obtained [@Mih18; @Mih19a; @Ghe00].
Waves in plasmas are investigated in Section \[waves\], using particle-in-cell (PIC) simulations. Caustic solitary density waves are discussed for microparticles levitated in electrodynamic traps [@Syr19a; @Syr19b]. Analytical and numerical simulations are performed, in order to better explain the physical mechanisms and phenomena observed. The interaction between an acoustic wave and levitated microparticles is also investigated [@Sto11]. Section \[Sec6\] presents the physics of multipole traps and their associated advantages, along with the latest experimental techniques and results [@Mih16a; @Mih16b]. Analytical and numerical simulations are supplied that support the experimental data and results.
Sections \[Sec7\] and \[Sec9\] review mass spectrometry methods and techniques to investigate single particles levitated in electrodynamic or optical traps [@Gong18; @Puro19]. Emphasis is placed on mass spectrometry techniques [@Werth09; @March05; @March17a], as well as on novel methods based on elastic and inelastic scattering of light, using the Mie theory [@Bast18; @Mie1908; @Herg12], or Raman and fluorescence spectroscopy [@Signo11; @Wrie09; @Mish09; @Red15]. Recent experiments seem to indicate that inelastic scattering can be successfully applied to the detection of biological particles such as pollen, bacteria, aerosols, traces of explosives or synthetic polymers [@Wang15].
Despite their micro and nanometer range size aerosols have a major impact on global climate and health, but the underlying mechanisms and subsequent effects are still far from being explained. Section \[Sec8\] brings new perspectives and sheds new light on aerosol and nanoparticle diagnosis [@Sande14; @Kulma11; @Krieg12; @Sulli18], with a special emphasis on the optical properties associated with them [@Kul11; @Rama18; @Sein16; @Islam18; @Tom17]. Novel methods and techniques are described, aimed at investigating and characterizing these particles.
Section \[Sec10\] focuses on laser plasma accelerated physics and the use of electrodynamic traps as targets for high power lasers. Two decades ago, it was suggested to employ the analogy between the dynamics of charged particles in the non-homogenous magnetic quadrupole field of accelerators or storage rings, and that of an electric quadrupole trap excited by square voltage pulses [@Kjae01]. Instead of using typical sinusoidal voltages, controllable, pulsed RF voltages are applied to the trap electrodes with an aim to investigate collective beam dynamics in various lattice structures [@Kelli15]. Moreover, the collective dynamics of a one-component plasma (OCP) in a RF quadrupole trap is physically equivalent to that of a charged-particle beam that propagates throughout a periodic magnetic lattice [@Fuku14; @Oht10]. Such an interesting feature allows one to investigate various beam-dynamics issues using compact setups, without the need for large-scale accelerators. Theoretical and experimental investigations with respect to laser plasma acceleration of particles also use isolated mass-limited targets, that result in increased ion energies and peak spectral distributions of interest for a large number of applications [@Bul08; @Klu10]. In order to lift all target restrictions as well as unintended interactions, Paul traps have been used in combination with high-power laser systems in an attempt to position isolated micron-sized targets in the laser focus with high accuracy [@Ost19; @Hilz18].
Acknowledgements
================
B. M. is grateful to G[ü]{}nther Werth, Cristina Stan (Polytechnic Univ. Bucharest) and Kenneth Libbrecht, for valuable comments and suggestions on the manuscript, as well as for their kindness. B. M. gratefully acknowledges support from the Romanian Space Agency (ROSA) - Contract Nr. 136/2017 *Quadrupole Ion Trap Mass Spectrometry* (QITMS), and Contract No. 3N/2018 (Project PN 18 13 01 03, Romanian National Authority for Scientific Research and Innovation). Part of the activities and work reported in Section \[Sec10\] was supported through the European Space Agency (ESA) - Contract. No. 4000121912/17/NL/CBi - [*Laser Plasma Accelerators as tools for Radiation Hardness Assessment (RHA). Studies and Tests in support of ESA space missions*]{} and ROSA Contract Nr. 53/2013 LEOPARD. B. M. also acknowledges support along the many years from his colleagues at INFLPR, and especially M. Ganciu, O. Stoican, [Ş]{}t. R[ă]{}dan, and V. Gheorghe who introduced me to ion traps and the wonderful physics associated with them.
V. F., R. S., and L. S. acknowledge stimulating discussions with Professors V. E. Fortov and O. F. Petrov. The work has been carried out under financial support of the Russian Foundation for Basic Research (RFBR) via grant 18-08-00350.
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abstract: 'Transmission lines are essential components in various signal and power distribution systems. In addition to their main use as connecting elements, transmission lines can also be employed as continuous sensors for the measurement and detection of external influences such as mechanical strains and deformations. The measuring principle is based on deformation-induced changes of the characteristic impedance. Reflections of an injected test signal at resulting impedance mismatches can be used to infer applied deformations. To determine the effect of deformations on the characteristic impedance, we develop a numerical framework that allows us to solve Maxwell’s equations for any desired transmission-line geometry over a wide frequency range. The proposed framework utilizes a staggered finite-difference Yee method on non-uniform grids to efficiently solve a set of decoupled partial differential equations that we derive from the frequency domain Maxwell equations. To test our framework, we compare simulation results with analytical predictions and corresponding experimental data. Our results suggest that the proposed numerical framework is able to capture experimentally observed deformation effects and may therefore be used in transmission-line-based deformation and strain sensing applications. Furthermore, our framework can also be utilized to simulate and study the electromagnetic properties of complex arrangements of conductor, insulator, and shielding materials.'
address:
- 'Institute for Theoretical Physics, ETH Zurich, 8093, Zurich, Switzerland'
- 'LEONI Studer AG, Hohlstrasse 190, 8004, Zurich, Switzerland'
- 'Institute for Theoretical Physics, ETH Zurich, 8093, Zurich, Switzerland'
- 'Center of Economic Research, ETH Zurich, 8092, Zurich, Switzerland'
- 'LEONI Studer AG, Hohlstrasse 190, 8004, Zurich, Switzerland'
author:
- 'Stefan H. Strub'
- Lucas Böttcher
title: Modeling Deformed Transmission Lines for Continuous Strain Sensing Applications
---
[*Keywords*]{}: transmission line, numerical analysis, finite-difference method, skin effect, Bessel functions, Maxwell’s equations, distributed sensing
Introduction
============
The transmission of electromagnetic waves is at the heart of many signal and power distribution applications. For wavelengths that are small compared to the conductor length, transmission lines such as coaxial cables or twisted pairs are used to limit radiative losses and reflections. In addition to their application in signal and power transmission systems, it is also possible to use transmission lines as continuous sensors for the measurement and detection of different environmental influences [@LiThAb; @zhu2019truly]. One important application is the measurement of mechanical strains and deformations to monitor the condition of civil engineering structures such as houses, bridges, and damns [@LiThAb]. In contrast to discrete sensors (*e.g.*, strain and displacement gauges), the advantage of continuous transmission-line sensors is the possibility to perform measurements at any location along a sensing line. The measuring principle in electrical transmission lines is based on two mechanisms: (i) Localized mechanical strains lead to transmission-line deformations and corresponding changes of the characteristic impedance. (ii) An injected test signal gets reflected at deformation-induced impedance mismatches and the information contained in signal reflections can be used to infer applied deformations.
Electrical transmission lines are, however, not the only possibility to realize continuous strain and deformation sensors in practice. Another approach is based on optical fibres and Bragg grating and Brillouin scattering methods. The idea behind the Bragg grating method is to use optical gratings within optical fibres and measure wave-length changes that correspond to certain deformations. The disadvantage of this method is that it is only quasi-continuous and requires a large number of gratings to achieve good resolution [@sirkis1998using]. In the Brillouin scattering technique, frequency shifts of the scattered light relative to the incident light are used to infer deformations. Despite the fact that this method does not require discrete gratings, the possible resolution is limited to about $10~\text{cm}$ [@brown1999brillouin; @murayama2004distributed]. In addition to the low spatial resolution, another drawback of optical methods is that they may not be suitable for cost-sensitive applications. Electrical transmission lines, on the other hand, can be fabricated more cost-efficiently and allow for resolutions of greater than $1~\text{cm}$ [@LiThAb].
To measure deformations with electrical transmission lines, it is necessary to determine the effect of deformations on the characteristic impedance. However, existing models only capture concentric transmission-line deformations and thus neglect any further shape-specific information [@LiThAb]. To accurately describe the effect of arbitrary deformations on the characteristic impedance, we develop a numerical framework to solve Maxwell’s equations in the frequency domain for any desired transmission-line geometry over a wide frequency range. Our framework is based on a staggered finite-difference Yee method on non-uniform grids and solves a set of decoupled partial differential equations (PDEs) that we derive from the frequency domain Maxwell equations. We test the ability of the proposed framework to characterize deformation effects in transmission lines by comparing simulation results with analytical predictions and corresponding experimental data.
Transmission-line theory {#sec:Telegraphers}
========================
{width="50.00000%"}
Before focusing on the development of our numerical framework for the study of deformed transmission lines, we summarize some concepts from transmission-line theory. In most transmission-line applications, the transversal electromagnetic (TEM) mode can be regarded as the dominant signal propagation mode [@paul2008analysis]. In this mode, electric and magnetic fields lie in the transverse $(x,y)$ plane orthogonal to the direction of propagation $z$ of the incident electromagnetic wave. Based on the TEM assumption, it is possible to derive the telegrapher’s equations from Maxwell’s theory and describe the TEM wave propagation along a transmission line [@chicone2016invitation; @paul2008analysis]. In the frequency domain, these equations describe the transmission-line voltage $V(z)$ and current $I(z)$ at position $z$ and are given by
$$\begin{aligned}
\frac{\mathrm{d} V\left(z\right)}{\mathrm{d} z}=-\left(R+ i \omega L\right) I\left(z\right)\,, \label{eq:telegraphers1} \\
\frac{\mathrm{d} I\left(z\right)}{\mathrm{d} z}=-\left(G+ i \omega C\right) V\left(z\right) \label{eq:telegraphers2}\,,\end{aligned}$$
where $\omega=2 \pi f$ and $f$ is the corresponding frequency. The transmission-line parameters are $R=R(z,\omega)$, $L=L(z,\omega)$, $C=C(z,\omega)$, and $G=G(z,\omega)$ and denote resistance, inductance, capacitance, and conductance per unit length, respectively. Some important factors that influence these parameters are transmission-line geometry, conductivity and dielectric properties of the used materials, and frequency of the incident wave.
The telegrapher’s equations may also be interpreted in terms of a sequence of equivalent circuits as we show in figure \[fig:transmissionline\]. Each segment is specified by the corresponding local transmission-line parameters and when put together they describe the whole transmission line. Instead of solving Maxwell’s equations for a given transmission line and incident wave in three dimensions, the telegrapher’s equations provide a way to describe the same problem by solving \[eq:telegraphers1,eq:telegraphers2\]. This approach permits a substantial reduction of computational effort. However, it also relies on the assumption of a TEM mode wave propagation that is only valid for uniform transmission lines consisting of perfect conductors surrounded by a homogeneous medium [@paul2008analysis]. Still, even in the cases of lossy conductors, inhomogeneous surrounding media, and nonuniform cross-sections, the telegrapher’s equations are used under the assumption that such effects only lead to negligible deviations from the TEM mode description (quasi-TEM assumption) [@paul2008analysis]. We therefore simulate deformed transmission lines according to \[eq:telegraphers1,eq:telegraphers2\].
In a transmission line, electromagnetic waves propagate with a phase velocity of $v=1/\sqrt{L C}$. If the surrounding medium is homogeneous with permeability $\mu$ and permittivity $\epsilon=\epsilon^\prime-i\,\epsilon^{\prime \prime}=\epsilon_0 \left(\epsilon_{\text{r}}^\prime-i\,\epsilon_{\text{r}}^{\prime\prime}\right)$, the phase velocity is $v=1/\sqrt{\mu \epsilon^\prime}$ [@paul2008analysis]. The imaginary part of the permittivity accounts for dielectric losses and $\epsilon_0$ is the vacuum permittivity. In most transmission line applications, the permeability $\mu=\mu_0 \mu_{\text{r}}$ is equal to the vacuum permeability $\mu_0$, because common conductors (*e.g.*, copper and aluminum) exhibit a relative permeability of $\mu_{\text{r}}\approx 1$.
For each transmission-line segment, the characteristic impedance is [@paul2008analysis] $$Z_0\left(\omega\right) = \sqrt{\frac{R+ i \omega L}{G+i \omega C}}
\label{eq:impedance}$$ and therefore a function of the transmission-line parameters. At an interface between a line segment with impedance $Z_1$ and another one with impedance $Z_2$, the corresponding reflection coefficient [@paul2008analysis] $$\Gamma_{12} = \frac{Z_2-Z_1}{Z_2+Z_1}
\label{eq:reflection_coeff}$$ describes the ratio of the reflected and the incident wave. No reflections occur for transmission lines with equal impedances along the line (matched transmission lines). To describe the influence of transmission-line deformations on characteristic impedance and signal propagation, we have to determine the corresponding transmission-line parameters. Only in certain cases such as for coaxial geometries, is it possible to find closed analytical solutions. For general deformations, it is necessary to utilize numerical methods.
Under the quasi-TEM assumption, we first determine the electric and magnetic fields $\mathbf{E}\left( z,\omega\right)$ and $\mathbf{B}\left(z,\omega\right)$ for a transmission-line segment at position $z$. Based on the obtained electromagnetic fields, we then compute the transmission-line parameters for the considered segment. Specifically, let $\Omega \subseteq \mathbb{R}^2$ be the cross section of a certain transmission-line segment at position $z$. The cross section $\Omega$ shall include the surrounding medium up to a certain distance where the magnetic field is sufficiently small. We denote the current density in $z$ direction by $J_z\left(z,\omega\right)$ and describe resistive losses by [@jackson1999classical] $$P\left(z,\omega\right) = \int_\Omega \frac{{J_z\left(z,\omega\right)}^2}{\sigma} \mathrm{d} A\,,
\label{eq:jouleheating}$$ where $\sigma$ denotes the electric conductivity. Based on \[eq:jouleheating\], the resistance per unit length is $$R\left(z,\omega\right) = \frac{P\left(z,\omega\right)}{I^2}\,,
\label{eq:R}$$ where $I$ is the current that flows through the transmission line. Next, we compute the inductance with the help of the total magnetic energy averaged over one cycle [@jackson1999classical] $$\bar{W}_m\left(z,\omega\right) = \frac{1}{4} \int_{\Omega} \mathbf{H}\left(z,\omega\right)\cdot \mathbf{B}^\ast\left(z,\omega\right)\, \mathrm{d} A\,,
\label{eq:magneticenergy}$$ where $\mathbf{H}\left(z,\omega\right)=\mathbf{B}\left(z,\omega\right)/\mu $. The inductance per unit length is [@jackson1999classical] $$L\left(z,\omega\right) = \frac{4 \bar{W}_m\left(z,\omega\right)}{I^2}\,.
\label{eq:L}$$ Similarly, we determine the capacitance by considering the total electric energy averaged over one cycle [@jackson1999classical] $$\bar{W}_e\left(z,\omega\right) = \frac{1}{4} \int_{\Omega} \mathbf{E}\left(z,\omega\right)\cdot \mathbf{D}^*\left(z,\omega\right) \mathrm{d} A\,,
\label{eq:electricenergy}$$ where $\mathbf{D}\left(z,\omega\right)=\mathbf{E}\left(z,\omega\right)/\epsilon^\prime$. We obtain the capacitance per unit length through [@jackson1999classical] $$C\left(z,\omega\right) = \frac{4 \bar{W}_e\left(z,\omega\right)}{V^2}\,,
\label{eq:C}$$ where $V$ is the voltage difference between inner and outer conductor. We note that electric field and capacitance are frequency-independent until currents that flow through the dielectric material become relevant. This is only the case for frequencies that are not captured by the quasi-TEM approximation. Another possibility to compute the capacitance is to consider the total charge $Q$. Therefore, let $\Omega' \subseteq \mathbb{R}^2$ be the area that covers all conductors of the same voltage level. The total charge is $$\begin{aligned}
\begin{split}
Q\left(z,\omega\right) &= \epsilon^\prime \int_{\Omega'} \rho \, \mathrm{d}A \\
&= \epsilon_0 \int_{\partial \Omega'} \epsilon_{\text{r}}^\prime \mathbf{E}\cdot \mathbf{n}' \, \mathrm{d} l\,,
\end{split}
\label{eq:total_charge}\end{aligned}$$ where $\rho$ is the charge density. The relative permittivity $\epsilon_{\text{r}}^\prime$ may be position-dependent. The second step in \[eq:total\_charge\] follows from Gauss’ law, where $\mathbf{n}'$ is perpendicular to the boundary $\partial \Omega'$ of $\Omega'$. The capacity per unit length is $$C\left(z,\omega\right) = \frac{Q\left(z,\omega\right)}{V}\,.$$ For a homogeneous dielectric medium, the conductance per unit length is [@paul2008analysis] $$G\left(z,\omega\right) = \omega \tan\left(\delta \right) C\left(z,\omega\right)\,,
\label{eq:Gapprox}$$ where $$\tan \left(\delta\right)=\frac{\omega \epsilon^{\prime \prime}- \sigma_{\text{dielectric}}}{\omega \epsilon^\prime}$$ denotes the loss tangent. In practice, dielectric losses dominate and $\tan\left(\delta\right)=\epsilon^{\prime\prime}/\epsilon^\prime$. To fully describe the dielectric material, it is necessary to determine the frequency dependence of $\tan\left(\delta\right)$. For typical dielectric materials that are used in transmission line applications, the frequency dependence is almost constant over a certain frequency range [@paul2008analysis]. In the more general case of a non-homogeneous dielectric medium, the conductance per unit length is [@paul2008analysis] $$G\left(z,\omega\right) = \frac{I_t\left(z,\omega\right)}{V}\,,$$ where $I_t$ is the total transverse conduction current per unit line length between the conductors. Integrating the current density $\mathbf{J} = \left(\sigma - \omega \epsilon^{\prime \prime}\right) \mathbf{E}+\mathbf{J}^s$ over $\partial \Omega'$ yields the total transverse conduction current and the conductance per unit length $$G\left(z,\omega\right) = \frac{1}{V} \int_{\partial \Omega'} \mathbf{J}\cdot \mathbf{n}' \, \mathrm{d}l\,,
\label{eq:G}$$ where $\mathbf{J}^s$ in $\mathbf{J}$ is the externally applied source current.
Potential formulation of Maxwell’s equations {#sec:Formulation}
============================================
After having outlined the basic strategy of how to describe transmission lines of arbitrary cross sections in section \[sec:Telegraphers\], we now determine the electromagnetic fields of a given transmission-line segment [@OldenburgFast; @haber2001fast]. Based on \[eq:R,eq:L,eq:C,eq:G\], it is then possible to obtain the corresponding transmission-line parameters.
We begin with a brief summary of the potential formulation of Maxwell’s equations as proposed in [@OldenburgFast; @haber2001fast]. Maxwell’s equations in the frequency domain are $$\begin{aligned}
\nabla\times\mathbf{E} - i \omega \mu \mathbf{H} &= 0\,, \label{eq:max1}\\
\nabla\times\mathbf{H} - \left(\sigma - i \omega \epsilon\right) \mathbf{E} &= \mathbf{J}^s\,, \label{eq:max2}\\
\nabla \cdot \left(\epsilon \mathbf{E}\right) &= \rho\,, \label{eq:max3}\\
\nabla \cdot \left(\mu \mathbf{H}\right) &= 0\,. \label{eq:max4}\end{aligned}$$ We divide \[eq:max1\] by $\mu$, take the curl, and substitute \[eq:max2\] into the resulting expression to obtain $$\nabla\times \left(\mu^{-1} \nabla\times\mathbf{E} \right) - i \omega \hat{\sigma} \mathbf{E} = i \omega \mathbf{J}^s\,,
\label{eq:PDE E}$$ where $$\hat{\sigma} \coloneqq \sigma - i \omega \epsilon$$ is the generalized conductivity. Moreover, a decomposition of $\mathbf{E}$ into the vector potential $\mathbf{A}$ and the scalar potential $\Phi$ yields $$\mathbf{E} = \mathbf{A} + \nabla \Phi \label{eq:decompose E}\,.$$ The vector potential satisfies the Coulomb gauge condition $$\nabla\cdot \mathbf{A} = 0\,.
\label{eq:gauge}$$ Substituting \[eq:decompose E\] into \[eq:PDE E\] leads to $$\nabla\times\nabla\times\mathbf{A} - i \omega \mu \hat{\sigma} \left(\mathbf{A} + \nabla \Phi \right)= i \omega \mu \mathbf{J}^s\,,
\label{eq:PDE_A_complex}$$ where we assumed a constant permeability $\mu$. Using the Coulomb gauge condition of \[eq:gauge\], simplifies \[eq:PDE\_A\_complex\] to $$\nabla^2\mathbf{A} + i \omega \mu \hat{\sigma} \left(\mathbf{A} + \nabla \Phi \right)= - i \omega \mu \mathbf{J}^s
\label{eq:PDE A phi}$$ and taking the divergence of \[eq:PDE A phi\] yields $$\nabla\cdot \left[\hat{\sigma} \left(\mathbf{A} + \nabla \Phi \right) \right] = - \nabla\cdot \mathbf{J}^s\,.
\label{eq:PDE all}$$ As described in section \[sec:Telegraphers\], under the quasi-TEM assumption we consider the transmission line to be composed of individual segments whose electromagnetic fields lie in the transverse $(x,y)$ plane. Therefore, the solution of \[eq:PDE A phi,eq:PDE all\] can be obtained with a substantial reduction in computational effort. Under the quasi-TEM assumption, the resulting PDEs are
$$\begin{aligned}
\left(\partial_x^2 + \partial_y^2\right) A_x + i \omega \mu \hat{\sigma} \left(A_x + \partial_x{\Phi} \right) &= - i \omega \mu J^{s}_x\,, \label{eq:A_x1}\\
\left(\partial_x^2 + \partial_y^2\right) A_y + i \omega \mu \hat{\sigma} \left(A_y + \partial_y{\Phi} \right)&= - i \omega \mu J^{s}_y\,, \label{eq:A_x2}\\
\left(\partial_x^2 + \partial_y^2\right) A_z + i \omega \mu \hat{\sigma} A_z &= - i \omega \mu J^{s}_z\,, \label{eq:A_z}\\
\partial_x{\left[\hat{\sigma} \left( A_x + \partial_x{\Phi} \right)\right]} + \partial_y{\left[\hat{\sigma} \left( A_y + \partial_y{\Phi}\right)\right]} &= -\partial_x J^{s}_x \nonumber \\
&- \partial_y J^{s}_y\,. \label{eq:A_x3}\end{aligned}$$
\[eq:PDE split\]
We note that $\Phi$ dropped out of \[eq:A\_z\]. As a result $A_z$ is decoupled from $A_x, A_y$, and $\Phi$. To determine capacitance and conductance for different frequencies, we have to solve \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\] only once since the capacitance as defined in \[eq:C\] exhibits no frequency dependence and the conductance scales linearly with $\omega$ according to \[eq:G\] [@paul2008analysis]. In section \[sec:electr\_magn\_fields\], we outline that it is sufficient to solve \[eq:A\_z\] to determine the frequency dependence of the magnetic field, and thus of the resistance and inductance of a certain transmission-line segment. This again leads to a substantial reduction of computational effort compared to the case in which we have to solve the whole set of PDEs given by \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\]. According to [@OldenburgFast], possible boundary conditions for the solution of \[eq:PDE A phi\] are
$$\begin{aligned}
\left(\nabla \times \mathbf{A} \right) \times \mathbf{n}|_{\partial \Omega} &= 0\,, \\
\mathbf{A} \cdot \mathbf{n}|_{\partial \Omega} &= 0\,, \\
\nabla \Phi |_{\partial \Omega}\cdot \mathbf{n}&= 0\,, \label{eq:Phi1}\\
\int_{\Omega} \Phi \mathrm{d}A &= 0 \label{eq:Phi2}\,,\end{aligned}$$
\[eq:BC\]
where $\mathbf{n}$ is the unit normal vector on the boundary $\partial \Omega$. In the case of transmission lines, the inner and outer conductors can be assumed to exhibit potentials of the same absolute value relative to the respective area, but with opposite signs. Therefore, the boundary conditions given by \[eq:Phi1,eq:Phi2\] are already satisfied.
Discretization {#sec:Discretization}
==============
![**Electromagnetic field discretization.** Position of the discretized electromagnetic field components according to [@YeeNum].[]{data-label="fig:yee_grid"}](tikz_Yee_grid2.pdf){width="37.00000%"}
To solve \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\] for arbitrary transmission-line cross sections, we employ a staggered finite-difference scheme on non-uniform grids [@YeeNum; @haber2001fast; @OldenburgFast]. Specifically, we use a method that is based on central differences at midpoints. That is, we compute derivatives between two adjacent grid points using a forward-difference scheme and consider the resulting derivative values to be located at the corresponding midpoints. In this way, boundaries between different conductors and dielectric materials can be resolved well.
We discretize the electromagnetic fields according to the Yee method (see figure \[fig:yee\_grid\]) [@YeeNum]. Here and in the subsequent sections, we use $F_x^{i,j}$ as a shorthand notation for the the $x$-component $F_x \left(x_i, y_j\right)$ of a vector field $\mathbf{F}$ at position $\left(x_i, y_j\right)$, and similarly for the $y$ and $z$ components of $\mathbf{F}$. Instead of solving the three-dimensional problem (see figure \[fig:yee\_grid\]), we consider a two-dimensional representation of the Yee grid (see figure \[fig:yee\_grid\_planes\]).
This reduction to two dimensions is a consequence of the quasi-TEM assumption that we described in section \[sec:Formulation\]. We show the discretization of the electromagnetic potentials and corresponding parameters in figure \[fig:yee\_grid\_potentials\].
Non-uniform grid
----------------
One possibility to numerically solve \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\], is to consider equal grid spacings in a two-dimensional Yee grid (see figure \[fig:yee\_grid\_planes\]). For perfectly coaxial geometries, such an approach is suitable to resolve the electromagnetic fields since they vanish outside the transmission line. However, for arbitrary geometries, it is important to resolve far field contributions of the magnetic field to correctly compute the inductance according to \[eq:magneticenergy,eq:L\]. We therefore employ a non-uniform Yee discretization of the computational domain $\Omega\subseteq \mathbb{R}^2$ (see figure \[fig:tikz\_x\_line\]) and denote the difference between grid points $x^{i}$ ($y^{i}$) and $x^{i+1}$ ($y^{i+1}$) by $h_x^{i}$ ($h_y^{i}$).
We first focus on the non-uniform Yee discretization of all derivatives occurring in \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\] and then present the fully discretized version of the considered PDEs. We only describe the discretization along the $x$-axis bearing in mind that the same steps also apply in $y$-direction. For the discretization of $\partial_x \Phi$, we consider central differences at midpoints and thus compute the derivative between $x^i$ and $x^{i+1}$ at position $x^{i+\frac{1}{2}}$ (see figure \[fig:tikz\_x\_line\]). We obtain
$$\left( \partial_x \Phi \right)^{i+\frac{1}{2},j} = \frac{\Phi^{i+1,j} - \Phi^{i,j}}{h_x^i}\,.$$
In the next step, we discretize $\hat{\sigma} A_x$ and consider $\hat{\sigma}^{i,j}$ to be located at the same position as $\Phi^{i,j}$ and $J_z^{i,j}$ (see figure \[fig:yee\_grid\_potentials\]). We have to determine $\hat{\sigma}^{i+\frac{1}{2},j}$ since the vector field component $A_x^{i+\frac{1}{2}}$ is located at $x^{i+\frac{1}{2}}$. To do so, we compute the harmonic mean $$\hat{\sigma}^{i+\frac{1}{2},j} = 2 \left( \frac{1}{\hat{\sigma}^{i,j}} + \frac{1}{\hat{\sigma}^{i+1,j}} \right)^{-1}$$ and obtain $$\hat{\sigma}^{1+\frac{1}{2},j} A_x^{i+\frac{1}{2},j} = 2 \left( \frac{1}{\hat{\sigma}^{i,j}} + \frac{1}{\hat{\sigma}^{i+1,j}} \right)^{-1} A_x^{i+\frac{1}{2},j}$$ and $$\left[\partial_x \left( \hat{\sigma} A_x \right) \right]^{i,j} = \frac{\hat{\sigma}^{i+\frac{1}{2},j} A_x^{i+\frac{1}{2},j}- \hat{\sigma}^{i-\frac{1}{2},j} A_x^{i-\frac{1}{2},j}}{h_x^{i-\frac{1}{2}}}\,.$$ For computing the second derivatives of the electromagnetic potentials, we consider the Taylor expansions $$f^{i-1,j} = f^{i,j} - h_x^{i-1} \partial_x f^{i,j} + \frac{1}{2!}\left(h_x^{i-1}\right)^2 \partial^2_x f^{i,j} - \mathcal{O}\left(\left(h_x^{i-1}\right)^3\right)$$ and $$f^{i+1,j} = f^{i,j} + h_x^{i} \partial_x f^{i,j} + \frac{1}{2!}\left(h_x^{i}\right)^2 \partial^2_x f^{i,j}+ \mathcal{O}\left(\left(h_x^{i}\right)^3\right)$$ of a function $f$. We eliminate $\partial_x f^{i,j}$ to obtain $$\begin{aligned}
\begin{split}
\left(\partial^2_x f \right)^{i,j} &= \frac{2 f^{i-1,j}}{h_x^{i-1} \left(h_x^{i-1}+h_x^{i}\right)} - \frac{2 f^{i,j}}{h_x^{i-1} h_x^{i}} \\
&+ \frac{2 f^{i+1,j}}{h_x^{i} \left(h_x^{i-1}+h_x^{i}\right)} + \mathcal{O}\left(h_x^{i-1}-h_x^i\right)
\end{split}\end{aligned}$$ and similarly for a vector potential component $$\begin{split}
\left(\partial^2_x A_y \right)^{i,j+\frac{1}{2}} &= \frac{2 A_y^{i-1,j+\frac{1}{2}}}{h_x^{i-1} \left(h_x^{i-1}+h_x^{i}\right)} - \frac{2 A_y^{i,j+\frac{1}{2}}}{h_x^{i-1} h_x^{i}} \\
&+ \frac{2 A_y^{i+1,j+\frac{1}{2}}}{h_x^{i} \left(h_x^{i-1}+h_x^{i}\right)} + \mathcal{O}\left(h_x^{i-1}-h_x^i\right)\,.
\end{split}
\label{eq:secondDerivative}$$ Since the vector potentials are located between grid points, we replace $h_x^{i,j}$ by $h_x^{i+\frac{1}{2},j} = \left(h_x^{i,j}+h_x^{i+1,j}\right)/2$ and, for the second derivative of the vector potential, obtain $$\begin{aligned}
\begin{split}
\left(\partial^2_x A_x \right)^{i+\frac{1}{2},j} &= \frac{2 A_x^{i-\frac{1}{2},j}}{h_x^{i-\frac{1}{2}} \left(h_x^{i-\frac{1}{2}}+h_x^{i+\frac{1}{2}}\right)} - \frac{2 A_x^{i+\frac{1}{2},j}}{h_x^{i-\frac{1}{2}} h_x^{i+\frac{1}{2}}} \\
&+ \frac{2 A_x^{i+\frac{3}{2},j}}{h_x^{i+\frac{1}{2}} \left(h_x^{i-\frac{1}{2}}+h_x^{i+\frac{1}{2}}\right)}\,.
\end{split}\end{aligned}$$ Furthermore, we compute the term $\partial_x{\left(\hat{\sigma} \partial_x{\Phi} \right)}$ in \[eq:A\_x3\] by considering central differences at midpoints: $$\begin{aligned}
\begin{split}
\partial_x{\left(\hat{\sigma} \partial_x{\Phi} \right)} &= \frac{\hat{\sigma}^{i-\frac{1}{2},j} \left(\Phi^{i-1,j} - \Phi^{i,j}\right)}{h_x^{i-\frac{1}{2}} h_x^{i-1}} \\
&- \frac{\hat{\sigma}^{i+\frac{1}{2},j} \left(\Phi^{i,j}- \Phi^{i+1,j}\right)}{h_x^{i-\frac{1}{2}}h_x^{i}}\,.
\end{split}\end{aligned}$$ Finally, we can formulate \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\] in terms of a staggered finite-difference Yee discretization on a non-uniform grid: $$\begin{aligned}
\begin{split}
&\frac{2 A_x^{i-\frac{1}{2},j}}{h_x^{i-\frac{1}{2}} \left(h_x^{i-\frac{1}{2}}+h_x^{i+\frac{1}{2}}\right)} - \frac{2 A_x^{i+\frac{1}{2},j}}{h_x^{i-\frac{1}{2}} h_x^{i+\frac{1}{2}}} \\
&+ \frac{2 A_x^{i+\frac{3}{2},j}}{h_x^{i+\frac{1}{2}} \left(h_x^{i-\frac{1}{2}}+h_x^{i+\frac{1}{2}}\right)} + \frac{2 A_x^{i+\frac{1}{2},j-1}}{h_y^{j-1} \left(h_y^{j-1}+h_y^{j}\right)} \\
&- \frac{2 A_x^{i+\frac{1}{2},j}}{h_y^{j-1} h_y^{j}} + \frac{2 A_x^{i+\frac{1}{2},j+1}}{h_y^{j} \left(h_y^{j-1}+h_y^{j}\right)} \\
& + i \omega \mu \hat{\sigma}^{i+\frac{1}{2},j} \left( A_{x}^{i+\frac{1}{2},j} + \frac{\Phi^{i,j} + \Phi^{i+1,j}}{h_x^i}\right) = -i \omega \mu J_{s,x}^{i+\frac{1}{2},j},
\label{eq:discr_1}
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
&\frac{2 A_y^{i-1,j+\frac{1}{2}}}{h_x^{i-1} \left(h_x^{i-1}+h_x^{i}\right)} - \frac{2 A_y^{i,j+\frac{1}{2}}}{h_x^{i-1} h_x^{i}} \\
&+ \frac{2 A_y^{i+1,j+\frac{1}{2}}}{h_x^{i} \left(h_x^{i-1}+h_x^{i}\right)}+ \frac{2 A_y^{i,j-\frac{1}{2}}}{h_y^{j-\frac{1}{2}} \left(h_y^{j-\frac{1}{2}}+h_y^{j+\frac{1}{2}}\right)} \\
&- \frac{2 A_y^{i,j+\frac{1}{2}}}{h_y^{j-\frac{1}{2}} h_y^{j+\frac{1}{2}}} + \frac{2 A_y^{i,j+\frac{3}{2}}}{h_y^{j+\frac{1}{2}} \left(h_y^{j-\frac{1}{2}}+h_y^{j+\frac{1}{2}}\right)} \\
&+ i \omega \mu \hat{\sigma}^{i,j+\frac{1}{2}} \left( A_{y}^{i,j+\frac{1}{2}} + \frac{\Phi^{i,j} + \Phi^{i,j+1}}{h_y^j}\right) = -i \omega \mu J_{s,y}^{i,j+\frac{1}{2}},
\label{eq:discr_2}
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
&\frac{2 A_z^{i-1,j}}{h_x^{i-1} \left(h_x^{i-1}+h_x^{i}\right)} - \frac{2 A_z^{i,j}}{h_x^{i-1} h_x^{i}} + \frac{2 A_z^{i+1,j}}{h_x^{i} \left(h_x^{i-1}+h_x^{i}\right)} \\
&+ \frac{2 A_z^{i,j-1}}{h_y^{j-1} \left(h_y^{j-1}+h_y^{j}\right)} - \frac{2 A_z^{i,j}}{h_y^{j-1} h_y^{j}} + \frac{2 A_z^{i,j+1}}{h_y^{j} \left(h_y^{j-1}+h_y^{j}\right)} \\
&+ i \omega \mu \hat{\sigma}^{i,j} A_{z}^{i,j} = -i \omega \mu J_{s,z}^{i,j}\,,
\label{eq:discr_3}
\end{split}\end{aligned}$$ and $$\begin{aligned}
\begin{split}
&\frac{\hat{\sigma}^{i+\frac{1}{2},j} A_x^{i+\frac{1}{2},j}- \hat{\sigma}^{i-\frac{1}{2},j} A_x^{i-\frac{1}{2},j}}{h_x^{i-\frac{1}{2}}} + \frac{\hat{\sigma}^{i,j+\frac{1}{2}} A_y^{i,j+\frac{1}{2}}- \hat{\sigma}^{i,j-\frac{1}{2}} A_y^{i,j-\frac{1}{2}}}{h_y^{j-\frac{1}{2}}} \\
&+\frac{\hat{\sigma}^{i+\frac{1}{2},j} \left(\Phi^{i+1,j} - \Phi^{i,j}\right)}{h_x^{i-\frac{1}{2}}h_x^{i}} -
\frac{\hat{\sigma}^{i-\frac{1}{2},j} \left(\Phi^{i,j} - \Phi^{i-1,j}\right)}{h_x^{i-\frac{1}{2}} h_x^{i-1}} \\
&+ \frac{\hat{\sigma}^{i,j+\frac{1}{2}} \left(\Phi^{i,j+1} - \Phi^{i,j}\right)}{h_y^{j-\frac{1}{2}}h_y^{j}} -
\frac{\hat{\sigma}^{i,j-\frac{1}{2}} \left(\Phi^{i,j} - \Phi^{i,j-1}\right)}{h_y^{j-\frac{1}{2}} h_y^{j-1}} \\
& = \frac{J_{s,x}^{i-\frac{1}{2},j} - J_{s,x}^{i+\frac{1}{2},j}}{h_x^{i-\frac{1}{2}}} + \frac{J_{s,y}^{i,j-\frac{1}{2}} - J_{s,y}^{i,j+\frac{1}{2}}}{h_y^{j-\frac{1}{2}}}\,.
\label{eq:discr_4}
\end{split}\end{aligned}$$ Our approach extends [@OldenburgFast] by considering a non-uniform discretization of the potential formulation of Maxwell’s equations as defined by \[eq:A\_x1,eq:A\_x2,eq:A\_z,eq:A\_x3\]. To numerically solve \[eq:discr\_1,eq:discr\_2,eq:discr\_3,eq:discr\_4\], we rewrite the discretized PDEs in terms of a linear system of equations $$D \mathbf{A}'=\mathbf{J}'\,,
\label{eq:system}$$ where $\mathbf{J}'=- i \omega \mu\left(J_{s,x}, J_{s,y}, J_{s,z}, \nabla \mathbf{J_s}/(i \omega \mu)\right)$, $\mathbf{A}'=\left(A_x,A_y,A_z,\Phi\right)$, and
$$D=
\begin{pmatrix}
\Delta + i \omega \mu \hat{\sigma} & 0 & 0 & i \omega \mu \hat{\sigma} \partial_x \\
0 & \Delta + i \omega \mu \hat{\sigma} & 0 & i \omega \mu \hat{\sigma} \partial_y \\
0 & 0 & \Delta + i \omega \mu \hat{\sigma} & 0\\\partial_x\hat{\sigma} & \partial_y\hat{\sigma} & 0 & \nabla \hat{\sigma} \nabla
\end{pmatrix}\,.$$
We solve \[eq:system\] with a sparse linear system solver and consider $J_{s,z}$ to be the only non-vanishing source current component.
Computing electromagnetic fields {#sec:electr_magn_fields}
--------------------------------
After having determined the electromagnetic potentials by solving \[eq:system\], we now compute the corresponding electromagnetic fields. The electric field $\mathbf{E}$ as defined by \[eq:decompose E\] is $$\mathbf{E}^{i,j} = \mathbf{A}^{i,j} + \begin{pmatrix}
\frac{\Phi^{i+1,j} - \Phi^{i,j}}{h_x^i} \\ \frac{\Phi^{i,j+1} - \Phi^{i,j}}{h_y^j} \\ 0
\end{pmatrix}\,.
\label{eq:E field}$$ Based on \[eq:max1\], the magnetic field is $$\mathbf{B} = \mu \mathbf{H} = \frac{1}{i \omega} \nabla\times\mathbf{E} = \frac{1}{i \omega}
\begin{pmatrix}
\partial_y E_z \\ - \partial_x E_z \\ \partial_x E_y - \partial_y E_x
\end{pmatrix}$$ and thus $$\mathbf{B}^{i,j} = \frac{1}{i \omega}
\begin{pmatrix}
\frac{E_z^{i,j+1} - E_z^{i,j}}{h_y^j} \\ - \frac{E_z^{i+1,j} - E_z^{i,j}}{h_x^i} \\ \frac{E_y^{i+1,j} - E_y^{i,j}}{h_x^i} - \frac{E_x^{i,j+1} - E_x^{i,j}}{h_y^j}
\end{pmatrix}\,.
\label{eq:B field}$$ We only have to compute the electric field for one frequency, because the capacitance is frequency-independent and the conductance scales linearly with $\omega$ according to \[eq:G\]. For all other frequencies, it is sufficient to numerically solve $$\left( \Delta + i \omega \mu \hat{\sigma} \right) A_z = - i \omega \mu J_{s,z}$$ to compute resistance and inductance as defined in \[eq:R,eq:L\]. The vector potential component $A_z$ fully determines the magnetic of a transmission-line segment field since
$$\begin{aligned}
B_x &= \frac{1}{i \omega} \partial_y E_z = \frac{1}{i \omega} \partial_y A_z\,, \\
B_y &= - \frac{1}{i \omega} \partial_y E_z = - \frac{1}{i \omega} \partial_x A_z\,.
\label{eq:Bfield}\end{aligned}$$
We note that the outlined numerical framework can be used to simulate the electromagnetic fields (see \[eq:E field,eq:B field\]) and transmission-line parameters (see \[eq:R,eq:L,eq:C,eq:G\]) of arbitrary arrangements of conductor, insulator, and shielding materials.
Results {#sec:Results}
=======
Based on the numerical framework that we introduced in sections \[sec:Formulation\] and \[sec:Discretization\], we now examine the effect of deformations on transmission-line parameters. We first study the convergence characteristics of the proposed framework by considering a single copper strand and comparing our numerical results with the corresponding analytical theory. After this initial test, we simulate the frequency-dependence of transmission-line parameters for undeformed coaxial geometries and draw a comparison to the deformed case. Finally, we compare our simulations of deformed transmission lines with corresponding experimental data.
Single copper strand
--------------------
To test the proposed numerical framework, we consider a single copper strand of radius $r_1$ and compare the numerically obtained values of the magnetic field with the ones predicted by analytic theory. For low frequencies, the current density within the wire is uniform and using Ampère’s law yields $$|B(r)| =
\begin{cases}
\frac{\mu I r}{2 \pi r^2_1}\,, & r\leq r_1\,, \\
\frac{\mu I}{2 \pi r}\,, & r > r_1\,, \\
\end{cases}
\label{eq:B strain low freq}$$ where $r$ is the distance to the center of the strand, $I$ is the current flowing through the strand, and $\mu = \mu_0$. For high frequencies, the current density is not uniformly distributed in the wire, but exhibits a higher density on its surface (*skin effect*). In this case, the electric field is [@Bessel] $$E_z \left(r\right)= \frac{k I_0\left(k r \right)}{2 \pi \sigma r_1 I_1\left(k r_1 \right)} I\,,$$ where $k = \sqrt{i \omega \mu \sigma}$, and $I_0$ and $I_1$ are the modified Bessel functions of first order and first and second kind, respectively. The current density is thus $$J_z \left(r\right) = \frac{k I_0\left(k r \right)}{2 \pi r_1 I_1\left(k r_1 \right)} I
\label{eq:bessel_J}$$ and by applying Ampère’s law to \[eq:bessel\_J\] we obtain $$|B(r)| =
\begin{cases}
\frac{\mu I_1\left(k r \right)}{2 \pi r_1 I_1\left(k r_1 \right)} I\,, & r \leq r_1\,, \\
\frac{\mu I}{2 \pi r}\,, & r > r_1\,. \\
\end{cases}
\label{eq:B strain}$$ We now compare our simulation results with the analytical predictions of \[eq:B strain\] for a single copper strand of radius $r_1=\SI{0.48}{\milli\metre}$ that is surrounded by air. We show the results in figures \[fig:single wire x line\] and \[fig:single wire x line high freq\], and find good agreement between our simulations and analytical theory. For a frequency of $f=\SI{1}{\mega\hertz}$, we see the influence of the skin effect on the magnetic field. To study the convergence characteristics of our method for different numbers of grid points along one axis $N$, we define the error $$\Delta\left(N\right) = \frac{1}{M} \sum_{(i,j) \in S}|B_{\text{simulation}}^{i,j}\left(N\right)-B_{\text{analytical}}^{i,j}\left(N\right)|\,,$$ where $S=\left\{(i,j)| r^{i,j} \leq r_c\right\}$ and $r^{i,j}=\sqrt{\left(x^i\right)^2+\left(y^j\right)^2}$. The cut-off radius is denoted by $r_c$, and $M$ is the number of considered points. We use the convention that the value of $r^{i,j}=0$ corresponds to the center of the cooper strand. In our subsequent analysis, we set $r_c= \SI{2}{\milli\metre}$. The analytical magnetic field values $B_{\text{analytical}}^{i,j}$ correspond to the ones of \[eq:B strain\]. According to \[eq:B field,eq:secondDerivative\], the global error is expected to be of order $\mathcal{O}\left(h\right)$, where $h$ is the distance between the nodes of the grid. We find that the numerically determined error of our method agrees well with the expected scaling (see figures \[fig:single wire error\] and \[fig:single wire error high freq\]).
Undeformed coaxial transmission line
------------------------------------
To examine the effect of deformations on transmission-line parameters, we first simulate an undeformed coaxial cable (see figure \[fig:cross section coax\]). In this way, we can later compare our results on deformed transmission lines with those obtained for an undeformed reference. We denote the radius of the inner conductor by $r_1$ and $r_2$ is the inner radius of the outer conductor of thickness $t$. Therefore, the outer radius of the outer conductor is $r_3=r_2+t$. Similarly to the magnetic field of a single copper strand (see \[eq:B strain\]), we find for an undeformed coaxial transmission line $$|B(r)| =
\begin{cases}
\frac{\mu I_1\left(k r \right)}{2 \pi r_1 I_1\left(k r_1 \right)} I\,, & r \leq r_1\,, \\
\frac{\mu I}{2 \pi r}\,, & r_1\leq r \leq r_2\,, \\
\frac{\mu I_1\left(k \left(r_3-r\right) \right)}{2 \pi r_2 I_1\left(k \left(r_3-r_2\right) \right)} I\,, & r_2\leq r \leq r_3\,, \\
0 \,,& r >r_3\,. \\
\end{cases}
\label{eq:B coax}$$ The expressions for $r\leq r_2$ and $r\geq r_3$ are exact. The magnetic field vanishes for $r\geq r_3$, because the currents in the inner and outer conductors are oriented in the opposite directions. For the magnetic field in the outer conductor, we use an approximation that assumes that the current density distribution is the same as for a wire of radius $t$. In figures \[fig:coax wire x line\] and \[fig:coax x line high freq\], we see that the simulated magnetic field values agree well with the predictions of \[eq:B coax\]. For large enough frequencies, we recognize the influence of the skin effect (see figure \[fig:coax x line high freq\]). As for the single copper strand, we examine the convergence properties of our method for the coaxial geometry (see figures \[fig:coax wire error\] and \[fig:coax error high freq\]). Due to the skin effect, a larger number of grid cells is required for higher frequencies than for lower ones to resolve the electromagnetic fields.
Next, we determine the transmission-line parameters of the undeformed coaxial cable according to \[eq:R,eq:L,eq:C,eq:G\], and compare the obtained values with the analytical low and high frequency approximations. The analytical expressions for capacitance and conductance per unit length are [@paul2008analysis] $$\begin{aligned}
C &= \frac{2 \pi \epsilon_0 \epsilon^\prime_{\text{r}}}{\ln\left( \frac{r_2}{r_1}\right)}\,,\label{eq:C_analyt} \\
G &= \omega \tan\left(\delta\right) C \label{eq:G_analyt}\,.\end{aligned}$$ We set $\epsilon^\prime_\text{r}=2.25$ and $\tan\left(\delta\right)=10^{-3}$. According to \[eq:C\_analyt,eq:G\_analyt\], the capacitance exhibits no frequency dependence and the conductance scales linearly with $\omega$. This behavior is captured by our simulations (see figure \[fig:coax2\]). Instead of setting $\tan\left(\delta\right)$ equal to a constant, it is also possible to use empirically determined frequency dependencies of $\tan\left(\delta\right)$.
The current densities in the conductors are frequency-dependent. For low frequencies, the current density is uniformly distributed within the conductor (DC approximation), and for high frequencies the current density is concentrated at the conductor surface (HF approximation). The skin effect is approximated by describing the conductors as hollow cylinders with their thickness being determined by the corresponding skin depth. In the low frequency regime, the resistance and conductance per unit length are [@Hayt] $$\begin{aligned}
R &= \frac{1}{\sigma \pi}\left( \frac{1}{r_1^2} + \frac{1}{r_3^2-r_2^2} \right)\,, \\
L &= \frac{\mu}{2 \pi} \left[ \ln\left( \frac{r_2}{r_1}\right) + \frac{1}{4} \right. \nonumber \\
&\left.+\frac{1}{4 \left( r_3^2-r_2^2 \right)} \left(r_2^2-3 r_3^2+\frac{4 r_3^2}{r_3^2-r_2^2} \ln\left( \frac{r_3}{r_2} \right)\right) \right] \label{eq:L_analytic}\end{aligned}$$ and for high frequencies $$\begin{aligned}
R &= \sqrt{\frac{\mu \omega}{2 \sigma}} \frac{1}{2 \pi} \left( \frac{1}{r_1} + \frac{1}{r_2} \right)\,, \\
L &= \frac{\mu}{2 \pi} \ln\left( \frac{r_2}{r_1}\right) + \frac{\mu}{4 \pi} \frac{2}{\mu \omega \sigma} \left( \frac{1}{r_1} + \frac{1}{r_2} \right)\,.\end{aligned}$$ We show the analytical approximations of $R$ and $L$ and the corresponding simulation results in figure \[fig:coax2\]. We find that the simulated transmission-line parameters are in good agreement with the analytic DC and HF approximations. In addition, our numerical results also describe the analytically inaccessible transition region between the DC and HF regime.
For large frequencies, deviations of the simulated resistance values from analytic theory occur. The reason is that very small grid spacings are necessary to resolve the current density and compute the resistance per unit length according to \[eq:jouleheating\]. Another more computationally efficient possibility is to extrapolate the resistance according to the scaling of $R\left(\omega\right)\sim \sqrt{\omega}$ for $\omega \rightarrow \infty$. If one is only interested in the impedance, it is also possible to neglect resistance-related deviations since $$Z_0\left(\omega\right) = \sqrt{\frac{L}{C}}\quad \text{for} \quad \omega \rightarrow \infty\,.
\label{eq:HF_Z0}$$
Deformed coaxial transmission line
----------------------------------
To study the influence of deformations on transmission-line parameters, we now compress the coaxial transmission line between two rigid parallel plates (see figure \[fig:cross section coax\]). The circumference is invariant under the applied deformation, so $$2 \pi r_2 = 2 \pi r_4 + 4 h\,,$$ where $r_2$ is the inner radius of the outer conductor of the undeformed cable, $r_4$ is the radius of the upper and lower half circles of the deformed cable, and $h$ is the distance of the centers of the half circles from the center of the inner conductor (see figure \[fig:cross section deformed coax\]).
We consider a deformation with $r_4=\SI{0.8}{\milli\metre}$ and $h=\SI{1.02}{\milli\metre}$, and compare the transmission-line parameters of this deformed coaxial cross-section with those of the undeformed reference. We show the corresponding results in figure \[fig:coax deformed\]. We find that resistance and conductance are almost unaffected by the deformation. However, capacitance and inductance differ significantly when compared to the undeformed reference. In the high frequency regime, the impedance of the undeformed coaxial cable is about 50 % larger than the impedance of the deformed one. According to \[eq:reflection\_coeff\], this corresponds to a reflection coefficient of $\Gamma=-0.2$ at the interface between the undeformed and deformed segments. We outline in section \[sec:experiment\], that such reflections are detectable with a time-domain reflectometer (TDR). In addition to variations in the cross section, it is also possible to incorporate spatial permittivity variations that result from applied mechanical strains.
Comparison with experimental data {#sec:experiment}
---------------------------------
Our simulations show that the impedance of a deformed coaxial transmission line is significantly smaller as compared to the undeformed case (see figure \[fig:coax deformed\]). To test our numerically obtained transmission-line parameters and impedance values, we experimentally analyze the effect of deformations on the TDR profile of a deformation-sensitive coaxial cable. TDR measurements are based on the injection of a test signal (*e.g.*, a step function) into a transmission line. Reflections of the injected signal occur at impedance discontinuities. Each reflection in the transmission line leads to a wave which travels back to the TDR where the resulting voltage differences and corresponding time stamps are monitored. These monitored quantities contain information about the spatial distribution of reflection coefficients and impedances along the transmission line.
$d \left[\SI{}{\milli\metre}\right]$ $r_4 \left[\SI{}{\milli\metre}\right]$ $h \left[\SI{}{\milli\metre}\right]$
-------------------------------------- ---------------------------------------- --------------------------------------
2.48(3) 0.95(1) 0.79(1)
2.22(3) 0.85(1) 0.93(1)
1.94(3) 0.75(1) 1.10(1)
: For each measurement and simulation in figures \[fig:L45 TDR\] and \[fig:L45 TDR zoom\], we summarize the corresponding cable geometry parameters. Parameters $r_4$ and $h$ determine the simulated deformation according to figure \[fig:cross section deformed coax\] and $d$ is the distance between the two parallel plates of the used vise.[]{data-label="tab:geometry"}
For our experiments, we use a Campbell Scientific TDR100. The considered coaxial transmission line has a length of $\SI{9.70(1)}{\metre}$ and its dielectric material is a thermoplastic elastomer with a relative permittivity of $\epsilon_{\text{r},\text{dielectric}}^{\prime} = 2.14$. The number in parentheses is the uncertainty in the last digit. With a vise of length $\SI{6.0(1)}{\centi\metre}$, we apply a deformation $\SI{6.98(1)}{\metre}$ away from the TDR. We denote the distance between the two parallel plates of the vise by $d$. We show an illustration of the deformed transmission lines that we considered in our simulations and experiments in figures \[fig:L45 simulation\] and \[fig:L45 microscope\]. For the considered transmission line, we did not observe any traces of plastic-deformation effects in the TDR curves. However, depending on the used dielectric and cable jacket materials, plastic deformations may occur and affect measurements. Under the quasi-TEM assumption, the transmission line is composed of undeformed and deformed segments. We have to compute the frequency dependence of the transmission-line parameters for both cross sections only once to simulate the corresponding TDR profile according to the telegraphers equations as defined in \[eq:telegraphers1,eq:telegraphers2\]. To solve \[eq:telegraphers1,eq:telegraphers2\], we employ a method as described in [@d2017transmission]. We again note that we only consider geometric effects in our simulations, and that we do not account for strain-induced spatial variations of the permittivity within the dielectric material. To test the validity of our simulation approach, we compare the simulated TDR profiles with experimentally obtained ones. We summarize the parameters that we use in our simulations in table \[tab:geometry\]. In figures \[fig:L45 TDR\] and \[fig:L45 TDR zoom\], we show the comparison between our simulations and corresponding experiments. These results clearly show that our framework is able to reproduce the observed TDR profiles. Determining the dependence of the voltage minimum on the plate distance $d$ in figure \[fig:L45 TDR zoom\] requires to numerically solve \[eq:telegraphers1,eq:telegraphers2\] for a given set of transmission-line parameters/impedances. A closed-form analytical expression can only be obtained for the impedance of concentrically-deformed transmission lines (*i.e.*, coaxial lines of different diameter) [@LiThAb]. In the case of concentric deformations, we can obtain the characteristic impedance from \[eq:C\_analyt,eq:G\_analyt,eq:L\_analytic,eq:HF\_Z0\]: $$Z_0(\omega)=\frac{1}{2 \pi} \sqrt{\frac{\mu_0 \mu_r}{\epsilon_0 \epsilon_r}} \ln\left(\frac{r_2}{r_1}\right)\quad \text{for} \quad \omega \rightarrow \infty\,.$$ To describe the effect of general deformations on $Z$, it is necessary to use a numerical framework such as the one we propose in this work. In particular, we find that the simulations are able to capture the functional dependence of voltage minimum and plate distance $d$ in the deformed regions. The experimentally observed TDR profiles are therefore consistent with our numerical framework for the simulation of arbitrarily-shaped transmission-line cross sections.
Conclusion {#sec:Discussion}
==========
We have introduced a numerical framework to simulate arbitrarily-shaped transmission lines over a wide frequency range. Our method is based on solving a sparse system of linear equations and thus is suitable for fast computations. The numerically computed transmission-line parameters and electromagnetic field values agree well with corresponding analytical predictions. We considered the effect of geometric deformations, but our framework can be extended to also account for strain-induced permittivity variations. We tested the proposed framework by comparing simulation results with corresponding experimental data and found good agreement between the model predictions and the experimentally observed behavior.
Our study may improve continuous sensing applications for the measurement of deformations and mechanical strains based on strain-sensitive transmission lines. Instead of mapping observed impedance values to concentric deformations and thus neglecting any further structural information as suggested in [@LiThAb], our method also captures changes in the cross-section geometry. Furthermore, our framework is also useful to determine analytically inaccessible transmission-line parameters and impedances for complex arrangements of conductor, insulator, and shielding materials.
A possible extension of our work is to combine our framework with an inversion algorithm and determine estimates of the most likely deformations and mechanical strains given a certain TDR observation. This can contribute to improvements in corrosion detection [@liu2002corrosion] and monitoring fracture propagation in concrete structures [@LiThAb; @chen2003continuous]. When embedding transmission-line sensors in concrete structures, it is important to examine strain transfer characteristics at the interface between cable jacket and surrounding material. Similar to fibre-optic sensors [@santos2014handbook], the mechanical and physicochemical properties of the materials at the interface between sensor line and surrounding material (*e.g.*, cable jacket material) may substantially influence the sensor performance. Recent advances in strain-transfer theory [@wang2018strain] may provide a possibility to correctly interpret measurement data, even in the presence of strain-transfer loss at the interface.
To enhance the long-term sensing performance, it is important to limit material degradation by using materials with physicochemical properties that are suitable for a given application [@santos2014handbook]. For instance, photochemical and oxidative degradation (*i.e.*, chemical aging) can be limited by lowering the influence of air and light on the sensor. In addition, one should also consider the effect of the change of physical properties over time (*i.e.*, physical aging) that may occur in the used polymer blends [@hodge1995physical]. If transmission-line sensors are tailored to a specific application, previous studies suggest that they can stay in service for decades [@LiThAb]. Depending on the used dielectric material, it may also be important to account for the influence of temperature variations on the measurement results if the permittivity temperature coefficient is large enough. The dielectric material of the transmission line we consider has a temperature coefficient of about $10^{-4}~\text{K}^{-1}$ [@harrop1969temperature], so the effect of temperature variations on the impedance is negligible. In addition to permittivity changes, temperature variations may also affect the coating material and strain transfer characteristics [@wang2019strain].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Dani Or and Daniel Breitenstein for the possibility to use their Campbell Scientific TDR100 time-domain reflectometer and Joshua LeClair for helpful comments and discussions. We also thank the LEONI AG for providing a sample of a pressure-sensitive coaxial transmission line.
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abstract: 'The problem of a random walk on a finite triangular lattice with a single interior source point and zig-zag absorbing boundaries is solved exactly. This problem has been previously considered intractable.'
address:
- ' Department of Theoretical Physics, Research School of Physical Sciences and Engineering, and Centre for Mathematics and its Applications, School of Mathematical Sciences, The Australian National University, Canberra ACT 0200, Australia'
- ' Department of Applied Mathematics, School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia'
author:
- M T Batchelor and B I Henry
title: Exact solution for random walks on the triangular lattice with absorbing boundaries
---
\#1
epsf.sty
Introduction
============
The problem of a random walk on a two-dimensional lattice with a single interior source point and absorbing boundaries was first considered by Courant *et al.* [@C28] in 1928 where general properties of the solution were discussed. This problem was solved exactly in 1940 [@MW40] for the case of random walks on a square lattice with rectangular absorbing boundaries. The problem on a triangular lattice with finite absorbing boundaries was considered in 1963 [@KM63] where the exact solution was given for an approximation of the problem using straight boundaries rather than the true zig-zag boundaries of the triangular lattice. Indeed the authors of this paper remarked that “An explicit solution of the difference equation can hardly be obtained if these boundary conditions are used.” Other variants of the problem on the square lattice have been solved exactly [@KM63; @M94] however the problem on the triangular lattice with true zig-zag triangular lattice boundaries has remained unsolved. In this paper we give the exact solution for this problem.
The problem of random walks on finite lattices with absorbing boundaries is fundamental to the theory of stochastic processes [@ML79] and has numerous applications. These include potential theory [@D53], electrical networks [@DS84], surface diffusion [@BCMO99] and Diffusion-Limited Aggregation (DLA) [@WS81]. For example, the exact results for the square lattice problem [@MW40] have been used to expedite the growth of large DLA clusters on the square lattice [@BHR95] and the formulae derived in this paper could similarly be used on the triangular lattice.
Field equations, boundary conditions and absorption probabilities
=================================================================
A schematic illustration showing random walk pathways (dashed lines) on an equi- angular triangular lattice is shown in Fig. 1. A natural co-ordinate system for the triangular lattice is to label the lattice vertices by the intersection points $(p,q)$ of horizontal straight lines $p=0,1,2,\ldots m+1$ and slanted straight lines $q=0,1,2,\ldots n+1$ – parallel to the $p$ axis shown in Fig. 1. In this co-ordinate system the expectation that a random walk on the lattice visits a site $(p,q)$, before exiting at a finite boundary, is given by the non-separable difference equation $$\begin{aligned}
F(p,q)&=&\frac{1}{6}\left[F(p,q-1)+F(p,q+1)+F(p-1,q)\right.\\
& &\quad \left. +F(p+1,q)+F(p+1,q-1) +F(p-1,q+1)\right].\end{aligned}$$ In this paper we adopt a different co-ordinate system, which may at first appear less natural but it has the advantage that the governing equations are separable.
In the co-ordinate system used here we label the vertices of the triangular lattice by the intersection points $(p,q)$ of horizontal straight lines $p=0,1,2,\ldots m+1$ and vertical zig-zag lines $q=0,1,2,\ldots n+1$ (see Fig. 2). In the absence of any source terms the expectation that a random walker visits a lattice site $(p,q)$ is given by the coupled homogeneous difference equations $$\begin{aligned}
F(p,q)&=&\frac{1}{6}\left[\hat F(p-1,q-1)+F(p,q-1)+\hat F(p+1,q-1)\right.\nonumber\\
& &\quad \left. +\hat F(p+1,q) +F(p,q+1)+\hat F(p-1,q)\right],\label{Feq}\\
\hat F(p,q)&=&\frac{1}{6}\left[ F(p-1,q)+\hat F(p,q-1)+F(p+1,q)\right.\nonumber\\
& &\quad \left. + F(p+1,q+1) +\hat F(p,q+1)+ F(p-1,q+1)\right],\label{Fhateq}\end{aligned}$$ where $F(p,q)$ denotes the field value at an even $p$ co-ordinate and $\hat F(p,q)$ denotes the field value at an odd $p$ co-ordinate. This separation of even and odd field equations, which is not necessary for the problem on the square lattice, is central to our solution below.
Now consider a source term at $(a,b)$ and absorbing boundaries at $p=0,m+1$ and $q=0,n+1$. In Section 3 we give the solution for the case where $m>1$ is odd. The same methods can be used with slightly altered equations to obtain the solution for $m$ even. The special case of the strip with $m=1$ has been detailed elsewhere [@BH02]. Following McCrea and Whipple [@MW40] we construct separate solutions, $F_I(p,q), \hat F_I(p,q)$ for $q\le b$ and $F_{II}(p,q), \hat F_{II}(p,q)$ for $q\ge b$. The absorbing boundary conditions are thus given by $$\begin{aligned}
F(0,q)&=&0,\\
F(m+1,q)&=&0,\\
F_I(p,0)&=&0,\\
\hat F_I(p,0)&=&0,\\
F_{II}(p,n+1)&=&0,\\
\hat F_{II}(p,n+1)&=&0.\end{aligned}$$ The ommision of a subscript $I$ or $II$ in the above indicates that the same equations are satisfied by both $F_I$ and $F_{II}$.
At $q=b$ we have the matching conditions $$\begin{aligned}
F_I(p,b)&=&F_{II}(p,b),\\
\hat F_I(p,b)&=&\hat F_{II}(p,b).\end{aligned}$$ Finally, with the inclusion of the source term, we have the coupled inhomogeneous field equations at $q=b$, $$\begin{aligned}
6F_I(p,b)&=&6\delta_{p,a}((a+1) \mbox{mod} 2) \nonumber\\
& &+\left[\hat F_I(p-1,b-1)+F_I(p,b-1)+\hat F_I(p+1,b-1)\right.\nonumber\\
& &\left. \quad + \hat F_I(p+1,b)+F_{II}(p,b+1)+\hat F_I(p-1,b)\right],\\
6\hat F_I(p,b)&=&6\delta_{p,a}(a \mbox{mod} 2)\nonumber\\
& &+\left[F_I(p-1,b)+\hat F_I(p,b-1)+F_I(p+1,b)\right.\nonumber\\
& &\left. \quad +F_{II}(p+1,b+1) +\hat F_{II}(p,b+1)+F_{II}(p-1,b+1)\right].\end{aligned}$$ In Section III we present the solutions to Eqs. (1)–(12).
The probabilities for the random walking particle to be absorbed at any specified point on one of the four boundaries, $p=0, p=m+1, q=0, q=n+1$, are readily obtained by averaging over the nearest neighbour expectation values subject to the boundary conditions, Eqs. (3)–(8). For example, $$\begin{aligned}
P(2k-1,0)&=&\frac{1}{6}\left[\hat F_I(2k-1,1)+F_I(2k-2,1)
+F_I(2k,1)\right];\nonumber\\
& &\qquad\quad k=1,\ldots \frac{m+1}{2}.\end{aligned}$$
Solution
========
Homogeneous equations – Separation of variables
-----------------------------------------------
We begin by solving the homogenous equations, Eqs. (1),(2), subject to the absorbing boundary conditions, Eqs. (3),(4). The homogenous field equations separate, on using $$\begin{aligned}
F(p,q)&=&P(p)Q(q),\label{sepFeq}\\
\hat F(p,q)&=&\hat P(p)\hat Q(q)\label{sepFhateq},\end{aligned}$$ into $$\begin{aligned}
\frac{6Q(q)-Q(q+1)-Q(q-1)}{\hat Q(q-1)+\hat Q(q)}&=&
\frac{\hat P(p-1)+\hat P(p+1)}{P(p)}=\lambda,\\
& &\nonumber\\
\frac{6\hat Q(q)-\hat Q(q+1)-\hat Q(q-1)}{Q(q+1)+ Q(q)}&=&
\frac{P(p-1)+ P(p+1)}{\hat P(p)} =\kappa,\end{aligned}$$ where $\kappa$ and $\lambda$ are separation constants. The two coupled equations for $P$ and $\hat P$ can be readily decoupled into separate equations $$\begin{aligned}
P(2k+2)+(2-\lambda\kappa)P(2k)+P(2k-2)=0,&&\\
\hat P(2k+3)+(2-\lambda\kappa)\hat P(2k+1)+\hat P(2k-1)=0,&&\end{aligned}$$ where $k$ is an integer. This second order linear difference equation for $P, (\hat P)$ on a lattice of even (odd) integers has the solution $$P(2k)=A\mu^k+B\mu^{-k},$$ where $$\mu=\frac{-(2-\lambda\kappa)+\sqrt{(2-\lambda\kappa)^2-4}}{2},$$ and $A$ and $B$ are arbitrary. It follows that the solution of $P(p)$ (with $p$ even) corresponding to the boundary conditions, Eqs. (3),(4), is $$P(p)=c\sin(\frac{\pi j p}{m+1})\label{Psol}$$ with $$\lambda\kappa=4\cos^2(\frac{\pi j}{m+1}).$$ The solution for $\hat P(p)$ (with $p$ odd) now follows from Eq. (17) as $$\hat P(p)=\frac{2c}{\kappa}\cos(\frac{\pi j}{m+1})\sin(\frac{\pi j p}{m+1}).
\label{Phatsol}$$
Equations (16) and (17) can also be decoupled for $Q(q)$ and $\hat Q(q)$ resulting in the same fourth order linear difference equation in each case; $$\begin{aligned}
Q(q+4)-(12+\lambda\kappa)Q(q+3)+(38-2\lambda\kappa)Q(q+2)&&\nonumber\\
\qquad\qquad -(12+\lambda\kappa)Q(q+1)+Q(q)=0.&&\end{aligned}$$ The zeroes of the corresponding characteristic quartic polynomial are in reciprocal pairs leading to the solutions $$\begin{aligned}
Q(q)&=&c_1{\mathrm{e}}^{\alpha q}+c_2{\mathrm{e}}^{-\alpha q}+c_3{\mathrm{e}}^{\beta q}+c_4{\mathrm{e}}^{-\beta q},
\label{Qeq}\\
\hat Q(q)&=&\hat c_1{\mathrm{e}}^{\alpha q}+\hat c_2{\mathrm{e}}^{-\alpha q}+\hat c_3{\mathrm{e}}^{\beta q}
+\hat c_4{\mathrm{e}}^{-\beta q},\label{Qhateq}\end{aligned}$$ where $$\begin{aligned}
\cosh\alpha&=&3+\frac{\lambda\kappa}{4}+\frac{1}{4}\sqrt{(\lambda\kappa)^2
+32\lambda\kappa},\\
\cosh\beta&=&3+\frac{\lambda\kappa}{4}-\frac{1}{4}\sqrt{(\lambda\kappa)^2
+32\lambda\kappa}.\end{aligned}$$ The coefficients $c_1, c_2, c_3, c_4$ may be chosen arbitrarily with the coefficients $\hat c_1, \hat c_2, \hat c_3, \hat c_4$ then determined by substituting Eqs. (\[Qeq\]), (\[Qhateq\]) into Eq. (16) or Eq. (17). From Eq. (17) we obtain $$\hat c_1=c_1 f(\alpha),\, \hat c_2=c_2 f(-\alpha),\, \hat c_3=c_3 f(\beta),\,
\hat c_4=c_4 f(-\beta)$$ where $$f(\omega)=\frac{\kappa ({\mathrm{e}}^{\omega}+1)}{2(3-\cosh\omega)}.$$
By combining Eqs. (14),(15),(20),(22),(24),(25),(28),(29) we can write the solutions to the field equations, Eqs. (\[Feq\]), (\[Fhateq\]), that satisfy the boundary conditions, Eqs. (3),(4), in the form $$\begin{aligned}
& &F(p,q;A,B,C,D)=\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)\nonumber\\
& &\quad\times
\left[A_j{\mathrm{e}}^{\alpha q}(3-\cosh\alpha)
+B_j{\mathrm{e}}^{-\alpha q}(3-\cosh\alpha)\right.\nonumber\\
& &\qquad\left.+ C_j{\mathrm{e}}^{\beta q}(3-\cosh\beta)+D_j{\mathrm{e}}^{-\beta q}(3-\cosh\beta)
\right],\\
& &\nonumber\\
& &\hat F(p,q;A,B,C,D)=\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)
\cos\left(\frac{\pi j}{m+1}\right)\nonumber\\
& &\quad\times\left[
A_j{\mathrm{e}}^{\alpha q}({\mathrm{e}}^\alpha+1)
+ B_j{\mathrm{e}}^{-\alpha q}({\mathrm{e}}^{-\alpha} +1)\right.\nonumber\\
& &\qquad\left. + C_j{\mathrm{e}}^{\beta q}({\mathrm{e}}^\beta+1)+
D_j{\mathrm{e}}^{-\beta q}({\mathrm{e}}^{-\beta}+1)\right],\end{aligned}$$ where $A_j,B_j,C_j,D_j$ are arbitrary.
Before imposing the remaining absorbing boundary conditions, Eqs. (5)–(8), we represent the solutions in the two regions: $I: q\le b$; $II: q\ge b$ by $$\begin{aligned}
F_I(p,q)&=&F(p,q;A,B,C,D),\\
\hat F_I(p,q)&=&\hat F(p,q;A,B,C,D),\\
F_{II}(p,q)&=&F(p,q;\tilde A,\tilde B,\tilde C,\tilde D),\\
\hat F_{II}(p,q)&=&\hat F(p,q;\tilde A,\tilde B,\tilde C,\tilde D).\end{aligned}$$ The absorbing boundary conditions, Eqs. (5)–(8), can be used to eliminate $C,D,\tilde C,\tilde D$. After an appropriate re-normalization of the coefficients $A,B,\tilde A, \tilde B$ this yields $$\begin{aligned}
F_I(p,q)&=&\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)\nonumber\\
& &\times\left[A_I(q,\alpha,\beta)A_j +A_I(q,-\alpha,\beta)B_j\right],\label{FI}
\\
& &\nonumber\\
\hat F_I(p,q)&=&\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)
\cos\left(\frac{\pi j}{m+1}\right)\nonumber\\
& &\times\left[\hat A_I(q,\alpha,\beta)A_j+\hat A_I(q,-\alpha,\beta)B_j\right],
\label{FIhat}\\
& &\nonumber\\
F_{II}(p,q)&=&\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)\nonumber\\
& &\times\left[A_{II}(q,\alpha,\beta)\tilde A_j+A_{II}(q,-\alpha,\beta)
\tilde B_j\right],\label{FII}\\
& &\nonumber\\
\hat F_{II}(p,q)&=&\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)
\cos\left(\frac{\pi j}{m+1}\right)\nonumber\\
& &\times\left[\hat A_{II}(q,\alpha,\beta)\tilde A_j+
\hat A_{II}(q,-\alpha,\beta)\tilde B_j\right],\label{FIIhat}\end{aligned}$$ where $$\begin{aligned}
A_I(q,\alpha,\beta)&=&(3-\cosh\alpha)(3-\cosh\beta)2\sinh\beta {\mathrm{e}}^{\alpha q}
\nonumber\\
&& -(3-\cosh\beta)\gamma(-\alpha,-\beta){\mathrm{e}}^{\beta q}\nonumber\\
&& +(3-\cosh\beta)\gamma(-\alpha,\beta){\mathrm{e}}^{-\beta q},\label{AI}\\
& &\nonumber\\
\hat A_I(q,\alpha,\beta)&=&2{\mathrm{e}}^{\alpha q}({\mathrm{e}}^\alpha+1)(3-\cosh\beta)\sinh\beta
\nonumber\\
&& -\gamma(-\alpha,-\beta)({\mathrm{e}}^\beta+1){\mathrm{e}}^{\beta q}
+\gamma(-\alpha,\beta)({\mathrm{e}}^{-\beta}+1){\mathrm{e}}^{-\beta q},\label{AIhat}\\
& &\nonumber\\
A_{II}(q,\alpha,\beta)&=&(3-\cosh\alpha)(3-\cosh\beta)2\sinh\beta {\mathrm{e}}^{\alpha q}
\nonumber\\
&& -{\mathrm{e}}^{(\alpha-\beta)(n+1)}\gamma(-\alpha,-\beta)(3-\cosh\beta){\mathrm{e}}^{\beta q}
\nonumber\\
&& +{\mathrm{e}}^{(\alpha+\beta)(n+1)}\gamma(-\alpha,\beta)(3-\cosh\beta){\mathrm{e}}^{-\beta q},
\label{AII}\\
& &\nonumber\\
\hat A_{II}(q,\alpha,\beta)&=&({\mathrm{e}}^\alpha+1)(3-\cosh\beta)2\sinh\beta
{\mathrm{e}}^{\alpha q}\nonumber\\
&& -{\mathrm{e}}^{(\alpha-\beta)(n+1)}\gamma(-\alpha,-\beta)({\mathrm{e}}^\beta+1){\mathrm{e}}^{\beta q}
\nonumber\\
&& +{\mathrm{e}}^{(\alpha+\beta)(n+1)}\gamma(-\alpha,\beta)({\mathrm{e}}^{-\beta}+1)
{\mathrm{e}}^{-\beta q},
\label{AIIhat}\end{aligned}$$ and the function $$\begin{aligned}
\gamma(\alpha,\beta)&=&4\cosh\alpha-4\cosh\beta-(3-\cosh\beta)
\sinh\alpha\nonumber\\
& &\quad-(3-\cosh\alpha)\sinh\beta\label{gamma}.\end{aligned}$$ The coefficients defined in Eq. (36) which are shown as implicit functions of $\alpha$ and $\beta$ are also functions of $j$ via Eqs. (21),(26),(27). Equations (\[FI\])–(\[FIIhat\]), are general solutions to the coupled homogeneous field equations, Eqs. (1),(2), that satisfy all of the absorbing boundary conditions, Eqs. (3)–(8).
Inhomogeneous equations – Matching conditions
---------------------------------------------
By using the two matching conditions, Eqs. (9),(10), the four arbitrary constants, $A, B, \tilde A, \tilde B$, can be reduced to two arbitrary constants, $A,B$ say. The solutions in Region II can then be written as $$\begin{aligned}
F_{II}(p,q)&=&\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)
\left[A_{II}^\star(q,\alpha,\beta)A_j+A_{II}^\star(q,-\alpha,\beta)B_j \right],\\
& &\nonumber\\
\hat F_{II}(p,q)&=&\sum_{j=1}^m\sin\left(\frac{\pi j p}{m+1}\right)
\cos\left(\frac{\pi j}{m+1}\right)\nonumber\\
& &\times\left[\hat A_{II}^\star(q,\alpha,\beta)A_j +\hat A_{II}^\star(q,-\alpha
,\beta)B_j\right],\end{aligned}$$ where $$\begin{aligned}
A_{II}^\star(q,\alpha,\beta)=A_{II}(q,\alpha,\beta)\Gamma_1+A_{II}(q,-\alpha,\beta)
\Gamma_2,&&\\
&&\nonumber\\
\hat A_{II}^\star(q,\alpha,\beta)=\hat A_{II}(q,\alpha,\beta)\Gamma_1+\hat A_{II
}(q,-\alpha,\beta)\Gamma_2,&&\end{aligned}$$ and $$\begin{aligned}
\Gamma_1=\frac{\hat A_{II}(b,-\alpha,\beta)A_I(b,\alpha,\beta)-A_{II}(b,-\alpha,
\beta)\hat A_I(b,\alpha,\beta)}
{A_{II}(b,\alpha,\beta)\hat A_{II}(b,-\alpha,\beta)-\hat A_{II}(b,\alpha,\beta)
A_{II}(b,-\alpha,\beta)},&&\\
&&\nonumber\\
\Gamma_2=\frac{\hat A_{II}(b,\alpha,\beta)A_I(b,\alpha,\beta)-A_{II}(b,\alpha,
\beta)\hat A_I(b,\alpha,\beta)}
{A_{II}(b,-\alpha,\beta)\hat A_{II}(b,\alpha,\beta)-A_{II}(b,\alpha,\beta)
\hat A_{II}(b,-\alpha,\beta)},&&\end{aligned}$$
Finally the remaining arbitrary constants $A,B$ are determined from the requirement that the solutions satisfy the coupled inhomogeneous field equations, Eqs. (11),(12). This step is facilitated using the identity $$\delta_{p,a}=\frac{2}{m+1}\sum_{j=1}^m\sin\left(\frac{\pi j a}{m+1}\right)
\sin\left(\frac{\pi j p}{m+1}\right).$$ Explicitly we find:\
i) $a$ even, $$\begin{aligned}
&&A_j=\frac{T_1(j)}{T_2(j)}B_j,\\
&&B_j=\frac{12}{m+1}\sin\left(\frac{\pi j a}{m+1}\right)\frac{T_2(j)}{T_1(j)T_3(
j)-T_4(j)T_2(j)};\end{aligned}$$ ii) $a$ odd, $$\begin{aligned}
&&A_j=\frac{T_4(j)}{T_3(j)}B_j,\\
&&B_j=\frac{12}{m+1}\sin\left(\frac{\pi j a}{m+1}\right)\frac{T_3(j)}{T_2(j)T_4(
j)-T_1(j)T_3(j)},\end{aligned}$$ and $$\begin{aligned}
T_1(j)&=&2A_I(b,-\alpha,\beta)+2A_{II}^\star(b+1,-\alpha,\beta)
+\hat A_I(b-1,-\alpha,\beta)\nonumber\\
& &+\hat A_{II}^\star(b+1,-\alpha,\beta)
-6\hat A_I(b,-\alpha,\beta),\\
& &\nonumber\\
T_2(j)&=&
6\hat A_I(b,\alpha,\beta)-2A_I(b,\alpha,\beta)-2A_{II}^\star(b+1,\alpha,\beta)
\nonumber\\
& &-\hat A_I(b-1,\alpha,\beta)-\hat A_{II}^\star(b+1,\alpha,\beta),\\
& &\nonumber\\
T_3(j)&=&6A_I(j,b)-2\cos^2(\frac{\pi j}{m+1})\hat A_I(b-1,\alpha,\beta)\nonumber
\\
& &-2\cos^2(\frac{\pi j}{m+1})\hat A_I(b,\alpha,\beta)-A_I(b-1,\alpha,\beta)
\nonumber\\
& &-A_{II}^\star(b+1,\alpha,\beta),\\
& &\nonumber\\
T_4(j)&=&2\cos^2(\frac{\pi j}{m+1})\hat A_I(b-1,-\alpha,\beta)\nonumber\\
& &+2\cos^2(\frac{\pi j}{m+1})\hat A_I(b,-\alpha,\beta)
+A_I(b-1,-\alpha,\beta)\nonumber\\
& &+A_{II}^\star(b+1,-\alpha,\beta)-6A_I(b,-\alpha,\beta).\end{aligned}$$
Our solution for the triangular lattice site expectation values with absorbing boundary conditions and a source point is finally given by Eqs. (32),(33) in Region I and Eqs. (41),(42) in Region II. The relevant quantities appearing in these equations are defined through the series of equations, Eqs. (21),(26),(27),(36)–(40),(43)–(46),(48)–(50).
Example
=======
Consider the case of a triangular lattice with $m=7, n=7$ and a source at $a=4, b=4$ (Fig. 2). In the approximation to this problem using straight edge boundaries and the $(p,q)$ co-ordinate system of Keberle and Montet [@KM63] this problem corresponds to; $m=16, n=7, a=8, b=4$.
We have calculated expectation values at the nearest neighbour lattice sites around the source and absorption probabilities at the boundaries; (a) using our exact results above, and (b) using the results of Keberle and Montet [@KM63] for the approximation to the problem – their equations (7a), (7b), (7c). In this small model lattice system we found that the straight line boundary approximation provides reasonable results for expectation values near the source (accurate to within a few percent) but provides poor results for the absorption probabilities (Table 1). Note that the ‘absorption probabilities’ for the zig-zag boundary co-ordinates which are inside the straight line boundary of Keberle and Montet [@KM63] are not true probabilities in their solution.
[@\*[5]{}[l]{}]{} $\0\0$(a)&&&(b)&$P(0,1)$&.012247&&$P(2,0)$&.019609\
$P(0,2)$&.035646&&$P(4,0)$&.039949\
$P(0,3)$&.054827&&$P(6,0)$&.057904\
$P(0,4)$&.063296&&$P(8,0)$&.065834\
$P(0,5)$&.056686&&$P(10,0)$&.059209\
$P(0,6)$&.039811&&$P(12,0)$&.042817\
$P(0,7)$&.020737&&$P(14,0)$&.024583\
$P(0,8)$&.005743&&$P(16,0)$&.007903\
$P(1,0)$&.026040&&$P(1,1)$&.033624\
$P(2,0)$&.013797&&$P(0,2)$&.036214\
$P(3,0)$&.069523&&$P(1,3)$&.088151\
$P(4,0)$&.021476&&$P(0,4)$&.053888\
$P(1,8)$&.005743&&$P(17,1)$&.015276\
$P(2,8)$&.040253&&$P(16,2)$&.051612\
$P(3,8)$&.015039&&$P(17,3)$&.038562\
$P(4,8)$&.059725&&$P(16,4)$&.075927\
Conclusion
==========
Although our solution is somewhat unwieldly, we have nevertheless solved the underlying field equations for random walks on a finite triangular lattice with a single interior source point and zig-zag absorbing boundaries and have thus calculated the associated absorption probabilities. This problem was previously considered to be intractable [@KM63]. We hope that our result will inspire further work in this area.
We thank Michael Barber for first drawing our attention to [@MW40] and Wolfgang Schief for his warm oral translation of [@C28]. MTB has been supported by The Australian Research Council.
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abstract: |
Motivated by the usefulness of boundaries in the study of $\delta$-hyperbolic and CAT(0) groups, Bestvina introduced a general axiomatic approach to group boundaries, with a goal of extending the theory and application of boundaries to larger classes of groups. The key definition is that of a $\mathcal{Z}$-structure on a group $G$. These $\mathcal{Z}$-structures, along with several variations, have been studied and existence results obtained for a variety of new classes groups. Still, relatively little is known about the general question of which groups admit any of the various $\mathcal{Z}$-structures—aside from the (easy) fact that any such $G$ must have type F, i.e., $G$ must admit a finite K($G,1$). In fact, Bestvina has asked whether *every* type F group admits a $\mathcal{Z}$-structure or at least a weak $\mathcal{Z}$-structure.
In this paper we prove some general existence theorems for weak $\mathcal{Z}$-structures. The main results are as follows.
**Theorem A.** *If* $G$ *is an extension of a nontrivial type F group by a nontrivial type F group, then* $G$ *admits a weak* $\mathcal{Z}$*-structure.*
**Theorem B.** *If* $G$ *admits a finite K(*$G,1$*) complex* $K$ *such that the* $G$-*action on* $\widetilde{K}$ *contains* $1\neq j\in G$ *properly homotopic to* $\operatorname{id}_{\widetilde{K}}$, *then* $G$ *admits a weak* $\mathcal{Z}$*-structure.*
**Theorem C.** *If* $G$ *has type F and is simply connected at infinity, then* $G$ *admits a weak* $\mathcal{Z}$*-structure.*
As a corollary of Theorem A or B, every type F group admits a weak $\mathcal{Z}$-structure after stabilization; more precisely: if $H$ has type F, then $H\times\mathbb{Z}
$ admits a weak $\mathcal{Z}$-structure. As another corollary of Theorem B, every type F group with a nontrivial center admits a weak $\mathcal{Z}$-structure.
address: 'Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201'
author:
- 'Craig R. Guilbault'
date: 'July 19, 2013'
title: 'Weak $\mathcal{Z}$-structures for some classes of groups'
---
[^1]
Introduction\[Section: Introduction\]
=====================================
Several lines of investigation in geometric topology and geometric group theory seek ‘nice’ compactifications of contractible manifolds or complexes (or ERs/ARs) on which a given group $G$ acts cocompactly as covering transformations. Bestvina [@Be] has defined a $\mathcal{Z}$-structure and a weak $\mathcal{Z}$-structure on a group $G$ as follows:
A $\mathcal{Z}$*-structure* on a group $G$ is a pair $\left(
\overline{X},Z\right) $ of spaces satisfying:
1. $\overline{X}$ is a compact ER,
2. $Z$ is a $\mathcal{Z}$-set[^2] in $\overline{X}$,
3. $X=\overline{X}-Z$ admits a proper, free, cocompact action by $G$, and
4. (nullity condition) For any open cover $\mathcal{U}$ of $\overline{X}$, and any compactum $K\subseteq X$, all but finitely many $G$-translates of $K$ lie in some element $U$ of $\mathcal{U}$.
If only conditions 1)-3) are satisfied, $\left( \overline
{X},Z\right) $ is called a *weak* **** $\mathcal{Z}$*-structure* on $G$.
An additional condition that can be added to conditions 1)-3), with or without condition 4), is:
1. The action of $G$ on $X$ extends to an action of $G$ on $\overline{X}$.
Farrell and Lafont [@FL] refer to a pair $\left( \overline
{X},Z\right) $ satisfying 1)-5) as an $E\mathcal{Z}$*-structure*. Others have considered pairs that satisfy 1)-3) and 5); we call those *weak* $E\mathcal{Z}$*-structures.* Depending on the set of conditions satisfied, $Z$ is referred to generically as a *boundary* for $G$; or more specifically as a $\mathcal{Z}$*-boundary*, a *weak* $\mathcal{Z}$*-boundary,* an $E\mathcal{Z}$*-boundary*, or a *weak* $E\mathcal{Z}$*-boundary*.
A torsion-free group acting geometrically on a finite-dimensional CAT(0) space $X$ admits an $E\mathcal{Z}$-structure—one compactifies $X$ by adding the visual boundary. Bestvina and Mess [@BM] have shown that each torsion-free word hyperbolic group $G$ admits an $E\mathcal{Z}$-structure $\left(
\overline{X},\partial G\right) $, where $\partial G$ is the Gromov boundary and $X$ is a Rips complex for $G$. Osajda and Przytycki have shown that systolic groups admit $E\mathcal{Z}$-structures. [@Be] contains a discussion of $\mathcal{Z}$-structures and weak $\mathcal{Z}$-structures on a variety of other groups, not all of which satisfy condition 5).
*Some authors (see [@Dr]) have extended the above definitions by allowing non-free* $G$*-actions (thus allowing for groups with torsion) and by loosening the ER requirement on* $\overline{X}$ *to that of AR, i.e., allowing* $\overline{X}$ *to be infinite-dimensional. Here we stay with the original definitions, but note that some analogous results are possible in the more general settings.*
A group $G$ has *type F* if it admits a finite K($G,1$) complex. The following proposition narrows the field of candidates for admitting any sort of $\mathcal{Z}$-structure to those groups of type F.
If there exists a proper, free, cocompact $G$-action on an AR $Y$, then $G$ has type F.
The quotient $q:Y\rightarrow G\backslash Y$ is a covering projection, so $G\backslash Y$ is aspherical and locally homeomorphic to $Y$. By the latter, $G\backslash Y$ is a compact ANR, and thus (by Theorem \[Theorem: West’s Theorem\]) homotopy equivalent to a finite complex. Any such complex is a K($G,1$).
In [@Be], Bestvina asked the following pair of questions:
**Bestvina’s Question.** *Does every type F group admit a* $\mathcal{Z}$*-structure?*
**Weak Bestvina Question.** *Does every type F group admit a weak* $\mathcal{Z}$*-structure?*
The Weak Bestvina Question was also posed by Geoghegan in [@Ge2 p.425]. Farrell and Lafont [@FL] have asked whether every type F group admits an $E\mathcal{Z}$-structure, and the question of which groups admit weak $E\mathcal{Z}$-structures appear in both [@BM] and [@Ge2]. Although interesting special cases abound, a general solution to any of these questions seems out of reach at this time.
As one would expect, the more conditions a $\mathcal{Z}$-structure or its corresponding boundary satisfies, the greater the potential applications. For example, Bestvina has shown that the topological dimension of a $\mathcal{Z}$-boundary is an invariant of the group—it is one less than the cohomological dimension of $G$; this is not true for weak $\mathcal{Z}$-boundaries. But a weak $\mathcal{Z}$*-*boundary carries significant information about $G$. For example, the Čech cohomology of a weak $\mathcal{Z}$*-*boundary reveals the group cohomology of $G$ with $\mathbb{Z}
G$-coefficients, and the $\operatorname*{pro}$-homotopy groups of a weak $\mathcal{Z}$-boundary are directly related to the corresponding end invariants (such as simple connectivity at infinity) of $G$. A weak $\mathcal{Z}$*-*boundary, when it exists, is well-defined up to shape and can provide a first step toward obtaining a stronger variety of $\mathcal{Z}$-structure on $G$. $E\mathcal{Z}$*-* and weak ** $E\mathcal{Z}$-boundaries, when they exist, carry the potential for studying $G$ by analyzing its action on the compactum Z. More about these topics can be found in [@Ge1], [@Ge2], [@GM1], [@Be], [@FL] and [@Gu3].
In this paper we prove the existence of weak $\mathcal{Z}$-structures for a variety of groups. A notable special case provides a stabilized solution to the Weak Bestvina Question. It asserts that, if $H$ has type F, then $H\times\mathbb{Z}
$ admits a weak $\mathcal{Z}$-structure. That result is an easy consequence of either of the following more general theorems, to be proven here.
\[Theorem 1\]If $G$ is an extension of a nontrivial type F group by a nontrivial type F group, that is, if there exists a short exact sequence $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow1$ where $N$ and $Q$ are nontrivial and type F, then $G$ admits a weak $\mathcal{Z}$-structure. More generally, if a type F group $G$ is virtually an extension of a nontrivial type F group by a nontrivial type F group, then $G$ admits a weak $\mathcal{Z}$-structure.
\[Theorem 2\]Suppose $G$ admits a finite K($G,1$) complex $K$, and the corresponding $G$-action on the universal cover $\widetilde{K}$ contains a $1\neq j\in G$ that is properly homotopic to $\operatorname{id}_{\widetilde
{K}}$. Then $G$ admits a weak $\mathcal{Z}$-structure.
*For finite K(*$G,1$*) complexes* $K$ *and* $L$*, or more generally, compact aspherical ANRs with* $\pi_{1}\left( K\right) \cong
G\cong\pi_{1}\left( L\right) $*, there is a* $G$*-equivariant proper homotopy equivalence* $\widetilde{f}:\widetilde{K}\rightarrow
\widetilde{L}$*. If* $j\in G$ *satisfies the hypothesis of Theorem \[Theorem 2\], then so does* $\widetilde{f}\circ j$*. Hence, the existence of such a* $j$ *can be viewed as a property of* $G$*, itself.*
For a closed, orientable, aspherical $n$-manifold $M^{n}$ with $\widetilde
{M}^{n}\cong\mathbb{R}
^{n}$ (e.g., $M^{n}$ a Riemannian manifold of nonpositive sectional curvature) every element of $\pi_{1}\left( M^{n}\right) $ satisfies the hypothesis of Theorem \[Theorem 1\]. On the other hand, for finitely generated free groups, no elements do. Of course, weak $\mathcal{Z}$-structures for both of these classes of groups are known for other reasons.
\[Corollary: Stable weak Z-structures\]If $H$ is type F, then $H\times\mathbb{Z}
$ admits a weak $\mathcal{Z}$-structure.
This corollary is immediate from Theorem \[Theorem 1\]. Alternatively, it may be obtained from Theorem \[Theorem 2\]. Let $K$ be a finite K($H,1$) and $H\times\mathbb{Z}
$ act diagonally on $\widetilde{K}\times\mathbb{R}
$. The nontrivial elements of $\mathbb{Z}
$ satisfy the hypotheses of that theorem.
A more general application of Theorem \[Theorem 2\] is the following.
If $G$ is a type F group with a nontrivial center, then $G$ admits a weak $\mathcal{Z}$-structure.
The point here is that, when $K$ is a finte K($G$,1), each nontrivial $j\in
Z\left( G\right) $ satisfies the hypothesis of Theorem \[Theorem 2\]. A complete proof of that fact can be deduced from [@Go Th.II.7]. We sketch an alternative argument.
Let $K^{\ast}=K\cup A$ where $A$ is an arc with terminal point identified to a vertex $p$ of $K$; let $p^{\ast}$ be the initial point of $A$. For arbitrary $j\in\pi_{1}\left( K,p\right) $ define $f_{j}:\left( K^{\ast},p^{\ast
}\right) \rightarrow\left( K^{\ast},p^{\ast}\right) $ to be the identity on $K$; to stretch the initial half of $A$ onto the image copy of $A$; and to send the latter half of $A$ around a loop corresponding to $j$. The induced homomorphism on $\pi_{1}\left( K^{\ast},p^{\ast}\right) $ is conjugation by $j$. If $j\in Z\left( G\right) $ that homomorphism is the identity, so $f_{j}$ is homotopic (rel $p^{\ast}$) to $\operatorname*{id}_{K^{\ast}}$. Since $K^{\ast}$ is compact, that homotopy lifts to a proper homotopy $\widetilde{F}:\widetilde{K^{\ast}}\times\left[ 0,1\right] \rightarrow
\widetilde{K^{\ast}}$. Collapse out the preimage of $A\times\left[
0,1\right] $ in the domain and the preimage of $A$ in the range to get a proper homotopy between $\operatorname*{id}_{\widetilde{K}}$ and the covering translation corresponding to $j$.
Theorems \[Theorem 1\] and \[Theorem 2\] will be obtained from a pair of more general results, with hypotheses more topological than group-theoretic.
\[Theorem 3\]Suppose $G$ admits a finite K($G,1$) complex $K$ with the property that $\widetilde{K}$ is proper homotopy equivalent to a product $X\times Y$ of noncompact ANRs, then $G$ admits a weak $\mathcal{Z}$-structure.
\[Theorem 4\]Suppose $G$ admits a finite K($G,1$) complex $K$ for which $\widetilde{K}$ is proper homotopy equivalent to an ANR $X$ that admits a proper $\mathbb{Z}
$-action generated by a homeomorphism $h:X\rightarrow X$ that is properly homotopic to $\operatorname*{id}_{X}$. Then $G$ admits a weak $\mathcal{Z}$-structure.
Note that neither the product structure in Theorem \[Theorem 3\] nor the $\mathbb{Z}
$-action in Theorem \[Theorem 4\] are required to have any relationship to the $G$-action on $\widetilde{K}$.
A third variety of existence theorem for weak $\mathcal{Z}$-structures has, as its primary hypothesis, a condition on the end behavior of $G.$
\[Theorem 5\]If $G$ is type F, 1-ended, and has pro-monomorphic fundamental group at infinity, then $G$ admits a weak $\mathcal{Z}$-structure.
\[Corollary: simply connected at infinity\]If a type F group $G$ is simply connected at infinity, then $G$ admits a weak $\mathcal{Z}$-structure.
Results found in [@Ja], [@Mi], [@Pr], and [@CM] show that simple connectivity at infinity is a common property for certain types of group extensions. By applying those results, some interesting overlap can be seen in the collections of groups covered by Corollary \[Corollary: simply connected at infinity\] and those covered by Theorems \[Theorem 1\] and \[Theorem 2\].
In the next section, we introduce some terminology and review a number of established results that are fundamental to later arguments. In §\[Section: Topological results\] we prove a variety topological theorems related to end properties of ANRs, complexes, and Hilbert cube manifolds. Most importantly, we prove $\mathcal{Z}$-compactifiability for a variety of spaces. Several results obtained there are more general than required for the group-theoretic applications in this paper, and may be of independent interest. In §\[Section: Proofs of main theorems\], we prove the main theorems stated above. In an appendix, we provide an alternative proof, based on the theory of approximate fibrations, of a crucial lemma from §\[Section: Proofs of main theorems\].
The author wishes to acknowledge Mike Mihalik and Ross Geoghegan for helpful conversations that led to significant improments in this paper.
Terminology and background\[Section: Background\]
=================================================
Inverse sequences of groups
---------------------------
Throughout this subsection all arrows denote homomorphisms, while arrows of the type $\twoheadrightarrow$ or $\twoheadleftarrow$ denote surjections and arrows of the type $\rightarrowtail$ and $\leftarrowtail$ denote injections.
Let $$G_{0}\overset{\lambda_{1}}{\longleftarrow}G_{1}\overset{\lambda_{2}}{\longleftarrow}G_{2}\overset{\lambda_{3}}{\longleftarrow}\cdots$$ be an inverse sequence of groups. A *subsequence* of $\left\{
G_{i},\lambda_{i}\right\} $ is an inverse sequence of the form $$\begin{diagram}
G_{i_{0}} & \lTo^{\lambda_{i_{0}+1}\circ\cdots\circ\lambda_{i_{1}}
} & G_{i_{1}} & \lTo^{\lambda_{i_{1}+1}\circ\cdots\circ
\lambda_{i_{2}}} & G_{i_{2}} & \lTo^{\lambda_{i_{2}+1}\circ
\cdots\circ\lambda_{i_{3}}} & \cdots.
\end{diagram}$$ In the future we denote a composition $\lambda_{i}\circ\cdots\circ\lambda_{j}$ ($i\leq j$) by $\lambda_{i,j}$.
Sequences $\left\{ G_{i},\lambda_{i}\right\} $ and $\left\{ H_{i},\mu
_{i}\right\} $ are *pro-isomorphic* if, after passing to subsequences, there exists a commuting ladder diagram: $$\begin{diagram} G_{i_{0}} & & \lTo^{\lambda_{i_{0}+1,i_{1}}} & & G_{i_{1}} & & \lTo^{\lambda_{i_{1}+1,i_{2}}} & & G_{i_{2}} & & \lTo^{\lambda_{i_{2}+1,i_{3}}}& & G_{i_{3}}& \cdots\\ & \luTo & & \ldTo & & \luTo & & \ldTo & & \luTo & & \ldTo &\\ & & H_{j_{0}} & & \lTo^{\mu_{j_{0}+1,j_{1}}} & & H_{j_{1}} & & \lTo^{\mu_{j_{1}+1,j_{2}}}& & H_{j_{2}} & & \lTo^{\mu_{j_{2}+1,j_{3}}} & & \cdots \end{diagram} \label{basic ladder diagram}$$ Clearly an inverse sequence is pro-isomorphic to any of its subsequences. To avoid tedious notation, we sometimes do not distinguish $\left\{
G_{i},\lambda_{i}\right\} $ from its subsequences. Instead we assume that $\left\{ G_{i},\lambda_{i}\right\} $ has the desired properties of a preferred subsequence—prefaced by the words after passing to a subsequence and relabeling.
An inverse sequence $\left\{ G_{i},\lambda_{i}\right\} $ is called *pro-monomorphic* if it is pro-isomorphic to an inverse sequence of monomorphisms and *pro-epimorphic* (more commonly called *semistable* or *Mittag-Leffler*) if it is pro-isomorphic to an inverse sequence of epimorphisms. It is *stable* if it is pro-isomorphic to a constant inverse sequence $\left\{ H,\operatorname{id}_{H}\right\} $, or equivalently, to an inverse sequence of isomorphisms. It is a standard fact that $\left\{ G_{i},\lambda_{i}\right\} $ is stable if and only if it is both pro-monomorphic and pro-epimorphic.
A few more special classes of inverse sequences will be of interest in this paper. A sequence that is pro-isomorphic to the trivial sequence $1\leftarrow1\leftarrow1\leftarrow\cdots$ is called *pro-trivial*; a sequence pro-isomorphic to an inverse sequence of finitely generated groups is called *pro-finitely generated*; and a sequence that is pro-isomorphic to an inverse sequence of free groups is called *pro-free*. A sequence that is both pro-finitely generated and pro-free is easily seen to be pro-isomorphic to an inverse sequence of finitely generated free groups. We call such a sequence *pro-finitely generated free*.
The *inverse limit* of a sequence $\left\{ G_{i},\lambda_{i}\right\} $ is the subgroup of $\prod G_{i}$ defined by $$\underleftarrow{\lim}\left\{ G_{i},\lambda_{i}\right\} =\left\{ \left.
\left( g_{0},g_{1},g_{2},\cdots\right) \in\prod_{i=0}^{\infty}G_{i}\right\vert \lambda_{i}\left( g_{i}\right) =g_{i-1}\right\} .$$
In the special case where $\left\{ G_{i},\lambda_{i}\right\} $ is an inverse sequence of abelian groups, we also define the *derived limit*[^3] to be the following quotient group:$$\underleftarrow{\lim}^{1}\left\{ G_{i},\lambda_{i}\right\} =\left(
\prod\limits_{i=0}^{\infty}G_{i}\right) /\left\{ \left. \left(
g_{0}-\lambda_{1}g_{1},g_{1}-\lambda_{2}g_{2},g_{2}-\lambda_{3}g_{3},\cdots\right) \right\vert \ g_{i}\in G_{i}\right\}$$
It is a standard fact that pro-isomorphic inverse sequences of groups have isomorphic inverse limits and, pro-isomorphic inverse sequences of abelian groups have isomorphic derived limits.
Absolute neighborhood retracts
------------------------------
Throughout this paper, all spaces are assumed to be separable metric. A locally compact space $X$ is an ANR (*absolute neighborhood retract*) if it can be embedded into $\mathbb{R}
^{n}$ or, if necessary, $\mathbb{R}
^{\infty}$ (a countable product of real lines) as a closed set in such a way that there exists a retraction $r:U\rightarrow X$, where $U$ is a neighborhood of $X$. If the entire space $\mathbb{R}
^{n}$ or $\mathbb{R}
^{\infty}$ retracts onto $X$, we call $X$ an AR (absolute retract). If $X$ is finite-dimensional, all mention of $\mathbb{R}
^{\infty}$ can be omitted. A finite-dimensional ANR is called an ENR (*Euclidean neighborhood retract*) and a finite-dimensional AR an ER. With a little effort it can be shown that an AR \[resp., ER\] is simply a contractible ANR \[resp., ENR\].
A space $X$ is *locally contractible* if every neighborhood $U$ of a point $x\in X$ contains a neighborhood $V$ of $x$ that contracts within $U$. It is a standard fact that every ANR is locally contractible. For finite-dimensional spaces, that property characterizes ANRs. In other words, a locally compact, finite-dimensional space $X$ is an ANR (and hence an ENR) if and only if it is locally contractible. It follows that every finite-dimensional locally finite polyhedron or CW complex is an ENR; if it is contractible, it is an ER.
The following famous result will be used in this paper.
\[West, [@We]\]\[Theorem: West’s Theorem\]Every ANR is homotopy equivalent to a locally finite polyhedron. Every compact ANR is homotopy equivalent to a finite polyhedron.
Proper maps and homotopies
--------------------------
When working with noncompact space, the notion of ‘properness’ is crucial. A map $f:X\rightarrow Y$ is *proper* if $f^{-1}\left( C\right) $ is compact whenever $C\subseteq Y$ is compact. Maps $f_{0},f_{1}:X\rightarrow Y$ are *properly homotopic*, denoted $f_{0}\overset{p}{\simeq}f_{1}$ if there exists a proper map $H:X\times\lbrack0,1]\rightarrow Y$ with $H_{0}=f_{0}$ and $H_{1}=f_{1}$. Spaces $X$ and $Y$ are *proper homotopy equivalent*, denoted $X\overset{p}{\simeq}Y$, if there exist proper maps $f:X\rightarrow Y$ and $g:Y\rightarrow X$ with $gf\overset{p}{\simeq
}\operatorname*{id}_{X}$ and $fg\overset{p}{\simeq}\operatorname*{id}_{Y}$.
Ends of spaces and the fundamental group at infinity
----------------------------------------------------
A subset $N$ of a space $X$ is a *neighborhood of infinity* if $\overline{X-N}$ is compact. A standard argument shows that, when $X$ is an ANR and $C$ is a compact subset of $X$, $X-C$ has at most finitely many unbounded components, i.e., finitely many components with noncompact closures. If $X-C$ has both bounded and unbounded components, the situation can be simplified by letting $C^{\prime}$ consist of $C$ together with all bounded components. Then $C^{\prime}$ is compact, and $X-C^{\prime}$ consists entirely of unbounded components.
We say that $X$ *has* $k$ *ends* if there exists a compactum $C\subseteq X$ such that, for every compactum $D$ with $C\subset D$, $X-D$ has exactly $k$ unbounded components. When $k$ exists, it is uniquely determined; if $k$ does not exist, we say $X$ has *infinitely many ends*. Thus, a space is $0$-ended if and only if $X$ is compact, and $1$-ended if and only if it contains arbitrarily small connected neighborhoods of infinity.
A nested sequence $N_{0}\supseteq N_{1}\supseteq N_{2}\supseteq\cdots$ of neighborhoods of infinity, with each $N_{i}\subseteq\operatorname*{int}N_{i-1}$, is *cofinal* if $\bigcap_{i=0}^{\infty}N_{i}=\varnothing$. Such a sequence is easily obtained: choose an exhaustion of $X$ by compacta $C_{0}\subseteq C_{1}\subseteq C_{2}\subseteq\cdots$, with $C_{i-1}\subseteq\operatorname*{int}C_{i}$; then let $N_{i}=X-C_{i}$. When closed neighborhoods of infinity are required, let $N_{i}=\overline{X-C_{i}}$.
Given a nested cofinal sequence $\left\{ N_{i}\right\} _{i=0}^{\infty}$ of neighborhoods of infinity, base points $p_{i}\in N_{i}$, and paths $r_{i}\subset N_{i}$ connecting $p_{i}$ to $p_{i+1}$, we obtain an inverse sequence: $$\pi_{1}\left( N_{0},p_{0}\right) \overset{\lambda_{1}}{\longleftarrow}\pi_{1}\left( N_{1},p_{1}\right) \overset{\lambda_{2}}{\longleftarrow}\pi_{1}\left( N_{2},p_{2}\right) \overset{\lambda_{3}}{\longleftarrow}\cdots.\medskip\label{Defn: pro-pi1}$$ Here, each $\lambda_{i+1}:\pi_{1}\left( N_{i+1},p_{i+1}\right)
\rightarrow\pi_{1}\left( N_{i},p_{i}\right) $ is the homomorphism induced by inclusion followed by the change of base point isomorphism determined by $r_{i}$. The proper ray $r:[0,\infty)\rightarrow X$ obtained by piecing together the $r_{i}$ in the obvious manner is referred to as the *base ray* for the inverse sequence, and the pro-isomorphism class of the inverse sequence is called the *fundamental group at infinity of* $X$ *based at* $r$ and is denoted $\operatorname{pro}$-$\pi_{1}\left( \varepsilon
(X),r\right) $. It is a standard fact that $\operatorname{pro}$-$\pi
_{1}\left( X,r\right) $ is independent of the sequence of neighborhoods $\left\{ N_{i}\right\} $ or the base points—provided those base points tend to infinity along the ray $r$, and corresponding subpaths of $r$ are used in defining the $\lambda_{i}$. More generally, $\operatorname{pro}$-$\pi
_{1}\left( \varepsilon(X),r\right) $ depends only upon the proper homotopy class of $r$. If $X$ is 1-ended and $\operatorname{pro}$-$\pi_{1}\left(
\varepsilon(X),r\right) $ is semistable for some proper ray $r$, it can be shown that all proper rays in $X$ are properly homotopic; in that case we say that $X$ is *strongly connected at infinity*. When $X$ is strongly connected at infinity, it is safe to omit mention of the base ray and to speak generally of the *fundamental group at infinity of* $X$, and denote it by $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(X)\right) $. If $X$ is 1-ended and $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(X),r\right) $ is pro-trivial, we call $X$ *simply connected at infinity*.
The fundamental group at infinity is clearly not a homotopy invariant of a space, but it is a proper homotopy invariant. More precisely, if $f:X\rightarrow Y$ is a proper homotopy equivalence, then $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(X),r\right) $ is pro-isomorphic to $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(Y),f\circ r\right) $.
For a group $G$ of type F, the universal cover $\widetilde{K}$ of a finite K$\left( G,1\right) $ complex $K$ is well-defined up to proper homotopy type. So the number of ends of $G$ is well-defined; and if $\widetilde{K}$ is 1-ended, except for the issue of a base ray, we may view $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(\widetilde{K}),r\right) $ as an invariant of $G$. The base ray issue goes away when $\operatorname{pro}$-$\pi_{1}\left(
\varepsilon(\widetilde{K}),r\right) $ is semistable, so there is no ambiguity in defining a 1-ended $G$ to have semistable, stable, or trivial fundamental group at infinity, according to whether $\operatorname{pro}$-$\pi_{1}\left(
\varepsilon(\widetilde{K}),r\right) $ has the corresponding property. With some additional work, it can be shown that the property of $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(\widetilde{K}),r\right) $ being pro-monomorphic is also independent of base ray and, thus, attributable to $G$. See [@GG §2] for further discussion.
Although not needed for this paper, the requirement in the previous paragraph, that $G$ have type F can be significantly weakened. In particular, if $G$ is finitely presented, and $L$ is any finite complex with fundamental group $G$, then the number of ends of $\widetilde{L}$ and the properties of $\operatorname{pro}$-$\pi_{1}\left( \varepsilon(\widetilde{L}),r\right) $ discussed above, are invariants of $G$. Thus, for example, a finitely presented group $G$ is called *simply connected at infinity* if $\widetilde{L}$ has that property. For more information about the fundamental group at infinity of spaces and groups, including proofs of the made in this section, see [@Ge2] or [@Gu3].
Finite domination and inward tameness
-------------------------------------
A space $Y$ has *finite homotopy type* if it is homotopy equivalent to a finite CW complex; it is *finitely dominated* if there is a finite complex $K$ and maps $u:Y\rightarrow K$ and $d:K\rightarrow Y$ such that $d\circ u\simeq\operatorname*{id}_{Y}$. If $Y$ is an ANR, then $Y$ is finitely dominated if and only if there exists a self-homotopy that ‘pulls $Y$ into a compact subset’, i.e., $H:Y\times\lbrack0,1]\rightarrow Y$ such that $H_{0}=\operatorname*{id}_{Y}$ and $\overline{H_{1}\left( Y\right) }$ is compact. This equivalence is easily verified when (for example) $K$ is a locally finite polyhedron; a discussion of the general case can be found in [@Gu3 §3.4].
The following clever observation will be used later.
\[Mather, [@Ma]\]\[Theorem: Mather’s theorem\]If a space $Y$ is finitely dominated, then $Y\times\mathbb{S}^{1}$ has finite homotopy type.
An ANR $X$ is *inward tame* if, for every closed neighborhood of infinity $N$ in $X$, there is a homotopy $K:N\times\left[ 0,1\right] \rightarrow N$ with $K_{0}=\operatorname*{id}_{N}$ and $\overline{K_{1}\left( N\right) }$ compact (a homotopy pulling $N$ into a compact subset). By an easy application of Borsuk’s Homotopy Extension Property, this is equivalent to the existence of a cofinal sequence $\{N_{i}\}$ of closed neighborhoods of infinity, each of which can be pulled into a compact set. If $X$ contains a cofinal sequence $\left\{ N_{i}\right\} $ of closed ANR neighborhoods[^4] of infinity, then inward tameness is equivalent to each of those (hence, all closed ANR neighborhoods of infinity) being finitely dominated. [@Gu3 §3.5] provides additional details.
Inward tameness is an invariant of proper homotopy type. Roughly speaking, if $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are proper homotopy inverses and $H$ is a homotopy that pulls a neighborhood of infinity of $X$ into a compact set, then $f\circ H_{t}\circ g$ pulls a neighborhood of $Y$ into a compact set. More details can be found in [@Gu3 §3.5].
Some basic K-theory
-------------------
An important result from [@Wa] asserts that, for each finitely dominated, connected space $Y$, there is a well-defined obstruction $\sigma\left(
Y\right) $, lying in the *reduced projective class group* $\widetilde
{K}_{0}\!\left(
\mathbb{Z}
\left[ \pi_{1}\left( Y\right) \right] \right) $, which vanishes if and only if $Y$ has finite homotopy type.
A related algebraic construction is the *Whitehead group*. If $\left(
A,B\right) $ is a pair of connected, finite CW complexes and $B\hookrightarrow A$ is a homotopy equivalence, then there is a well-defined obstruction $\tau\left( B\right) $, lying in an abelian group $\operatorname*{Wh}\!\left( \pi_{1}\left( B\right) \right) $ that vanishes if and only if $B\hookrightarrow A$ is a simple homotopy equivalence. Definitions and details can be found in [@Co].
Both of the above algebraic constructs act as functors in the sense that, if $\lambda:G\rightarrow H$ is a group homomorphism, there are naturally induced homomorphims $\lambda_{\ast}:\widetilde{K}_{0}\!\left(
\mathbb{Z}
\left[ G\right] \right) \rightarrow\widetilde{K}_{0}\!\left(
\mathbb{Z}
\left[ G\right] \right) $ and $\lambda_{\ast}:\operatorname*{Wh}\!\left(
G\right) \rightarrow\operatorname*{Wh}\!\left( H\right) $.
For the purposes of this paper, the main thing we need to know about $\widetilde{K}_{0}\!\left(
\mathbb{Z}
\left[ \pi_{1}\left( Y\right) \right] \right) $ or $\operatorname*{Wh}\!\left( \pi_{1}\left( B\right) \right) $ is contained in a famous result of Bass, Heller and Swan [@BHS].
\[Theorem: Bass-Heller-Swan\]If $G$ is a finitely generated free group, then both $\widetilde{K}_{0}\!\left(
\mathbb{Z}
\left[ G\right] \right) $ and $\operatorname*{Wh}\!\left( G\right) $ are the trivial group.
Mapping cylinders, mapping tori, and mapping telescopes
-------------------------------------------------------
For any map $f:K\rightarrow L$ and closed interval $\left[ a,b\right] $, the *mapping cylinder* $\mathcal{M}_{\left[ a,b\right] }\left( f\right) $ is the quotient space $L\sqcup\left( K\times\left[ a,b\right] \right)
/\!\sim$, where $\sim$ is the equivalence relation generated by the rule $\left( x,a\right) \sim f\left( x\right) $ for all $x\in K$. Let $q_{\left[ a,b\right] }:L\sqcup(K\times\left[ a,b\right] )\rightarrow
\mathcal{M}_{\left[ a,b\right] }\left( f\right) $ be the quotient map. Then, for each $r\in(a,b]$, $q_{\left[ a,b\right] }$ restricts to an embedding of $K\times\left\{ r\right\} $ into $\mathcal{M}_{\left[
a,b\right] }\left( f\right) $; denote the image of $K\times\left\{
r\right\} $ by $K_{r}$. The quotient map is also an embedding when restricted to $L$; let $L_{a}\subseteq\mathcal{M}_{\left[ a,b\right] }\left( f\right)
$ be that copy of $L$. We call $K_{b}$ the *domain end* and $L_{a}$ the *range end* of $\mathcal{M}_{\left[ a,b\right] }\left( f\right) $. Note the existence of a projection map $p_{\left[ a,b\right] }:\mathcal{M}_{\left[ a,b\right] }\left( f\right) \rightarrow\left[
a,b\right] $ for which $p_{\left[ a,b\right] }^{-1}\left( r\right)
=K_{r}$ is a copy of $K$ for each $r\in(a,b]$ and $p_{\left[ a,b\right]
}^{-1}\left( a\right) =L_{a}$ is a copy of $L$. Note also that, when $K=L$, i.e., $f$ maps $K$ to itself, all of the above still applies. In that case, each point preimage of $p_{\left[ a,b\right] }$ is a copy of $K$, but the copy $K_{a}$ differs from the others, in that it is not necessarily parallel to neighboring copies.
*Clearly the topological type of* $\mathcal{M}_{\left[ a,b\right]
}\left( f\right) $ *does not depend on the interval* $\left[
a,b\right] $*, and for most purposes can be taken to be* $\left[
0,1\right] $*. But in the treatment that follows, it will be useful to allow the interval to vary.*
The following standard application of mapping cylinders will be used several times in this paper. A proof, in which properness is not mentioned, can be found in [@Du p.372]. For our purposes, it is only the easy (converse) direction of the proper assertion that will be used.
\[Lemma: mapping cylinders of homotopy equivalences\]A map $f:K\rightarrow
L$ between ANRs is a homotopy equivalence if and only if there exists a strong deformation retraction of $\mathcal{M}_{\left[ a,b\right] }\left( f\right)
$ onto $M_{b}$. It is a proper homotopy equivalence if and only if there exists a proper strong deformation retraction of $\mathcal{M}_{\left[
a,b\right] }\left( f\right) $ onto $K_{b}$.
The *bi-infinite mapping telescope* of a map $f:K\rightarrow K$ is obtained by gluing together infinitely many mapping cylinders. More precisely, $$\operatorname*{Tel}\nolimits_{f}\left( K\right) =\cdots\cup\mathcal{M}_{\left[ -2,-1\right] }\left( f\right) \cup\mathcal{M}_{\left[
-1,0\right] }\left( f\right) \cup\mathcal{M}_{\left[ 0,1\right] }\left(
f\right) \cup\mathcal{M}_{\left[ 1,2\right] }\left( f\right)
\cup\mathcal{M}_{\left[ 2,3\right] }\left( f\right) \cup\cdots$$ where the gluing is accomplished by identifying the domain end of each $\mathcal{M}_{\left[ n-1,n\right] }\left( f\right) $ with the range end of $\mathcal{M}_{\left[ n,n+1\right] }\left( f\right) $. Notationally, this works well since, under the convention described above, each is denoted $K_{n}$. Projection maps may be pieced together to obtain a projection $p:\operatorname*{Tel}\nolimits_{f}\left( K\right) \rightarrow\mathbb{R}
$, for which $p^{-1}\left( r\right) =K_{r}$ is a copy of $K$, for each $r\in\mathbb{R}
$. A schematic of $\operatorname*{Tel}\nolimits_{f}\left( K\right) $ is contained in Figure \[Fig 1\] of §\[Subsection: Mapping tori of self-homotopy equivalences\].
The *mapping torus* of $f:K\rightarrow K$ is obtained from $\mathcal{M}_{\left[ 0,1\right] }\left( f\right) $ by identifying $K_{0}$ with $K_{1}$. It may also be defined more directly as the quotient space$$\operatorname*{Tor}\nolimits_{f}\left( K\right) =K\times\left[ 0,1\right]
/\sim$$ where $\sim$ is the equivalence relation generated by $\left( x,0\right)
\sim\left( f\left( x\right) ,1\right) $ for each $x\in K$. The following facts about mapping mapping tori are standard.
Let $K$ be a connected ANR, $f:\left( K,p\right) \rightarrow\left(
K,q\right) $ a map that induces an isomorphism $\varphi:\pi_{1}\left(
K,p\right) \rightarrow\pi_{1}\left( K,q\right) $, and $\lambda$ a path in $K$ from $q$ to $p$. Then
1. $\pi_{1}\left( \operatorname*{Tor}\nolimits_{f}\left( K\right)
,(p,0\right) )\allowbreak\cong\allowbreak\pi_{1}\left( K,p\right)
\rtimes_{\varphi}\left\langle t\right\rangle $, where $t$ is an infinite order element represented by the loop $\left( \left\{ p\right\} \times\left[
0,1\right] \right) \cdot\lambda$, and
2. the infinite cyclic cover of $\operatorname*{Tor}\nolimits_{f}\left(
K\right) $ corresponding to the projection $\pi_{1}\left( K,p\right)
\rtimes_{\varphi}\left\langle t\right\rangle \rightarrow\left\langle
t\right\rangle $ is the bi-infinite mapping telescope $\operatorname*{Tel}\nolimits_{f}\left( K\right) $.
The following fact about mapping tori can be found in [@GG], where it plays a crucial role. We will make significant use of it here as well.
\[Lemma: mapping torus/Z-action\]Let $X$ be a connected ANR that admits a proper $\mathbb{Z}
$-action generated by a homeomorphism $j:X\rightarrow X$. Then $(\left\langle
j\right\rangle \backslash X)\times\mathbb{R}$ is homeomorphic to $\operatorname*{Tor}\nolimits_{j}\left( X\right) $.
Hilbert cube manifolds
----------------------
The *Hilbert cube* is the infinite product $$\mathcal{Q=}\prod_{i=1}^{\infty}\left[ -1,1\right] \text{, with metric
}d\left( \left( x_{i}\right) ,\left( y_{i}\right) \right) =\sum
_{i=1}^{\infty}\frac{\left\vert x_{i}-y_{i}\right\vert }{2^{i}}$$ A *Hilbert cube manifold* is a space $X$ with the property that each $x\in X$ has a neighborhood homeomorphic to $\mathcal{Q}$. Although we are primarily interested in finite-dimensional spaces, Hilbert cube manifolds play a key role in this paper. A pair of classical results will allow us to move between the categories of ANRs and locally finite polyhedra through the use of Hilbert cube manifolds.
\[Edwards, [@Ed]\]\[Theorem: Edwards HCM Theorem\]If $A$ is an ANR, then $A\times\mathcal{Q}$ is a Hilbert cube manifold.
\[Chapman, [@Ch]\]\[Theorem: Chapmans triangulation of HCMs\]If $X$ is a Hilbert cube manifold, then there is a locally finite polyhedron $K$ for which $X\approx K\times\mathcal{Q}$.
$\mathcal{Z}$-sets and $\mathcal{Z}$-compactifications\[Subsection: Z-sets\]
----------------------------------------------------------------------------
A closed subset $A$ of an ANR $Y$ is a $\mathcal{Z}$*-set* if either of the following equivalent conditions is satisfied:
- There exists a homotopy $H:Y\times\left[ 0,1\right] \rightarrow Y$ such that $H_{0}=\operatorname*{id}_{Y}$ and $H_{t}\left( X\right) \subseteq
Y-A$ for all $t>0$. (We say that $H$ *instantly homotopes* $Y$ off from $A$.)
- For every open set $U$ in $Y$, $U-A\hookrightarrow U$ is a homotopy equivalence.
A $\mathcal{Z}$*-compactification* of a space $X$ is a compactification $\overline{X}=X\sqcup Z$ with the property that $Z$ is a $\mathcal{Z}$-set in $\overline{X}$. In this case, $Z$ is called a $\mathcal{Z}$*-boundary* for $X$. Implicit in this definition is the requirement that $\overline{X}$ be an ANR; moreover, since an open subset of an ANR is an ANR, $X$ itself must be an ANR to be a candidate for $\mathcal{Z}$-compactification. Hanner’s Theorem [@Ha] ensures that every compactification $\overline{X}$ of an ANR $X$, for which $\overline{X}-X$ satisfies either of the negligibility conditions in the definition of $\mathcal{Z}$-set, is necessarily an ANR; hence, it is a $\mathcal{Z}$-compactification. By a similar (but much easier) result in dimension theory, $\mathcal{Z}$-compactification does not increase dimension; so, if $X$ is an ENR, so is $\overline{X}$.
The compactification of $\mathbb{R}
^{n}$ obtained by adding the $\left( n-1\right) $-sphere at infinity is the prototypical $\mathcal{Z}$-compactification. More generally, addition of the visual boundary to a proper CAT(0) space is a $\mathcal{Z}$-compactification. In [@BM], it is shown that, for a torsion-free $\delta$-hyperbolic group $G$, an appropriately chosen Rips complex can be $\mathcal{Z}$-compactified by adding the Gromov boundary $\partial G$.
The purely topological question of when an ANR, an ENR, or even a locally finite polyhedron admits a $\mathcal{Z}$-compactification is an open question (see [@Gu1]). However, Chapman and Siebenmann [@CS] have given a complete classification of $\mathcal{Z}$-compactifiability for Hilbert cube manifolds. That result, in combination with Theorem \[Theorem: Edwards HCM Theorem\], has substantial implications for the general case.
Here we provide a slightly simplified version of the Chapman-Siebenmann theorem. We state the result only for 1-ended Hilbert cube manifolds $X$, since that is all we need in this paper. We also simplify the definitions of $\sigma_{\infty}\left( X\right) $ and $\tau_{\infty}\left( X\right) $ by basing them on a prechosen nested, cofinal sequence of nice neighborhoods of infinity. It is true, but would take some time, to explain why the resulting obstructions do not depend on that choice.
A particularly nice variety of closed neighborhood of infinity $N\subseteq X$ is one that is, itself, a Hilbert cube manifold and whose topological boundary $\operatorname*{Bd}_{X}N$ is a Hilbert cube manifold with a neighborhood in $X$ homeomorphic to $\operatorname*{Bd}_{X}N\times\lbrack-1,1]$. Call such neighborhoods of infinity *clean*. By applying Theorems \[Theorem: Edwards HCM Theorem\] and \[Theorem: Chapmans triangulation of HCMs\], clean neighborhoods of infinity are easily found.
\[Chapman-Siebenmann\]\[CS Theorem\]Let $X$ be a 1-ended Hilbert cube manifold and $\left\{ N_{i}\right\} $ a nested cofinal sequence of connected clean neighborhoods of infinity. Then $X$ admits a $\mathcal{Z}$-compactification if and only if each of the following conditions holds:
1. $X$ is inward tame.
2. $\sigma_{\infty}(X)\in\underleftarrow{\lim}\left\{ \widetilde
{K}_{0}(\mathbb{Z}
\pi_{1}(N_{i})),\lambda_{i\ast}\right\} $ is zero.
3. $\tau_{\infty}\left( X\right) \in\underleftarrow{\lim}^{1}\left\{
\operatorname*{Wh}(\pi_{1}(N_{i})),\lambda_{i\ast}\right\} $ is zero.
The inverse sequences $\left\{ \widetilde{K}_{0}(\mathbb{Z}
\pi_{1}(N_{i})),\lambda_{i\ast}\right\} $ and $\left\{ \operatorname*{Wh}(\pi_{1}(N_{i})),\lambda_{i\ast}\right\} $ in conditions b) and c) are obtained by applying the $\widetilde{K}_{0}$-functor and the $\operatorname*{Wh}$-functor to sequence (\[Defn: pro-pi1\]). The obstruction $\sigma_{\infty}(X)$ is just the sequence $\left( \sigma\left(
N_{0}\right) ,\sigma\left( N_{1}\right) ,\sigma\left( N_{2}\right)
,\cdots\right) $ of Wall finiteness obstructions of the $N_{i}$. Condition a) ensures that each $N_{i}$ is finitely dominated, so the individual obstructions are all defined; without condition a), there is no condition b). Similarly, condition c) requires condition b). It is related to the Whitehead torsion of inclusions $\operatorname*{Bd}_{X}N_{i}\hookrightarrow
\overline{N_{i}-N_{i+1}}$, after the $N_{i}$ have been improved significantly so that those inclusions are homotopy equivalences. The reader should consult [@CS] for details or [@Gu3 §8.2] for a less formal discussion of Theorem \[CS Theorem\]. For our purposes, those details are not so important since the obstructions arising here will be shown to vanish by straightforward applications of Theorem \[Theorem: Bass-Heller-Swan\].
*Condition a) makes sense for an arbitrary ANR* $X$*. If* $X$ *satisfies a) and is sharp at infinity, then condition b) also makes immediate sense; it is satisfied if and only if* $X$ *contains arbitrarily small closed ANR neighborhoods of infinity having finite homotopy type. Condition c) is more problematic; even when a) and b) are satisfied, if* $X$ *is not a Hilbert cube manifold, it may be impossible to find neighborhoods of infinity* $N_{i}$ *with each* $\operatorname*{Bd}_{X}N_{i}\hookrightarrow
\overline{N_{i}-N_{i+1}}$ *a homotopy equivalence—an example can be found in [@GT]. The solution to this problem is to* define *the obstructions for an ANR* $X$ *to be the corresponding obstructions for the Hilbert cube manifold* $X\times\mathcal{Q}$. *Then a)-c) are necessary for* $\mathcal{Z}$*-compactifiability of* $X$*; unfortunately, they are not sufficient. [@Gu1] exhibits a locally finite 2-dimensional polyhedron* $K$ *that satisfies a)-c), but is not* $\mathcal{Z}$*-compactifiable. A suitable characterization of* $\mathcal{Z}$*-compactifiable ANRs is an open question.*
For an ANR $X$, Theorem \[CS Theorem\] allows us to determine whether $X\times\mathcal{Q}$ is $\mathcal{Z}$-compactifiable. The following result, with $\mathbb{I=}\left[ -1,1\right] $, frequently allows us to restore finite-dimensionality.
\[Ferry, [@Fe]\]\[Theorem: Ferry’s stabilization theorem\]If $K$ is a finite-dimensional locally finite polyhedron and $K\times\mathcal{Q}$ is $\mathcal{Z}$-compactifiable, then $K\times\mathbb{I}^{2\cdot\dim K+5}$ is $\mathcal{Z}$-compactifiable.
Topological results\[Section: Topological results\]
===================================================
In this section we prove a variety of topological results that are primary ingredients in the proofs of our main theorems. We have broken the section into three parts: the first contains results about product spaces; the second deals with spaces that admit a proper $\mathbb{Z}
$-action generated by a homeomorphism properly homotopic to the identity; and the third looks at spaces that are simply connected at infinity.
Products of noncompact spaces
-----------------------------
\[Lemma: f.d. cross R is inward tame\]Let $X$ be an ANR that is finitely dominated. Then $X\times\mathbb{R}
$ is inward tame.
Since inward tameness is an invariant of proper homotopy type, we may use Theorem \[Theorem: West’s Theorem\] to reduce to the case that $X$ is a locally finite polyhedron. For that case, the proof given in [@Gu2 Prop. 3.1] for open manifolds is valid, with only minor modifications. With a few additional modifications, the appeal to Theorem \[Theorem: West’s Theorem\] can be eliminated.
The next lemma requires a new definition. We say that an ANR $X$ is *movably finitely dominated* if, for every neighborhood of infinity $N\subseteq X$, there is a self-homotopy of $X$ that pulls $X$ into a compact subset of $N$, i.e., $H:X\times\lbrack0,1]\rightarrow X$ such that $H_{0}=\operatorname*{id}_{X}$ and $\overline{H_{1}\left( X\right) }$ is compact and contained in $N$. The motivation for this definition becomes immediately clear in the following lemma. The most important examples are the simplest—every contractible ANR is movably finitely dominated, since it is dominated by each singleton subset.
\[Lemma: inward tameness of products\]Let $X$ and $Y$ be connected, noncompact, movably finitely dominated ANRs. Then $X\times Y$ is inward tame.
Let $A\subseteq X$ and $B\subseteq Y$ be compact and $N=\overline{(X\times
Y)-(A\times B)}$ the corresponding closed neighborhood of infinity. It suffices to prove:
**Claim.** *There exits a homotopy* $J:N\times\left[
0,1\right] \rightarrow N$ *with* $J_{0}=\operatorname*{id}_{N}$ *and* $\overline{J_{1}\left( N\right) }$ *compact.*
Choose compacta $A^{\prime}\subseteq X$ and $B^{\prime}\subseteq Y$ such that $A\subseteq\operatorname*{int}_{X}A^{\prime}$ and $B\subseteq
\operatorname*{int}_{X}B^{\prime}$, and let $\lambda:X\rightarrow\left[
0,1\right] $ and $\mu:Y\rightarrow\left[ 0,1\right] $ be Urysohn functions with $\lambda\left( A\right) =0=\mu\left( B\right) $ and $\lambda\left(
\overline{X-A^{\prime}}\right) =1=\mu\left( \overline{Y-B^{\prime}}\right)
$. Then choose compact $K\subseteq X-A^{\prime}$ and $L\subseteq Y-B^{\prime}$ along with homotopies $F:X\times\left[ 0,1\right] \rightarrow X$ such that $F_{0}=\operatorname*{id}_{X}$ and $F_{1}\left( X\right) \subseteq K$ and $G:Y\times\left[ 0,1\right] \rightarrow Y$ such that $G_{0}=\operatorname*{id}_{Y}$ and $G_{1}\left( X\right) \subseteq L$.
We will build a homotopy $H$ that pulls $X\times Y$ into a compact subset while fixing $A\times B$. By arranging that tracks of points from $N$ stay in $N$, the restriction of this homotopy will satisfy the claim.
Define $\widehat{F}:X\times Y\times\left[ 0,1\right] \rightarrow X\times Y$ by $\widehat{F}\left( x,y,t\right) =\left( F\left( x,\mu\left( y\right)
\cdot t\right) ,y\right) $ and note that:
- $\widehat{F}_{1}\left( X\times Y\right) \subseteq(X\times B^{\prime
})\cup\left( K\times Y\right) $,
- $\left. \widehat{F}_{t}\right\vert _{X\times B}=\operatorname*{id}$ for all $t$, and
- tracks of points in $N$ stay in $N$.
Similarly, let $\widehat{G}:X\times Y\times\left[ 0,1\right] \rightarrow
X\times Y$ by $\widehat{G}\left( x,y,t\right) =\left( x,G\left(
y,\lambda\left( x\right) \cdot t\right) \right) $ and note that:
- $\widehat{G}_{1}\left( X\times Y\right) \subseteq(A^{\prime}\times
Y)\cup\left( X\times L\right) $,
- $\left. \widehat{G}\right\vert _{A\times Y}=\operatorname*{id}$, and
- tracks of points in $N$ stay in $N$.
Now define $H:X\times Y\times\left[ 0,1\right] \rightarrow X\times Y$ by the rule.$$H_{t}=\left\{
\begin{array}
[c]{cc}\widehat{F}_{3t} & 0\leq t\leq\frac{1}{3}\\
\widehat{G}_{3t-1}\circ\widehat{F}_{1} & \frac{1}{3}\leq t\leq\frac{2}{3}\\
\widehat{F}_{3t-2}\circ\widehat{G}_{1}\circ\widehat{F}_{1} & \frac{2}{3}\leq
t\leq1
\end{array}
\right.$$ A quick check shows that $H_{1}\left( X\times Y\right) $ is contained in the compact set $\widehat{F}_{1}\widehat{G}_{1}(A^{\prime}\times B^{\prime})\cup\left( K\times L\right) $; moreover, since the tracks of $H$ are all concatenations of tracks of $\widehat{F}$ and $\widehat{G}$, $A\times B$ stays fixed and tracks of points from $N$ stay in $N$. Letting $J$ be the restriction of $H$ completes the claim.
\[Corollary: products of noncompact ANRs are inward tame\]The product of any two noncompact ARs is inward tame.
\[Lemma: products that are fg free at infinity\]Let $X$ and $Y$ be noncompact, simply connected ANRs. Then $X\times Y$ contains arbitrarily small path-connected neighborhoods of infinity, each with a fundamental group that is finitely generated and free.
Let $U\subseteq X$ and $V\subseteq Y$ be open neighborhoods of infinity, consisting of finitely many unbounded path-connected components $\left\{
U_{i}\right\} _{i=1}^{k_{1}}$ and $\left\{ V_{j}\right\} _{j=1}^{k_{2}}$, respectively. Then the corresponding rectangular neighborhood of infinity $R=(U\times Y)\cup\left( X\times V\right) $ may be covered by the finite collection of path-connected open sets $\left\{ U_{i}\times Y\right\}
_{i=1}^{k_{1}}\cup\left\{ X\times V_{j}\right\} _{j=1}^{k_{2}}$ in which each of the two subcollections is pairwise disjoint, and each $U_{i}\times Y$ intersects each $X\times V_{j}$ in the path-connected set $U_{i}\times V_{j}$. The nerve of this cover is the complete bipartite graph $K_{k_{1},k_{2}}$ and the connectedness of this graph implies the path-connectedness of $R$. A straight-forward application of the Generalized van Kampen Theorem to the corresponding generalized graph of groups (see [@Ge2 Th.6.2.11]) shows that the fundamental group of $R$ is free on $\left( k_{1}-1\right) \left(
k_{2}-1\right) $ generators, the key observation being that each element of a vertex group $\pi_{1}\left( U_{i}\times Y\right) $ can be represented by a loop in $U_{i}\times V_{j}$ which then contracts in $X\times V_{j}$, and similarly for elements of vertex groups $\pi_{1}\left( X\times V_{j}\right)
$.
\[Theorem: products of HCMs\]Let $X$ and $Y$ be noncompact, simply connected, movably finitely dominated Hilbert cube manifolds. Then $X\times Y$ is $\mathcal{Z}$-compactifiable.
By a combination of Corollary \[Corollary: products of noncompact ANRs are inward tame\], Lemma \[Lemma: products that are fg free at infinity\], and Theorem \[Theorem: Bass-Heller-Swan\], $X\times Y$ satisfies all conditions of Theorem \[CS Theorem\].
Let $P_{1}$ and $P_{2}$ be noncompact, simply connected, moveably finitely dominated, finite-dimensional, locally finite polyhedra. Then $P_{1}\times
P_{2}\times\mathbb{I}^{2(\dim P_{1}+\dim P_{2})+5}$ is $\mathcal{Z}$-compactifiable.
Apply Theorems \[Theorem: Edwards HCM Theorem\] and \[Theorem: products of HCMs\] to $P_{1}\times P_{2}\times\mathcal{Q}$; then use Theorem \[Theorem: Ferry’s stabilization theorem\].
Spaces admitting homotopically simple $\mathbb{Z}
$-actions
-------------------------------------------------
In this section we consider spaces $X$ that admit a proper $\mathbb{Z}
$-action generated by a homeomorphism properly homotopic to $\operatorname*{id}_{X}$. Under the right circumstances, that hypothesis has significant implication for the topology of $X$.
\[Lemma: inward tameness of spaces admitting nice Z-actions\]Let $X$ be an ANR that admits a proper $\mathbb{Z}
$-action generated by a homeomorphism $j:X\rightarrow X$ that is properly homotopic to $\operatorname*{id}_{X}$. Then
1. if the action is cocompact, $X$ is 2-ended;
2. if the action is not cocompact, $X$ is 1-ended; and
3. if $X$ is finitely dominated, then $X$ is inward tame.
By Lemma \[Lemma: mapping torus/Z-action\], $(\left\langle j\right\rangle
\backslash X)\times\mathbb{R}
\approx\operatorname*{Tor}_{j}\left( X\right) $, and since $j\overset
{p}{\simeq}\operatorname*{id}_{X}$, the latter space is proper homotopy equivalent to $X\times\mathbb{S}^{1}$. Now $(\left\langle j\right\rangle
\backslash X)\times\mathbb{R}
$ has either two or one ends, according to whether $\left\langle
j\right\rangle \backslash X$ is compact or noncompact, and since the number of ends is a proper homotopy invariant, the same is true for $X\times
\mathbb{S}^{1}$. Since $X\times\mathbb{S}^{1}$ has the same number of ends as $X$, the first two assertions follow.
Next assume that $X$ is finitely dominated. By Theorem \[Theorem: Mather’s theorem\], $X\times\mathbb{S}^{1}$ has finite homotopy type, so by the above equivalences, $\left\langle j\right\rangle \backslash X$ also has finite homotopy type. By Lemma \[Lemma: f.d. cross R is inward tame\], $(\left\langle j\right\rangle
\backslash X)\times\mathbb{R}
$ is inward tame, and since inward tameness is an invariant of proper homotopy type, $X\times\mathbb{S}^{1}$ is inward tame. It follows that $X$ is inward tame since, if $N$ is a closed neighborhood of infinity in $X$, then $N\times\mathbb{S}^{1}$ is a closed neighborhood of infinity in $X\times
\mathbb{S}^{1}$; and a homotopy that pulls $N\times\mathbb{S}^{1}$ into a compact subset projects to a homotopy that pulls $N$ into a compact subset.
\[Lemma: pro-pi1 of spaces admitting Z-actions\]Let $X$ be a simply connected ANR that admits a proper $\mathbb{Z}
$-action generated by a homeomorphism $j:X\rightarrow X$ that is properly homotopic to $\operatorname{id}_{X}$. Then
1. if the action is cocompact, $X$ is simply connected at each of its two ends, and
2. if the action is not cocompact, $X$ is strongly connected at infinity and $\operatorname*{pro}$-$\pi_{1}\left( \varepsilon\left( X\right)
\right) $ is pro-finitely generated free.
The proof is primarily an application of work done in [@GM2]; we add a few observations to make those results fit our situation more precisely. For both assertions, we again use the equivalences:$$(\left\langle j\right\rangle \backslash X)\times\mathbb{R}
\approx\operatorname*{Tor}\nolimits_{j}\left( X\right) \overset{p}{\simeq
}X\times\mathbb{S}^{1}. \label{Equivalences: Quotient x R versus X x S1}$$
First assume that $\left\langle j\right\rangle \backslash X$ is compact. Then $(\left\langle j\right\rangle \backslash X)\times\mathbb{R}
$ is 2-ended and the natural choices of base rays: $r_{-}=\left\{ p\right\}
\times(-\infty,0]$ and $r_{+}=\left\{ p\right\} \times\lbrack0,\infty)$, along with the natural choice of neighborhoods of infinity $(\left\langle
j\right\rangle \backslash X)\times(-\infty,-n]\cup\lbrack n,\infty)$ yield representations of $\operatorname*{pro}$-$\pi_{1}\left( \varepsilon\left(
(\left\langle j\right\rangle \backslash X)\times\mathbb{R}
\right) ,r_{\pm}\right) $ of the form $\mathbb{Z}
\overset{\operatorname*{id}}{\longleftarrow}\mathbb{Z}
\overset{\operatorname*{id}}{\longleftarrow}\mathbb{Z}
\overset{\operatorname*{id}}{\longleftarrow}\cdots$. The proper homotopy equivalence promised above implies the same for the two ends of $X\times
\mathbb{S}^{1}$. Clearly, that can happen only if $X$ is simply connected at each of its two ends.
In the non-cocompact case, $(\left\langle j\right\rangle \backslash X)\times\mathbb{R}
$ is 1-ended and by [@GM2 Prop. 3.12], with an appropriate choice of base ray, $\operatorname*{pro}$-$\pi_{1}\left( \varepsilon\left( (\left\langle
j\right\rangle \backslash X)\times\mathbb{R}
\right) ,r\right) $ may be represented by an inverse sequence$$F_{0}\times\left\langle a\right\rangle \overset{\lambda_{1}\times
\operatorname*{id}}{\twoheadleftarrow}F_{1}\times\left\langle a\right\rangle
\overset{\lambda_{2}\times\operatorname*{id}}{\twoheadleftarrow}F_{2}\times\left\langle a\right\rangle \overset{\lambda_{3}\times\operatorname*{id}}{\twoheadleftarrow}\cdots\label{Inverse sequence: frees cross Z}$$ where each $F_{i}$ is a finitely generated free group, $\lambda_{i}$ takes $F_{i+1}$ onto $F_{i}$, and $\left\langle a\right\rangle $ is an infinite cyclic group corresponding to a ‘copy’ of $\pi_{1}\left( (\left\langle
j\right\rangle \backslash X)\times\left\{ r_{i}\right\} \right) $, for increasingly large $r_{i}$. Semistability of this sequence implies that $(\left\langle j\right\rangle \backslash X)\times\mathbb{R}
$, and hence $X\times\mathbb{S}^{1}$, is strongly connected at infinity. This allows us to dispense with mention of base rays. It also implies that $X$ is strongly connected at infinity, so $\operatorname*{pro}$-$\pi_{1}\left(
\varepsilon(X\right) )$ is semistable and may be represented by an inverse sequence of surjections $H_{0}\overset{\mu_{1}}{\twoheadleftarrow}H_{1}\overset{\mu_{2}}{\twoheadleftarrow}H_{2}\overset{\mu_{3}}{\twoheadleftarrow}\cdots$. It follows that $\operatorname*{pro}$-$\pi
_{1}(\varepsilon(X\times\mathbb{S}^{1}))$ may be represented by $$H_{0}\times\left\langle t\right\rangle \overset{\mu_{1}\times
\operatorname*{id}}{\twoheadleftarrow}H_{1}\times\left\langle t\right\rangle
\overset{\mu_{2}\times\operatorname*{id}}{\twoheadleftarrow}H_{2}\times\left\langle t\right\rangle \overset{\mu_{3}\times\operatorname*{id}}{\twoheadleftarrow}\cdots\label{Inverse sequence: H-sequence cross Z}$$ where each $\left\langle t\right\rangle $ is the infinite cyclic group corresponding to the $\mathbb{S}^{1}$-factor.
The equivalences of (\[Equivalences: Quotient x R versus X x S1\]) ensure a ladder diagram between subsequences of (\[Inverse sequence: frees cross Z\]) and (\[Inverse sequence: H-sequence cross Z\]). After relabeling to avoid messy subsequence notation, that diagram has the form: $$\begin{diagram} H_{0}\times\left\langle t\right\rangle & & \lTo^{\mu_{1}\times\operatorname*{id}} & & H_{1}\times\left\langle t\right\rangle & & \lTo^{\mu_{2}\times\operatorname*{id}} & & H_{2}\times\left\langle t\right\rangle & & \lTo^{\mu_{3}\times\operatorname*{id}} & & H_{3}\times\left\langle t\right\rangle& \cdots\\ & \luTo ^{u_{0}} & & \ldTo^{d_{1}} & & \luTo ^{u_{1}} & & \ldTo^{d_{2}} & & \luTo^{u_{2}} & & \ldTo^{d_{3}} &\\ & & F_{0}\times\left\langle a\right\rangle & & \lTo^{\lambda_{1}\times\operatorname*{id}} & & F_{1}\times\left\langle a\right\rangle & & \lTo^{\lambda_{2}\times\operatorname*{id}} & & F_{2}\times\left\langle a\right\rangle & & \lTo^ {\lambda_{3}\times\operatorname*{id}} & & \cdots \end{diagram} \label{Diagram: Big ladder}$$ A close look at the homeomorphism between $(\left\langle j\right\rangle
\backslash X)\times\mathbb{R}
$ and $\operatorname*{Tor}\nolimits_{j}\left( X\right) $, as described in [@GG §8], shows that, with appropriate choice of base rays, we may arrange that each $u_{i}$ takes $a$ to $t$. Then, by commutativity, each $d_{i}$ takes $t$ to $a$, each $u_{i}$ takes $F_{i}$ into $H_{i}$, and each $d_{i}$ takes $H_{i}$ into $F_{i-1}$. So diagram (\[Diagram: Big ladder\]) restricts to a diagram of the form $$\begin{diagram} H_{0} & & \lTo^{\mu_{1}} & & H_{1} & & \lTo^{\mu_{2}} & & H_{2} & & \lTo^{\mu_{3}} & & H_{3} & \cdots\\ & \luTo & & \ldTo & & \luTo & & \ldTo & & \luTo & & \ldTo &\\ & & F_{0} & & \lTo^{\lambda_{1}} & & F_{1} & & \lTo^{\lambda_{2}} & & F_{2} & & \lTo^ {\lambda_{3}} & & \cdots \end{diagram} \label{Diagram: Small ladder}$$ which verifies that $\operatorname*{pro}$-$\pi_{1}\left( \varepsilon\left(
X\right) \right) $ is pro-finitely generated free.
\[Theorem: HCMs admitting z-actions\]If a simply connected and finitely dominated Hilbert cube manifold $X$ admits a proper $\mathbb{Z}
$-action generated by a homeomorphism $j:X\rightarrow X$ that is properly homotopic to $\operatorname*{id}_{X}$, then $X$ is $\mathcal{Z}$-compactifiable.
If the action is not cocompact, the previous two lemmas together with Theorem \[Theorem: Bass-Heller-Swan\], ensure that $X$ satisfies the conditions of Theorem \[CS Theorem\]. In the cocompact case, the same lemmas imply that $X$ is inward tame and 2-ended, and that each of those ends is simply connected. In order to use the 1-ended version of Theorem \[CS Theorem\] provided here, split $X$ into a pair of 1-ended Hilbert cube manifolds and apply the theorem to each end individually.
If a simply connected, locally finite polyhedron $P$ is finitely dominated and finite-dimensional, and admits a proper $\mathbb{Z}
$-action generated by a homeomorphism $j:P\rightarrow P$ that is properly homotopic to $\operatorname*{id}_{P}$, then $P\times\mathbb{I}^{2\cdot\dim
P+5}$ is $\mathcal{Z}$-compactifiable.
By Theorem \[Theorem: Edwards HCM Theorem\], $j\times\operatorname*{id}_{\mathcal{Q}}:P\times\mathcal{Q\ }\mathcal{\rightarrow}P\times\mathcal{Q}$ satisfies the hypotheses of Theorem \[Theorem: HCMs admitting z-actions\]. An application of Theorem \[Theorem: Ferry’s stabilization theorem\] completes the proof.
Spaces that are simply connected at infinity
--------------------------------------------
The key result about Hilbert cube manifolds that are simply connected at infinity is our easiest application of Theorem \[CS Theorem\]; it can be found in Chapman and Siebenmann’s original work. For completeness, we include a sketch of their proof.
\[[[@CS Cor. to Th.8]]{}\]\[Theorem: 1-conn at infinity Z-compactifiablility theorem\]If $X$ is a Hilbert cube manifold that is simply connected at infinity and $H_{\ast
}\left( X;\mathbb{Z}
\right) $ is finitely generated, then $X$ is $\mathcal{Z}$-compactifiable.
\[Sketch of Proof\]Due to the triviality of $\operatorname*{pro}$-$\pi
_{1}\left( \varepsilon\left( X\right) \right) $, we need only show that $X$ is inward tame. If $N$ is a clean neighborhood of infinity, then $\operatorname*{Bd}_{X}N$ is homotopy equivalent to a finite complex. Since $H_{i}\left( \operatorname*{Bd}\nolimits_{X}N;\mathbb{Z}
\right) $ and $H_{i}\left( X;\mathbb{Z}
\right) $ are both finitely generated for all $i$, and eventually trivial, the Mayer-Vietoris sequence $$\cdots\rightarrow H_{i}\left( \operatorname*{Bd}\nolimits_{X}N;\mathbb{Z}
\right) \rightarrow H_{i}\left( \overline{X-N};\mathbb{Z}
\right) \oplus H_{i}\left( N;\mathbb{Z}
\right) \rightarrow H_{i}\left( X;\mathbb{Z}
\right) \rightarrow\cdots$$ shows that the same is true for $H_{i}\left( N;\mathbb{Z}
\right) $. Furthermore, the simple connectivity at infinity of $X$, together with standard techniques from Hilbert cube manifold topology, ensure the existence of arbitrarily small simply connected $N$. Since a simply connected complex with finitely generated $\mathbb{Z}
$-homology necessarily has finite homotopy type (see [@Sp p.420]), it follows that $X$ is inward tame.
If $P$ is a finite-dimensional, locally finite polyhedron that is simply connected at infinity and $H_{\ast}\left( P;\mathbb{Z}
\right) $ is finitely generated, then $P\times\mathbb{I}^{2\cdot\dim P+5}$ is $\mathcal{Z}$-compactifiable.
Apply Theorems \[Theorem: Edwards HCM Theorem\], \[Theorem: 1-conn at infinity Z-compactifiablility theorem\], \[Theorem: Ferry’s stabilization theorem\].
Proofs of the main theorems\[Section: Proofs of main theorems\]
===============================================================
We now provide proofs of the unverified theorems from §\[Section: Introduction\]. Theorems \[Theorem 3\] and \[Theorem 4\] require only an assemby of ingredients from §2 and §3, so we begin there.
\[Proof of Theorem \[Theorem 3\]\]Since $\widetilde{K}$ is contractible, both $X$ and $Y$ are also contractible. By Lemmas \[Lemma: inward tameness of products\] and \[Lemma: products that are fg free at infinity\], $X\times Y$ is inward tame and 1-ended with $\operatorname*{pro}$-$\pi_{1}\left( X\times Y,r\right) $ that is pro-finitely generated free, and since $\widetilde{K}\overset
{p}{\simeq}X\times Y$, each of these properties is inherited by $\widetilde
{K}$. Applying Theorems \[Theorem: Edwards HCM Theorem\], \[Theorem: Bass-Heller-Swan\], and \[CS Theorem\] in the usual way provides a $\mathcal{Z}$-compactification of $\widetilde{K}\times\mathcal{Q}$, and since $\dim\widetilde{K}=\dim K<\infty$, Theorem \[Theorem: Ferry’s stabilization theorem\] provides a $\mathcal{Z}$-compactification of the ER $\widetilde{K}\times\mathbb{I}^{2\cdot\dim K+5}$.
\[Proof of Theorem \[Theorem 4\]\]By Lemmas \[Lemma: inward tameness of spaces admitting nice Z-actions\] and \[Lemma: pro-pi1 of spaces admitting Z-actions\], $X$ is inward tame, and either: 2-ended and simply connected at each end; or 1-ended with pro-finitely generated free fundamental group at infinity. By proper homotopy invariance, the same is true for $\widetilde{K}$, so by the usual argument, $\widetilde
{K}\times\mathcal{Q}$ is $\mathcal{Z}$-compactifiable. Another application of Theorem \[Theorem: Ferry’s stabilization theorem\] provides a $\mathcal{Z}$-compactification of $\widetilde{K}\times\mathbb{I}^{2\cdot\dim K+5}$.
*In the special case, where* $X$ *(or* $\widetilde{K}$*) admits a cocompact* $\mathbb{Z}
$*-action, the above argument is overkill. There, since* $X$ *is contractible,* $\left\langle j\right\rangle \backslash X$ *is homotopy equivalent to a circle; and since* $\left\langle j\right\rangle \backslash
X$ *is compact, a homotopy equivalence* $f:\left\langle j\right\rangle
\backslash X\rightarrow S^{1}$ *lifts to a proper homotopy equivalence* $X\overset{p}{\simeq}\mathbb{R}
$*. It is then straightforward to show that the 2-point compactifications of* $X$ *and* $\widetilde{K}$ *are themselves* $\mathcal{Z}$*-compactifications.*
To obtain the full strength of Theorem \[Theorem 5\], we require a new ingredient from [@GG].
\[Proof of Theorem \[Theorem 5\]\]Since $G$ is type $F$, each nontrivial element has infinite order; so we may apply [@GG Th.1.4] to conclude that $G$ is either simply connected at infinity or $G$ is virtually a surface group. In other words, if $K$ is a finite K($G,1$) complex, then $\widetilde{K}$ is either simply connected at infinity, or $\widetilde{K}$ is the universal cover of the corresponding finite K($H,1$) complex $H\backslash\widetilde{K}$, where $H$ is a finite index subgroup of $G$ and $H\cong\pi_{1}\left( S\right) $, where $S$ is a closed surface with infinite fundamental group. (**Note.** By [@CJ] or [@Ga] a torsion-free virtual surface group is, in fact, a surface group; but that fact is not needed here.)
In the case where $\widetilde{K}$ is simply connected at infinity, we may apply Theorem \[Theorem: 1-conn at infinity Z-compactifiablility theorem\] to conclude that $\widetilde{K}\times\mathcal{Q}$ is $\mathcal{Z}$-compactifiable, and hence $\widetilde{K}\times\mathbb{I}^{2\cdot\dim
K+5}$, admits the desired $\mathcal{Z}$-compactification.
In the second case, we may conclude that $\widetilde{K}\overset{p}{\simeq
}\widetilde{S}\approx\mathbb{\mathbb{R}
}^{2}$. It follows that $\widetilde{K}$ is 1-ended and inward tame, with $\operatorname*{pro}$-$\pi_{1}\left( \varepsilon\left( \widetilde{K}\right)
\right) $ stably isomorphic to $\mathbb{Z}
$. By Theorem \[Theorem: Bass-Heller-Swan\], $\widetilde{K}\times
\mathcal{Q}$ satisfies the hypotheses of Theorem \[CS Theorem\], and is therefore $\mathcal{Z}$-compactifiable. Another application of Theorem \[Theorem: Ferry’s stabilization theorem\] completes the proof.
Theorems \[Theorem 2\] is a special case of Theorem \[Theorem 4\], so a proof of Theorem \[Theorem 1\] is all that remains. It is a consequence of Theorem \[Theorem 3\] and the following crucial lemma.
\[Lemma: properties of classifying spaces for group extensions\]Let $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow1$ be a short exact sequence of groups where both $N$ and $Q$ have type F, then $G$ also has type F. Moreover, if $Y$ and $Z$ are finite classifying spaces for $N$ and $Q$, respectively, then $G$ admits a finite classifying space $W^{\prime}$ with the property that $\widetilde{W}^{\prime}$ is proper homotopy equivalent to $\widetilde{Y}\times\widetilde{Z}$.
The first sentence of this lemma follows from techniques laid out in §6.1, §7.1, and §7.2 of [@Ge2]; the second sentence is essentially a restatement of Proposition 17.3.1 of [@Ge2]. Due to their importance in this paper, we provide a guide to those arguments in the following outline. Since Proposition 17.3.1 in [@Ge2] is light on details (and since we had worked out an alternative approach prior to discovering that proposition), we have included an appendix with an alternative proof. A novel aspect of the approach presented there is its use of approximate fibrations.
\[Proof of Lemma \[Lemma: properties of classifying spaces for group extensions\] (outline)\]The construction of a finite K($G,1$) complex is obtained by an application of the Borel construction followed by the Rebuilding Lemma (see [@Ge2 §6.1]). For the Borel construction, begin with a (not necessarily finite) K($G,1$) complex $X$ and let $G$ act diagonally on $\widetilde
{X}\times\widetilde{Z}$ , where the (nonfree) action of $G$ on $\widetilde{Z}$ is induced by the quotient map $G\rightarrow Q$. Since the diagonal action itself is free, the quotient $W=G\backslash\left( \widetilde{X}\times\widetilde{Z}\right) $ is a K($G,1$) complex—probably not finite. Inspection of this quotient space reveals a natural projection map $q:W\rightarrow Z$ that is a fiber bundle with fiber the aspherical CW complex $N\backslash\widetilde{X}$.
Next is the rebuilding stage of the argument. Here the K($G,1$) complex $W$ is rebuilt by replacing each fiber $N\backslash\widetilde{X}$ of the map $q:W\rightarrow Z$ with the homotopy equivalent (but finite) complex $Y$. This is done inductively over the skeleta of $Z$: first a copy of $Y$ is placed over each vertex of $Z$, then over each edge $e$ of $Z$ a copy of $Y\times\left[ 0,1\right] $ is attached with $Y\times0$ being glued to the copy of $Y$ lying over the initial vertex of $e$ and $Y\times1$ glued to the copy of $Y$ lying over the terminal vertex of $e$. From there we move to the 2-cells of $Z$, and so on. At each step, the bundle map $q$ provides instructions for the gluing maps. At the end we have a bundle-like stack of CW complexes $q^{\prime}:W^{\prime}\rightarrow Z$ with each point preimage a copy of $Y$ and a homotopy equivalence $k:W^{\prime}\rightarrow W$. Since both $Z$ and $Y$ are finite complexes, $W^{\prime}$ is a finite complex, so $G$ has type F.
Obtaining a proper homotopy equivalence $h:\widetilde{W}^{\prime}\rightarrow\widetilde{Y}\times\widetilde{Z}$ is an interesting and delicate task. A proof can be found in the appendix; otherwise, the reader is referred to [@Ge2 Prop.17.3.1].
An alternative approach to Lemma \[Lemma: properties of classifying spaces for group extensions\]
=================================================================================================
In this appendix we take a closer look at the proper homotopy equivalence promised in Lemma \[Lemma: properties of classifying spaces for group extensions\] and offer an alternative to the proof suggested in [@Ge2]. Begin with a short exact sequence of groups $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow1$ where both $N$ and $Q$ have type F. Then, as described in the sketched proof of Lemma \[Lemma: properties of classifying spaces for group extensions\], $G$ also has type F. More specifically, if $Y$ is a finite K($N,1$) complex and $Z$ is a finite K($Q,1$) complex, then there is a finite K($G,1$) complex $W^{\prime}$, obtained by an application of the Borel construction followed by the Rebuilding Lemma. As a corollary of the construction, $W^{\prime}$ comes equipped with a map $q:W^{\prime}\rightarrow Z$ for which each point preimage is a copy of $Y$. In fact, for each open $k$-cell $\mathring{e}^{k}$ of $Z$, $q^{-1}\left( \mathring{e}^{k}\right) \approx\mathring{e}^{k}\times Y$.
Although $q:W^{\prime}\rightarrow Z$ is not necessarily a fiber bundle, it exhibits many properties of a fiber bundle; it is a stack of CW complexes over $Z$ with fiber $Y$. If we let $\widehat{W}^{\prime}$ be the intermediate cover of $W^{\prime}$ corresponding to $N\trianglelefteq G$, we get another stack of CW complexes $\widehat{q}:\widehat{W}^{\prime}\rightarrow\widetilde{Z}$ over the contractible space $\widetilde{Z}$. Given the standard fact that a fiber bundle over a contractible space is always a product bundle, it is reasonable to hope that, in the case at hand, $\widehat{W}^{\prime}$ is approximately a product. By using the aptly named theory of approximate fibrations, we will eventually arrive at the following main result of this appendix.
\[Prop: Main result of Appendix\]Given the above setup, $\widehat
{W}^{\prime}$ is proper homotopy equivalent to $Y\times\widetilde{Z}$.
This result is stronger than needed to complete Lemma \[Lemma: properties of classifying spaces for group extensions\], and may be of interest in its own right. Lemma \[Lemma: properties of classifying spaces for group extensions\] is obtained from Proposition \[Prop: Main result of Appendix\] by lifting the promised proper homotopy equivalence to the universal covers. It is worth noting that $W^{\prime}$, itself, is typically not homotopy equivalent to $Y\times Z$.
In this appendix, we first provide a constructive proof of the special case where $Q$ is infinite cyclic; in that case $G$ is a semidirect product $G\cong
N\rtimes_{\varphi}\mathbb{Z}
$. The special case motivates the work to be done later and also illustrates the subtleties that are overcome with the general theory. After completing the special case, we will provide a brief overview of the theory of approximate fibrations. Then we employ that theory to prove Proposition \[Prop: Main result of Appendix\] in full generality.
Mapping tori of self-homotopy equivalences\[Subsection: Mapping tori of self-homotopy equivalences\]
----------------------------------------------------------------------------------------------------
In this section we focus on the special case of Proposition \[Prop: Main result of Appendix\], where $G$ is an extension of the form $1\rightarrow N\rightarrow G\rightarrow\mathbb{Z}
\rightarrow1$; equivalently, $G\cong N\rtimes_{\varphi}\mathbb{Z}
$ for some automorphism $\varphi:G\rightarrow G$. In this case, the Borel/Rebuilding procedure yields a finite K($G$,1) complex that is the mapping torus of a map $f:Y\rightarrow Y$, with $f_{\#}=\varphi$. Since $Y$ is a K($H$,1), $f$ is necessarily a homotopy equivalence. The goal of this section then becomes:
\[Lemma: Universal cover of a mapping torus\]If $K$ is a compact connected ANR and $f:K\rightarrow K$ is homotopy equivalence, then the canonical infinite cyclic cover, $\operatorname*{Tel}\nolimits_{f}\left( K\right) $, of $\operatorname*{Tor}\nolimits_{f}\left( K\right) $ is proper homotopy equivalent to $K\times\mathbb{R}
$.
Let $g:K\rightarrow K$ be a homotopy inverse for $f$ and $B:K\times\left[
0,1\right] \rightarrow K$ with $B_{0}=\operatorname*{id}_{K}$ and $B_{1}=fg$. In accordance with Lemma \[Lemma: mapping cylinders of homotopy equivalences\], our goal is to define a map $u:K\times\mathbb{R}
\rightarrow\operatorname*{Tel}\nolimits_{f}\left( K\right) $, such that there is a proper strong deformation retraction of $\mathcal{M}_{\left[
0,1\right] }\left( u\right) $ onto the domain copy of $K\times\mathbb{R}
$. For each integer $n$, define a function $u_{n}:K\times\left[ n,n+1\right]
\rightarrow\mathcal{M}_{\left[ n,n+1\right] }\left( f\right) $ by the rule:$$u_{n}\left( x,r\right) =q_{\left[ n,n+1\right] }(B_{r-n}\left(
g^{n}\left( x\right) \right) ,r)\text{, when }n\geq0$$ and$$u_{n}\left( x,r\right) =q_{\left[ n,n+1\right] }\left( f^{-n}\left(
x),r\right) \right) \text{, when }n<0\text{.\smallskip}$$ Here it is understood that $g^{0}=\operatorname*{id}_{K}$.
Note that $$u_{-1}\left( x,0\right) \allowbreak=\allowbreak q_{[-1,0]}\left( f\left(
x),0\right) \right) \text{, while }u_{0}\left( x,0\right) \allowbreak
=\allowbreak q_{\left[ 0,1\right] }\left( B_{0}(x),0\right) \allowbreak
=\allowbreak q_{\left[ 0,1\right] }\left( x,0\right)$$ and for each integer $n\geq1$, $$u_{n-1}\left( x,n\right) \allowbreak=\allowbreak q_{[n-1,n]}\left(
B_{1}\left( g^{n-1}\left( x\right) \right) \right) \allowbreak
=\allowbreak q_{[n-1,n]}\left( fgg^{n-1}\left( x\right) ,n\right)
\allowbreak=\allowbreak q_{[n-1,n]}\left( fg^{n}\left( x\right) ,n\right)$$ and $$u_{n}\left( x,n\right) \allowbreak=\allowbreak q_{[n,n+1]}\left(
B_{0}\left( g^{n}\left( x\right) \right) ,n\right) \allowbreak
=\allowbreak q_{[n,n+1]}\left( g^{n}\left( x\right) ,n\right) .$$ Similarly, for each for each integer $n\leq-1$, $$u_{n-1}\left( x,n\right) \allowbreak=\allowbreak q_{[n-1,n]}\left(
f^{-(n-1)}\left( x),n\right) \right) \allowbreak=\allowbreak q_{[n-1,n]}\left( f(f^{-n}\left( x),n\right) \right)$$ and $$u_{n}\left( x,n\right) \allowbreak=\allowbreak q_{[n,n+1]}\left(
f^{-n}\left( x),n\right) \right) \text{.}$$ It follows that the $u_{n}$ can be glued together to obtain a proper map $u:K\times\mathbb{R}
\rightarrow\operatorname*{Tel}\nolimits_{f}\left( K\right) $. See Figure \[Fig 1\].
\[ptb\]
[WeakZ-fig.eps]{}
<span style="font-variant:small-caps;">Claim.</span> **** *There is a proper strong deformation retraction of* $\mathcal{M}_{\left[ 0,1\right] }\left( u\right) $ *onto* $K\times\mathbb{R}
$.
First note that, since $u$ respects $\mathbb{R}
$-coordinates, the natural projections $K\times\mathbb{R}
\rightarrow\mathbb{R}
$ and $p:\operatorname*{Tel}\nolimits_{f}\left( K\right) \rightarrow\mathbb{R}
$ can be extended to a projection $\widehat{p}:\mathcal{M}_{\left[
0,1\right] }\left( u\right) \rightarrow\mathbb{R}
$ with the property that each point preimage $\widehat{p}\,^{-1}\left(
r\right) $ is a mapping cylinder $C_{r}$ of a map from $K\times\left\{
r\right\} $ to $K_{r}$. Indeed, for an integer $n\geq0$, $C_{n}$ is the mapping cylinder of $fg^{n}$ and for an integer $n<0$, $C_{n}$ is the mapping cylinder of $f^{-\left( n-1\right) }$. So each $C_{n}$ is a mapping cylinder of a homotopy equivalence—a fact that will be useful later. (In fact, each $C_{r}$ is a mapping cylinder of a homotopy equivalence, but this fact will only be used for integral values of $r$.) Note also that $\mathcal{M}_{\left[
0,1\right] }\left( u\right) $ may be viewed as a countable union $\bigcup_{n\in\mathbb{Z}
}\mathcal{M}_{[0,1]}\left( u_{n}\right) $, where each $\mathcal{M}_{[0,1]}\left( u_{n}\right) $ intersects $\mathcal{M}_{[0,1]}\left(
u_{n-1}\right) $ in $C_{n}$.
<span style="font-variant:small-caps;">Subclaim.</span> **** *For each* $n$, $\mathcal{M}_{[0,1]}\left( u_{n}\right) $ *strong deformation retracts onto the subset* $C_{n}\cup\left( K\times\left[ n,n+1\right] \right)
_{1}\cup C_{n+1}$.
It suffices to show that $C_{n}\cup\left( K\times\left[ n,n+1\right]
\right) _{1}\cup C_{n+1}\hookrightarrow\mathcal{M}\left( u_{n}\right) $ is a homotopy equivalence. Since $C_{n}$ and $C_{n+1}$ are mapping cylinders of homotopy equivalences, each strong deformation retracts onto its domain end, so $\left( K\times\left[ n,n+1\right] \right) _{1}\hookrightarrow
C_{n}\cup\left( K\times\left[ n,n+1\right] \right) _{1}\cup C_{n+1}$ is a homotopy equivalence; therefore, it is enough to show that $\left(
K\times\left[ n,n+1\right] \right) _{1}\hookrightarrow\mathcal{M}_{\left[
0,1\right] }\left( u_{n}\right) $ is a homotopy equivalence. Note that the inclusions $K_{n}\hookrightarrow C_{n}$, $K_{n}\hookrightarrow\mathcal{M}_{\left[ n,n+1\right] }\left( f\right) $ and $\mathcal{M}_{\left[
n,n+1\right] }\left( f\right) \hookrightarrow\mathcal{M}_{\left[
0,1\right] }\left( u_{n}\right) $ are all homotopy equivalences, since each subspace is the range end of a corresponding mapping cylinder. It follows that $C_{n}\hookrightarrow\mathcal{M}_{\left[ 0,1\right] }\left( u_{n}\right) $ is a homotopy equivalence, and since $K\times\left\{ n\right\}
\hookrightarrow C_{n}$ is a homotopy equivalence it follows that $K\times\left\{ n\right\} \hookrightarrow\mathcal{M}_{\left[ 0,1\right]
}\left( u_{n}\right) $, and hence, $\left( K\times\left[ n,n+1\right]
\right) _{1}\hookrightarrow\mathcal{M}_{\left[ 0,1\right] }\left(
u_{n}\right) $ is a homotopy equivalence. The subclaim follows.
To complete the claim, first properly strong deformation retract $\mathcal{M}_{\left[ 0,1\right] }\left( u\right) $ onto $(K\times\mathbb{R}
)_{1}\cup\left( \bigcup\nolimits_{n\in\mathbb{Z}
}C_{n}\right) $ using the union of the strong deformation retractions provided by the subclaim. Follow that by a strong deformation of $(K\times\mathbb{R}
)_{1}\cup\left( \bigcup\nolimits_{n\in\mathbb{Z}
}C_{n}\right) $ onto $\left( K\times\mathbb{R}
\right) _{1}$ obtained by individually strong deformation retracting each $C_{n}$ onto its domain end.
*The delicate nature of defining* $u:K\times\mathbb{R}
\rightarrow\operatorname*{Tel}\nolimits_{f}\left( K\right) $*, in the above proof, hints at the subtelty of Lemma \[Lemma: properties of classifying spaces for group extensions\].*
Approximate fibrations
----------------------
We now review the main definitions and a few fundmental facts from the theory of approximate fibrations—a theory developed by Coram and Duvall [@CD1],[@CD2] to generalize the notions of fibration and fiber bundle.
A proper surjective map $p:E\rightarrow B$ between (locally compact, metric) ANRs is an *approximate fibration* if it satisfies the following *approximate lifting property*:
*For every homotopy* $H:X\times\left[ 0,1\right] \rightarrow
B$*, map* $h:X\rightarrow E$ *with* $\pi h=H_{0}$*, and open cover* $\mathcal{U}$ *of* $B$*, there exists* $\overline{H}:X\times\left[ 0,1\right] \rightarrow E$ *such that* $\overline{H}_{0}=h$ *and* $p\overline{H}$ *is* $U$*-close to* $H$ *(that is, for each* $\left( x,t\right) \in X\times\left[ 0,1\right]
$ *there exists* $U\in\mathcal{U}$ *containing both* $H\left(
x,t\right) $ *and* $p\overline{H}\left( x,t\right) $*).*
For each $b\in B$, $F_{b}:=$ $p^{-1}\left( b\right) $ is called a *fiber*. Approximate fibrations allows for fibers with bad local properties; however, the theory is easier, but still rich, when fibers are ANRs. Since fibers of the maps considered in this paper are always ANRs (in fact, finite CW complexes), we will focus on that special case. In this context, there is a particularly nice criterion for recognizing an approximate fibration.
Suppose $p:E\rightarrow B$ is a proper map between connected ANRs with ANR fibers. Then, for each fiber $F_{b}$, some neighborhood $U_{b}$ retracts onto $F_{b}$, and for points $b^{\prime}$ sufficiently close to $b$, this induces a map of $F_{b^{\prime}}$ to $F_{b}$. By [@CD2], $p:E\rightarrow B$ is an approximate fibration if and only if each $b\in B$ has a neighborhood over which each of these induced fiber maps is a homotopy equivalence.
If $f:K\rightarrow K$ is homotopy equivalence of a compact connected ANR to itself, the above criterion is easily applied to show that quotient maps $p_{1}:\mathcal{M}_{\left[ a,b\right] }\left( f\right) \rightarrow\left[
a,b\right] $, $p_{2}:\operatorname*{Tor}\nolimits_{f}\left( K\right)
\rightarrow\mathbb{S}^{1}$, and $p_{3}:\operatorname*{Tel}\nolimits_{f}\left(
K\right) \rightarrow\mathbb{R}
$ are all approximate fibrations. In the simple case where $K$ is an arc and $f$ is a constant map, these projections are not actual fibrations. In the case where $K$ is a bouquet of circles and $f$ is an arbitrary map inducing a $\pi_{1}$-isomorphism, the examples are already of group theoretic interest.
As with the case of ordinary fibrations, approximate fibrations give rise to homotopy long exact sequences [@CD1]. When no restrictions are placed on the fibers, these sequence involve the shape (or Čech) homotopy groups of the fiber; when the fibers are ANRs that technicality vanishes and we have:
Let $p:E\rightarrow B$ be an approximate fibration between connected ANRs with connected ANR fibers. Then, for any $b\in B$, there is a long exact sequence $$\cdots\rightarrow\pi_{k+1}\left( B\right) \rightarrow\pi_{k}\left(
F_{b}\right) \overset{i_{\#}}{\rightarrow}\pi_{k}\left( E\right)
\overset{p_{\#}}{\rightarrow}\pi_{k}\left( B\right) \rightarrow\pi
_{k-1}\left( F_{b}\right) \rightarrow\cdots$$ where $i$ is the inclusion map.
We now prove a general fact about approximations that is almost tailor-made for proving Proposition \[Prop: Main result of Appendix\].
\[Theorem: approx. fibrations over contractible base\]Let $p:E\rightarrow
B$ be an approximate fibration between connected ANRs with connected ANR fibers and let $b\in B$. If $B$ is contractible then $E$ is proper homotopy equivalent to $F_{b}\times B$.
By the homotopy long exact sequence and an application of the Whitehead Theorem, $F_{b}\overset{i}{\hookrightarrow}E$ is a homotopy equivalence. Let $r_{b}:E\rightarrow F_{b}$ be a homotopy inverse that retracts $E$ onto $F_{b}$. We will observe that $r_{b}\times p:E\rightarrow F_{b}\times B$ is a proper homotopy equivalence.
Clearly $r_{b}\times p$ is proper, and by contractibility of $B$, it is a homotopy equivalence. We will show that it is a proper homotopy equivalence by exhibiting cofinal sequences of neighborhoods of infinity in the domain and range, respectively, such that $r_{b}\times p$ restricts to homotopy equivalences between corresponding entries. An application of the Proper Whitehead Theorem [@Ge2 Th.17.1.1] or [@Si Prop.IV] (applied to the mapping cylinder of $r_{b}\times p$) completes the proof.
Let $\left\{ V_{i}\right\} _{i=0}^{\infty}$ be a cofinal nested sequence of neighborhoods of infinity in $B$. For convenience, assume that $B$ is 1-ended and that each $V_{i}$ is chosen to be connected; we will return to the general case momentarily. For each $i$, let $U_{i}=p^{-1}\left( V_{i}\right) $. Then $\left\{ U_{i}\right\} $ and $\left\{ F_{b}\times V_{i}\right\} $ are nested cofinal sequence of neighborhoods of infinity in $E$ and $F_{b}\times
B$, respectively. Morevover, $r_{b}\times p$ retricts to a map of $U_{i}$ to $F_{b}\times V_{i}$, for each $i$.
Note that the restriction $p_{i}:U_{i}\rightarrow V_{i}$ is, itself, an approximate fibration. By choosing $b$ to lie in $V_{i}$, and recalling that the composition $F_{b}\hookrightarrow U_{i}\hookrightarrow E$ induces $\pi
_{k}$-isomorphisms, for all $k$, we see that the long exact sequence for $p_{i}:U_{i}\rightarrow V_{i}$ yields a short exact sequence$$1\rightarrow\pi_{k}\left( F_{b}\right) \overset{i_{\#}}{\rightarrow}\pi
_{k}\left( U_{i}\right) \overset{p_{i\#}}{\rightarrow}\pi_{k}\left(
V_{i}\right) \rightarrow1$$ for each $k$. Since $U_{i}$ retracts onto $F_{b}$, these sequences split; so $\pi_{k}\left( U_{i}\right) \cong\pi_{k}\left( F_{b}\right) \times\pi
_{k}\left( V_{i}\right) $. From there it is easy to see that each restriction of $r_{b}\times p$ induces isomorphisms $\pi_{k}\left(
U_{i}\right) \rightarrow\pi_{k}\left( F_{b}\times V_{i}\right) $, completing the proof.
If $B$ has more than one end, one simply applies the above argument to individual components of the $V_{i}$.
\[Proof of Proposition \[Prop: Main result of Appendix\]\]By Theorem \[Theorem: approx. fibrations over contractible base\], it suffices to show that the stack of CW complexes $\widehat{q}:\widehat{W}^{\prime}\rightarrow\widetilde{Z}$ is an approximate fibration. Since each point preimage is a copy of the finite complex $Y$, we need only check that these fibers line up homotopically in the sense of the approximate fibration recognition criterion described above.
From the initial Borel/Rebuilding construction, it is clear that the inclusion of each fiber into $\widehat{W}^{\prime}$ indicies a $\pi_{1}$-isomorphism; and, since both $Y$ and $\widehat{W}^{\prime}$ are aspherical, these inclusions are homotopy equivalences. So any retraction of $\widehat
{W}^{\prime}$ onto a fiber $Y$ restricts to homotopy equivalences between fibers. The recognition criterion follows.
[999]{}
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[^1]: Work on this project was aided by a Simons Foundation Collaboration Grant.
[^2]: The definition of $\mathcal{Z}$-set, along with definitions of numeous other terms used in the introduction, can be found in §\[Section: Background\].
[^3]: The definition of derived limit can be generalized to include nonableian groups (see [@Ge2 §11.3]), but that will not be needed in this paper.
[^4]: In this case, $X$ is called *sharp at infinity*. Most commonly arising ANRs, for example: locally finite polyhedra, manifolds, proper CAT(0) spaces, and Hilbert cube manifolds) are sharp at infinity.
|
---
bibliography:
- 'main.bib'
title: 'From Safety To Termination And Back: SMT-Based Verification For Lazy Languages'
---
[ ]{}
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---
abstract: |
We present simultaneous [[Chandra]{}]{}-[High Energy Transmission Gratings]{} ([[HETG]{}]{}) and [Rossi X-ray Timing Explorer]{} ([[RXTE]{}]{}) observations of a moderate flux ‘soft state’ of the black hole candidate [[4U 1957+11]{}]{}. These spectra, having a minimally discernible hard X-ray excess, are an excellent test of modern disk atmosphere models that include the effects of black hole spin. The [[HETG]{}]{} data show, by modeling the broadband continuum and direct fitting of absorption edges, that the soft disk spectrum is only very mildly absorbed with ${\rm N_H} =
1$–$2 \times 10^{21}~{\rm cm}^{-2}$. These data additionally reveal $\lambda\lambda13.449$ absorption consistent with the warm/hot phase of the interstellar medium. The fitted disk model implies a highly inclined disk around a low mass black hole rapidly rotating with normalized spin $a^* \approx 1$. We show, however, that pure Schwarzschild black hole models describe the data extremely well, albeit with large disk atmosphere “color-correction” factors. Standard color-correction factors can be attained if one additionally incorporates mild Comptonization. We find that the [[Chandra]{}]{} observations do not uniquely determine spin, even with this otherwise extremely well-measured, nearly pure disk spectrum. Similarly, [XMM-Newton]{}/[[RXTE]{}]{} observations, taken only six weeks later, are equally unconstraining. This lack of constraint is partly driven by the unknown mass and unknown distance of [[4U 1957+11]{}]{}; however, it is also driven by the limited bandpass of [[Chandra]{}]{} and [XMM-Newton]{}. We therefore present a series of 48 [[RXTE]{}]{} observations taken over the span of several years and at different brightness/hardness levels. These data prefer a spin of $a^* \approx
1$, even when including a mild Comptonization component; however, they also show evolution of the disk atmosphere color-correction factors. If the rapid spin models with standard atmosphere color-correction factors of ${\rm h_d} = 1.7$ are to be believed, then the [[RXTE]{}]{} observations predict that [[4U 1957+11]{}]{} can range from a 3${{\rm M_\odot}}$ black hole at 10kpc with $a^*\approx 0.83$ to a 16${{\rm M_\odot}}$ black hole at 22kpc with $a^* \approx 1$, with the latter being statistically preferred at high formal significance.
author:
- |
Michael A. Nowak, Adrienne Juett, Jeroen Homan, Yangsen Yao,\
Jörn Wilms, Norbert S. Schulz, Claude R. Canizares
title: |
Disk Dominated States of [[4U 1957+11]{}]{}: [*Chandra*]{}, [*XMM*]{}, and [*RXTE*]{} Observations\
of Ostensibly the Most Rapidly Spinning Galactic Black Hole
---
Introduction {#sec:intro}
============
Despite having an X-ray brightness comparable to and occasionally exceeding that of [[LMC X-1]{}]{} or [[LMC X-3]{}]{}, and despite also being one of the few persistent black hole candidates (BHC), [[4U 1957+11]{}]{} has received relatively little attention. Similar to [[LMC X-3]{}]{} [see, e.g., @wilms:01a], [[4U 1957+11]{}]{} is usually in a soft state that shows long term (hundreds of days) variations [@nowak:99d; @wijnands:02c]. Long term variations also have been seen in its optical lightcurve: modulation over its 9.33hr orbital period has ranged from $\pm 10\%$ and sinusoidal [@thorstensen:87a] to $\pm 30\%$ and complex [@hakala:99a]. @hakala:99a interpret these changes in the optical lightcurve as evidence for an accretion disk with a large outer rim, possibly due to a warp, being nearly edge-on and partly occulted by the secondary. The lack of any X-ray evidence for binary orbital modulation [@nowak:99d; @wijnands:02c] indicates that the orbital inclination cannot exceed $\approx 75^\circ$.
Observations with [Ginga]{} showed a soft spectrum that could be fit with a multi-temperature disk blackbody [[ diskbb]{}; @mitsuda:84a] plus a power law tail with photon index $\Gamma \approx 2$–3 [@yaqoob:93a]. The power-law component in those observations comprised up to 25% of the 1-18keV flux. Later [[RXTE]{}]{} observations also were consistent with a disk blackbody model, but showed no evidence of either a hard component or any X-ray variability above background fluctuations [@nowak:99d]. @wijnands:02c conducted a series of Target of Opportunity observations designed to catch [[4U 1957+11]{}]{} at the high end of its count rate as determined by the [All Sky Monitor]{} ([[ASM]{}]{}) on-board of [[RXTE]{}]{}. They found that at its highest flux, [[4U 1957+11]{}]{} exhibited both a hard tail and mild X-ray variability.
For all of the above cited X-ray observations, the disk parameters had two attributes in common: the best fit inner disk temperatures were high (up to 1.7keV), and the fitted normalizations were low ($\sim
10$). Nominally, the disk blackbody normalization corresponds to $(R_{\rm km}/D_{\rm 10})^2 \cos\theta$, where $R_{\rm km}$ is the disk inner radius in [km]{}, $D_{\rm 10}$ is the source distance in units of 10kpc, and $\theta$ is the inclination of the disk. In physical models, such large temperatures with such low normalizations normally can be achieved only by a combination of large distance, high inclination, low black hole mass, high accretion rate, and possibly rapid black hole spin. Disk temperatures increase as the 1/4 power of fractional Eddington luminosity, but decrease as the 1/4 power of black hole mass. Thus, lower mass serves to increase the temperature and decrease the normalization via decreasing $R_{\rm
km}$. Large distances and high inclination also serve to reduce the normalization. (Again, the lack of X-ray eclipses limits $i{\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}75^\circ$.) Finally, rapid spin yields higher efficiency in the accretion disk (and thus higher temperatures), and a decreased inner disk radius.
As pointed out by a number of authors, however, the [diskbb]{} model is not a self-consistent description of a physical accretion disk [see, for example, @li:05a; @davis:05a; @davis:06a]. Several authors have developed more sophisticated models that incorporate a torque (or lack thereof) on the inner edge of the disk, atmospheric effects on the emerging disk radiation field, the effects of limb darkening and returning radiation (due to gravitational light bending), and the effects of black hole spin. Two of these models are the [kerrbb]{} model of @li:05a and the [bhspec]{} model of @davis:05a. Using these models on soft-state BHC spectra with minimal hard tails, various claims have been made as to their ability to observationally constrain black hole spin [e.g., @shafee:06a; @davis:06a; @mcclintock:06a; @middleton:06a]. Given the apparent dominance of a soft spectrum in [[4U 1957+11]{}]{}, and the typical weakness of any hard tail, [[4U 1957+11]{}]{} becomes an excellent testbed for assessing the ability of these models to constrain physical parameters.
The outline of this paper is as follows. First, we describe the analysis procedure for the [[Chandra]{}]{}, [XMM-Newton]{}, and [[RXTE]{}]{} data (§\[sec:data\]). We then discuss fits to the [[Chandra]{}]{} data with both phenomenological (i.e., [diskbb]{}) and more physically motivated (i.e., [kerrbb]{}) models (§\[sec:broad\]). We then discuss the [XMM-Newton]{} data, and specifically look at the absorption edge regions in both the [[Chandra]{}]{} gratings spectra and the [XMM-Reflection Gratings Spectrometer]{} ([RGS]{}) spectra (§\[sec:edge\]). We then discuss spectral fits to 48 observations from the [[RXTE]{}]{} archives (§\[sec:rxte\_spectra\]), and we consider the variability properties of these observations (§\[sec:rxte\_vary\]). Finally, we present our conclusions and summary (§\[sec:discuss\]).
Data Preparation {#sec:data}
================
Chandra {#subsec:chandra}
-------
[[4U 1957+11]{}]{} was observed by [[Chandra]{}]{} on 2004 Sept. 7 (ObsID 4552) for 67ksec (two binary orbital periods). The [High Energy Transmission Gratings]{} [[[HETG]{}]{}; @canizares:05a] were inserted. The [[HETG]{}]{} is comprised of the [High Energy Gratings]{} ([[HEG]{}]{}), with coverage from $\approx 0.7$–8keV, and the [Medium Energy Gratings]{} ([[MEG]{}]{}), with coverage from $\approx 0.4$–8keV. The data readout mode was Timed Exposure-Faint. To minimize pileup in the gratings spectra, a 1/2 subarray was applied to the CCDs, and the aimpoint of the gratings was placed closer to the CCD readout. This configuration reduces the readout time to 1.741sec, without any loss of the dispersed spectrum.
We used [CIAO v3.3]{} and [CALDB v3.2.0]{} to extract the data and create the spectral response files[^1]. The location of the center of the [$0^{\rm th}$]{} order image was determined using the [ findzo.sl]{} routine[^2], which provides $\approx 0.1$ pixel ($\approx 0.001$Å, for [[MEG]{}]{}) accuracy. The data were reprocessed with pixel randomization turned off, but pha randomization left on. We applied the standard grade and bad pixel file filters, but we did not destreak the data. (We find that for sources as bright as [[4U 1957+11]{}]{}, destreaking the data can remove real photon events, while order sorting is already very efficient at removing streak events.)
Although our instrumental set up was designed to minimize pileup, it is still present in both the [[MEG]{}]{} and [[HEG]{}]{} spectra. Pileup serves to reduce the peak of the [[MEG]{}]{} spectra, which occurs near 2keV, by approximately 13%, while it reduces the peak of the [[HEG]{}]{} spectra by approximately 3%. In the Appendix we describe how we incorporate the effects of pileup in our model fits to the [[Chandra]{}]{} data.
All analyses, figures, and tables presented in this work were produced using the [Interactive Spectral Interpretation System]{} ([[ISIS]{}]{}; @houck:00a). (Note: all ‘unfolded spectra’ shown in the figures were generated using the model-independent definition provided by [[ISIS]{}]{}; see @nowak:05a for further details.) For our $\chi^2$-minimization fits to [[Chandra]{}]{} data, we take the statistical variance to be the predicted model counts, as opposed to the observed data counts. (Choosing the former aids in our Bayesian line searchs; see §\[sec:edge\].) Additionally, we evaluated the model for each individual gratings arm, but calculated the fit statistic using the [[ISIS]{}]{} [combine\_datasets]{} function to add the data from the gratings arms[^3]. The [[Chandra]{}]{} data show very uniform count rates and spectral colors over the course of the observation, therefore throughout we use data from the entire 67ksec observation.
XMM-Newton {#subsec:xmm}
----------
[[4U 1957+11]{}]{} was observed by [[XMM-Newton]{}]{} on 2004 Oct. 16 (ObsID 206320101) for 45ksec. [[XMM-Newton]{}]{} carries three different instruments, the European Photon Imaging Cameras [[EPIC]{}; @struder:01a; @turner:01a] the Reflection Grating Spectrometers [[RGS]{}; @herder:01a], and the Optical Monitor [[OM]{}; @mason:01a]. The [OM]{} was not used in this analysis.
The [EPIC]{} instruments consist of 3 CCD cameras, [MOS-1]{}, [MOS-2]{}, and [pn]{}, each of which provides imaging, spectral and timing data. The [pn]{} and [MOS-1]{} cameras were run in timing mode which provides high time resolution event information by sacrificing 1-dimension in positional information. The [MOS-2]{} camera was run in full-frame mode. The [EPIC]{} instruments provide good spectral resolution ($\Delta E$$=$50–200eV FWHM) over the 0.3–12.0 keV range. There are two [RGS]{} detectors onboard [[XMM-Newton]{}]{} which provide high-resolution spectra ($\lambda/\Delta\lambda$$=$100–500 FWHM) over the 5–38Å range. The grating spectra are imaged onto CCD cameras similar to the [EPIC-MOS]{} cameras which allows for order sorting of the high-resolution spectra.
The [[XMM-Newton]{}]{} data were reduced using [SAS version 7.0]{}. Standard filters were applied to all [[XMM-Newton]{}]{} data. The [EPIC-pn]{} data were reduced using the procedure [epchain]{}. We reduced the [EPIC-MOS]{} data using [emchain]{}. Source and background spectra for the [pn]{} and [MOS-1]{} data were extracted using filters in the one spatial coordinate, [RAWX]{}. Response files were created using the [SAS]{} tools [rmfgen]{} and [ arfgen]{}. The [MOS-2]{} data were found to be considerably piled up and were therefore not used in this analysis.
The [RGS]{} data were reduced using [rgsproc]{}, which produced standard first order source and background spectra and response files for both detectors.
RXTE {#subsec:rxte}
----
During the past ten years, [[RXTE]{}]{} has observed [[4U 1957+11]{}]{} numerous times. The first of these observations was presented in @nowak:99d. A series of observations, specifically triggered to observe high flux states, were presented in @wijnands:02c. One [[RXTE]{}]{} observation was scheduled to occur simultaneously with the [[Chandra]{}]{} observation (PI: Nowak), while another was scheduled to occur simultaneously with the [[XMM-Newton]{}]{} observation (PI: Homan). The remaining observations come from different monitoring campaigns[^4], conducted by various groups. We obtained all of these data from the archives – 108 ObsIDs, 4 of which we exclude as they occurred during a brief period when [[RXTE]{}]{} experienced poor attitude control. We combine ObsIDs wherein the observations were performed within a few days of one another and color-intensity diagrams indicate little or no evolution of the source spectra, yielding 48 spectra.
The data were prepared with the tools from the HEASOFT v6.0 package. We used standard filtering criteria for data from the [*Proportional Counter Array*]{} ([[PCA]{}]{}). Specifically, we excluded data from within 30 minutes of South Atlantic Anomaly (SAA) passage, from whenever the target elevation above the limb of the earth was less than $10^\circ$, and from whenever the electron ratio (a measure of the charged particle background) was greater than 0.15. Owing to the very modest flux of [[4U 1957+11]{}]{}, we used the background models appropriate to faint data. We also extracted data from the [*High Energy X-ray Timing Experiment*]{} ([[HEXTE]{}]{}), but as [[4U 1957+11]{}]{} is both faint and very soft, none of these data were of sufficient quality to use for further analysis.
We applied 0.5% systematic errors to all [[PCA]{}]{} channels, added in quadrature to the errors calculated from the data count rate. For all [[PCA]{}]{} fits, we grouped the data, starting at $\ge 3$keV, with the criteria that the signal-to-noise (after background subtraction, but excluding systematic errors) in each bin had to be $\ge 4.5$. We then only considered data for which the lower boundary of the energy bin was $\ge 3$keV, and the upper boundary of the energy bin was $\le
18$keV. This last criterion yielded upper cutoffs ranging from 12–18keV.
Spectra Viewed with Chandra and XMM-Newton {#sec:broad}
==========================================
Broadband Spectra
-----------------
Our main goals in modeling the [[Chandra]{}]{} data are to describe the broad band continuum, accurately fit the edge structure due to interstellar and/or local-system absorption, and to search for narrow emission and absorption line features. To accurately describe the absorption due to the interstellar medium, we use [tbnew]{}, an updated version of the absorption model of @wilms:00a, which models the Fe $L_2$ and $L_3$ edges, and includes narrow resonance line structure in the Ne and O edges, as have been observed at high spectral resolution with [[Chandra]{}]{}-[[HETG]{}]{} [@juett:04a; @juett:06a abundances and depletion factors used here have been set to be consistent with these previous studies, i.e., they take on the default parameter values of the model, with the exception of the Fe abundance parameter being set to 0.647, and the Fe and O depletion parameters both being set to 1].
We began by jointly modeling both the [[HEG]{}]{} and the [[MEG]{}]{} spectra; however, as seen in Fig. \[fig:combo\], we find systematic differences (up to $\approx 5$%, which is less than the statistical noise level shown in the figure) between the gratings arms, especially at low energies, which cannot be accounted for via the pileup modeling. Specifically, the [[HEG]{}]{} spectra show a greater soft excess, such that fits to solely those data in the 0.7–8keV range with an absorbed [diskbb]{} model yield no absorption, contrary to the edge structure detected by [[MEG]{}]{} below 0.7keV. Given the disagreements with [[MEG]{}]{}, the lack of a fittable column in [[HEG]{}]{}, the lack of apparent lines in the 1–8keV region (see below), and the fact that the [[MEG]{}]{} allows us to model line and edge structure in the 0.5–1keV region, we shall only consider [[MEG]{}]{} data.
Systematic disagreements were also found between the simultaneous [[Chandra]{}]{} and [[RXTE]{}]{} data. The [[RXTE]{}]{} data were slightly softer in the regions of overlap ($\approx 4$–7keV), and owing to the very large effective area of [[RXTE]{}]{}, we found that the latter instrument dominated any joint fits. The two instruments fit qualitatively similar models with comparable parameters; however, no joint model, even when applying systematic errors to the [[PCA]{}]{} data, produced formally acceptable fits. The differences between the instruments are likely attributable to calibration uncertainties for both, and therefore we chose to analyze the [[Chandra]{}]{} and [[RXTE]{}]{} data independently.
For our subsequent broadband spectral fits to the [[MEG]{}]{} data, we grouped the $\pm1^{\rm st}$ orders to a combined signal-to-noise of $\ge 6$ and a minimum of two wavelength channels (i.e., 0.01Å) per bin. We then fit the data in the 0.45–7.75keV region. Results are presented in Table \[tab:meg\_kerrbb\]. Overall, the spectra are well-described by a very mildly absorbed (${\rm N_H} \approx
1$–$1.2\times 10^{21}~{\rm cm^{-2}}$) [diskbb]{} spectrum, with a fairly high temperature (${\rm kT_{in}} = 1.75$keV), but low normalization (${\rm A_{dbb}} =7.2$), with little need for additional features such as a narrow or broad FeK$\alpha$ line (cf. §\[sec:edge\]). Formally, if we add a line with fixed energy and width of 6.4keV and 0.5keV, respectively, the 90% confidence upper limit the line flux is $7\times10^{-5}$ phcm$^{-2}$s$^{-1}$.
The low disk model normalization could correspond to a combination of low black hole mass, high inclination, and large distance. The fitted temperature is on the high side for a ‘soft state’ BHC. Fits to soft states of LMC X-1 and LMC X-3, for example, have consistently found ${\rm kT_{in}} {\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}1.25$keV [@nowak:01a; @wilms:01a; @davis:06a], while our own fits to [[RXTE]{}]{}and [[ASCA]{}]{} data of [[4U 1957+11]{}]{} have found ${\rm kT_{in}} {\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}1.6$keV. Such a high temperature might be taken as a hallmark of some combination of high accretion rate, low black hole mass, and/or high black hole spin. Another possibility is the presence of a Comptonizing corona, serving to harden any disk spectrum.
To show, at least qualitatively, how the presence of a corona would affect our disk fit parameters, we added the [comptt]{} Comptonization model [@titarchuk:94a], to the [diskbb]{} model, even though the disk fit residuals do not indicate the necessity of an extra component. Given the quality of the simple disk fit, we chose to reduce the number of free parameters by tying the temperature of the seed photons input to the corona to the temperature of the disk inner edge. Furthermore, we froze the temperature of the corona itself to 50keV. The additional fit parameters were then the coronal plasma optical depth, $\tau_{\rm p}$ (with its lower limit set to 0.01), and the normalization of the coronal spectrum. Results of this fit are shown in Fig. \[fig:comptt\] and are presented in Table \[tab:meg\_kerrbb\].
table1
Although the improvement to the fit is only very mild ($\Delta
\chi^2=11$), we see that the temperature of the [diskbb]{} component drops to 1.25keV, while its normalization constant increases by more than a factor of 2. The [comptt]{} component now comprises $\approx
30\%$ of the implied bolometric flux. Furthermore, at energies ${\mathrel{ \rlap{\raise 0.511ex \hbox{$>$}}{\lower 0.511ex \hbox{$\sim$}}}}3$keV it is consistent with a $\Gamma \approx 2$ photon index power law; however, this latter fact is tempered by the limited bandpass of [[Chandra]{}]{} at these energies. In a “physical interpretation” of these results, we see that a very weak, mild corona can qualitatively replace the need for an unusually hot, low normalization disk.
As an alternative for explaining the high disk temperature, we can consider spin of the black hole. Although the simple, two parameter [diskbb]{} model does an extremely good job of modeling the [[MEG]{}]{}spectra, it has a number of shortcomings from a theoretical perspective [see @davis:06a]. Its temperature profile is too peaked at the inner edge, its implied radiative efficiency does not match realistic expectations, it incorporates no atmospheric physics, and it includes no (special or general) relativistic effects. In response to these model shortcomings, several authors have developed more sophisticated disk models, specifically the [kerrbb]{} model of @li:05a and the [bhspec]{} model of @davis:06a. Here we consider the former model.
The [kerrbb]{} model increases the number of disk parameters from two to seven, and it allows other effects to be included, such as limb darkening and returning radiation to the disk from gravitational light bending (both turned on for fits described in this paper). The fit parameters are the mass, spin ($a^*$), and distance to the black hole, the mass accretion rate through the disk (${\rm \dot M}$), the inclination of the accretion disk to the line of sight, a torque applied to the inner edge of the disk (here set to zero), and a spectral hardening factor (${\rm h_d}$). The latter is to absorb uncertainties of the disk atmospheric physics, and represents the ratio of the disk’s color temperature to its effective temperature. Preferred values have been ${\rm h_d} \sim 1.7$ [@li:05a; @shafee:06a; @mcclintock:06a], while other models (e.g., [bhspec]{}) attempt to calculate color corrections from first principles [@davis:06a].
To reduce the fit complexity of the [kerrbb]{} model to match the two parameters of the [diskbb]{} model, one typically invokes knowledge obtained from other observations (i.e., for system mass, distance, and inclination) and from theoretical expectations (i.e., for inner edge torque and ${\rm h_d}$). The former is lacking for [[4U 1957+11]{}]{}. Here we fix the inclination to $75^\circ$, to be consistent with the both the optical and X-ray lightcurve behavior [@hakala:99a; @nowak:99d]. Furthermore, we fix the mass to 3${{\rm M_\odot}}$ and the distance to 10kpc. We discuss these choices further in §\[sec:rxte\_spectra\] and §\[sec:discuss\], but here we note that these latter choices allow the closest possible distance to [[4U 1957+11]{}]{}, while still maintaining a minimum inferred bolometric luminosity of $> 3\%\,{\rm L_{Edd}}$. Finally, we set the torque parameter to 0, as its value is essentially subsumed by degeneracies with the fitted mass accretion rate [@li:05a; @shafee:06a].
Allowing the spectral hardening factor, ${\rm h_d}$ to remain free, we obtain a [kerrbb]{} fit to the [[MEG]{}]{} spectra that is of comparable quality to the simple [diskbb]{} fit. We note, however, that the fitted value of ${\rm h_d}=1.22$ is smaller than the commonly preferred value of 1.7 [see @done:08a], and that the black hole spin is found to be near unity ($a^* = 0.9999$). Again, the latter is being driven by the fact that the fitted [diskbb]{} temperature is itself rather high for a soft state BHC system. Freezing ${\rm h_d}=1.7$, we obtain a slightly worse, but still reasonable, fit ($\Delta \chi^2=71$), with a large inferred black hole spin ($a^*=0.95$). If instead we freeze the spin $a^*=0$, but allow the spectral hardening factor to be free, we obtain a $\chi^2$ intermediate to those of the previous two fits ($\Delta \chi^2 = 30$ from the best [kerrbb]{} fit), but with ${\rm h_d}=3.41$. This rather large value is well-outside the normal range of variation (${\rm h_d}=$1.4–1.8; @done:08a). As shown in Fig. \[fig:kerrbb\], however, these fits are virtually indistinguishable from one another.
We next consider broadband fits to [[XMM-Newton]{}]{} data. These [[XMM-Newton]{}]{} observations occured only 6 weeks after the [[Chandra]{}]{} observations, with apparently little source variation as measured by the [[RXTE]{}]{}-[[ASM]{}]{} in the intervening period. As we further discuss in §\[sec:rxte\], the [[XMM-Newton]{}]{} spectra showed a flux and spectral hardness comparable to the [[Chandra]{}]{} spectra. Thus, we expect the [[XMM-Newton]{}]{} spectra to be qualitatively and quantitatively similar to the [[Chandra]{}]{} spectra. We have found that in practice, the [[XMM-Newton]{}]{} spectra are somewhat difficult to fit over their entire spectral range, and furthermore, the [EPIC-pn]{} and [-MOS]{} data do not completely agree with one another. For a variety of models that we tried, strong, sharply peaked residuals occurred for both detectors near 0.5keV (the O K edge), although these residuals were most pronounced in the [pn]{} data. Above 8keV, the two detectors also show (different from each other, and from [[RXTE]{}]{}) spectral deviations from simple models. Accordingly, we only consider data in the 0.7–8keV range. Additionally, we add uniform 0.5% ([pn]{}) and 1% ([MOS]{}) systematic errors to each spectrum. (These levels were required to obtain reduced $\chi^2
\approx 1$–2 with ‘simple’ spectral models.)
The only simple models that we found to adequately describe the data required an additional spectral component below 2keV, which here we model as a *second* [diskbb]{} component with ${\rm kT_{in}}
\approx 0.3$keV and ${\rm A_{dbb}} \approx 1300$ ([MOS]{}) and $\approx 2600$ ([pn]{}). The required ${\rm N_h}\approx 3 \times
10^{21}\,{\rm cm^{-2}}$ for both detectors was somewhat larger than that found for [[MEG]{}]{}. The dominant disk component, primarily required to fit the 2-8keV spectra, had comparable parameters to the [ diskbb]{} fit to the [[MEG]{}]{} data; namely, ${\rm A_{dbb}} \approx 8$ and ${\rm kT_{in}} \approx 1.6$keV.
Significant residuals clearly remain in the 0.7–2keV range for both the [pn]{} and [MOS]{} data, and the [pn]{} and [MOS]{} spectra quantitatively disagree with each other. Given the fact that the [[MEG]{}]{} spectra, as well as prior [ASCA]{} spectra [@nowak:99d] were well described with a simple absorbed [diskbb]{} model, with lower ${\rm N_H}$, we are inclined to ascribe a large fraction of the additional required $\approx 0.3$keV disk component and the low energy residuals to systematic errors in the [[XMM-Newton]{}]{} calibration.
With the above caveats in mind, the [[XMM-Newton]{}]{} data, however, confirm that in the 2–8keV range, the characteristic disk temperature for [[4U 1957+11]{}]{} is indeed rather large. Furthermore, the [[XMM-Newton]{}]{} spectra show no evidence for any narrow, or moderately broad, Fe K$\alpha$ line. Similar to the [[Chandra]{}]{} data, adding a 6.4keV line with 0.5keV width (both fixed) yields a 90% confidence upper limit to the line flux of $6\times10^{-5}$phcm$^{-2}$s$^{-1}$.
table2
Edge and line fits {#sec:edge}
------------------
We have employed a variety of techniques to search for lines in the [[4U 1957+11]{}]{} data, ranging from direct fits of narrow components at known line locations to a “blind search” using a Bayesian Blocks technique. The latter involves fitting a continuum model to the binned data, using the predicted model counts as the statistical variance in the fits, and then unbinning the data and comparing the likelihood of the observed counts to the predicted counts. (For a successful application of this technique to [[Chandra]{}]{}-[[HETG]{}]{} data of the low luminosity active galactic nuclei, M81\*, see @young:07a.) No candidate features that appear to be local to the [[4U 1957+11]{}]{} system were found. This is in contrast to, for example, [[Chandra]{}]{} observations of GROJ1655$-$40, which at a similar source luminosity in a spectrally soft, disk-dominated state showed very strong absorption features associated with the disk atmosphere [@miller:06a]. Comparable equivalent width features would have been *very easily* detected in our observation of [[4U 1957+11]{}]{}. We note, however, that GROJ1655$-$40 itself has not always shown disk atmosphere absorption features. Two weeks prior to the observation described by @miller:06a, despite the source being only slightly fainter while also being in a soft, disk-dominated state, [[Chandra]{}]{}-[[HETG]{}]{} observations revealed *no* narrow emission or absorption lines [@miller:06a; @miller:08a]. The ‘duty cycle’ with which such lines are prominent in disk-dominated BHC spectra comparable to the spectra of both GROJ1655$-$40 and [[4U 1957+11]{}]{}, remains an open observational question.
On the other hand, the above Bayesian Blocks procedure yielded a few candidate features in the [[Chandra]{}]{} data that are likely attributable to absorption by the Interstellar Medium (ISM). The strongest such feature is the absorption line at 13.44Å. This feature is consistent with absorption from the hot phase of the interstellar medium [@juett:06a]. Accordingly, we also simultaneously fit expected features from and absorption at 14.61Å and 14.51Å, respectively.
We again choose an absorbed disk model, albeit with the pileup fraction fixed to the values from our broad-band models, and only consider the 13–15Å region. We group the spectrum to a minimum signal-to-noise of 5 in each bin, and we let the equivalent widths and wavelengths of the and lines be free parameters while fixing the wavelength of the absorption to 14.508Å. Results from this fit are presented in Fig. \[fig:lines\] and Table \[tab:lines\]. The fitted wavelengths of the and lines agree well with the [[MEG]{}]{} studies of @juett:06a, although we find the line wavelength to be redshifted by 0.005Å (i.e., 111km s$^{-1}$). Our fits, however, are within 0.002Å of the theoretical value of 14.447Å.
Formally, the fit improves with addition of any of the three Ne lines, although the region residual is far from the strongest in the overall spectrum. The and absorption lines are more clearly detected. Their equivalent widths are near the mid-level to higher end[^5] of the sample discussed by @juett:06a. As we discuss in greater detail elsewhere [@yao:08a], the [[4U 1957+11]{}]{} equivalent width is consistent with the source’s probable location outside the disk of the Galaxy. Specifically, presuming that [[4U 1957+11]{}]{} is greater than 5kpc distant, and choosing a model where the hot phase of the ISM lies in layers predominantly outside the galactic plane, the equivalent width of the line in [[4U 1957+11]{}]{} is consistent with the equivalent width of the same feature observed in the Active Galactic Nuclei Mkn 421, appropriately scaled for that sightline through our Galaxy. Therefore it is likely that [[4U 1957+11]{}]{} is sampling a large fraction of the hot phase interstellar gas that lies within and directly outside the galactic plane [@yao:07a].
Consistent with this picture, both the [[RGS]{}]{} and [[MEG]{}]{} spectra show evidence of longer wavelength features that we also associate with absorption by the interstellar medium (Table \[tab:lines\]; see also @yao:08a). For the [[RGS]{}]{} data, we grouped both the [[RGS1]{}]{}(fit between 18–25Å) and [[RGS2]{}]{} (fit between 18–19.9Å) data to have a signal-to-noise of 5 and a minimum of four channels per bin. The [[MEG]{}]{} data were grouped to either 16 channels per bin, or 32 channels per bin. The [[MEG]{}]{} data were fit with an absorbed disk blackbody, while the [[RGS]{}]{} data required additional continuum structure (which we modeled here with a broken power-law). For all these model fits we added absorption lines expected from the warm and hot phases of the interstellar medium. The line from the warm phase of the ISM (not included in the [tbnew]{} model; @juett:04a) is clearly detected by the [[RGS]{}]{} data, and consistent limits are found with the [[MEG]{}]{} data.
The hot phase of the ISM is detected with both the [[RGS]{}]{} and [[MEG]{}]{} via K$\alpha$ absorption (18.96Å) measurements. The presence of K$\beta$ (18.63Å) is not strictly required, but the limits on its equivalent width are consistent with the measured K$\alpha$ line [@yao:08a]. In addition to , we also find evidence for K$\alpha$ absorption. Interestingly, an even stronger feature is detected at 21.8Å. We have no good identification for this feature (it is 3000${\rm
km~s^{-1}}$ redshifted from ). Its equivalent width might change between the [[MEG]{}]{} and [[RGS]{}]{} observations, and therefore it could be a candidate for a line intrinsic to the source.
It is interesting to note that for the 18–25Å fits discussed above, we find that the implied neutral column is somewhat larger than that found with the broadband fits. Specifically, we find $N_H =
1.9\pm0.2$, $1.8\pm0.5$, and $2.0\pm0.2$ $\times 10^{21}~{\rm
cm^{-2}}$ for the [[RGS]{}]{} and [[MEG]{}]{} (16 channels per bin, 32 channels per bin) data. These values are primarily driven by the O edge region of the fits, although we have found for our broadband fits that merely changing the O abundance is insufficient to bring their fitted neutral columns into agreement with the 18–25Å fits. It is possible that an additional very low energy component added to the broad band fits might allow for a larger column that is consistent with these narrow band fits.
RXTE Observations {#sec:rxte}
=================
In the previous sections we have shown that the [[4U 1957+11]{}]{} spectrum is consistent with a disk having a high peak temperature. One could reduce the need for this large peak temperature by including a hard component, e.g., a Comptonization spectrum; however, this added component only becomes dominant at photon energies above the bandpasses of [[Chandra]{}]{} and [[XMM-Newton]{}]{}. Thus, to distinguish between the possibilities of high black hole spin or an additional hard spectral component leading to the high fitted temperatures, we turn toward the [[RXTE]{}]{} data, which can provide spectral information up to $\approx
20$keV for [[4U 1957+11]{}]{}.
table3
Spectra {#sec:rxte_spectra}
-------
In Fig. \[fig:lightcurve\] we present the [[RXTE]{}]{}-[[ASM]{}]{} lightcurve, with 6day bins, and also show the scaled 3–18keV flux from the pointed [[RXTE]{}]{} observations. This figure highlights the times of the [[Chandra]{}]{} and [[XMM-Newton]{}]{} observations (which were simultaneous with [[RXTE]{}]{} ObsIDs 90123-01-03 and 90063-01-01, respectively). Generally speaking, [[4U 1957+11]{}]{} exhibits an X-ray lightcurve with a factor of 3–4 variability on a few hundred day time scale. @nowak:99d hypothesized that this long term variability was quasi-periodic; however, our longer lightcurve does not show any clear super-orbital periodicities. Note that the [[Chandra]{}]{} and [[XMM-Newton]{}]{} observations occur during the same local peak in the [[ASM]{}]{} lightcurve and have very similar 3–18keV fluxes as measured by the [[PCA]{}]{}. We shall present detailed results for these two observations, as well as for the faintest [[RXTE]{}]{} observation (ObsID 70054-01-11-00) and the brightest [[RXTE]{}]{} observation (ObsID 40044-01-02-01). This latter observation has been presented previously by @wijnands:02c.
As discussed by @nowak:99d and @wijnands:02c, the [[RXTE]{}]{} data of [[4U 1957+11]{}]{} can be well-described by a model that consists of a combination of a multi-temperature disk spectrum, a power-law, and a gaussian line. For most of the [[RXTE]{}]{} observations the line peaks at an amplitude of ${\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}2$–$3\%$ of the local continuum level, and it is statistically required by the data. No such line was present in either the [[Chandra]{}]{} or the [[XMM-Newton]{}]{} observation (line flux ${\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}7\times10^{-5}$phcm$^{-2}$s$^{-1}$). There are several possibilities for this discrepancy, and perhaps some combination of all are at play. With its $\approx 1\,{\rm deg}^2$ effective field of view, the [[PCA]{}]{} will detect diffuse galactic emission which shows a prominent 6.7keV line. Based upon scalings between the [[PCA]{}]{} line flux and the DIRBE 4.9$\mu$m infrared flux [@revnivtsev:06a], one expects an [[RXTE]{}]{}-detected line amplitude of $\approx
4\times10^{-5}$phcm$^{-2}$s$^{-1}$ (i.e., approximately 10% of the observed line) at the Galactic coordinates of [[4U 1957+11]{}]{} ($l=51.307^\circ$, $b=-9.330^\circ$). Remaining systematic uncertainties in the [[PCA]{}]{} response matrix may be as large as 1% of the continuum level in the Fe line region, which could account for a line amplitude of ${\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}1.5\times10^{-4}$phcm$^{-2}$s$^{-1}$ for the observations simultaneous with [[Chandra]{}]{} and [[XMM-Newton]{}]{}. The remaining 50% of the line flux for these latter two observations, which is still three times larger than the [[Chandra]{}]{} and [[XMM-Newton]{}]{} upper limits, may be real and could be indicative of the greater sensitivity of [[PCA]{}]{} to weak and *broad* line features. Such a broad line would have been detectable by [[Chandra]{}]{} and [[XMM-Newton]{}]{} only if their own systematic uncertainties were at the 1% level, and we have already noted larger broad band discrepancies between the [[HEG]{}]{} and [[MEG]{}]{}.
table4
For the disk plus power-law fits, we constrained the power-law photon index, $\Gamma$, to lie between 1–3, but let the power-law normalization freely vary. These fits were fairly successful overall (reduced $\chi^2$ ranged from 0.5–2, and averaged 1). Parameter trends with observed flux are shown in Fig. \[fig:rxte\_flux\], and show several features of interest. First, the vast majority of the observations are consistent with having a nearly constant disk normalization, which one could take as a proxy for disk radius. At the low end of 3–18keV flux, the disk normalization/radius turns upward. Phenomenologically, such observed increases in disk radius have been associated with low luminosity transitions to the hard state [e.g., see the similar behavior of LMC X-3; @wilms:01a], which we expect to occur near 3% of the Eddington luminosity [@maccarone:03a]. It is therefore possible that our lowest luminosity observations are near such a transition at 3% ${\rm
L_{Edd}}$. (On the other hand, @saito:06a note a similar increase in fitted disk radius for GRO J1655$-$40, without any associated transition, or near transition, to a low/hard state.)
Second, we see that as the 3–18keV flux increases, the fraction of the flux contributed by the power-law slightly increases. Half a dozen points, however, show a much more significant contribution by the power-law (Fig. \[fig:rxte\_flux\]). At the same time, those observations show a dramatic decrease in the disk normalization. Each of those observations is associated with peaks in the ASM lightcurve, and several are associated with high variability, including one instance of “rapid state transitions” (see §\[sec:rxte\_vary\]). We identify those observations with what @remillard:06a refer to as the “steep power-law” state (or, the “very high state”; @miyamoto:93a).
Even though the disk normalization drops and the power-law flux increases for the disk plus power-law fits, the fitted disk temperature does not show any significant deviations from a trend of 3–18keV flux $\propto (kT)^5$ (not shown in Fig. \[fig:rxte\_flux\]). For a disk spectrum with constant disk radius, we expect the bolometric flux to scale $\propto (kT)^4$, and, for these temperatures, to scale approximately as $(kT)^5$ in the 3–18keV bandpass. This same relation between disk temperature and 3–18keV flux holds if we instead fit the spectra with a disk plus Comptonization spectrum, as shown in Fig. \[fig:rxte\_flux\].
For these latter fits, we model the spectrum in the same manner as in §\[sec:broad\]: we use a [diskbb]{} [+]{} [comptt]{} model and tie the seed photon temperature to the disk peak temperature, freeze the corona temperature to 50keV, but let the coronal spectrum normalization and optical depth (constrained by $0.01 < \tau_{\rm p} <
5$) be fit parameters. Results for several of these fits are presented in Table \[tab:rxtedbb\]. The majority of the observations again show a nearly constant disk radius, and a 3–18keV flux that scales approximately $\propto (kT)^5$. The Comptonization models, however, now show the half dozen ‘steep power-law state’ observations as being associated with dramatic drops in the disk/seed photon temperatures (Fig. \[fig:rxte\_flux\]).
We now consider fits where the unphysical [diskbb]{} component is replaced with the [kerrbb]{} model. As for the fits to the [[Chandra]{}]{} data, we fix the disk inclination to 75$^\circ$, and set the source distance to 10kpc and the black hole mass to 3${{\rm M_\odot}}$. The latter choices are motivated by our faintest [[RXTE]{}]{} observation for which we measure a 3–18keV absorbed flux of $3.1\times10^{-10}~{\rm
ergs~sec^{-1}~cm^{-2}}$. Taking the [diskbb]{} model at face value, with an inclination of $75^\circ$, this result translates to an unabsorbed bolometric luminosity of 3%${\rm L_{Edd}}$ for this presumed mass and distance. The black hole is unlikely to be less massive than 3${{\rm M_\odot}}$, and low black hole masses lead to higher disk temperatures. Thus, we see these parameter choices as *minimizing* the need for high spin parameters in the [ kerrbb]{} fits, with 10kpc then being a *lower limit* to the source distance.
The remaining [kerrbb]{} fit parameters are the accretion rate, $\dot {\rm M}$, the black hole spin, $a^*$, and the disk atmosphere color-correction factor, ${\rm h_d}$. The [kerrbb]{} model does not fit a disk temperature per se; therefore, we set the input seed photon temperature of the [comptt]{} model equal to the [diskbb]{} peak temperature from our previous fits [see @davis:06a]. As before, we also freeze the coronal temperature to 50keV, and constrain the optical depth to lie between 0.01–5. Selected fit results are presented in Table \[tab:rxtekrra\] and Fig. \[fig:kerr\_fits\].
Even with choosing a very low black hole mass, and thus naturally favoring higher disk temperatures, the [kerrbb]{} models uniformly prefer to fit $a^*\approx 1$. On the other hand, the disk atmosphere spectral hardening remains low, with ${\rm h_d} \approx 1.1$. There is a slight trend for ${\rm h_d}$ to decrease with increasing flux, as shown in Fig. \[fig:kerrbb\_hd\]. For some of the observations where the disk contribution to the total flux dramatically decreased, ${\rm
h_d}$ drops slightly lower still. This is not surprising as the inclusion of a coronal component can qualitatively replace the need for hardening from the disk atmosphere.
The [kerrbb]{} model has a number of parameter degeneracies which we can exploit to search for fits with the more usual color-correction factor of ${\rm h_d} = 1.7$. Specifically, the apparent temperature increases $\propto {\rm h_d}$, and $\propto (\dot {\rm M}/{\rm
M}^2)^{1/4}$. If we increase ${\rm h_d}$, then in order to retain roughly the same spectral shape we must decrease $({\rm \dot M}/{\rm
M}^2)^{1/4}$. Solely decreasing ${\rm \dot M}$, however, makes our faintest observation less than 3%${\rm L_{Edd}}$. Additionally, the source distance would need to be reduced $\propto {\rm \dot M}^{1/2}$ in order to retain the same observed flux. We would have expected to detect a transition to a hard spectral state if ${\rm \dot M}$ were lower than we have assumed. In order to keep our faintest observation at $\approx 3\%$${\rm L_{Edd}}$, we must instead scale ${\rm M}
\propto {\rm h_d}^4$, $\dot {\rm M} \propto {\rm h_d}^4$, and the source distance as $\propto {\rm h_d}^{2}$. A consistent picture may be obtained with ${\rm h_d} = 1.7$, if $M \approx 16$${{\rm M_\odot}}$ and the distance to the source is $\approx 22$kpc.
We searched for a set of [kerrbb+comptt]{} fits where we froze the disk color-correction factor at ${\rm h_d}=1.7$, the black hole mass at 16${{\rm M_\odot}}$, but let the disk accretion rate, $\dot {\rm M}$, spin parameter, $a^*$, and source distance all be free parameters. Clearly, the source distance cannot truly be physically changing in a perceptible way; however, the degree to which we obtain a uniform set of fitted distances might provide a self-consistency check on the use of the [kerrbb]{} model. Selected results from these fits are presented in Table \[tab:rxtekrra\], and the fitted distances vs. 3–18keV flux are presented in Fig. \[fig:kerrbb\_d\]. In general, these fits work every bit as well as the 3${{\rm M_\odot}}$, 10kpc fits. The lower flux observations cluster about a fitted distance of $\approx 22$kpc. There is a trend, however, for the fitted distances to decrease with higher fluxes, and for several of the ‘steep power-law state’ observations to fit distances as low as 15kpc.
We have considered another class of fit degeneracy: disk atmosphere hardening factor and spin. Here we again freeze the black hole mass to 3${{\rm M_\odot}}$ and the distance to 10kpc, but freeze the hardening factor to ${\rm h_d} = 1.7$. As more of the peak in the apparent temperature is being attributed to the color-correction factor, the disk spin must be decreased, thereby decreasing the accretion efficiency and increasing the radius of the emitting area near the peak temperatures. To maintain the same observed flux, the accretion rate must be increased above that for the models with lower ${\rm
h_d}$. Formally, these are significantly worse fits than our previous ones, with a mean $\Delta \chi^2 = 22$. I.e., an increased accretion rate and spectral hardening factor are not acting as simple proxies for high spin in these models. (In terms of fractional residuals, however, the fits are somewhat reasonable.)
As shown in Fig. \[fig:kerrbb\_a\], these fits yield a relatively uniform spin of $a^* \approx 0.84$, with a slight trend for fitted spin to increase with increasing flux. Notable exceptions to this behavior are seen, however, as some of the ‘steep power-law state’ observations fit much reduced black hole spins. Again, this is a contribution from a Comptonization component, which is strong for these observations, replacing some of the need for spectral hardening due to rapid spin.
Note that we did search for a set of disk atmosphere model fits where we kept the source mass low ($3\,{\rm M}_\odot$) and moved the distance further out (17kpc) while fitting higher accretion rates, to intrinsically increase the disk temperatures and raise the implied fractional Eddington luminosities by a factor of $\approx 3$. These models *failed completely, with best fit $\chi^2$ values increasing by factors of almost 3 over high-spin models*. We again find that high-spin is *not* simply degenerate with increased accretion rate in the models. Essentially this is because high-spin adds an inner disk region with high temperature and high flux (approximately tripling the flux when going from non-spinning to Kerr solutions), which leads to a strong, broad component in the high energy spectrum. High accretion, low spin models, fail to reproduce such broad, high energy [[PCA]{}]{} spectra.
Variability {#sec:rxte_vary}
-----------
The study of the rapid variability properties of [[4U 1957+11]{}]{} was performed using high time resolution data from the [[PCA]{}]{}. We created power spectra from the entire [[PCA]{}]{} energy band ($\approx$2–60 keV), covering the 2$^{-7}$–2048 Hz frequency range. Observations with high disk fractions ($>$80%, as defined by Fig. \[fig:rxte\_flux\]) typically showed weak or no significant variability. Combining these observations resulted in a power spectrum that could be fitted reasonably well by a single power-law component ($\chi^2$/dof=101.0/79). This fit improved significantly ($\chi^2$/dof=73.1/74) by adding two Lorentzians, one (with Q-value fixed at 0 Hz) for a weak band-limited noise component around 2Hz and the other for a QPO at 25Hz. The resulting parameters from the fit are listed in Table \[tab:pds\_fit\]. Note, that although the improvement in the fit is statistically significant, the two added components are only marginally significant themselves ($\approx3\sigma$).
Observations with lower disk fractions showed an increase in variability, with maximum of 12.1$\pm$0.6% root mean square (rms) variability (0.01–100 Hz) in the observation with the lowest disk fraction (ObsID 50128-01-09-00; disk fraction of $\approx$30%; see Fig. \[fig:qpo\]). Combining the power spectra of all observations with disk fractions lower than 80% and fitting with the same model as above, we find that all variability components increased in strength (see Table \[tab:pds\_fit\]).
The power spectral properties of the high- and low-disk fraction observations are consistent with the soft state and the soft end of the transition between the hard and soft states, respectively. Observation of such variability strengthens the argument that [[4U 1957+11]{}]{} is indeed a black hole, and not a neutron star. The frequencies of the QPOs are similar to that seen in a few other black hole systems when they were in or close to their soft state, e.g. XTE J1550$-$564 [@homan:01a] and GRO J1655$-$40 [@sobczak:00a]. The comparable frequencies observed in these latter two sources have ranged from 15–22Hz, and futhermore the spectra of these sources have been fit with disk models that imply black hole spins in the range of $a^* \approx 0.1$–0.8 [@davis:06a; @shafee:06a]. Thus, if such variability features are related to black hole spin, a consistent picture arises between their spectral and variability results.
In two of the low-disk fraction observations (ObsIDs 40044-01-02-01 and 70054-01-04-00) we observed fast changes in the count rate, which resemble the dips/flip-flops observed in other black hole X-ray binaries during transition from the hard to the soft state [e.g., @miyamoto:91a]. In [[4U 1957+11]{}]{}, these changes in the count rate were accompanied by moderate changes in the spectral variability properties, but count rates were too low to classify the power spectra in the high and low count rate phases independently.
Discussion {#sec:discuss}
==========
We have presented observations of the black hole candidate [[4U 1957+11]{}]{} performed with the [[Chandra]{}]{}, [[XMM-Newton]{}]{}, and [[RXTE]{}]{} X-ray observatories. All of these observations point toward a relatively simple and soft spectrum, indicative of a classic BHC disk-dominated soft state. The [[Chandra]{}]{} and [[XMM-Newton]{}]{} spectra, especially at high resolution, indicate a remarkably unadulterated disk spectrum. That is, the absorption of the spectrum is very low at only ${\rm N_H} =
1$–$2\times 10^{21}\,{\rm cm}^{-2}$, and with the possible exception of an unidentified line at 21.8Å, there is very little evidence of spectral complexity intrinsic to the source. From that vantage point, [[4U 1957+11]{}]{} may be the cleanest disk spectrum with which to study modern disk atmosphere models.
table5
Perhaps the one indication of additional, unmodeled spectral complexity is the fact that the broadband fits and the more localized, high resolution edge and line fits yield (low) neutral columns that differ from each other by a factor of two. This, along with the 21.8Å absorption feature might indicate the presence of other weak, unmodeled spectral components at very soft X-ray energies. Even if this were the case, this is to be compared to other BHC to which such atmosphere models have been applied, e.g., GRO J1655$-$40 [@shafee:06a], where there is both a larger neutral column (${\rm
N_H} = 7 \times 10^{21}\,{\rm cm^{-2}}$), a complex Fe K$\alpha$ region (modeled with both emission and smeared edges by @shafee:06a) and high resolution observations of intermittently present, complex X-ray spectral features that are local to the system [i.e., @miller:06a; @miller:08a]. In comparison, the [[4U 1957+11]{}]{} spectrum, even with its uncertainties, is much simpler.
The observations presented here strengthen the case that [[4U 1957+11]{}]{} is a black hole system. The spectrum closely matches a classic, soft state disk spectrum (well-modeled by [diskbb]{}), with a contribution from a harder, non-disk component that generally increases as the flux increases. Occasionally (a half dozen observations), the contribution from this hard component increases, the X-ray variability increases, a possible QPO is observed, and we even observe lightcurve ‘dipping’ and ‘flip-flop’ behavior [@miyamoto:91a; @miyamoto:93a]. All of this is behavior familiar from studies of other BHC [@homan:01a and references therein]. Unfortunately, observation of these behaviors provides little strong input to estimates of the system parameters, i.e., mass, distance, and inclination. ‘Flip-flop’ behavior has often been associated with luminosities of a few tens of percent [@miyamoto:91a; @miyamoto:93a; @homan:01a], which is higher than we have presumed for the fits described here. The threshold luminosity for such behavior, however, is not firm, and we further note that high accretion rate/large distance fits to the [[PCA]{}]{} spectra failed (§\[sec:rxte\_spectra\]). The fact that we do not see a transition to a BHC hard state— although we may see some indications from increases in the fitted disk radius at low luminosity— strongly suggest that the mass is $>3\,{{\rm M_\odot}}$ and the distance is $>10$kpc.
Despite the fact that we do not have a firm estimate of mass and distance for [[4U 1957+11]{}]{}, the relativistic disk models still strongly statistically prefer rapid spin solutions. Essentially, this is because the [diskbb]{} models fit very high temperatures – as high as 1.7keV – with fairly low normalizations. We saw, however, that for the [[Chandra]{}]{} observation, or for the [[RXTE]{}]{} observations where the power-law/Comptonization component strengthens, the need for spectral hardening of the disk, either via a color-correction factor (${\rm h_d}$) or rapid spin, was greatly reduced. Phenomenologically, a coronal component can mimic the effects of rapid spin. This naturally leads to the question of whether or not one is sure that any residual coronal component completely vanishes at low flux. Does one ever observe a “bare” disk spectrum?
It is a rather remarkable fact that the [[Chandra]{}]{} observation, and the low flux [[RXTE]{}]{} observations, are mostly described via three parameters: absorption, [diskbb]{} temperature, and [diskbb]{} normalization. Even more remarkably, changes in the spectrum, both in terms of amplitude and overall shape, are mainly driven by variations of only *one* parameter: [diskbb]{} temperature. The relativistic disk models, on the other hand, must ascribe essentially this one parameter to a combination of four parameters: $a^*$, disk inclination (which affects the appearance of relativistic features), $({\rm \dot M/M}^2)$, and ${\rm h_d}$. In principle, any of the latter three can vary among observations. (Disk inclination can vary due to warping; see @pringle:96a.) Unfortunately, there is no truly unique, sharp feature in the high resolution spectrum that completely breaks degeneracies among these parameters. The overall magnitude of the [diskbb]{} temperature and shape of the [[4U 1957+11]{}]{} spectra, however, seem best reproduced with the most rapid spin models; simply increasing accretion rates or spectral hardening factors is insufficient to model the spectra fully.
We do not see the essential question as being whether or not [[4U 1957+11]{}]{} is rapidly spinning. Instead, we view the more observationally motivated and perhaps more fundamental question as being: why is the characteristic disk temperature of [[4U 1957+11]{}]{} so high? Rapid spin is one hypothesis. Another would be that there is indeed a residual, low temperature corona, even at low flux. The possibly high inclination of this source ($\approx 75^\circ$ to be consistent with optical lightcurve variations, while being consistent with the lack of X-ray eclipses) would mean that we are viewing the disk through a larger scattering depth than would be usual for most BHC in the soft state. We also note that there exists very little, self-consistent theoretical modeling of low temperature Comptonizing coronae with high temperature (1–2keV) seed photon temperatures. Most energetically balanced, self-consistent models [e.g., @dove:97a] have focused on high coronal temperatures (50–200keV) with low seed photon temperatures (100–300eV).
If one is to take the rapid spin hypothesis for the high [diskbb]{} temperature at face value, then these observations offer something of a prediction. Statistically, they do prefer rapid spin, and based upon observations of other BHC we would have expected to observe a hard state if the faintest observations were ${\mathrel{ \rlap{\raise 0.511ex \hbox{$<$}}{\lower 0.511ex \hbox{$\sim$}}}}3\%\,{\rm
L_{Edd}}$. If the theoretically preferred value of the spectral hardening factor is indeed ${\rm h_d}=1.7$, future measurements should find [[4U 1957+11]{}]{} to be an $\approx 16\,{{\rm M_\odot}}$ black hole at $\approx
22$kpc. Whether or not this turns out to be the case, given the nature of [[4U 1957+11]{}]{} as perhaps the simplest, cleanest example of a BHC soft state, observations to independently determine this system’s parameters (mass and distance) are urgently needed.
It is a pleasure to acknowledge useful conversations with Manfred Hanke, David Huenemoerder, John Houck, and John Davis. This work has been supported by NASA grant SV3-73016 and DLR grant 50OR0701.
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In this appendix, we describe how we model the mild pile up that is present in the [[MEG]{}]{} and, to lesser degree, the [[HEG]{}]{} data. Our model follows the discussions of @davis:01a and @davis:03a, although we do not attempt to arrive at an [*ab initio*]{} description of pile up. Instead, we present a simple, phenomenological description where the absolute amplitude of pile up is left as a fit parameter (albeit one that can be determined roughly with empirical knowledge and then frozen at a fixed value, or fitted with a limited range of “acceptable” values). In contrast to spectral pile up solely with the CCD detector, where piled events can reappear in the spectrum at higher implied energies, gratings pile up (at least in [$1^{\rm st}$]{} order spectra) can be thought of as a straight loss of events. Whether the piled events ‘migrate’ to bad grades, or are read as a single event of higher energy and are thus removed from the ‘order sorting window’, it is still a simple loss term [@davis:03a].
The degree of loss due to pile up scales with $C_i$, the number of expected incident events in a given detector region per detector frame integration time. The detected number of events is then reduced by a factor $\exp({\cal A} C_i)$, for a suitably chosen region and where ${\cal A}$ is a ‘fudge factor’ of order unity. The proper detector region to consider is approximately 3 pixels in the ‘cross dispersion’ direction, by a 3–5 pixel length along the dispersion direction of the gratings arm being analyzed. (Events in a single frame that land in adjacent pixels will always be piled, while events that are removed from one another by four pixels will only be piled if their charge clouds extend toward one another; see @davis:03a.) Given that the peak effective area of the [[MEG]{}]{} is approximately twice that of the [[HEG]{}]{}, and the fact that detector pixels cover twice the wavelength range in the [[MEG]{}]{}, we expect the pile up to be approximately 4 times larger in the [[MEG]{}]{}.
It is important to note that one needs to consider *all* events in a given detector region, i.e., all spectral orders, background events, contributions from the wings of the [$0^{\rm th}$]{} order point spread function, etc., whether or not these events are otherwise rejected by order sorting. To partly account for this necessity, the pile up model described here uses information from the [$2^{\rm nd}$]{} and [$3^{\rm rd}$]{} order ancillary response functions ([[arfs]{}]{}). Specifically, we create a ‘convolution model’ where, given an input spectrum, we define $C_i(\lambda)$ as the first order [[arf]{}]{} multiplied by the unpiled model spectrum (yielding counts per detector bin). We further increase $C_i(\lambda)$ by multiplying the unpiled model by the [$2^{\rm nd}$]{}order [[arf]{}]{}, then shifting the [$2^{\rm nd}$]{} order wavelengths by a factor of two, and then rebinning and adding this contribution to $C_i(\lambda)$. A similar procedure is used for the [$3^{\rm rd}$]{} order contribution. We then use the input pile up fraction fit parameter, $p_f$, to normalize the exponential reduction of the spectrum. Specifically, we multiply the counts per bin predicted for the unpiled model by $\exp[\log(1-p_f)[C_i(\lambda)/\max(C_i)]]$.
There are two important subtleties to address. First, in certain regions of the spectrum, specifically those wavelength regions that are dithered over chip gaps or detector bad pixels, the counts are lower not due to a low intrinsic flux, but rather due to a fractionally reduced exposure. The [FRACEXPO]{} column found in the [[arf]{}]{} FITS files for [[Chandra]{}]{} gratings data provides this information; therefore, we can approximately correct for this effect. Second, for [[Chandra]{}]{} gratings spectra, a portion of the detector area information is contained within the response matrix files ([[rmf]{}]{}), which are *not* normalized to unity. In practice, before fitting the pile up model, we renormalize the gratings [$1^{\rm st}$]{}order [[arfs]{}]{} using the [[ISIS]{}]{} functions [factor\_rsp]{}, [get\_arf]{}, and [put\_arf]{}. Assuming [a\_id]{} is the data index of the [$1^{\rm st}$]{} order [[arf]{}]{}, and [r\_id]{} is the data index of the [$1^{\rm st}$]{}order [[rmf]{}]{}, we proceed as follows in [[ISIS]{}]{}:
isis> a = get_arf(a_id); % Read the existing arf into a structure
isis> na_id = factor_rsp(r_id); % Factor the rmf into normalized rmf*arf
isis> na = get_arf(na_id); % Read the new, factored arf component
isis> a.value *= na.value; % Multiply original arf by factored value
isis> put_arf(a_id, na); % Replace old arf value with rescaled value
isis> delete_arf(na_id); % Delete the factored arf component
It is precisely because [[ISIS]{}]{} has the ability to read and manipulate information from all input data files, and then easily mathematically manipulate this information to create new functionality, that we are able to define a relatively simple pile up correction model.
Ideally, we should incorporate additional information, such as the variation of the line spread function (LSF; i.e., the cross dispersion profile) along the gratings arms, the 5–10% of the events that are typically excluded by order sorting[^6], etc. For these reasons we cannot assign an absolute estimate of the pile up fraction. However, to the extent that we have correctly captured the *relative* changes of *total* count rate along the dispersion direction, and have chosen a suitable peak pile up fraction, we should have a reasonably accurate pile up correction as a function of wavelength. More sophisticated [*ab initio*]{} models currently are under development (J. Davis, priv. comm.).
For our data, the peak of the detected [[MEG]{}]{} count rate (including contributions from $2^{\rm nd}$ and $3^{\rm rd}$ order events) occurs in the 6–8Å region, and peaks at a value of $\approx$ 0.08 counts/frame/3 pixels. Thus, making a simple first order correction since this rate already includes pile up, and including up to 5 pixels, we expect a peak pile up fraction in the 0.09–0.16 range. We could have chosen a value in this range and have frozen it in the fits, but the fitted values fall within the middle of these estimates. Likewise, for the [[HEG]{}]{} spectra we estimate a peak pile up fraction of 0.025–0.036. Such fractions are comparable to or smaller than existing systematic uncertainties in the [[Chandra]{}]{} calibration. Additionally, we found a great deal of degeneracy in any attempts to fit the [[HEG]{}]{} value directly, thus we froze peak the pile up fraction at a value of 0.03.
The [S-lang]{} code that we used to define a simple gratings pile up model as an [[ISIS]{}]{} user model is presented in the electronic version of this manuscript.
define simple_gpile_fit(lo,hi,par,fun)
{
% Peak pileup correction goes as:
% exp(log(1-pfrac)*[counts/max(counts)])
variable pfrac = par[0];
% Pileup scales with model counts from *data set* indx ...
variable indx = typecast(par[1],Integer_Type);
if( indx == 0 or pfrac == 0. )
{
return fun; % Quick escape for no changes ...
}
% ... but the arf index could be a different number, so get that
variable arf_indx = get_data_info(indx).arfs;
% Get arf information
variable arf = get_arf(arf_indx[0]);
% In dither regions (or bad pixel areas), counts are down not
% from lack of area, but lack of exposure. Pileup fraction
% therefore should scale with count rate assuming full exposure.
% Use the arf "fracexpo" column to correct for this effect
variable fracexpo = get_arf_info(arf_indx[0]).fracexpo;
if( length(fracexpo) > 1 )
{
fracexpo[where(fracexpo ==0)] = 1.;
}
else if( fracexpo==0 )
{
fracexpo = 1;
}
% Rebin arf to input grid, correct for fractional exposure, and
% multiply by "fun" to get ("corrected") model counts per bin
variable mod_cts;
mod_cts = fun * rebin( lo, hi,
arf.bin_lo, arf.bin_hi,
arf.value/fracexpo*(arf.bin_hi-arf.bin_lo) )
/ (hi-lo);
% Go from cts/bin/s -> cts/angstrom/s
mod_cts = mod_cts/(hi-lo);
% Use 2nd and 3rd order arfs to include their contribution.
% Will probably work best if one chooses a user grid that extends
% from 1/3 of the minimum wavelength to the maximum, and has
% at least 3 times the resolution of the first order grid.
variable mod_ord;
if(par[2] > 0)
{
indx = typecast(par[2],Integer_Type);
arf = get_arf(indx);
fracexpo=get_arf_info(indx).fracexpo;
if( length(fracexpo) > 1 )
{
fracexpo[where(fracexpo ==0)] = 1.;
}
else if( fracexpo==0 )
{
fracexpo = 1;
}
mod_ord = arf.value/fracexpo*
rebin(arf.bin_lo,arf.bin_hi,lo,hi,fun);
mod_ord = rebin(lo,hi,2*arf.bin_lo,2*arf.bin_hi,mod_ord)/(hi-lo);
mod_cts = mod_cts+mod_ord;
}
if(par[3] > 0)
{
indx = typecast(par[3],Integer_Type);
arf = get_arf(indx);
fracexpo=get_arf_info(indx).fracexpo;
if( length(fracexpo) > 1 )
{
fracexpo[where(fracexpo ==0)] = 1.;
}
else if( fracexpo==0 )
{
fracexpo = 1;
}
mod_ord = arf.value/fracexpo*
rebin(arf.bin_lo,arf.bin_hi,lo,hi,fun);
mod_ord = rebin(lo,hi,3*arf.bin_lo,3*arf.bin_hi,mod_ord)/(hi-lo);
mod_cts = mod_cts+mod_ord;
}
variable max_mod = max(mod_cts);
if(max_mod <= 0){ max_mod = 1.;}
% Scale maximum model counts to pfrac
mod_cts = log(1-pfrac)*mod_cts/max_mod;
% Return function multiplied by exponential decrease
fun = exp(mod_cts) * fun;
return fun;
}
add_slang_function("simple_gpile",
["pile_frac","data_indx","arf2_indx","arf3_indx"]);
set_function_category("simple_gpile", ISIS_FUN_OPERATOR);
[^1]: We did, however, use a pre-release version of the [Order Sorting and Integrated Probability]{} ([OSIP]{}) file from CALDB v3.3.0.
[^2]: [ http://space.mit.edu/ASC/analysis/findzo/]{}
[^3]: Owing to the fact that the pileup correction is a convolution model that requires knowledge of the individual response functions for each gratings arm, the dataset combination performed here is distinct from adding the PHA and response files before fitting. (As a result, this combined data analysis is, in fact, impossible to reproduce in [XSPEC]{}.)
[^4]: Some of these observations were scheduled simultaneously with ground-based optical observations. No optical data are included in this work.
[^5]: The line equivalent width measured in [[4U 1957+11]{}]{} is a factor of two less than that observed in GX 339$-$4; however, as discussed by @miller:04a and @juett:06a the latter source’s column is likely dominated by a warm absorber intrinsic to the binary.
[^6]: LSF variations and the exclusion of events that fall outside of the order sorting windows is, of course, accounted for in calculation of the gratings [[arfs]{}]{}.
|
6.5 in 9.0in -10mm
-0.25 in -0.25 in
[**Proposed Measurement of an Effective Flux Quantum in the Fractional Quantum Hall Effect**]{}
J.K. Jain$^{1}$, S.A. Kivelson$^{2}$, and D.J. Thouless$^3$
1\. [*Department of Physics, State University of New York at Stony Brook, Stony Brook, New York, 11794-3800*]{}
2\. [*Department of Physics, University of California at Los Angeles, Los Angeles, California 90024*]{}
3\. [*Department of Physics, University of Washington, Seattle, Washington, 98195*]{}
We consider a channel of an incompressible fractional-quantum-Hall-effect (FQHE) liquid containing an island of another FQHE liquid. It is predicted that the resistance of this channel will be periodic in the flux through the island, with the period equal to an odd integer multiple of the fundamental flux quantum, $\phi_{0}=hc/e$. The multiplicity depends on the quasiparticle charges of the two FQHE liquids.
Since the seminal works of Laughlin [@laughlin] and Halperin [@halperin], it has been recognized that the elementary excitations (quasiparticles) in the fractional quantum Hall effect (FQHE) [@fqhe] have fractional charge and obey fractional statistics. These fractional quantum numbers essentially follow from the incompressibility at fractional filling factors, and their values can be determined from rather general principles [@su]. For the principal FQHE liquids at filling factors [@ff] \_[n]{}, the charge of a quasihole is e\_[n]{}= , \[charge\] while its statistics is \_[n]{}= , \[statistics\] defined so that an exchange of two quasiholes produces a phase factor of $e^{i\pi\theta}$ [@general]. It has been argued that the fractionally quantized Hall resistance itself is a measurement of the charge of the quasiparticles [@laughlinbook], but, on the other hand, the Hall resistance is a property of the condensate and therefore does not [*directly*]{} probe the excitations [@kp]. The observation of the ‘hierarchy fractions’ has been cited as evidence for the fractional statistics of quasiparticles [@laughlin3], but it is clear that all fractions [*can be*]{} understood without reference to quasiparticles at all [@jain89]. Several experiments have reported evidence for the fractional charge [@simmons]. However, their theoretical interpretation is either not unique, or not completely understood. A [*definitive*]{} and [*direct*]{} observation of the fractional charge or the fractional statistics of the quasiparticles is therefore lacking.
In order to illustrate the basic conceptual difficulty with the measurement of the fractional charge, consider the Aharonov-Bohm (AB) geometry in Fig.1a. In the FQHE regime, the current is carried by fractionally charged quasiparticles, so it is tempting to expect that the properties of the system, such as the resistance, will be periodic in the flux with period $\phi_{0}^*=hc/e_{n}$, in analogy with the argument of Byers and Yang (BY) [@by]. However, in any true AB geometry, the period must always be $\phi_{0}=hc/e$. The reason is that while the quasiparticles may provide an [*effective*]{} description, the fundamental particles are still electrons [@kr]. In fact, periods [*greater*]{} than $\phi_{0}$ are ruled out by the BY argument (while smaller periods are, of course, possible and do occur, e.g. in the case of superconductors).
In this Letter, we consider a resonant tunneling experiment and predict that, under certain conditions, the resistance will exhibit approximate periodicity in flux with period equal to an odd integer multiple of $\phi_{0}$. An observation of this periodicity should provide direct and unambiguous evidence of the existence of fractional quantum numbers in the FQHE. There have been other proposals for the observation of the fractional quantum numbers [@kivelson], but they deal with [*non-equilibrium*]{} situations. The experiment proposed in the present work, on the other hand, probes an [*equilibrium*]{} property of the system.
We consider the geometry of Fig.1b, in which a (narrow) channel of $\nu'=p'/q'$ FQHE liquid (where $p'$ and $q'$ are relatively prime integers) contains an island of area $A$ of the $\nu=p/q$ FQHE liquid. This could be produced experimentally by creating a gentle potential hill or valley with the help of an external gate. The chemical potential at the edges of the sample is assumed to be fixed externally. The BY argument clearly does not apply in this situation, since electrons occupy the entire sample. It is possible for a quasiparticle to tunnel from one edge of the channel to the other, which is actually a tunneling between two many-body configurations, one in which the quasiparticle is on one edge, and the other in which it is on the other. The tunneling amplitude determines the longitudinal resistance, as was shown in a Landauer-type formulation of the QHE [@streda]. The longitudinal resistance exhibits peaks whenever there is [*resonant*]{} tunneling from one edge of the sample to the other through a quasi-bound state on the potential island [@jain88]. The main conclusion of this work is that successive peaks occur when the flux through the island changes by j\_[0]{}=\_[0]{}, \[tpd\] where $s$ is an integer, equal to the highest common factor of $q$ and $q'$. Since $q$ and $q'$ are, in general, odd integers, $j$ is also an odd integer. Note that $j$ depends only on $q$ and $q'$, i.e., only on the quasiparticle charges of the two FQHE liquids.
To give the simplest derivation of this result, let us change the flux through the $\nu=p/q$ island liquid in a way that no quasiparticles (quasiholes or quasielectrons) are created in the bulk. This can be achieved by spreading the additional flux over a sufficiently large area of the island. The additional flux $j\phi_{0}$ contracts the island liquid, so that an additional charge $jep/q$ is required to restore the edge of the island FQHE liquid to its original state. Since the charge must be supplied by $j'$ quasiparticles of the channel ($\nu'$) FQHE liquid, we must have j=j’, which leads to the period $j\phi_{0}$ given by Eq. (\[tpd\]). In particular, if $\nu=0$, i.e. if the island is charge free, the period is $\phi_{0}$, since the channel FQHE liquid can return to its original state by the transfer of $p'$ quasiparticles from the outer edge to the inner edge. (Thus, $\nu=0$ is to be interpreted as $\nu=0/1$ for the purpose of Eq. \[tpd\].) This is equivalent to a gauge transformation of the original wave function.
Let us now give a more microscopic description, which takes account of the internal structure of the various FQHE liquids. We use the framework of the composite fermion (CF) theory [@jain89], in which the the wave function of the $\nu_{n}$ FQHE liquid is given by \_[n/(2n+1)]{}=\_[j<k]{}(z\_[j]{}-z\_[k]{})\^[2]{}\_[n]{}, where $\Psi_{n}$ is the wave function of $n$ filled Landau levels (LLs), and $z_{j}=x_{j}+iy_{j}$ denotes the position of the $j$th electron. Consider the situation when the island FQHE liquid is $\nu_{n-1}$ and the channel FQHE liquid is $\nu_{n}$. This state corresponds to an integer quantum Hall effect (IQHE) state which has $n$ filled LLs everywhere except in an island where the filling factor is $n-1$. An integer number ($K$) of electrons have been removed from the $n$th LL to create the island [@semiclassical]. In the IQHE state $\Psi$, each hole has an excess charge $e$ associated with it. Upon multiplication by the Jastrow factor, $\prod_{j<k}(z_{j}-z_{k})^{2}$, which converts each electron into a CF, each hole in the $n$th LL of $\Psi$ becomes a quasihole of the $\nu_{n}$ liquid, with an excess charge $e_{n}=e/(2n+1)$ associated with it [@charge]. Therefore, for $K$ quasiholes, there is a net deficiency of charge $Ke_{n}$ in the island region. This deficiency is related to the difference between the filling factors outside and inside the island as: Ke\_[n]{}=(\_[n]{}-\_[n-1]{})(/\_[0]{}), where $\Phi=AB$ is the flux through the island. Thus, for $K$ quasiholes, the flux through the island is given by =K (2n-1)\_[0]{}. \[flux\] Addition or removal of a single quasihole requires a flux change of $(2n-1)\phi_{0}$ through the island, which gives the period =(2n-1)\_[0]{}. \[period\] When the island liquid is $\nu_{n+1}$ (and the channel liquid is $\nu_{n}$), Ke\_[n]{}=(\_[n+1]{}-\_[n]{}) (/\_[0]{}), and the period is given by =(2n+3)\_[0]{}. In both cases, the periods are in agreement with the general formula, Eq. (\[tpd\]).
It is instructive to consider this problem from yet another perspective. We write pseudo wave functions in terms of the coordinates of the quasiparticles, treating them as point particles [@halperin]. First consider the situation when the channel liquid is $\nu_{n}$ and the island liquid is $\nu_{n-1}$. Since the low-energy states contain quasiholes in the topmost level only (i.e., related to holes only in the $n$th LL of $\Psi_{n}$), they fill a lowest LL of their own. The most energetically favorable situation is when they completely fill the LL. The wave function is then \_[j<k]{}(\_[j]{}-\_[k]{})\^ , \[qhwf\] where $\eta_{j}$ denote the positions of the quasiholes, and $l_{n}^2=\hbar c/e_{n} B$. The area of the island is given by (neglecting irrelevant corrections of order unity) \[14a\] A=K . \[ratio\] With $\theta=\theta_{n}$, given by Eq. (\[statistics\]), this is identical to Eq. (\[flux\]), and gives a period of $(2n-1)\phi_{0}$. In the other case, when the island liquid is $\nu_{n+1}$, we write the quasielectron wave function [@halperin; @khd1] \_[j<k]{}(\_[j]{}-\_[k]{})\^[-]{} , \[qewf\] where now $\eta_{j}$ are the quasielectron coordinates. In this case, one is tempted to choose the quasiparticle statistics $\theta=\theta_{n}$. However, in order for the quasielectron wave function to be regular as two quasielectrons approach one another, which is required by the hermiticity of the Hamiltonian [@khd], we must choose the statistics to be [@ma] =\_[n]{}-2= -. (The resulting quasielectron wave function can also be interpreted as a FQHE liquid of quasielectrons of statistics $\theta_{n}$ [@halperin].) The period from Eq. (\[ratio\]) is $(2n+3)\phi_{0}$, as expected. From this perspective, the period can be interpreted as a measure of the [*ratio*]{} of the statistics to charge of the quasiparticles of the channel FQHE liquid (see Eq. \[ratio\]).
We close with the following remarks.
\(i) The above arguments actually show that for a consistent description in terms of quasiparticles, they [*must*]{} be assigned fractional statistics. Similar arguments had originally led Halperin to discover that quasiparticles obey fractional statistics [@halperin].
\(ii) It is interesting to see how the BY result is obtained from the perspective of the quasiparticles. This pertains to the situation when charge is completely depleted from the island region. In the CF theory, this relates to the IQHE state in which all $n$ LLs are empty in the island region. In the quasihole language, $n$ LLs of quasiholes are occupied. In analogy with the CF theory, the wave function of this quasihole state is given by [@ma] \_[j<k]{}(\_[j]{}-\_[k]{})\^[\_[n]{}-1]{} \_[n]{}. The size of the droplet described by this wave function is such that the flux through it is given by =. In this case, the number of quasiholes increases in units of $n$ (since, whenever it is possible to add a quasihole in one level, it is possible in other levels as well), and we recover the BY period of $\phi_{0}$.
\(iii) We have so far assumed that the $\nu=p/q$ FQHE liquid in the island is ideal. It is easy to see that the presence of a [*fixed*]{} number of quasielectrons or quasiholes in this liquid will not alter the period. Whenever a [*new*]{} quasiparticle is created, the periodic sequence will suffer a phase shift. The same will be true when there are lakes of other FQHE liquids inside the island; the period will remain $j\phi_{0}$ except when a new quasiparticle is created in one of the lakes. Thus, in general, we expect [*finite*]{} sequences of peaks in the longitudinal resistance with the predicted spacing. The larger the amount of the $\nu=p/q$ fluid in the island, the longer will be the length of the sequence.
\(iv) We have neglected the Coulomb blockade effects [@lee], which are expected to be small for sufficiently large islands. These are also well understood and may be subtracted out to reveal the effects discussed here. We note that the periodicity of the effect does not depend on the structure of the interface between the two FQHE liquids, so long as it is narrow compared to the regions of the FQHE liquids.
\(v) Any $j\phi_{0}$ periodicity ($j\neq 1$) in the situation when the island is completely depleted, as is presumably the case in the experiment of Simmons [*et al.*]{} [@simmons], [*must*]{} be a non-equilibrium effect [@kivelson]. This should be experimentally testable.
In conclusion, we predict conditions under which an interference between two FQHE liquids allows the observation of an effective flux quantum, which is equal to an odd integer multiple of the fundamental flux quantum. The period depends on the quasiparticle charges of the two FQHE liquids; in the case of two successive FQHE states of a sequence, it can be also interpreted as a measure of the ratio of the statistics to the charge of the quasiparticles of the channel FQHE liquid. This experiment should also serve as a probe into the internal structure of the FQHE liquids.
We thank V.J. Goldman for discussions and comments. This work was supported in part by the National Science Foundation under Grants Nos. DMR90-20637, DMR90-11803, and DMR92-20733. We acknowledge the hospitality of the Aspen Center for Physics.
[99]{}
R.B. Laughlin, Phys. Rev. Lett. [**50**]{}, 1395 (1983).
B.I. Halperin, Phys. Rev. Lett. [**52**]{}, 1583 (1984).
D.C. Tsui, H.L. Störmer, and A.C. Gossard, Phys. Rev. Lett. [**48**]{}, 1559 (1982).
W.P. Su, Phys. Rev. B [**34**]{}, 1031 (1986). This work shows that the quasiparticle charge and statistics (mod 2) can be determined completely by assuming that there is only one type of quasiparticle, and its charge assumes the largest allowed value.
The filling factor at magnetic field $B$ is defined by $\nu=\rho\phi_{0}/B$ where $\rho$ is the electron density, and $\phi_{0}=hc/e$.
In general, the charge of the quasihole of the $\nu=p/q$ liquid is given by $e/q$ and the statistics is given by $\theta=p'/q$ where $p'p=1$ mod $q$. This simple expression for the statistics (which is, of course, in agreement with those of Refs.[@halperin; @su]) was given by G. Moore and N. Read, Nucl. Phys. B [**360**]{}, 362 (1991).
R.B. Laughlin in [*The Quantum Hall Effect*]{}, Eds. R.E. Prange and S.M. Girvin, 2nd edition (Springer-Verlag, NY).
S.A. Kivelson and V.L. Pokrovsky, Phys. Rev. B [**40**]{}, 1373 (1989).
See, for example, R.B. Laughlin, Int. J. Mod. Phys. B [**5**]{}, 1507 (1991).
J.K. Jain, Phys. Rev. Lett. [**63**]{}, 199 (1989).
J.A. Simmons [*et al.*]{}, Phys. Rev. Lett. [**63**]{}, 1731 (1989); Phys. Rev. B [**44**]{}, 12933 (1991); R.G. Clark [*al.*]{}, Phys. Rev. Lett. [**60**]{}, 1747 (1988); A.M. Chang and J.E. Cunningham, Surf. Sci. [**229**]{}, 216 (1990).
N. Byers and C.N. Yang, Phys. Rev. Lett. [**7**]{}, 46 (1961).
This actually implies that the fractionally charged quasiparticles must obey fractional statistics. See, S.A. Kivelson and M. Rocek, Phys. Lett. B [**156**]{}, 85 (1989).
\(a) S.A. Kivelson, Phys. Rev. Lett. [**65**]{}, 3369 (1990); (b) D.J. Thouless, Phys. Rev. B [**40**]{}, 12034 (1989); D.J. Thouless and Y. Gefen, Phys. Rev. Lett. [**66**]{}, 806 (1991); Y. Gefen and D.J. Thouless, Phys. Rev. B [**47**]{}, 10423 (1993); Pryadko and V.L. Pokrovsky, unpublished.
P. Streda, S. Kucera, and A.H. MacDonald, Phys. Rev. Lett. [**59**]{}, 1973 (1987); J.K. Jain and S.A. Kivelson, Phys. Rev. B [**37**]{}, 4276 (1988).
J.K. Jain and S.A. Kivelson, Phys. Rev. Lett. [**60**]{}, 1542 (1988). Edge transport in the FQHE regime was demonstrated by J.K. Wang and V.J. Goldman, Phys. Rev. Lett. [**67**]{}, 749 (1991).
We will assume that the number of quasiparticles $K>>1$ so that order unity corrections can be neglected.
J.K. Jain, Phys. Rev. B [**41**]{}, 7653 (1990).
For a more detailed discussion of the quasielectron statistics, see A. Karlhede, S.A. Kivelson, and S.L. Sondhi, in [*Prodeedings of the Ninth Jerusalem Winter School on Theoretical Physics*]{}, January 1992.
S. Artz, T.H. Hansson, A. Karlhede, and T. Staab, Phys. Lett. B [**267**]{}, 389 (1991).
Also see, M. Ma and F.C. Zhang, Phys. Rev. Lett. [**66**]{}, 1769 (1991).
See, for example, P.A. Lee, Phys. Rev. Lett. [**65**]{}, 2206 (1990).
[**Figure Caption**]{}:
Figure 1. (a) Standard Aharonov Bohm geometry. (b) Schematic drawing of the proposed resonant tunneling experiment. The shaded area is the island of $\nu=p/q$ FQHE liquid surrounded by the $\nu'=p'/q'$ FQHE liquid. The dashed lines show the most probable tunneling paths.
|
---
abstract: 'It is known that a tilting generator on an algebraic variety $X$ gives a derived equivalence between $X$ and a certain non-commutative algebra. In this paper, we present a method to construct a tilting generator from an ample line bundle, and construct it in several examples.'
author:
- Yukinobu Toda and Hokuto Uehara
title: Tilting generators via ample line bundles
---
Introduction
============
Let $D^b(X)$ be the bounded derived category of coherent sheaves on an algebraic variety $X$. Modern algebraic geometers have often observed that $D^b(X)$ appears in a symmetry connecting two mathematical objects. For example, Beilinson [@Bei] finds an example of such phenomena: he discovers that the derived category $D^b(\mathbb{P}^n)$ on the projective space ${\ensuremath{\mathbb{P}}}^n$ is equivalent to the derived category $D^b({\operatorname{mod}}{\mathop{\mathrm{End}}\nolimits}_{\mathbb{P}^n}({\ensuremath{\mathcal{E}}}))$ of the abelian category of finitely generated right ${\mathop{\mathrm{End}}\nolimits}_{\mathbb{P}^n}({\ensuremath{\mathcal{E}}})$-modules, where ${\ensuremath{\mathcal{E}}}$ is the vector bundle $$\mathcal{O}_{\mathbb{P}^n}\oplus
\mathcal{O}_{\mathbb{P}^n}(-1) \oplus \cdots
\oplus \mathcal{O}_{\mathbb{P}^n}(-n).$$ We also have the so-called *McKay correspondence* ([@BKR], [@KaVa]), which is a symmetry between complex algebraic geometry and representation theory. We now understand the McKay correspondence as a derived equivalence between an algebraic variety and a non-commutative algebra.
Van den Bergh proposes a generalization of Beilinson’s theorem and the McKay correspondence through derived Morita theory [@Rickard].
***[[@MVB Theorem A]]{}***\[Intro:1\] Let $f\colon X\to Y={\operatorname{Spec}}R$ be a projective morphism between Noetherian schemes. Assume that $f$ has at most one-dimensional fibers and $\mathbb{R}f_{\ast}\mathcal{O}_X =\mathcal{O}_Y$. Then there is a vector bundle $\mathcal{E}$ on $X$ such that the functor $$\mathbb{R}{\mathop{\mathrm{Hom}}\nolimits}_X(\mathcal{E}, -)\colon
D^b(X) \to D^b({\operatorname{mod}}{\mathop{\mathrm{End}}\nolimits}_X(\mathcal{E})),$$ defines an equivalence of derived categories.
Such a vector bundle $\mathcal{E}$ is called a *tilting generator*. In the proof, Van den Bergh uses a globally generated ample line bundle ${\ensuremath{\mathcal{L}}}$ on $X$ and constructs ${\ensuremath{\mathcal{E}}}$ from ${\ensuremath{\mathcal{O}}}_X$ and ${\ensuremath{\mathcal{L}}}^{-1}$.
Recently, Kaledin [@Kale] proved the existence of a tilting generator étale locally on $Y$ when $f\colon X\to Y$ is a crepant resolution and $Y$ has symplectic singularities. He uses quite sophisticated tools such as mod $p$ reductions and deformation quantizations, but it seems difficult to apply his method when $Y$ does not have symplectic singularities.
The aim of this paper is to generalize Van den Bergh’s arguments using ample line bundles, and to construct a tilting generator in a more general setting. In particular, we relax the fiber dimensionality assumption. One of our main results is:
***[\[Theorem \[thm:rel.dim2\]\]]{}*** \[thm:main0\] Let $f\colon X\to Y={\operatorname{Spec}}R$ be a projective morphism between Noetherian schemes and $R$ be a ring of finite type over a field, or a Noetherian complete local ring. Assume that $f$ has at most two-dimensional fibers and ${\ensuremath{\mathbb{R}}}f_{\ast}{\ensuremath{\mathcal{O}}}_X ={\ensuremath{\mathcal{O}}}_Y$. Further assume that there is an ample globally generated line bundle $\mathcal{L}$ on $X$ that satisfies $\mathbb{R}^2 f_{\ast}\mathcal{L}^{-1}=0$. Then there is a tilting vector bundle generating the derived category $D^-(X)$.
Our method can apply to more general situations: for instance, we can show that there is a tilting generator on $X=T^*G(2,4)$, where $G(2,4)$ is the Grassmann manifold. The variety $X$ admits the Springer resolution $f\colon X\to {\operatorname{Spec}}R$, which has a 4-dimensional fiber.
The paper is organized as follows. In §\[section:ample\], we show some easy results on ample line bundles, which we use later. In §\[section:tilting\], we define tilting generators and explain their properties. In §\[section: construction\], we present our main construction of tilting generators and the assumptions behind it. In §\[section:heart\], we study the heart of a t-structure given in §\[section: construction\]. The results in §\[section:heart\] are not used in any other sections. In §\[section:2dimensional\], we prove Theorem \[thm:main0\] and find several examples where we can apply Theorem \[thm:main0\]. In §\[section:G(2,4)\], we find a tilting generator of the derived category of the cotangent bundle of the Grassmann manifold $G(2,4)$. In §\[section:auxiliary\], we show an auxiliary result which is needed in §\[subsection:gluing\]. To prove the result in §\[section:auxiliary\], we require the dualizing complex $D_R$ on $Y$ in Theorem \[thm:main0\]. This requirement is why we assume that $Y$ is a scheme of finite type over a field or a spectrum of a Noetherian complete local ring. In the appendix, we apply our result to prove the existence of non-commutative crepant resolutions in the sense of Van den Bergh ([@nonc]).
#### Notation and Conventions.
For a right (respectively, left) Noetherian (possibly non-commutative) ring $A$, ${\operatorname{mod}}A$ (respectively, $A{\operatorname{mod}}$) is the abelian category of finitely generated right (respectively, left) $A$-modules and we set $D^b(A)=D^b({\operatorname{mod}}A)$, $D^-(A)=D^-({\operatorname{mod}}A)$ etc. We denote by $A^\circ$ the opposite ring of a ring $A$.
For a Noetherian scheme $X$, we denote by $D(X)$ (respectively, $D^b(X)$, $D^-(X)$, $\ldots$.) the unbounded (respectively, bounded, bounded above, $\ldots$.) derived category of coherent sheaves. If ${\ensuremath{\mathcal{A}}}$ is a sheaf of ${\ensuremath{\mathcal{O}}}_X$-algebras, then we denote by ${\operatorname{Coh}}{\ensuremath{\mathcal{A}}}$ the category of right coherent ${\ensuremath{\mathcal{A}}}$-modules. Put $D^-({\ensuremath{\mathcal{A}}})=D^-({\operatorname{Coh}}{\ensuremath{\mathcal{A}}})$.
We also denote by $D_X$ the dualizing complex (if it exists) and by ${\ensuremath{\mathbb{D}}}_X$ the dualizing functor $${\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X(-,D_X)\colon D^-(X)\to D^+(X).$$
For a complex ${\ensuremath{\mathcal{K}}}$ of coherent sheaves on $X$, we denote by $\tau_{\le p}{\ensuremath{\mathcal{K}}}(=\tau_{<p+1}{\ensuremath{\mathcal{K}}})$ and $\tau_{> p}{\ensuremath{\mathcal{K}}}(=\tau_{\ge p+1}{\ensuremath{\mathcal{K}}})$ the following complexes: $$(\tau_{\le p}{\ensuremath{\mathcal{K}}})^n= \begin{cases}
{\ensuremath{\mathcal{K}}}^n & n < p \\
{\mathop{\mathrm{Ker}}\nolimits}d^p & n = p \\
0 & n > p
\end{cases}$$ $$(\tau_{> p}{\ensuremath{\mathcal{K}}})^n= \begin{cases}
0 & n < p \\
{\mathop{\mathrm{Im}}\nolimits}d^p & n = p \\
{\ensuremath{\mathcal{K}}}^n & n > p .
\end{cases}$$ Here, $d^{p}\colon \mathcal{K}^{p}\to \mathcal{K}^{p+1}$ is the differential. Similarly we denote by $\sigma_{\le p}{\ensuremath{\mathcal{K}}}(=\sigma_{< p+1}{\ensuremath{\mathcal{K}}})$ and $\sigma_{> p}{\ensuremath{\mathcal{K}}}(=\sigma_{\ge p+1}{\ensuremath{\mathcal{K}}})$ the following complexes: $$(\sigma_{\le p}{\ensuremath{\mathcal{K}}})^n= \begin{cases}
{\ensuremath{\mathcal{K}}}^n & n \le p \\
0 & n>p
\end{cases}$$ and $$(\sigma_{> p}{\ensuremath{\mathcal{K}}})^n= \begin{cases}
0 & n\le p \\
{\ensuremath{\mathcal{K}}}^n & n > p.
\end{cases}$$ Then there are distinguished triangles in $D(X)$: $$\tau_{\le p}{\ensuremath{\mathcal{K}}}\to {\ensuremath{\mathcal{K}}}\to \tau_{>p}{\ensuremath{\mathcal{K}}}\to \tau_{\le p}{\ensuremath{\mathcal{K}}}[1]$$ and $$\sigma_{> p}{\ensuremath{\mathcal{K}}}\to {\ensuremath{\mathcal{K}}}\to \sigma_{\le p}{\ensuremath{\mathcal{K}}}\to \sigma_{> p}{\ensuremath{\mathcal{K}}}[1].$$
We denote by $D(X)^{\le p}$ the full subcategory of $D(X)$: $$D(X)^{\le p} = \bigl\{
{\ensuremath{\mathcal{K}}}\in D(X) \bigm|
{\ensuremath{\mathcal{H}}}^i({\ensuremath{\mathcal{K}}})=0 \mbox{ for all } i>p \bigr\}.$$ We also define $D(A)^{\ge p}\ldots$ similarly.
#### Acknowledgement.
Y.T. is supported by J.S.P.S for Young Scientists (No.198007). H.U. is supported by the Grants-in-Aid for Scientific Research (No.17740012). H.U. thanks Hiraku Nakajima for useful discussions.
Results on ample line bundles {#section:ample}
=============================
In this section, we present some easy results on ample line bundles. Let $f \colon X\to Y={\operatorname{Spec}}R$ be a projective morphism from a Noetherian scheme to a Noetherian affine scheme. Suppose that $\mathbb{R}^i f_{\ast}\mathcal{O}_X=0$ for $i>0$ and the fibers of $f$ are at most $n$-dimensional ($n\ge 0$). Assume further that there is an ample, globally generated line bundle ${\ensuremath{\mathcal{L}}}$ on $X$, satisfying $$\label{eqn:ample4}
{\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^{-j}=0$$ for $i\ge 2, 0<j<n$.
Take general elements $H_k\in |{\ensuremath{\mathcal{L}}}|$, $1\le k\le n$, and put $H^k=H_1\cap\cdots\cap H_k, H^0=X$ and $H=H^1$. Below we often use the following exact sequence: $$\begin{aligned}
\label{eqn:Lj}
0\to{\ensuremath{\mathcal{L}}}^{l-1}|_{H^k}\to{\ensuremath{\mathcal{L}}}^{l}|_{H^k}\to{\ensuremath{\mathcal{L}}}^{l}|_{H^{k+1}}\to 0.\end{aligned}$$
\[lem:vanishing\] In the above situation, we have $${\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^j=0$$ for all $i>0, j\ge 0$.
We show the assertion by induction on $n$, the upper bound of the dimension of the fibers of $f$. The statement obviously holds when $f$ is quasi-finite, that is, $n=0$. Next, suppose that $n>0$ and the statement holds for $n-1$.
By (\[eqn:ample4\]) and (\[eqn:Lj\]), we see ${\ensuremath{\mathbb{R}}}^1 f_{\ast}(\mathcal{O}_H)=0$ and $${\ensuremath{\mathbb{R}}}^if_*({\ensuremath{\mathcal{L}}}^{-j}|_H)=0$$ for $i\ge 2, 0\le j<n-1$. Hence we can use the induction hypothesis, and conclude $${\ensuremath{\mathbb{R}}}^if_*({\ensuremath{\mathcal{L}}}^j|_H)=0$$ for all $i>0, j\ge 0$. Therefore, there is a surjection $
{\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^{j-1}\twoheadrightarrow {\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^j.
$ Since ${\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{O}}}_X= 0$ for $i>0$, we obtain the assertion.
In the application below, $X$ is always a smooth variety and $-K_X$ is $f$-nef and $f$-big. If, furthermore, $X$ is defined over ${\ensuremath{\mathbb{C}}}$, then Lemma \[lem:vanishing\] is automatically true by the vanishing theorem. (cf. [@KMM Theorem 1-2-5].) Next we see the following:
\[lem:properties\] In the above situation, we have $${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{L}}}^{-n},C)\in R{\operatorname{mod}}$$ for $C\in{\operatorname{Coh}}X$ with ${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X (\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{-i},C)=0$, and $${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{L}}}^{n},C)\in R{\operatorname{mod}}[-n]$$ for $C\in{\operatorname{Coh}}X$ with ${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X (\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{i},C)=0$.
Take $C\in{\operatorname{Coh}}X$ such that $${\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(X,\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{i}\otimes C)
\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X (\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{-i},C)=0.$$ Then we can show from (\[eqn:Lj\]) that $
{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(H^k,\bigoplus_{i=k}^{n-1}{\ensuremath{\mathcal{L}}}^{i}\otimes C|_{H^k})=0
$ for $k=0,\ldots,n-1$ inductively. Therefore we have $${\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(X,{\ensuremath{\mathcal{L}}}^{n}\otimes C)\cong {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(H,{\ensuremath{\mathcal{L}}}^{n}\otimes C|_H)
\cong \cdots \cong {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({H^n},{\ensuremath{\mathcal{L}}}^{n}\otimes C|_{H^n}).$$ Because $H^n$ is relative $0$-dimensional, we obtain $
{\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{L}}}^{-n},C)\in R {\operatorname{mod}}$ as required.
Take $C\in{\operatorname{Coh}}X$ such that $$\begin{aligned}
{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(X,\bigoplus_{i=-n+1}^{0}{\ensuremath{\mathcal{L}}}^{i}\otimes C)
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X(\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{i},C) =0.\end{aligned}$$ Then we can show from (\[eqn:Lj\]) that $
{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(H^k,\bigoplus_{i=-n+k+1}^{0}{\ensuremath{\mathcal{L}}}^{i}\otimes C|_{H^k})=0
$ for $k=0,\ldots,n-1$ inductively. Therefore we have $${\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(X,{\ensuremath{\mathcal{L}}}^{-n}\otimes C)\cong {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(H,{\ensuremath{\mathcal{L}}}^{-n+1}\otimes C|_H)[-1]
\cong \cdots \cong {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({H^n}, C|_{H^n})[-n].$$ Because $H^n$ is $0$-dimensional, we obtain ${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{L}}}^{n},C)
\in R{\operatorname{mod}}[-n]$
The following lemma is fundamental in this paper.
***[[@MVB Lemma 3.2.2]]{}***\[lem322\] Let $f\colon X\to Y$ be a projective morphism between Noetherian schemes with at most $n$-dimensional fibers. Assume that $Y$ is affine. Let ${\ensuremath{\mathcal{L}}}$ be a globally generated ample line bundle on $X$. Then $\bigoplus_{i=0}^{n} {\ensuremath{\mathcal{L}}}^{i}$ is a generator of $D^-(X)$ (see the definition of generators in Definition \[def:tilting\].)
Tilting generators {#section:tilting}
==================
In this section, we define tilting generators on algebraic varieties.
Let $f \colon X\to Y={\operatorname{Spec}}R$ be a projective morphism from a Noetherian scheme to an affine Noetherian scheme.
\[def:tilting\] Let ${\ensuremath{\mathcal{E}}}$ be a perfect complex on $X$: that is, locally ${\ensuremath{\mathcal{E}}}$ is quasi-isomorphic to a bounded complex of finitely generated free ${\ensuremath{\mathcal{O}}}_X$-modules.
1. ${\ensuremath{\mathcal{E}}}$ is said to be tilting if ${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}})=0$ for any $i\ne 0$.
2. ${\ensuremath{\mathcal{E}}}$ is called a generator of $D^-(X)$ if the vanishing ${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{K}}})=0$ for ${\ensuremath{\mathcal{K}}}\in D^-(X)$ implies ${\ensuremath{\mathcal{K}}}=0$.
The vector bundle $
{\ensuremath{\mathcal{E}}}=\bigoplus_{i=0}^n \mathcal{O}_{\mathbb{P}^n}(-i)
$ on ${\ensuremath{\mathbb{P}}}^n$ is a tilting generator by Lemma \[lem322\]. This fact was first observed by Beilinson [@Bei] .
For a tilting vector bundle ${\ensuremath{\mathcal{E}}}$ on $X$, we denote by $A$ the endomorphism algebra ${\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}})$ and define functors: $$\begin{aligned}
\Phi(-)
&={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},-) \colon D^-(X)\longrightarrow D^-(A), \\
\Psi(-)
&=-{\stackrel{{\ensuremath{\mathbb{L}}}}{\otimes}}_{A}{\ensuremath{\mathcal{E}}}\colon D^-(A)\longrightarrow D^-(X).\end{aligned}$$ Note that $\Psi$ is a left adjoint functor of $\Phi$ and $\Phi\circ\Psi \cong {\ensuremath{\mathop{\mathrm{id}}}}_{D^-(A)}$.
The following lemma explains a characteristic property of tilting generators. The statement is well-known, but for the reader’s convenience, we supply the proof.
\[lemma:tilting\] In the above setting, assume furthermore that ${\ensuremath{\mathcal{E}}}$ is a generator of $D^-(X)$. Then $\Phi$ and $\Psi$ define an equivalence of triangulated categories between $D^-(X)$ and $D^-(A)$. This equivalence restricts to an equivalence between $D^b(X)$ and $D^b(A)$.
The isomorphism $\Phi\circ\Psi \cong {\ensuremath{\mathop{\mathrm{id}}}}_{D^-(A)}$ implies that the cone $C$ of the adjunction morphism $\Psi\circ \Phi ({\ensuremath{\mathcal{F}}})\to {\ensuremath{\mathcal{F}}}$ for ${\ensuremath{\mathcal{F}}}\in D^-(X)$ is annihilated by $\Phi$. Since ${\ensuremath{\mathcal{E}}}$ is a generator of $D^-(X)$, $C$ is zero. In particular $\Psi\circ \Phi \cong {\ensuremath{\mathop{\mathrm{id}}}}_{D^-(X)}$: that is, $\Phi$ and $\Psi$ define an equivalence of triangulated categories between $D^-(X)$ and $D^-(A)$.
We can show that this equivalence restricts to an equivalence between $D^b(X)$ and $D^b(A)$. It is obvious that $\Phi ({\ensuremath{\mathcal{F}}})\in D^b(A)$ for ${\ensuremath{\mathcal{F}}}\in D^b(X)$, so we only need to check that $\Psi(M)\in D^b(X)$ for any $M\in D^b(A)$. To prove this fact, we may assume $M\in{\operatorname{mod}}A$. For a sufficiently small integer $m$, consider the map $$\phi\colon \tau_{<m}\Psi (M)\to \Psi (M)$$ induced by the canonical truncation $\tau$, and apply $\Phi$ to it; $$\Phi(\phi)\colon \Phi(\tau_{<m}\Psi (M))\to \Phi\circ\Psi (M)\cong M.$$ Then the map $\Phi(\phi)$ is zero by the choice of $m$. Hence the map $\phi$ is also zero, since $\Phi\colon D^-(X)\to D^-(A)$ gives an equivalence. This implies $\Psi (M)\in D^b(X)$.
Main construction {#section: construction}
=================
In this section, we show how to construct tilting generators from ample line bundles. The main result in this section is Theorem \[thm:main\].
Setting
-------
Let $f \colon X\to Y={\operatorname{Spec}}R$ be a projective morphism from a Noetherian scheme to an affine scheme of finite type over a field, or an affine scheme of a Noetherian complete local ring. Suppose that ${\ensuremath{\mathbb{R}}}f_*{\ensuremath{\mathcal{O}}}_X={\ensuremath{\mathcal{O}}}_Y$ and fibers of $f$ are at most $n$-dimensional. Assume furthermore that there is an ample, globally generated line bundle ${\ensuremath{\mathcal{L}}}$ on $X$, satisfying $$\label{eqn:ample2}
{\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^{-j}=0$$ for $i\ge 2, 0<j<n$. Then as shown in Lemma \[lem:vanishing\], we have $$\label{eqn:ample3}
{\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^j=0$$ for all $i>0, j \ge 0$. Furthermore, we know that $\bigoplus _{i=0}^n{\ensuremath{\mathcal{L}}}^{-i}$ is a generator of $D^-(X)$ by Lemma \[lem322\].
If we assume that (\[eqn:ample2\]) holds for $i\ge 1$ and $0<j\le n$, then $\bigoplus _{i=0}^n{\ensuremath{\mathcal{L}}}^{-i}$ is already a tilting generator, so there is nothing left to prove.
Orientation {#subsection:orientation}
-----------
For illustrative purposes, before explaining our construction, we sketch a proof of Theorem \[Intro:1\].\
[*Proof of Theorem \[Intro:1\].*]{} First take the extension corresponding to a set of a generators of the $R$-module $H^1(X,{\ensuremath{\mathcal{L}}}^{-1})$; $$\label{eqn:orientation}
0\to {\ensuremath{\mathcal{L}}}^{-1}\to {\ensuremath{\mathcal{N}}}\to {\ensuremath{\mathcal{O}}}_X^{\oplus r} \to 0.$$ Then by a direct calculation, we can show that ${\ensuremath{\mathcal{E}}}={\ensuremath{\mathcal{O}}}_X\oplus {\ensuremath{\mathcal{N}}}$ is a tilting object. We can also see that ${\ensuremath{\mathcal{E}}}$ is a generator of $D^-(X)$ by Lemma \[lem322\].\
In the following subsections, we construct tilting vector bundles ${\ensuremath{\mathcal{E}}}_k$ inductively as follows. First take ${\ensuremath{\mathcal{E}}}_0={\ensuremath{\mathcal{O}}}_X$, which is tilting by the assumption ${\ensuremath{\mathbb{R}}}f_*{\ensuremath{\mathcal{O}}}_X={\ensuremath{\mathcal{O}}}_Y$ (or (\[eqn:ample3\])). Giving a tilting vector bundle ${\ensuremath{\mathcal{E}}}_{k-1}$ with $0<k\le n-1$, take the extension (\[eqn:short\]) as (\[eqn:orientation\]) and define a new tilting vector bundle ${\ensuremath{\mathcal{E}}}_k$ as ${\ensuremath{\mathcal{E}}}_{k-1}\oplus{\ensuremath{\mathcal{N}}}_{k-1}$. To construct a tilting generator ${\ensuremath{\mathcal{E}}}_n$, we need a slightly more careful treatment, as explained in §\[subsection:gluing\].
Inductive construction of tilting vector bundles {#inductive}
------------------------------------------------
Under the setting in §\[setting\], we shall construct tilting vector bundles ${\ensuremath{\mathcal{E}}}_k$ for $0\le k\le n-1$ inductively.
Induction hypotheses.
Put ${\ensuremath{\mathcal{E}}}_0={\ensuremath{\mathcal{O}}}_X$ and fix an integer $k$ with $0<k\le n-1$. Assume that we have a tilting vector bundle ${\ensuremath{\mathcal{E}}}_{k-1}$ on $X$. Let us denote the endomorphism algebra ${\mathop{\mathrm{End}}\nolimits}_X {\ensuremath{\mathcal{E}}}_{k-1}$ by $A_{k-1}$. We also define the following functors: $$\begin{aligned}
\Phi_{k-1}(-)
&={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}}_{k-1},-) \colon D(X)\longrightarrow D(A_{k-1}), \\
\Psi_{k-1}(-)
&=-{\stackrel{{\ensuremath{\mathbb{L}}}}{\otimes}}_{A_{k-1}}{\ensuremath{\mathcal{E}}}_{k-1} \colon D^-(A_{k-1})
\longrightarrow D^-(X).\end{aligned}$$ Note that $\Phi_{k-1}$ restricts to the functor $\Phi_{k-1}\colon D^{-}(X) \to D^{-}(A)$, giving the right adjoint functor of $\Psi_{k-1}$.
As induction hypotheses, we assume the following.
- For any $i\ne 0, 1$ and any $l$ with $0< l\le n-1$, we have $$\label{eqn1}
{\mathop{\mathrm{Hom}}\nolimits}^i_X({\ensuremath{\mathcal{E}}}_{k-1},{\ensuremath{\mathcal{L}}}^{-l})=0.$$
- For any $i\ne 0$ and any $l$ with $k-1\le l\le n$, we have $$\label{eqn2}
{\mathop{\mathrm{Hom}}\nolimits}_X ^i({\ensuremath{\mathcal{L}}}^{-l},{\ensuremath{\mathcal{E}}}_{k-1})=0.$$
Note that if $k=1$, (\[eqn1\]) and (\[eqn2\]) hold by (\[eqn:ample2\]) and (\[eqn:ample3\]).
\[step:construction\] Construction of ${\ensuremath{\mathcal{E}}}_{k}$.
Take a free $A_{k-1}$ resolution of $\Phi_{k-1}({\ensuremath{\mathcal{L}}}^{-k})$ and denote it by $P_{k-1} $. Since ${\ensuremath{\mathcal{H}}}^{i}(\Phi_{k-1}({\ensuremath{\mathcal{L}}}^{-k}))=0$ unless $i=0, 1$ by (\[eqn1\]), we can take $P_{k-1}$ satisfying $P_{k-1}^i=0$ for $i\ge 2$. We obtain a natural morphism $\sigma_{\ge 1}(P_{k-1} )\to P_{k-1} $, and hence we have a morphism $\Psi_{k-1}(\sigma_{\ge 1}(P_{k-1} ))
\to {\ensuremath{\mathcal{L}}}^{-k}$, since $\Psi_{k-1}$ is a left adjoint functor of $\Phi_{k-1}$. Define an object ${\ensuremath{\mathcal{N}}}_{k-1}\in D^-(X)$ to be the cone of this morphism; $$\begin{aligned}
\label{eq:cone}
\Psi_{k-1}(\sigma_{\ge 1}(P_{k-1} )) \to {\ensuremath{\mathcal{L}}}^{-k}
\to {\ensuremath{\mathcal{N}}}_{k-1} \to \Psi_{k-1}
(\sigma_{\ge 1}(P_{k-1} ))[1],\end{aligned}$$ and we also define $${\ensuremath{\mathcal{E}}}_k={\ensuremath{\mathcal{E}}}_{k-1}\oplus {\ensuremath{\mathcal{N}}}_{k-1}.$$ Applying $\Phi_{k-1}$ to (\[eq:cone\]) and using the isomorphism, $$\Phi_{k-1}\circ\Psi_{k-1}\cong {{\ensuremath{\mathop{\mathrm{id}}}}} _{D^-(A_{k-1})},$$ we have $\Phi_{k-1}({\ensuremath{\mathcal{N}}}_{k-1})
\cong \sigma_{<1}(P_{k-1} )$. Furthermore, we know that $\Psi_{k-1}(\sigma_{\ge 1}(P_{k-1} ))$ is isomorphic to an object of the form ${\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r_{k-1}}[-1]$ for some $r_{k-1}\ge 0$. Hence, there is a short exact sequence of coherent sheaves; $$\label{eqn:short}
0\to {\ensuremath{\mathcal{L}}}^{-k} \to {\ensuremath{\mathcal{N}}}_{k-1} \to {\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r_{k-1}}\to 0.$$ Consequently, ${\ensuremath{\mathcal{N}}}_{k-1}$ and ${\ensuremath{\mathcal{E}}}_k$ are vector bundles on $X$.
${\ensuremath{\mathcal{E}}}_k$ satisfies the induction hypotheses.
We shall check below that ${\ensuremath{\mathcal{E}}}_k$ has similar properties to (\[eqn1\]) and (\[eqn2\]).
\[cla:tilting1\] ${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_k,{\ensuremath{\mathcal{L}}}^{-l})=0$ for any $i\ne 0,1$ and any $l$ with $0< l\le n-1$
Claim \[cla:tilting1\] follows from (\[eqn:ample2\]), (\[eqn:ample3\]), (\[eqn1\]) and the long exact sequence $$\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r},{\ensuremath{\mathcal{L}}}^{-l})\to
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{k-1},{\ensuremath{\mathcal{L}}}^{-l})\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-k},{\ensuremath{\mathcal{L}}}^{-l}) \to .$$
\[cla:tilting2\] ${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-l},{\ensuremath{\mathcal{E}}}_k)=0$ for any $i\ne 0$ and any $l$ with $k\le l\le n$.
Claim \[cla:tilting2\] follows from (\[eqn:ample3\]), (\[eqn2\]) and the long exact sequence $$\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-l},{\ensuremath{\mathcal{L}}}^{-k})\to
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-l}, {\ensuremath{\mathcal{N}}}_{k-1})\to
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-l}, {\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r}) \to.$$
\[cla:tilting3\] ${\ensuremath{\mathcal{E}}}_k$ is a tilting object.
From $\Phi_{k-1}({\ensuremath{\mathcal{N}}}_{k-1})\cong
\sigma_{<1}(P_{k-1} )$, we obtain $$\label{eqn:tilting1}
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_{k-1},{\ensuremath{\mathcal{N}}}_{k-1})={\ensuremath{\mathcal{H}}}^i(\Phi_{k-1}({\ensuremath{\mathcal{N}}}_{k-1}))=0$$ for all $i\ne 0$. By (\[eqn2\]) and the long exact sequence $$\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r},{\ensuremath{\mathcal{E}}}_{k-1})\to
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{k-1},{\ensuremath{\mathcal{E}}}_{k-1})\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-k},{\ensuremath{\mathcal{E}}}_{k-1}) \to,$$ we have $$\label{eqn:tilting2}
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{k-1},{\ensuremath{\mathcal{E}}}_{k-1})=0$$ for all $i\ne 0$. Finally, by Claim \[cla:tilting2\], (\[eqn:tilting1\]) and the long exact sequence $$\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r},{\ensuremath{\mathcal{N}}}_{k-1})\to
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{k-1},{\ensuremath{\mathcal{N}}}_{k-1})\to {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-k},{\ensuremath{\mathcal{N}}}_{k-1}) \to,$$ we have $$\label{eqn:tilting3}
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{k-1},{\ensuremath{\mathcal{N}}}_{k-1})=0$$ for all $i\ne 0$. The equalities (\[eqn:tilting1\]), (\[eqn:tilting2\]) and (\[eqn:tilting3\]) imply that ${\ensuremath{\mathcal{E}}}_k$ is a tilting object.
By induction on $k$, we can construct a tilting vector bundle ${\ensuremath{\mathcal{E}}}_{n-1}$.
\[rem:process\] We cannot apply our method in this subsection to construct ${\ensuremath{\mathcal{E}}}_n$. In Step \[step:construction\], we need the vanishing of ${\mathop{\mathrm{Hom}}\nolimits}^i_X({\ensuremath{\mathcal{E}}}_{n-1}, {\ensuremath{\mathcal{L}}}^{-n})$ for $i\ge 2$. However this is not guaranteed by the induction hypothesis (\[eqn1\]).
Gluing t-structures {#subsection:gluing}
-------------------
The vector bundle ${\ensuremath{\mathcal{E}}}_{n-1}$ does not generate the category $D^{-}(X)$ yet (see Lemma \[lem:generator\]), so we need one more step to construct a tilting generator ${\ensuremath{\mathcal{E}}}_{n}$. As we mentioned in Remark \[rem:process\], a similar method in §\[inductive\] does not work. In this subsection, we make some assumptions and construct a tilting generator ${\ensuremath{\mathcal{E}}}_{n}$ of $D^-(X)$.
As in §\[inductive\], we define as $A_{n-1}={\mathop{\mathrm{End}}\nolimits}_X {\ensuremath{\mathcal{E}}}_{n-1}$ and $$\begin{aligned}
\Phi_{n-1}(-)&={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}}_{n-1},-)
\colon D(X)\longrightarrow D(A_{n-1}), \notag\\
\Psi_{n-1}(-)&=-{\stackrel{{\ensuremath{\mathbb{L}}}}{\otimes}}_{A_{n-1}}{\ensuremath{\mathcal{E}}}_{n-1} \colon D^-(A_{n-1})
\longrightarrow D^-(X).\notag\end{aligned}$$ Take a free $A_{n-1}$ resolution $P_{n-1}$ of $\Phi_{n-1}({\ensuremath{\mathcal{L}}}^{-n})$. Then each ${\ensuremath{\mathcal{H}}}^i(P_{n-1} )$ vanishes for $i<0$ but does not necessarily vanish for $i\ge 2$. (cf. Remark \[rem:process\].) As in Step \[step:construction\] in §\[inductive\], define an object ${\ensuremath{\mathcal{N}}}_{n-1}\in D^b(X)$ such that ${\ensuremath{\mathcal{N}}}_{n-1}$ fits into a triangle $$\label{eqn:triangle}
\Psi_{n-1}(\sigma_{\ge 1}(P_{n-1} )) \to {\ensuremath{\mathcal{L}}}^{-n}
\to {\ensuremath{\mathcal{N}}}_{n-1} \to \Psi_{n-1}
(\sigma_{\ge 1}(P_{n-1} ))[1].$$ Note that ${\ensuremath{\mathcal{N}}}_{n-1}$ is a perfect complex, since so is $\Psi_{n-1}(\sigma_{\ge 1}(P_{n-1} ))$. We again define $${\ensuremath{\mathcal{E}}}_n={\ensuremath{\mathcal{E}}}_{n-1}\oplus{\ensuremath{\mathcal{N}}}_{n-1}.$$ Although we cannot conclude that ${\ensuremath{\mathcal{E}}}_n$ is tilting, we consider the functor $\Phi_n(-)={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}}_{n},-)$.
Let us define ${\ensuremath{\mathcal{C}}}_k$ for $0\le k\le n$ to be the full subcategory of the unbounded derived category $D(X)$, $${\ensuremath{\mathcal{C}}}_k =\{ {\ensuremath{\mathcal{K}}}\in D(X) \mid \Phi_{k}({\ensuremath{\mathcal{K}}}) =0\}.$$
\[lem:generator\] Let $k$ be an integer such that $0\le k \le n$. Then ${\ensuremath{\mathcal{K}}}\in{\ensuremath{\mathcal{C}}}_{k}$ if and only if $${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X(\bigoplus _{i=0}^k{\ensuremath{\mathcal{L}}}^{-i},{\ensuremath{\mathcal{K}}})=0.$$ In particular, ${\ensuremath{\mathcal{E}}}_{n}$ is a generator of $D^-(X)$.
The proof proceeds by induction on $k$. First, note that the statement is true for $k=0$, since ${\ensuremath{\mathcal{O}}}_X={\ensuremath{\mathcal{E}}}_0$. For $0<k\le n-1$, we obtain from (\[eqn:short\]) that $$\begin{aligned}
{\ensuremath{\mathcal{K}}}\in{\ensuremath{\mathcal{C}}}_k &\iff {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{E}}}_{k-1},{\ensuremath{\mathcal{K}}})
\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{L}}}^{-k},{\ensuremath{\mathcal{K}}})=0 \\
&\iff {\ensuremath{\mathcal{K}}}\in{\ensuremath{\mathcal{C}}}_{k-1} \mbox{ and } {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{L}}}^{-k},{\ensuremath{\mathcal{K}}})=0.\end{aligned}$$ For $k=n$, we have a similar conclusion by (\[eqn:triangle\]), since each term of the complex $\Psi_{n-1}(\sigma_{\ge 1}(P_{n-1} ))[1]$ is a direct sum of ${\ensuremath{\mathcal{E}}}_{n-1}$.
Suppose that $
{\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}}_n,{\ensuremath{\mathcal{K}}})=0
$ for ${\ensuremath{\mathcal{K}}}\in D^-(X)$. Then, the assertion we proved above and Lemma \[lem322\] imply that ${\ensuremath{\mathcal{K}}}=0$, which implies the last statement.
\[rem:easycase\]
1. In the setting in §\[setting\], assume furthermore $$\begin{aligned}
\label{eq:assume}
{\ensuremath{\mathbb{R}}}^if_*{\ensuremath{\mathcal{L}}}^{-n}=0\end{aligned}$$ for $i\ge 2$: that is, the vanishing in (\[eqn:ample2\]) for $j=n$. Then we can show that ${\ensuremath{\mathcal{E}}}_{n}$ is a tilting vector bundle that generates $D^-(X)$ as follows: In this case, we can show ${\mathop{\mathrm{Hom}}\nolimits}^i_X({\ensuremath{\mathcal{E}}}_{n-1}, {\ensuremath{\mathcal{L}}}^{-n})=0$ for $i\ge 2$ as Claim \[cla:tilting2\] and so the inductive construction in §\[inductive\] works for ${\ensuremath{\mathcal{E}}}_n$ (see Remark \[rem:process\]). By the lemma above, ${\ensuremath{\mathcal{E}}}_n$ is a generator.
In particular, in this extra condition (\[eq:assume\]) for $n=2$, our main Theorem \[thm:main0\] becomes rather obvious.
2. In (i), there is a short exact sequence of coherent sheaves $$\label{eqn:shortk}
0\to {\ensuremath{\mathcal{L}}}^{-k} \to {\ensuremath{\mathcal{N}}}_{k-1} \to {\ensuremath{\mathcal{E}}}_{k-1}^{\oplus r_{k-1}}\to 0$$ for all $k$ with $1\le k\le n$, and some $r_{k-1}\ge 0$. Moreover we have $${\ensuremath{\mathcal{E}}}_n={\ensuremath{\mathcal{O}}}_X\oplus\bigoplus_{k=0}^{n-1}{\ensuremath{\mathcal{N}}}_k.$$ We can easily see that the dual vector bundle ${\ensuremath{\mathcal{E}}}^\vee$ of ${\ensuremath{\mathcal{E}}}$ is also a tilting generator of $D^-(X)$.
Let us return to the situation in §\[setting\]. Instead of assuming (\[eq:assume\]), we shall work under the following assumption until the end of this section.
\[assum:induction\] For an object ${\ensuremath{\mathcal{K}}}\in D(X)$, if we have the equality $${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X(\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{-i},{\ensuremath{\mathcal{K}}})=0,$$ then the equality $${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X (\bigoplus_{i=0}^{n-1}{\ensuremath{\mathcal{L}}}^{-i},{\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}))=0$$ holds for all $k$.
In §\[section:2dimensional\] and §\[section:G(2,4)\], we will study the cases where Assumption \[assum:induction\] holds. Assumption \[assum:induction\] means that ${\ensuremath{\mathcal{K}}}\in{\ensuremath{\mathcal{C}}}_{n-1}$ implies ${\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}})\in {\ensuremath{\mathcal{C}}}_{n-1}$ for all $k$. Then we can define a t-structure on ${\ensuremath{\mathcal{C}}}_{n-1}$ induced by the standard one on $D(X)$. Next we introduce the triangulated category $$D^{\dag}(X)= \left\{ {\ensuremath{\mathcal{K}}}\in D(X) \bigm|
\Phi_{n-1}({\ensuremath{\mathcal{K}}})\in D^b( A_{n-1}) \right\}.$$ Note that $\Psi_{n-1}$ defines a functor from $D^b(A_{n-1})$ to $D^{\dag}(X)$, since $\Phi_{n-1}\circ \Psi_{n-1}\cong {\ensuremath{\mathop{\mathrm{id}}}}_{D^b(A_{n-1})}$. Hence it is a left adjoint functor of $\Phi_{n-1}\colon D^{\dag}(X)\to D^b(A_{n-1})$. The advantage of considering $D^{\dag}(X)$ is the existence of a right adjoint functor $$\Psi_{n-1}'\colon D^b(A_{n-1}) \to D^{\dag}(X)$$ of $\Phi_{n-1}\colon D^{\dag}(X)\to D^b(A_{n-1})$ (see Lemma \[lem:adjoint\]). Therefore, we can construct a new t-structure on $D^{\dag}(X)$ by gluing the t-structure on ${\ensuremath{\mathcal{C}}}_{n-1}$ (with perversity $p\in \mathbb{Z}$) and the standard t-structure on $D^b(A_{n-1})$ via the exact triple of triangulated categories [@GM pp. 286]; $${\ensuremath{\mathcal{C}}}_{n-1}\stackrel{i_{n-1}}{\to} D^{\dag}(X)\to D^b(A_{n-1}).$$ Let $i_{n-1}^{\ast}$, $i_{n-1}^{!}\colon D^{\dag}(X) \to {\ensuremath{\mathcal{C}}}_{n-1}$ be the left and right adjoint functors of the inclusion functor $i_{n-1}$ respectively, whose existence follows from the existence of the right and left adjoint functors of $\Phi_{n-1}$. Specifically, $(i_{n-1}^{\ast}, i_{n-1}^{!})$ are constructed so that there are distinguished triangles $$\begin{aligned}
& \Psi_{n-1}\circ\Phi_{n-1}(E) \to E \to i_{n-1}^{\ast}E, \\
& i_{n-1}^{!}(E) \to E \to \Psi_{n-1}'\circ\Phi_{n-1}(E)\end{aligned}$$ for any $E\in D^{\dag}(X)$. We, therefore, obtain the new t-structure on $D^{\dag}(X)$: $$\begin{aligned}
^{p}{\ensuremath{\mathcal{D}}}^{\le 0} &=\{ {\ensuremath{\mathcal{K}}}\in D^{\dag}(X) \mid
\Phi_{n-1}({\ensuremath{\mathcal{K}}}) \in D^b(A_{n-1})^{\le 0},\ i_{n-1}^{\ast}{\ensuremath{\mathcal{K}}}\in
{\ensuremath{\mathcal{C}}}_{n-1}^{\le p}\}, \\
^{p}{\ensuremath{\mathcal{D}}}^{\ge 0} &=\{ {\ensuremath{\mathcal{K}}}\in D^{\dag}(X) \mid
\Phi_{n-1}({\ensuremath{\mathcal{K}}}) \in D^b(A_{n-1})^{\ge 0},\ i_{n-1}^{!}{\ensuremath{\mathcal{K}}}\in
{\ensuremath{\mathcal{C}}}_{n-1}^{\ge p}\}.\end{aligned}$$ Here, $p$ is an integer that determines the *perversity* of the t-structure and we denote ${\ensuremath{\mathcal{C}}}_{n-1}^{\le p}={\ensuremath{\mathcal{C}}}_{n-1}\cap D(X)^{\le p}$ and ${\ensuremath{\mathcal{C}}}_{n-1}^{\ge p}={\ensuremath{\mathcal{C}}}_{n-1}\cap D(X)^{\ge p}$. The heart of the above t-structure is called the category of *perverse coherent sheaves* (cf. [@Br1]):
$${\!{\ }^{p} \! \mathop{\mathrm{Per}}}(X/A_{n-1})=
\left\{
{\ensuremath{\mathcal{K}}}\in D^{\dag}(X) \
\begin{array}{|l}
\Phi_{n-1}({\ensuremath{\mathcal{K}}})\in {\operatorname{mod}}A_{n-1} \mbox{ and } \\
i_{n-1}^{\ast}{\ensuremath{\mathcal{K}}}\in {\ensuremath{\mathcal{C}}}_{n-1}^{\le p},\
i_{n-1}^{!}{\ensuremath{\mathcal{K}}}\in {\ensuremath{\mathcal{C}}}_{n-1}^{\ge p}
\end{array}
\right\}.$$
\[rem:notsoeasy\] Note that since the functor $\Phi_{n-1}\colon D^b(X)\to D^b(A_{n-1})$ does not necessarily have a right adjoint functor, we cannot construct the perverse t-structure on $D^b(X)$ in a similar way. However we will see in §\[section:heart\] that ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$ is in fact the heart of a bounded t-structure on $D^b(X)$.
The condition $i_{n-1}^{\ast}{\ensuremath{\mathcal{K}}}\in {\ensuremath{\mathcal{C}}}_{n-1}^{\le p}$ (resp. $i_{n-1}^{!}{\ensuremath{\mathcal{K}}}\in {\ensuremath{\mathcal{C}}}_{n-1}^{\ge p}$) is equivalent to the condition $$\begin{aligned}
\label{eq:equiv}
{\mathop{\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{K}}}, C)=0 \quad (\mbox{ resp. }{\mathop{\mathrm{Hom}}\nolimits}_X(C, {\ensuremath{\mathcal{K}}})=0)\end{aligned}$$ for any $C\in {\ensuremath{\mathcal{C}}}_{n-1}^{\ge p+1}$ (resp. $C\in {\ensuremath{\mathcal{C}}}_{n-1}^{\le p-1}$). If ${\ensuremath{\mathcal{K}}}\in D^b(X)$, it is enough to check (\[eq:equiv\]) for $C\in {\ensuremath{\mathcal{C}}}_{n-1}\cap {\operatorname{Coh}}X[j]$ with $j<-p$ (resp. $j>-p$).
\[cla:N\_n0\] The object ${\ensuremath{\mathcal{N}}}_{n-1}$ belongs to ${\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1})$.
Since ${\ensuremath{\mathcal{N}}}_{n-1}\in D^b(X)$, it is enough to check the following; $$\begin{aligned}
\label{check1}& \Phi_{n-1}({\ensuremath{\mathcal{N}}}_{n-1}) \in {\operatorname{mod}}A_{n-1}, \\
& {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{n-1}, C)=0 \mbox{ for }i<0 \mbox{ and }
\label{check2}C\in {\ensuremath{\mathcal{C}}}_{n-1}\cap {\operatorname{Coh}}X,\\
\label{check3}& {\mathop{\mathrm{Hom}}\nolimits}_X^i(C, {\ensuremath{\mathcal{N}}}_{n-1})=0 \mbox{ for }i<0 \mbox{ and }
C\in {\ensuremath{\mathcal{C}}}_{n-1}\cap {\operatorname{Coh}}X. \end{aligned}$$ First let us check (\[check1\]). By the triangle (\[eqn:triangle\]), we have $\Phi_{n-1}({\ensuremath{\mathcal{N}}}_{n-1})\cong \sigma_{\le 0}P_{n-1}$, hence ${\ensuremath{\mathcal{H}}}^i(\Phi_{n-1}({\ensuremath{\mathcal{N}}}_{n-1}))=0$ for $i>0$. For $i<0$, we have $$\begin{aligned}
{\ensuremath{\mathcal{H}}}^{i}(\Phi_{n-1}({\ensuremath{\mathcal{N}}}_{n-1}))
&\cong {\ensuremath{\mathcal{H}}}^{i}(\Phi_{n-1}({\ensuremath{\mathcal{L}}}^{-n})) \\
&=0, \end{aligned}$$ since ${\ensuremath{\mathcal{E}}}_{n-1}$ and ${\ensuremath{\mathcal{L}}}^{-n}$ are vector bundles on $X$. Therefore (\[check1\]) holds. Next for $C\in {\ensuremath{\mathcal{C}}}_{n-1} \cap {\operatorname{Coh}}X$, we have $$\begin{aligned}
\label{eq:check2}
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{n-1}, C) \cong {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-n}, C)\end{aligned}$$ for any $i$ by the triangle (\[eqn:triangle\]). Therefore (\[check2\]) follows. Finally we check (\[check3\]). Since ${\ensuremath{\mathcal{L}}}^{-n}$ and $\Psi_{n-1}(\sigma_{\ge 1}(P_{n-1} )[1])$ belong to $D(X)^{\ge 0}$, we have $$\begin{aligned}
{\mathop{\mathrm{Hom}}\nolimits}_X ^i(C, {\ensuremath{\mathcal{L}}}^{-n})
&\cong {\mathop{\mathrm{Hom}}\nolimits}_X ^i(C, \Psi_{n-1}(\sigma_{\ge 1}(P_{n-1} )[1])) \\
&=0\end{aligned}$$ for $i<0$ and $C\in{\ensuremath{\mathcal{C}}}_{n-1}\cap{\operatorname{Coh}}X$. By the triangle (\[eqn:triangle\]), (\[check3\]) also follows.
\[cla:N\_n\] For $i>0$ and $B\in{\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1})$, we have $$\label{eqn:projective}
{\mathop{\mathrm{Hom}}\nolimits}_X ^i({\ensuremath{\mathcal{N}}}_{n-1},B)=0.$$ In particular, ${\ensuremath{\mathcal{N}}}_{n-1}$ is a projective object of ${\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1})$.
We have a triangle $$\begin{aligned}
\label{tri2}
\Psi_{n-1}\circ\Phi_{n-1}(B)\to B\to i_{n-1}^{\ast}B.\end{aligned}$$ By the definition of ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$, we have $i_{n-1}^{\ast}B\in{\ensuremath{\mathcal{C}}}_{n-1}^{\le 0}$. To see (\[eqn:projective\]), it suffices to show $$\label{eqn:projective2}
{\mathop{\mathrm{Hom}}\nolimits}_X ^i({\ensuremath{\mathcal{N}}}_{n-1}, i_{n-1}^{\ast}B)=0$$ and $$\label{eqn:projective2'}
{\mathop{\mathrm{Hom}}\nolimits}_X ^i({\ensuremath{\mathcal{N}}}_{n-1}, \Psi_{n-1}\circ\Phi_{n-1}(B))=0$$ for $i>0$. To prove (\[eqn:projective2\]), it is enough to show $${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{n-1}, C)=0$$ for any $i>0$ and $C\in{\ensuremath{\mathcal{C}}}_{n-1}\cap{\operatorname{Coh}}X$. Then the assertion follows from (\[eq:check2\]) and Lemma \[lem:properties\].
Next let us show (\[eqn:projective2’\]). By the triangle (\[eqn:triangle\]), it is enough to check the following; $$\begin{aligned}
\label{check4}
& {\mathop{\mathrm{Hom}}\nolimits}_X^i(\Psi_{n-1}(\sigma_{\ge 1}(P_{n-1})[1]),
\Psi_{n-1}\circ\Phi_{n-1}(B)) =0, \\
\label{check5}
& {\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{L}}}^{-n},\Psi_{n-1}\circ\Phi_{n-1}(B))=0\end{aligned}$$ for $i>0$. Note that $$\begin{aligned}
\label{eqn:projective3}
(\ref{check4})\cong
{\mathop{\mathrm{Hom}}\nolimits}_{A_{n-1}}^i(\sigma_{\ge 1}(P_{n-1})[1], \Phi_{n-1}(B)).
\end{aligned}$$ Since $\Phi_{n-1} (B)\in {\operatorname{mod}}A_{n-1}$, $\sigma_{\ge 1}(P_{n-1} )[1]\in D^b(A_{n-1})^{\ge 0}$ and each term of $\sigma_{\ge 1}(P_{n-1} )[1]$ is a projective $A_{n-1}$-module, we conclude $(\ref{eqn:projective3})=0$. In order to check (\[check5\]), let us take a free $A_{n-1}$ resolution ${\ensuremath{\mathcal{Q}}}=(\cdots \to {\ensuremath{\mathcal{Q}}}^{-1}
\to {\ensuremath{\mathcal{Q}}}^0 \to 0)$ of an $A_{n-1}$-module $\Phi_{n-1} (B)$. Then each term of $\Psi_{n-1} ({\ensuremath{\mathcal{Q}}}) $ is a direct sum of ${\ensuremath{\mathcal{E}}}_{n-1}$. Hence by Claim \[cla:tilting2\], we conclude (\[check5\]) holds.
We readily see that ${\ensuremath{\mathcal{E}}}_{n-1}\in {\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1})$, and therefore we have ${\ensuremath{\mathcal{E}}}_n\in{\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1})$.
${\ensuremath{\mathcal{E}}}_n$ is a tilting object.
(\[check1\]) yields $$\begin{aligned}
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_{n-1},{\ensuremath{\mathcal{N}}}_{n-1})=0\end{aligned}$$ for all $i\ne 0$. Also Claim \[cla:N\_n\] implies that $$\begin{aligned}
{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{n-1},{\ensuremath{\mathcal{N}}}_{n-1})&\cong{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{N}}}_{n-1},{\ensuremath{\mathcal{E}}}_{n-1})\\
&=0\end{aligned}$$ for all $i\ne 0$. Moreover recalling that ${\ensuremath{\mathcal{E}}}_{n-1}$ is a tilting vector bundle, we have ${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}_{n-1},{\ensuremath{\mathcal{E}}}_{n-1})$ vanishes for $i\ne 0$. Combining these equalities, we see that ${\ensuremath{\mathcal{E}}}_n={\ensuremath{\mathcal{E}}}_{n-1}\oplus {\ensuremath{\mathcal{N}}}_{n-1}$ is a tilting object in $D^b(X)$.
${\ensuremath{\mathcal{E}}}_n$ is a vector bundle.
It is enough to show that ${\ensuremath{\mathcal{N}}}_{n-1}$ is a vector bundle. By Lemma \[lemma:vector\], we know $\Phi_{n-1}({\ensuremath{\mathcal{O}}}_x) \in {\operatorname{mod}}A_{n-1}$ for any closed points $x\in X$, which implies ${\ensuremath{\mathcal{O}}}_x \in {\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1})$. Hence it follows from Claim \[cla:N\_n\] that ${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{N}}}_{n-1},{\ensuremath{\mathcal{O}}}_x)\in R{\operatorname{mod}}$, and in particular ${\ensuremath{\mathcal{N}}}_{n-1}$ is a vector bundle by Lemma \[lemma:vector\].
***[[@Br2 Lemma 4.3]]{}***\[lemma:vector\] For a Noetherian scheme $X$ and an object ${\ensuremath{\mathcal{E}}}\in D^b(X)$, the following are equivalent.
1. ${\ensuremath{\mathcal{E}}}$ is a vector bundle.
2. ${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}}, {\ensuremath{\mathcal{O}}}_x)=0$ for any points $x\in X$ and $i\ne 0$.
Combining the above argument and Lemma \[lem:generator\], we can prove the following theorem.
\[thm:main\] Let $f$ and ${\ensuremath{\mathcal{L}}}$ be as in §\[setting\]. Assume that Assumption \[assum:induction\] is satisfied. Then there is a vector bundle ${\ensuremath{\mathcal{E}}}$ such that ${\ensuremath{\mathcal{E}}}$ is a tilting generator of $D^-(X)$.
For $n=2$, we can show that Assumption \[assum:induction\] is always satisfied in §\[subsection:Main result\]. Since we have proved that ${\ensuremath{\mathcal{N}}}_1$ is a vector bundle, taking the cohomology of (\[eqn:triangle\]) yields ${\mathop{\mathrm{Ext}}\nolimits}_{X}^2({\ensuremath{\mathcal{E}}}_1, {\ensuremath{\mathcal{L}}}^{-2})\otimes_{A_1}{\ensuremath{\mathcal{E}}}_1=0$. As ${\mathop{\mathrm{Ext}}\nolimits}_X^2({\ensuremath{\mathcal{E}}}_1, {\ensuremath{\mathcal{L}}}^{-2})$ may be non-zero, this vanishing is not obvious.
The hearts of t-structures {#section:heart}
==========================
Let $f$ and ${\ensuremath{\mathcal{L}}}$ be as in §\[setting\] and furthermore assume that Assumption \[assum:induction\] holds. Below we use the same notation as in §\[section: construction\], but we omit the index $n$, for instance ${\ensuremath{\mathcal{E}}}={\ensuremath{\mathcal{E}}}_n$, $A=A_{n}={\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}}_n)$, etc. Recall that the equivalence $$\Phi={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},-)\colon D^-(X)\longrightarrow D^-(A)$$ induces an equivalence between $D^b(X)$ and $D^b(A)$ by Lemma \[lemma:tilting\].
The aim of this section is the following, which will not be used in any subsequent sections. Recall that ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$ is, by definition, the heart of the t-structure $({^0{\ensuremath{\mathcal{D}}}^{\le 0}},{^0{\ensuremath{\mathcal{D}}}^{\ge 0}})$ on $D^\dag(X)$.
\[prop:hearts\[0\]\] The abelian category ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$ is the heart of a bounded t-structure on $D^b(X)$, and $\Phi({\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1}))={\operatorname{mod}}A$.
We first show that ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})\subset D^b(X)$. For an object $E\in {^0{\ensuremath{\mathcal{D}}}^{\le 0}}$, we have the distinguished triangle in $D^{\dag}(X)$ $$\begin{aligned}
\label{tri3}
\Psi_{n-1}\circ\Phi_{n-1}(E) \to E \to
i_{n-1}^{\ast}(E). \end{aligned}$$ By the definition of ${^0{\ensuremath{\mathcal{D}}}^{\le 0}}$, we have $\Phi_{n-1}(E) \in D^b(A_{n-1})^{\le 0}$ and $i_{n-1}^{\ast}(E) \in {\ensuremath{\mathcal{C}}}_{n-1}^{\le 0}$. Therefore $\Psi_{n-1}\circ\Phi_{n-1}(E)$ and $i_{n-1}^{\ast}(E)$ are objects in $D(X)^{\le 0}$, hence (\[tri3\]) yields $E\in D(X)^{\le 0}$. In particular, we have ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1}) \subset D^{-}(X)$. On the other hand, Claim \[cla:N\_n\] implies that the equivalence $\Phi\colon D^{-}(X) \to D^{-}(A)$ takes ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$ to ${\operatorname{mod}}A$. Since $\Phi$ restricts to an equivalence between $D^b(X)$ and $D^b(A)$, we must have ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})\subset D^b(X)$.
Let $(\tau_{\le 0}^{0}, \tau_{\ge 0}^{0})$ be the truncation functors corresponding to the t-structure $({^0{\ensuremath{\mathcal{D}}}^{\le 0}}, {^0{\ensuremath{\mathcal{D}}}^{\ge 0}})$. In order to conclude that ${\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$ is the heart of a bounded t-structure of $D^b(X)$, it is enough to show that for any object $E\in D^b(X)$, we have $\tau_{\le -i}^{0}(E)=\tau_{\ge i}^{0}(E)=0$ for $i\gg 0$.
Since the functor $\Phi_{n-1}\colon D^{\dag}(X) \to D^b(A_{n-1})$ takes $({^0{\ensuremath{\mathcal{D}}}^{\le 0}}, {^0{\ensuremath{\mathcal{D}}}^{\ge 0}})$ to $(D^b(A_{n-1})^{\le 0}, D^b(A_{n-1})^{\ge 0})$, we have $$\begin{aligned}
\label{eq:tau}
\Phi_{n-1}(\tau_{\ge i}^{0}(E))\cong \tau_{\ge i}^{A}(\Phi_{n-1}(E)),\end{aligned}$$ where $(\tau_{\le 0}^{A}, \tau_{\ge 0}^{A})$ are the truncation functors with respect to the standard t-structure on $D^b(A_{n-1})$. Since $\Phi_{n-1}(E) \in D^b(A_{n-1})$, we have $(\ref{eq:tau})=0$ for $i\gg 0$. Therefore $\tau_{\ge i}^{0}(E) \in {\ensuremath{\mathcal{C}}}_{n-1}^{\ge i} \subset D(X)^{\ge i}$. On the other hand, since $E\in D^b(X)$, we have ${\mathop{\mathrm{Hom}}\nolimits}(E, F)=0$ for $F\in D(X)^{\ge i}$ for $i \gg 0$. Therefore the natural morphism $E\to \tau_{\ge i}^0(E)$ is zero, which implies $\tau_{\ge i}^0(E)=0$ for $i\gg 0$. By a similar argument, we have $\tau_{\le -i}^0(E)=0$ for $i\gg 0$.
Since both of $\Phi({\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1}))$ and ${\operatorname{mod}}A$ are the hearts of bounded t-structures on $D^b(A)$, and we also know $\Phi({\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1}))\subset {\operatorname{mod}}A$, we obtain $$\Phi({\!{\ }^{0} \! \mathop{\mathrm{Per}}}(X/A_{n-1}))={\operatorname{mod}}A.$$
Assume furthermore that the equality (\[eq:assume\]) holds. Then Remark \[rem:easycase\] implies that ${\ensuremath{\mathcal{E}}}$ and ${\ensuremath{\mathcal{E}}}^\vee$ are tilting generators of $D^-(X)$. We define the functor $$\Phi_k^\vee={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}}_k^\vee,-)\colon D(X)\longrightarrow D(A_k^\circ),$$ and then $\Phi^\vee=\Phi_{n}^\vee$ gives an equivalence between $D^b(X)$ and $D^b(A^\circ)$. Here, we identify $D(A_k^\circ)$ with $D({\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}}_k^\vee))$, using the isomorphism $A_k^\circ\cong {\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}}_k^\vee)$.
Define the full subcategories of the unbounded derived category $D(X)$ as $$\begin{aligned}
{\ensuremath{\mathcal{C}}}_{n-1}^\vee &=\{ {\ensuremath{\mathcal{K}}}\in D(X) \mid \Phi_{n-1}^\vee({\ensuremath{\mathcal{K}}}) =0\}\\
D^{\dag\dag}(X)&=\left\{ {\ensuremath{\mathcal{K}}}\in D(X) \bigm|
\Phi_{n-1}^\vee({\ensuremath{\mathcal{K}}})\in D^b( A_{n-1}^\circ) \right\}.\end{aligned}$$ It is easy to see from Lemma \[lem:generator\] that for an object ${\ensuremath{\mathcal{K}}}\in D(X)$, ${\ensuremath{\mathcal{K}}}$ belongs to ${\ensuremath{\mathcal{C}}}^\vee_{n-1}$ if and only if ${\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{L}}}^{\otimes -n+1}$ belongs to ${\ensuremath{\mathcal{C}}}_{n-1}$. Therefore we can check that ${\ensuremath{\mathcal{K}}}\in{\ensuremath{\mathcal{C}}}_{n-1}^\vee$ implies that ${\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}})\in{\ensuremath{\mathcal{C}}}_{n-1}^\vee$ for all $k$ by Assumption \[assum:induction\], and hence by the exact triple of triangulated categories $${\ensuremath{\mathcal{C}}}_{n-1}^\vee{\to} D^{\dag\dag}(X)\stackrel{\Phi_{n-1}^\vee}{\to} D^b(A_{n-1}^\circ),$$ we can define the category of perverse coherent sheaves ${\!{\ }^{p} \! \mathop{\mathrm{Per}}}(X/A_{n-1}^\circ)$ as ${\!{\ }^{p} \! \mathop{\mathrm{Per}}}(X/A_{n-1})$.
Note that (\[eqn:shortk\]) yields $
{\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{N}}}_{n-1}^\vee,C)={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{L}}}^{n},C)
$ for $C\in {\ensuremath{\mathcal{C}}}_{n-1}^\vee$, hence Lemma \[lem:properties\] implies $${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X ({\ensuremath{\mathcal{N}}}^{\vee}_{n-1},C)\in R{\operatorname{mod}}$$ for $C\in {\ensuremath{\mathcal{C}}}_{n-1}^\vee\cap {\operatorname{Coh}}X[n]$. In particular, we see $$\label{eqn:coh2}
\Phi^\vee({\ensuremath{\mathcal{C}}}_{n-1}^\vee\cap {\operatorname{Coh}}X[n])\subset {\operatorname{mod}}A^\circ,$$ The proof of the next proposition uses this fact. (By comparison, $$\Phi ({\ensuremath{\mathcal{C}}}_{n-1}\cap {\operatorname{Coh}}X)\subset {\operatorname{mod}}A$$ holds by Lemma \[lem:properties\] and (\[eq:check2\]). The proof of Claims \[cla:N\_n0\] and \[cla:N\_n\] relies on this fact.)
\[prop:hearts\[n\]\] In the setting of Proposition \[prop:hearts\[0\]\], assume furthermore the equality (\[eq:assume\]) holds. Then the abelian category ${\!{\ }^{-n} \! \mathop{\mathrm{Per}}}(X/A_{n-1}^\circ)$ is the heart of a bounded t-structure on $D^b(X)$, and $\Phi^\vee({\!{\ }^{-n} \! \mathop{\mathrm{Per}}} (X/A_{n-1}^\circ))={\operatorname{mod}}A^\circ$.
We outline the proof and leave the details to the reader. First we show that the object ${\ensuremath{\mathcal{N}}}_{n-1}^\vee$ belongs to ${\!{\ }^{-n} \! \mathop{\mathrm{Per}}} (X/A_{n-1}^\circ)$ as Claim \[cla:N\_n0\]. In the proof, we use (\[eqn:coh2\]).
Next, we mimic the proof of Claim \[cla:N\_n\] and show $${\mathop{\mathrm{Hom}}\nolimits}_X ^i({\ensuremath{\mathcal{N}}}_{n-1}^\vee,B)=0$$ for $i>0$ and $B\in{\!{\ }^{-n} \! \mathop{\mathrm{Per}}} (X/A_{n-1}^\circ)$. We again use (\[eqn:coh2\]) here.
From these facts, we can conclude $$\Phi^\vee({\!{\ }^{-n} \! \mathop{\mathrm{Per}}}(X/A_{n-1}^\circ))\subset {\operatorname{mod}}A^\circ$$ and then a similar argument to Proposition \[prop:hearts\[0\]\] works.
In this example, we show that tilting generators induce the derived equivalence between certain varieties connected by a Mukai flop. We also apply Propositions \[prop:hearts\[0\]\] and \[prop:hearts\[n\]\].
Let $X$ be the cotangent bundle $T^* {\ensuremath{\mathbb{P}}}^n$ of the projective space ${\ensuremath{\mathbb{P}}}^n$ $(n\ge 2)$ and $g\colon Z \to X$ a blow-up along the zero section of the projection $\pi\colon X \to {\ensuremath{\mathbb{P}}}^n$. The exceptional locus $E(\subset Z)$ of $g$ is the incidence variety in ${\ensuremath{\mathbb{P}}}^n \times ({\ensuremath{\mathbb{P}}}^n)^{\vee}$, where $({\ensuremath{\mathbb{P}}}^n)^{\vee}$ is the dual projective space. By contracting curves contained in fibers of the projection $E\to ({\ensuremath{\mathbb{P}}}^n)^{\vee}$, we obtain a birational contraction $g^{+}\colon
Z \to X^{+}=T^* (({\ensuremath{\mathbb{P}}}^n)^\vee)$. The resulting birational map $$\phi =g^{+}\circ g^{-1}\colon X \dashrightarrow X ^+$$ is so called a *Mukai flop*. Put $R={\operatorname{Spec}}H^0(X,\mathcal{O}_X)$. Then we have a birational contraction $$f\colon X\to Y={\operatorname{Spec}}R$$ which contracts only the zero section of $\pi$. In particular, $f$ has at most $n$-dimensional fibers.
We put ${\ensuremath{\mathcal{O}}}_X(1)=\pi^* {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}^n}(1)$. Then by direct calculations (refer to calculations in §\[section:G(2,4)\]) and Lemma \[lem322\] we know that $
{\ensuremath{\mathcal{E}}}=\bigoplus _{i=0}^n{\ensuremath{\mathcal{O}}}_X(-i)
$ is a tilting generator of $D^-(X)$. On the other hand, we can see that (\[eq:assume\]) holds for ${\ensuremath{\mathcal{L}}}={\ensuremath{\mathcal{O}}}_X(1)$. Apply the arguments in §\[inductive\] and Remark \[rem:easycase\]; we obtain tilting vector bundles ${\ensuremath{\mathcal{E}}}_k=\bigoplus _{i=0}^k{\ensuremath{\mathcal{O}}}_X(-i)$ for all $k$ with $0\le k\le n$ (in other words, $r_k=0$ in (\[eqn:shortk\]) for all $k$ with $0< k\le n$). We can also check that Assumption \[assum:induction\] holds. Therefore we can apply Propositions \[prop:hearts\[0\]\] and \[prop:hearts\[n\]\].
In what follows, we use the same notation as in the previous section, and we also use the superscript $+$ to denote the corresponding object on $X^+$ to the object on $X$. For instance, $
{\ensuremath{\mathcal{E}}}^+=\bigoplus _{i=0}^n{\ensuremath{\mathcal{O}}}_{X^+}(-i).
$ Since $\phi$ is isomorphic in codimension one, there is an equivalence between categories of reflexive sheaves on $X$ and $X^+$. Hence, we have a reflexive sheaf ${\ensuremath{\mathcal{E}}}'$ on $X^+$ corresponding to ${\ensuremath{\mathcal{E}}}$, satisfying ${\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}})\cong{\mathop{\mathrm{End}}\nolimits}_{X^+}({\ensuremath{\mathcal{E}}}')$. It is known that the corresponding reflexive sheaf on $X^+$ to ${\ensuremath{\mathcal{O}}}_X(-1)$ is ${\ensuremath{\mathcal{O}}}_{X^+}(1)$ (cf. [@Na1 Lemma 1.3], [@Na2 Lemma 2.3.1]). From these facts, we see that ${\ensuremath{\mathcal{E}}}'\cong({\ensuremath{\mathcal{E}}}^+)^\vee$ and so we have an isomorphism of rings, denoted by $\phi_*$: $$\phi _*\colon A={\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}})\cong{\mathop{\mathrm{End}}\nolimits}_X({\ensuremath{\mathcal{E}}}')\cong
{\mathop{\mathrm{End}}\nolimits}_X(({\ensuremath{\mathcal{E}}}^+)^\vee)\cong A^\circ.$$ In particular, we have an equivalence $D^b(A)\cong D^b(A^\circ)$ preserving the hearts of the standard t-structures. Compose this equivalence with equivalences given by tilting generators ${\ensuremath{\mathcal{E}}}$ and $({\ensuremath{\mathcal{E}}}^+)^\vee$, and then we obtain an equivalence $$\Xi \colon D^b(X)\to D^b(A)\to D^b(A^\circ)\to D^b(X^+),$$ which satisfies $\Xi({\!{\ }^{0} \! \mathop{\mathrm{Per}}} (X/A_{n-1}))={\!{\ }^{-n} \! \mathop{\mathrm{Per}}} (X^+/A_{n-1}^\circ)$ by Propositions \[prop:hearts\[0\]\] and \[prop:hearts\[n\]\]. Compare the results in [@Na1] and [@Kaw02 Corollary 5.7], where a similar derived equivalence is shown to exist by a very different method from ours.
The case of two-dimensional fibers {#section:2dimensional}
==================================
Main result {#subsection:Main result}
-----------
Let $f \colon X\to Y={\operatorname{Spec}}R$ be a projective morphism from a Noetherian scheme to an affine scheme of finite type over a field, or an affine scheme of a Noetherian complete local ring. Suppose that the fibers of $f$ are at most two-dimensional. Assume furthermore that ${\ensuremath{\mathbb{R}}}f_*{\ensuremath{\mathcal{O}}}_X={\ensuremath{\mathcal{O}}}_Y$ and there is an ample, globally generated line bundle ${\ensuremath{\mathcal{L}}}$ on $X$, satisfying ${\ensuremath{\mathbb{R}}}^2 f_{\ast}{\ensuremath{\mathcal{L}}}^{-1}=0$. The following is a main theorem in this paper.
\[thm:rel.dim2\] Under the above situation, there is a tilting vector bundle generating the derived category $D^-(X)$.
We have to show that Assumption \[assum:induction\] holds so that we apply Theorem \[thm:main\]. Take ${\ensuremath{\mathcal{K}}}\in D(X)$, which satisfies $$\begin{aligned}
\label{eqn:induction1}
{\ensuremath{\mathbb{R}}}f_{\ast}{\ensuremath{\mathcal{K}}}= {\ensuremath{\mathbb{R}}}f_{\ast}({\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{L}}})=0.\end{aligned}$$ Let $H\in \lvert {\ensuremath{\mathcal{L}}}\rvert$ be a general member. In what follows, we repeatedly use the fact that ${\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}|_{H})={\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}})|_{H}$ for any $k\in \mathbb{Z}$, since $H$ is a general member. We have the distinguished triangle $$\begin{aligned}
{\ensuremath{\mathcal{K}}}\to {\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{L}}}\to {\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}}.\end{aligned}$$ Applying ${\ensuremath{\mathbb{R}}}f_{\ast}$ and using (\[eqn:induction1\]), we obtain ${\ensuremath{\mathbb{R}}}f_{\ast}({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}})=0$. Since $f|_{H}\colon H\to f(H)$ has at most one-dimensional fibers, we have (cf. [@Br1 Lemma 3.1]) $$\begin{aligned}
\label{eqn:induction2}
{\ensuremath{\mathbb{R}}}f_{\ast}({\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}}))=0\end{aligned}$$ for any $k$. Similarly, applying ${\ensuremath{\mathbb{R}}}f_{\ast}$ to the triangle $${\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}) \to {\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{L}}}) \to
{\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}})$$ and using (\[eqn:induction2\]), we obtain $$\begin{aligned}
\label{eqn:induction3}
{\ensuremath{\mathbb{R}}}f_{\ast}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}))
\cong
{\ensuremath{\mathbb{R}}}f_{\ast}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{L}}})).\end{aligned}$$ Next let us consider the spectral sequence: $$E_2^{p,q}={\ensuremath{\mathbb{R}}}^p f_{\ast}({\ensuremath{\mathcal{H}}}^q({\ensuremath{\mathcal{K}}})) \Rightarrow
{\ensuremath{\mathbb{R}}}^{p+q}f_{\ast}{\ensuremath{\mathcal{K}}}.$$ Since $E_2^{p,q}=0$ unless $0\le p\le 2$, the above spectral sequence and (\[eqn:induction1\]) imply $$\begin{aligned}
\label{eqn:induction4}
{\ensuremath{\mathbb{R}}}^1 f_{\ast}({\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}))=0, \quad
f_{\ast}({\ensuremath{\mathcal{H}}}^{k+1}({\ensuremath{\mathcal{K}}}))
\cong
{\ensuremath{\mathbb{R}}}^2 f_{\ast}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}))\end{aligned}$$ for any $k$. By (\[eqn:induction3\]) and (\[eqn:induction4\]), if we show ${\ensuremath{\mathbb{R}}}^2 f_{\ast}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}))=0$ for any $k$, then the conclusion of Assumption \[assum:induction\] follows.
Suppose that ${\ensuremath{\mathbb{R}}}^2 f_{\ast}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}))\neq 0$ for some $k$. By the formal function theorem, there is a closed sub-scheme $E\subset X$ supported by a two-dimensional fiber of $f$, such that $H^2(E, {\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E})\neq 0$. By the Grothendieck duality, we have $$0\neq H^2(E, {\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E}) \cong
{\mathop{\mathrm{Hom}}\nolimits}_E({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E}, {\ensuremath{\mathcal{H}}}^{-2}(D_E))^{\vee}.$$ Let $u\colon {\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E} \to {\ensuremath{\mathcal{H}}}^{-2}(D_E)$ be a non-zero morphism, and consider its image ${\mathop{\mathrm{Im}}\nolimits}u \subset {\ensuremath{\mathcal{H}}}^{-2}(D_E)$. Then the support of ${\mathop{\mathrm{Im}}\nolimits}u$ is two-dimensional because $$0\neq {\mathop{\mathrm{Hom}}\nolimits}_E({\mathop{\mathrm{Im}}\nolimits}u, {\ensuremath{\mathcal{H}}}^{-2}(D_E)) \cong
H^2(E, {\mathop{\mathrm{Im}}\nolimits}u)^{\vee}$$ by the duality. Hence by the choice of $H\in \lvert {\ensuremath{\mathcal{L}}}\rvert$, we may assume that $({\mathop{\mathrm{Im}}\nolimits}u)|_{H} \neq 0$. We may also assume that $H$ does not contain any associated prime of ${\mathop{\mathrm{Coker}}\nolimits}u$. Then we can show that $u|_{H} \colon
{\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E\cap H} \to {\ensuremath{\mathcal{H}}}^{-2}(D_E)|_{H}$ is a non-zero morphism. By adjunction, we have $$D _{H\cap E}
\cong (D_E[-1]\otimes {\ensuremath{\mathcal{L}}})|_{H}.$$ Hence $u|_{H}$ induces the non-zero morphism in $${\mathop{\mathrm{Hom}}\nolimits}_{E\cap H}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E\cap H},
{\ensuremath{\mathcal{H}}}^{-1}(D_{H\cap E}\otimes {\ensuremath{\mathcal{L}}}^{-1})).$$ Then the duality on $E\cap H$ implies $$\begin{aligned}
\label{eqn:induction5}
0\neq
H^1(E\cap H, {\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}})|_{E\cap H}\otimes {\ensuremath{\mathcal{L}}}) \cong
H^1(E\cap H,{\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}})|_{E}).\end{aligned}$$ On the other hand, the surjection $${\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}})
\twoheadrightarrow {\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}})|_{E}$$ induces the surjection $$\mathbb{R}^1 f_{\ast}({\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}}))
\twoheadrightarrow \mathbb{R}^1 f_{\ast}({\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}}|_{H}\otimes {\ensuremath{\mathcal{L}}})|_{E}).$$ However this contradicts (\[eqn:induction2\]) and (\[eqn:induction5\]), hence it follows that ${\ensuremath{\mathbb{R}}}^2 f_{\ast}({\ensuremath{\mathcal{H}}}^{k}({\ensuremath{\mathcal{K}}}))=0$.
Crepant resolutions of three dimensional canonical singularities
----------------------------------------------------------------
Let $0\in Y={\operatorname{Spec}}R$ be a $3$-dimensional canonical singularity and $R$ be a Noetherian complete local ring. Suppose that there is a crepant resolution $f\colon X\to Y$ such that the exceptional locus is an irreducible divisor $E\subset X$ and $\mathbb{R}f_*{\ensuremath{\mathcal{O}}}_X={\ensuremath{\mathcal{O}}}_Y$. Then, $E$ is a generalized del Pezzo surface: that is, $\omega _E^{-1}$ is ample. We aim to construct a tilting generator of $D^-(X)$.
\[lem:extension\] If there is an ample, globally generated line bundle ${\ensuremath{\mathcal{L}}}_1$ on $E$ with $H^2(E, {\ensuremath{\mathcal{L}}}_1^{-1})=0$, then we have an ample, globally generated line bundle $\mathcal{L}$ on $X$ such that $\mathbb{R}^2 f_{\ast}\mathcal{L}^{-1}=0$.
Let $I_E \subset \mathcal{O}_X$ be the defining ideal of $E$ and $E_n \subset X$ the subscheme defined by $I_E ^n$ for $n>0$. Then the obstruction to extend a line bundle $\mathcal{L}_n \in {\mathop{\mathrm{Pic}}\nolimits}(E_n)$ to a line bundle $\mathcal{L}_{n+1}\in {\mathop{\mathrm{Pic}}\nolimits}(E_{n+1})$ lies in $H^2 (E, I_E ^n/I_E ^{n+1})$. We have $$\begin{aligned}
H^2 (E, I_E ^n/I_E ^{n+1}) & \cong H^2 (E, \mathcal{O}_E(-nE)) \\
& \cong H^0(E, \mathcal{O}_E((n+1)E))^{\vee} \\
& = 0.\end{aligned}$$ Here, the second isomorphism follows from the Serre duality, and the last isomorphism holds because $-E$ is $f$-ample. Hence, for a given line bundle ${\ensuremath{\mathcal{L}}}_1 \in {\mathop{\mathrm{Pic}}\nolimits}(E)$, we obtain an element $$\hat{\mathcal{L}}=\{ \mathcal{L}_n \}_{n\ge 1}
\in \lim _{\longleftarrow}{\mathop{\mathrm{Pic}}\nolimits}(E_n) \cong {\mathop{\mathrm{Pic}}\nolimits}(\hat{X}).$$ By the Grothendieck existence theorem, there is a line bundle $\mathcal{L}$ on $X$ such that $\mathcal{L}|_{\hat{X}}\cong \hat{\mathcal{L}}$. Take an ample, globally generated line bundle ${\ensuremath{\mathcal{L}}}_1$ on $E$ such that $H^2 (E, {\ensuremath{\mathcal{L}}}_1^{-1})=0$. Let $\mathcal{L} \in {\mathop{\mathrm{Pic}}\nolimits}(X)$ be its extension. We have $$\begin{aligned}
H^2 (E, {\ensuremath{\mathcal{L}}}_1^{-1}\otimes I_E ^n /I_E ^{n+1}) & \cong
H^0(E, {\ensuremath{\mathcal{L}}}_1\otimes \mathcal{O}_E (nE)\otimes \omega _E)^{\vee}\\
&= 0,\end{aligned}$$ since $-E$ is $f$-ample and $H^0(E, {\ensuremath{\mathcal{L}}}_1\otimes \omega _E)=0$. Hence $\mathbb{R}^2 f_{\ast}\mathcal{L}^{-1}=0$ by the formal function theorem. $\mathcal{L}$ is also globally generated by the basepoint free theorem, and clearly $\mathcal{L}$ is ample.
In particular, Theorem \[thm:rel.dim2\] implies the following.
\[thm:CY\] In the situation of Lemma \[lem:extension\], there is a tilting generator of $D^-(X)$.
There is a $3$-dimensional crepant resolution $f\colon X\to Y$ from a Calabi-Yau threefold $X$ defined over $\mathbb{C}$ whose exceptional locus is isomorphic to $E$ in (i), (ii) below ([@Kapu2], [@Kapu]). Replace $Y$ with its completion at the singular point and shrink $X$ accordingly.
We show the existence of tilting generators of $D^-(X)$. The key fact is that if we have a line bundle ${\ensuremath{\mathcal{L}}}_1$ on $E$, as in Lemma \[lem:extension\], then we can find a tilting generator of $D^-(X)$ by Theorem \[thm:CY\].
1. The first example is a quadric $E\subset \mathbb{P}^3$, that is, $E$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ or the cone over a conic. Then ${\ensuremath{\mathcal{L}}}_1=\mathcal{O}_{\mathbb{P}^3}(1)|_{E}$ satisfies $H^2 (E, {\ensuremath{\mathcal{L}}}_1^{-1})=0$.
2. For the second example, take the cone over a conic $S \subset \mathbb{P}^3$. Let $E$ be a surface obtained by the blowing-up $\pi \colon E\to S$ at a non-singular point in $S$. Note that $E$ is a singular del Pezzo surface. Denote by $C$ the exceptional curve of $\pi$ and put $\mathcal{O}(1)=\mathcal{O}_{{\ensuremath{\mathbb{P}}}^3}(1)|_S$. Then ${\ensuremath{\mathcal{L}}}_1=\pi^{\ast}\mathcal{O}(1) \otimes \mathcal{O}_E(-C)$ is an ample, globally generated line bundle satisfying $H^2(E,{\ensuremath{\mathcal{L}}}_1^{-1})=0$.
The cotangent bundle of $G(2,4)$ {#section:G(2,4)}
================================
In §\[sec:preliminary\], we cite and prove some results that §\[sub1\] uses. In §\[sub1\], we find tilting generators on a one-parameter deformation of the cotangent bundle $X_0=T^*G(2,4)$, where $G(2, 4)$ is the Grassmann manifold. We assume all varieties are defined over ${\ensuremath{\mathbb{C}}}$ in this section.
The Bott theorem {#sec:preliminary}
----------------
Let $G$ be the Grassmann manifold $G(k,V)$ of $k$-dimensional subspaces in an $n$-dimensional ${\ensuremath{\mathbb{C}}}$-vector space $V$. There is a non-split exact sequence $$0\to \Omega _G\to \widetilde{\Omega}_G \to {\ensuremath{\mathcal{O}}}_G \to 0$$ corresponding to a nonzero element of the $1$-dimensional space $H^ 1(G,\Omega _G)$. Put $\widetilde{T}_G=(\widetilde{\Omega}_G)^\vee$. We denote the total space of $\widetilde{\Omega}_G$ (resp. $\Omega_G$) by $X$ (resp. $X_0$). Then there is a one-parameter deformation ([@Na2], [@Kaw24]) $$\xymatrix{
X \ar[r]^{f}\ar[dr] & Y \ar[d] \\
& \mathbb{A}^1
}$$ of the Springer resolution $$f_0\colon X_0\to Y_0={\operatorname{Spec}}R_0.$$ We denote by $\pi\colon X \to G$ and $\pi_0\colon X_0\to G$ the projections.
Let ${\ensuremath{\mathcal{U}}}$ be the tautological $k$-dimensional sub-bundle of ${\ensuremath{\mathcal{O}}}_G\otimes V$. We also define ${\ensuremath{\mathcal{U}}}^\perp$ to be $(({\ensuremath{\mathcal{O}}}_G\otimes V)/ {\ensuremath{\mathcal{U}}})^\vee$, the dual of the quotient bundle. For a vector bundle ${\ensuremath{\mathcal{E}}}$ of rank $m$ on $G$, we consider the associated principal ${GL}(m,{\ensuremath{\mathbb{C}}})$-bundle and denote by $\Sigma ^\alpha {\ensuremath{\mathcal{E}}}$ the vector bundle associated with the ${GL}(m,{\ensuremath{\mathbb{C}}})$ representation of highest weight $\alpha\in {\ensuremath{\mathbb{Z}}}^m$. For $\alpha=(\alpha_1,\ldots,\alpha_m)\in {\ensuremath{\mathbb{Z}}}^m$ with $\alpha_1\ge\cdots\ge \alpha_m$ (such a sequence is called a *non-increasing* sequence), we have $$\Sigma ^\alpha ({\ensuremath{\mathcal{E}}}^\vee) =
\Sigma ^{(-\alpha_m,\ldots,-\alpha_1)} {\ensuremath{\mathcal{E}}}=
(\Sigma ^\alpha {\ensuremath{\mathcal{E}}})^\vee.$$
We have the following equality:
$$\begin{aligned}
\label{eqn:base2}
&{\mathop{\mathrm{Hom}}\nolimits}^i_G(\Sigma^{\alpha}{\ensuremath{\mathcal{U}}},\Sigma^{\beta}{\ensuremath{\mathcal{U}}}\otimes (\bigoplus_{n\ge 0}{\mathrm{Sym}}^n(T_G)))\notag\\
=&\bigoplus_{n\ge 0} \bigoplus_{|\lambda|=n}
H^i(G,\Sigma^{\alpha}{\ensuremath{\mathcal{U}}}^\vee\otimes\Sigma^{\beta}{\ensuremath{\mathcal{U}}}\otimes
\Sigma^{\lambda}{\ensuremath{\mathcal{U}}}^\vee\otimes \Sigma ^{\lambda}({\ensuremath{\mathcal{U}}}^{\perp})^\vee). \end{aligned}$$
Here $|\lambda|=\sum \lambda_l$ and all the $\lambda_l$’s are non-negative. For the proof of (\[eqn:base2\]), see [@F-H page 80] and use $T_G={\ensuremath{\mathcal{U}}}^\vee\otimes({\ensuremath{\mathcal{U}}}^{\perp})^\vee$.
\[lem:X\_0toX\] Suppose that the vector space in (\[eqn:base2\]) is 0-dimensional for fixed $i$, $\alpha$ and $\beta$. Then the vector space $
{\mathop{\mathrm{Hom}}\nolimits}^i_X(\pi^*\Sigma^{\alpha}{\ensuremath{\mathcal{U}}},\pi^*\Sigma^{\beta}{\ensuremath{\mathcal{U}}})
$ is also 0-dimensional.
The assertion follows from the equality $$\begin{aligned}
{\mathop{\mathrm{Hom}}\nolimits}^i_X(\pi^*\Sigma^{\alpha}{\ensuremath{\mathcal{U}}},\pi^*\Sigma^{\beta}{\ensuremath{\mathcal{U}}})
&\cong {\mathop{\mathrm{Hom}}\nolimits}^i_G(\Sigma^{\alpha}{\ensuremath{\mathcal{U}}},\Sigma^{\beta}{\ensuremath{\mathcal{U}}}\otimes \pi_*{\ensuremath{\mathcal{O}}}_X) \\
&\cong {\mathop{\mathrm{Hom}}\nolimits}^i_G(\Sigma^{\alpha}{\ensuremath{\mathcal{U}}},\Sigma^{\beta}{\ensuremath{\mathcal{U}}}\otimes (\bigoplus_{n\ge 0}{\mathrm{Sym}}^n(\widetilde{T}_G)))\end{aligned}$$ and the filtration $${\mathrm{Sym}}^n(\widetilde{T}_G)=F^0\supset F^1\supset \cdots
\supset F^n\supset F^{n+1}=0$$ with $F^l/F^{l+1}\cong {\mathrm{Sym}}^{n-l}(T_G)$.
Let $F(V)$ be the flag variety of ${GL}(V)$ and $${\ensuremath{\mathcal{U}}}_1\subset {\ensuremath{\mathcal{U}}}_2\subset\cdots\subset {\ensuremath{\mathcal{U}}}_{n-1}\subset
{\ensuremath{\mathcal{U}}}_{n}=
V\otimes {\ensuremath{\mathcal{O}}}_{F(V)}.$$ the sequence of the universal sub-bundles ${\ensuremath{\mathcal{U}}}_i$ of rank $i$. We put $${\ensuremath{\mathcal{O}}}(\delta_1,\ldots,\delta_n)=
{\ensuremath{\mathcal{U}}}_1^{-\delta_1}
\otimes({\ensuremath{\mathcal{U}}}_2/{\ensuremath{\mathcal{U}}}_1)^{-\delta_2}\otimes
\cdots \otimes( {\ensuremath{\mathcal{U}}}_{n}/ {\ensuremath{\mathcal{U}}}_{n-1})^{-\delta_n}.$$ The following lemma is taken from the proof of [@Kapranov Proposition 2.2].
\[lem:Kapranov\] For non-increasing sequences $\alpha\in {\ensuremath{\mathbb{Z}}}^k,\beta \in {\ensuremath{\mathbb{Z}}}^{n-k}$, we have $$H^i(G,\Sigma^{\alpha}{\ensuremath{\mathcal{U}}}^\vee\otimes\Sigma^{\beta}{\ensuremath{\mathcal{U}}}^\perp)
=H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta)),$$ where $\Delta=(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_{n-k})$.
By Lemma \[lem:X\_0toX\] and Lemma \[lem:Kapranov\], showing the vanishing of the vector space $${\mathop{\mathrm{Hom}}\nolimits}^i_X(\pi^*\Sigma^{\alpha}{\ensuremath{\mathcal{U}}},\pi^*\Sigma^{\beta}{\ensuremath{\mathcal{U}}})$$ is reduced to the dimension counting of the cohomology $H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))$ on the flag variety $F(V)$. Hence, we shall compute $H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))$ for $\Delta=(\delta_1,\ldots,\delta_n)\in {\ensuremath{\mathbb{Z}}}^n$. The permutation group $\mathfrak S _n$ naturally acts on ${\ensuremath{\mathbb{Z}}}^n$: $$\sigma (\delta_1,\ldots,\delta_n)
=(\delta_{\sigma(1)},\ldots,\delta_{\sigma(n)}).$$ We also define the tilde action of $\mathfrak S _n$ on ${\ensuremath{\mathbb{Z}}}^n$: $$\tilde{\sigma} (\Delta)=\sigma(\Delta+\rho)-\rho.$$ Here $\rho =(n-1,n-2,\ldots,0)$. For instance, when we put $\sigma_l=(l\ l+1)$, we obtain $$\tilde{\sigma} _l(\delta_1,\ldots,\delta_n)
=(\delta_1,\ldots,\delta_{l-1},\delta_{l+1}-1,\delta_l+1,\delta_{l+2},
\ldots,\delta_n).$$ The Bott theorem implies that:
\(1) If $\Delta$ is non-increasing, then we have $$H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))=
\begin{cases}
\Sigma^\Delta V & i=0\\
0 &i>0.
\end{cases}$$
\(2) If $\Delta$ is not non-increasing, then we apply the tilde action of $\frak S_n$ for transpositions like $\sigma_l=(l\ l+1)$, trying to move bigger numbers to the right past smaller numbers. Repeat this process. Then there are two possibilities:
- Suppose that eventually, we achieve $\delta_{l+1}=\delta_l+1$ for some $l$. Then $H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))=0$ for all $i$.
- Suppose that after applying $j$ times tilde actions of transpositions in $\frak S_n$, we can transform $\Delta$ into a non-increasing sequence $\Delta _0$. Then we have $$H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))=
\begin{cases}
\Sigma^{\Delta _0} V & i=j\\
0 &i\ne j.
\end{cases}$$
$G(2,4)$ {#sub1}
--------
Henceforth in this section, $G$ denotes $G(2,4)$. Let us find a tilting generator of $D^-(X)$ using Theorem \[thm:main\] in this subsection. Let ${\ensuremath{\mathcal{O}}}_G(1)=\Sigma^{(-1,-1)}{\ensuremath{\mathcal{U}}}$ be a line bundle on $G$ which gives the Plücker embedding $G\hookrightarrow {\ensuremath{\mathbb{P}}}^5$ and we denote $\pi^*{\ensuremath{\mathcal{O}}}_G(1)$ by ${\ensuremath{\mathcal{O}}}_X(1)$.
First we want to show (\[eqn:ample2\]) for ${\ensuremath{\mathcal{L}}}={\ensuremath{\mathcal{O}}}_X(1)$; namely $$\label{eqn:Xample2}
H^i(X,{\ensuremath{\mathcal{O}}}_X(-j))(=
{\mathop{\mathrm{Hom}}\nolimits}^i_X(\pi^*\Sigma^{(0,0)}{\ensuremath{\mathcal{U}}},\pi^*\Sigma^{(j,j)}{\ensuremath{\mathcal{U}}}))
=
0$$ for $0<j<4$ and $i\ge 2$. Putting $\alpha=(0,0)$ and $\beta=(j,j)$ in (\[eqn:base2\]) and using Lemma \[lem:Kapranov\], we obtain $$\begin{aligned}
\label{eqn:Xample2'}
&{\mathop{\mathrm{Hom}}\nolimits}^i_G(\Sigma^{(0,0)}{\ensuremath{\mathcal{U}}},\Sigma^{(j,j)}{\ensuremath{\mathcal{U}}}\otimes (\bigoplus_{n\ge 0}{\mathrm{Sym}}^n(T_G)))\notag\\
=&\bigoplus_{n\ge 0} \bigoplus_{|\lambda|=n}
H^i(G,
\Sigma^{(j-\lambda_2,j-\lambda_1)}{\ensuremath{\mathcal{U}}}\otimes
\Sigma^{(-\lambda_2,-\lambda_1)}{\ensuremath{\mathcal{U}}}^\perp)\notag\\
=&\bigoplus_{n\ge 0} \bigoplus_{|\lambda|=n}
H^i(F(V),{\ensuremath{\mathcal{O}}}(\lambda_1-j,\lambda_2-j,-\lambda_2,-\lambda_1)),\end{aligned}$$ where we put $\lambda =(\lambda_1,\lambda_2)$. For the proof of (\[eqn:Xample2\]), by Lemma \[lem:X\_0toX\], it is enough to see the vanishing of (\[eqn:Xample2’\]) for $0<j<4$ and $i\ge 2$.
Denote $$\Delta=(\lambda_1-j,\lambda_2-j,-\lambda_2,-\lambda_1)$$ below. The Bott theorem says that one of the following occurs:
- If $\lambda_2-j \ge -\lambda_2$, then $
H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))= 0
$ if and only if $i\ne 0$.
- If $\lambda_2-j+1= -\lambda_2$, then $
H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))= 0
$ for all $i$.
- If $\lambda_2-j+1< -\lambda_2$, then $\lambda_2=0$ and $j=2,3$, which implies $$\tilde{\sigma}_2\Delta=(\lambda_1-j,-1,-j+1,-\lambda_1).$$ In the case $\lambda_1-j\ge -1$, $
H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))\ne 0
$ implies $i= 1$. In the case $\lambda_1-j+1=-1$, $
H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))= 0
$ for all $i$. In the case $\lambda_1-j+1<-1$, we obtain $
\lambda_1=0
$ and $j=3$. Then it is easy to see that $
H^i(F(V),{\ensuremath{\mathcal{O}}}(\Delta))= 0
$ for all $i$.
Therefore we obtain (\[eqn:Xample2\]) as desired.
Next we want to check that Assumption \[assum:induction\] is true, i.e. ${\ensuremath{\mathcal{K}}}\in D(X)$ satisfies the equality $$\label{eqn:241}
{\ensuremath{\mathbb{R}}}f_{*}({\ensuremath{\mathcal{H}}}^k({\ensuremath{\mathcal{K}}})\otimes {\ensuremath{\mathcal{O}}}_X(j))=0$$ for any $k$ and $j$, $(0\le j\le 3)$ when we assume the equalities $$\label{eqn:242}
{\ensuremath{\mathbb{R}}}f_{*}({\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{O}}}_X(j))=0$$ for any $j$, $(0\le j \le 3)$. Because ${\ensuremath{\mathcal{O}}}_X(1)$ gives an embedding $h\colon X\hookrightarrow {\ensuremath{\mathbb{P}}}^5_R,$ we can say that (\[eqn:242\]) is equivalent to $$\label{eqn:243}
{\ensuremath{\mathbb{R}}}g_{*}(h_{*}{\ensuremath{\mathcal{K}}}\otimes {\ensuremath{\mathcal{O}}}(j))=0$$ for all $j$ with $0\le j\le 3$, where $g \colon {\ensuremath{\mathbb{P}}}_R^5 \to {\operatorname{Spec}}R$ is the structure morphism. On the other hand, $D({\ensuremath{\mathbb{P}}}_R^5)$ has a semi-orthogonal decomposition $$D({\ensuremath{\mathbb{P}}}_R^5)={\left<g^{*}D(R)\otimes {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5),
g^{*}D(R)\otimes {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-4),\ldots, g^{*}D(R)\right>},$$ and hence it follows from our assumption (\[eqn:243\]) that $$h_{*}{\ensuremath{\mathcal{K}}}\in {\left<g^{*}D(R)\otimes {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5), g^{*}D(R)
\otimes {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-4)\right>}.$$ Consequently, there is a triangle $$\cdots\to g^{*}W_{-4} \otimes _R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-4)
\to h_{*}{\ensuremath{\mathcal{K}}}\to g^{*}W_{-5} \otimes _R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5)
\to \cdots$$ for some $W_l \in D(R)$, and then we obtain a long exact sequence $$\cdots\to {\ensuremath{\mathcal{H}}}^k(W_{-4})\otimes _R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-4) \to {\ensuremath{\mathcal{H}}}^k(h_{*}{\ensuremath{\mathcal{K}}})
\to {\ensuremath{\mathcal{H}}}^k (W_{-5})\otimes _R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5) \to \cdots.$$ Because the support of ${\ensuremath{\mathcal{H}}}^k(h_{*}{\ensuremath{\mathcal{K}}})$ is contained in $X$ and the support of ${\ensuremath{\mathcal{H}}}^k(W_{-5})\otimes _R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5)$ is the inverse image of some closed subset on $Y$ by $g$, the morphism ${\ensuremath{\mathcal{H}}}^k(h_{*}{\ensuremath{\mathcal{K}}}) \to {\ensuremath{\mathcal{H}}}^k (W_{-5})\otimes _R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5)$ should be zero. Therefore we have a short exact sequence $$0 \to {\ensuremath{\mathcal{H}}}^{k-1}(W_{-5})\otimes_R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-5)
\to {\ensuremath{\mathcal{H}}}^{k}(W_{-4})\otimes_R {\ensuremath{\mathcal{O}}}_{{\ensuremath{\mathbb{P}}}_R^5}(-4)
\to {\ensuremath{\mathcal{H}}}^k(h_{*}{\ensuremath{\mathcal{K}}}) \to 0.$$ Then (\[eqn:241\]) follows. Now we can construct a tilting generator of $D^-(X)$ by Theorem \[thm:main\].
We have proved the following:
\[thm:G(2,4)\] The derived category $D^-(X)$ has a tilting generator which is a vector bundle on $X$.
\[cor:g(2,4)\] The derived category $D^-(X_0)$ has a tilting generator which is a vector bundle on $X_0$.
Let ${\ensuremath{\mathcal{E}}}$ be a tilting generator in $D^-(X)$ constructed above. Put ${\ensuremath{\mathcal{E}}}_0=i^*{\ensuremath{\mathcal{E}}}$, where $i\colon X_0\hookrightarrow X$ is the embedding. Since $X$ is a one-parameter deformation of $X_0$, there is an exact sequence $0\to {\ensuremath{\mathcal{O}}}_X\to {\ensuremath{\mathcal{O}}}_X \to {\ensuremath{\mathcal{O}}}_{X_0}\to 0$. Taking a tensor product with ${\ensuremath{\mathcal{E}}}$, we obtain an exact sequence $$\begin{aligned}
\label{eqn:seq}
0\to {\ensuremath{\mathcal{E}}}\to {\ensuremath{\mathcal{E}}}\to {\ensuremath{\mathcal{E}}}_{0}\to 0.\end{aligned}$$ Applying ${\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}({\ensuremath{\mathcal{E}}}, -)$ to (\[eqn:seq\]), we can conclude that ${\ensuremath{\mathcal{E}}}_0$ is a tilting object. We can directly check that ${\ensuremath{\mathcal{E}}}_0$ is a generator.
Auxiliary result: the existence of a right adjoint functor {#section:auxiliary}
==========================================================
In this section, we show the existence of a right adjoint functor of $\Phi_{n-1}$, which is needed in §\[subsection:gluing\]. Let $Y$ be a scheme of finite type over a field or a spectrum of a Noetherian complete local ring. This condition assures the existence of the dualizing complex on $Y$. Let us consider a projective morphism between schemes $f\colon X\to Y$. Then we know that $R=H^0(X,{\ensuremath{\mathcal{O}}}_X)$ has the dualizing complex $D_R$. For a vector bundle ${\ensuremath{\mathcal{E}}}$ on $X$, put $$\begin{aligned}
&{\ensuremath{\mathcal{A}}}={\mathop{\mathcal{E}nd}\nolimits}_X{\ensuremath{\mathcal{E}}},\quad A={\mathop{\mathrm{End}}\nolimits}_X {\ensuremath{\mathcal{E}}}, \\
&D_{\ensuremath{\mathcal{A}}}={\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{A}}},D_X), \quad D_A={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_R(A,D_R),\\
&{\ensuremath{\mathbb{D}}}_{\ensuremath{\mathcal{A}}}(-)={\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_{\ensuremath{\mathcal{A}}}(-,D_{\ensuremath{\mathcal{A}}})\colon D^-({\ensuremath{\mathcal{A}}})\to D^+({\ensuremath{\mathcal{A}}}^{\circ}),\\
&{\ensuremath{\mathbb{D}}}_A(-)={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_A(-,D_A)\colon D^-(A)\to D^+(A^{\circ}),\\
&\tilde{\Phi}(-)={\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}},-)\colon D^-(X)\to D^-({\ensuremath{\mathcal{A}}}),\\
&\Phi(-)={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},-)\colon D^-(X)\to D^-(A),\\
&\Psi(-)=(-){\stackrel{{\ensuremath{\mathbb{L}}}}{\otimes}}_A {\ensuremath{\mathcal{E}}}\colon D^{-}(A) \to D^{-}(X),\\
&{\ensuremath{\mathbb{D}}}_R={\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_R(-,D_R) \colon D^-(R)\to D^+(R).\end{aligned}$$ For the dual vector bundle ${\ensuremath{\mathcal{E}}}^\vee$ of ${\ensuremath{\mathcal{E}}}$, we put $$\begin{aligned}
&\tilde{\Phi}^{\circ}={\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}}^\vee,-)
\colon D^+(X)\to D^+({\ensuremath{\mathcal{A}}}^{\circ}).\end{aligned}$$ Lemma \[lemma:duality\] must be well-known to specialists. When ${\ensuremath{\mathcal{E}}}={\ensuremath{\mathcal{O}}}_X$, the lemma is a paraphrase of the Grothendieck duality for the natural projective morphism $g\colon X \to {\operatorname{Spec}}R$.
\[lemma:duality\] ${\ensuremath{\mathbb{D}}}_A\circ \Phi\cong \Phi\circ {\ensuremath{\mathbb{D}}}_X$.
We have a diagram: $$\label{eqn:cd}
\begin{CD}
D^-(X) @>{\tilde{\Phi}}>> D^-({\ensuremath{\mathcal{A}}})
@>{{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}}>> D^-(A) \\
@V{{\ensuremath{\mathbb{D}}}_X}VV @V{{\ensuremath{\mathbb{D}}}_{\ensuremath{\mathcal{A}}}}VV @V{{\ensuremath{\mathbb{D}}}_A}VV \\
D^+(X) @>{\tilde{\Phi}^{\circ}}>> D^+({\ensuremath{\mathcal{A}}}^{\circ})
@>{{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}}>> D^+(A^{\circ}).
\end{CD}$$ We note that there is an isomorphism $\Phi\cong {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}\circ \tilde{\Phi}$ and that $\tilde{\Phi}$ gives an equivalence of derived categories ([@Rickard]).
First, we show that the left diagram in (\[eqn:cd\]) is commutative. For ${\ensuremath{\mathcal{N}}}\in D^-(X)$, we have $$\begin{aligned}
{\ensuremath{\mathbb{D}}}_{\ensuremath{\mathcal{A}}}\circ\tilde{\Phi}({\ensuremath{\mathcal{N}}})
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_{\ensuremath{\mathcal{A}}}({\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{N}}}),D_{\ensuremath{\mathcal{A}}})\notag\\
&\cong
{\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_{\ensuremath{\mathcal{A}}}({\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{N}}}),{\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}}\otimes D_X))\notag\\
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{N}}},{\ensuremath{\mathcal{E}}}\otimes D_X) \label{eqn:Ecirc1} \\
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}}^{\vee}, {\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{N}}}, D_X)) \notag\\
&\cong \tilde{\Phi}^{\circ}\circ {\ensuremath{\mathbb{D}}}_X({\ensuremath{\mathcal{N}}})\notag.\end{aligned}$$ Here, the isomorphism (\[eqn:Ecirc1\]) comes from the Morita equivalence ${\operatorname{Coh}}U\cong {\operatorname{Coh}}{\ensuremath{\mathcal{A}}}|_U$ on every affine open set $U\subset X$.
Therefore, it remains to show that the right diagram in (\[eqn:cd\]) is commutative. The Grothendieck duality for $g\colon X\to {\operatorname{Spec}}R$ implies $${\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(D_{{\ensuremath{\mathcal{A}}}}) \cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_{R}({\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{A}}}), D_{R}).$$ Composing this isomorphism with the natural morphism $A \to {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{A}}})$, we obtain the morphism $$\label{eqn:omega}
{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(D_{{\ensuremath{\mathcal{A}}}})\to D_A.$$ Moreover since we have $${\mathop{\mathrm{Hom}}\nolimits}_A(M, {\mathop{\mathrm{Hom}}\nolimits}_R(A, N))\cong {\mathop{\mathrm{Hom}}\nolimits}_R(M, N)$$ for any $M\in {\operatorname{mod}}A$, $N\in R{\operatorname{mod}}$, we have the isomorphism, $$\label{eqn:AR}
{\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_A(M,D_A)\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_R(M,D_R)$$ in $D^-(R)$ for $M\in D^-(A)$.
For ${\ensuremath{\mathcal{M}}}\in D^{-}({\ensuremath{\mathcal{A}}})$, we have the following sequence of isomorphisms and natural morphisms, $$\begin{aligned}
{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}\circ {\ensuremath{\mathbb{D}}}_{{\ensuremath{\mathcal{A}}}}({\ensuremath{\mathcal{M}}})
&={\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}\circ {\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_{\ensuremath{\mathcal{A}}}({\ensuremath{\mathcal{M}}},D_{{\ensuremath{\mathcal{A}}}}) \notag \\
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_{\ensuremath{\mathcal{A}}}({\ensuremath{\mathcal{M}}},D_{{\ensuremath{\mathcal{A}}}}) \notag \\
&\to
{\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_{\ensuremath{\mathcal{A}}}(\tilde{\Phi}\circ\Psi\circ {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{M}}}),D_{{\ensuremath{\mathcal{A}}}}) \label{eqn:nat}
\\
& \cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_{A}({\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{M}}}), {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}(D_{{\ensuremath{\mathcal{A}}}})) \label{eqn:nat2}\\
& \to {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_{A}({\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{M}}}), D_{A}) \label{eqn:nat3} \\
&= {\ensuremath{\mathbb{D}}}_A \circ {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{M}}}) \notag.\end{aligned}$$ Here the morphism (\[eqn:nat\]) and the isomorphism (\[eqn:nat2\]) are obtained from the fact that $\tilde{\Phi}\circ\Psi$ is a left adjoint functor of ${\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}$, and moreover the morphism (\[eqn:nat3\]) comes from the morphism (\[eqn:omega\]). Consequently we obtain a morphism of functors $$\phi\colon {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}\circ {\ensuremath{\mathbb{D}}}_{{\ensuremath{\mathcal{A}}}}\to {\ensuremath{\mathbb{D}}}_A\circ{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}.$$
Next we want to check that $\phi$ is an isomorphism. Note that it is enough to check that $\phi$ is isomorphic after applying the forgetful functor $D^-(A)\to D^-(R)$. Take ${\ensuremath{\mathcal{N}}}\in D^-(X)$ such that $\tilde{\Phi}({\ensuremath{\mathcal{N}}})={\ensuremath{\mathcal{M}}}$. Then, because of the commutativity of the left diagram in (\[eqn:cd\]), we have $${\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}\circ {\ensuremath{\mathbb{D}}}_{{\ensuremath{\mathcal{A}}}}({\ensuremath{\mathcal{M}}})\cong
{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}\circ {\ensuremath{\mathbb{D}}}_X({\ensuremath{\mathcal{E}}}^\vee\otimes {\ensuremath{\mathcal{N}}}).$$ We also have $$\begin{aligned}
{\ensuremath{\mathbb{D}}}_A\circ{\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{M}}})
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_A({\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{N}}}),D_A)\\
&\cong {\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_R({\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{N}}}),D_R)\\
&\cong {\ensuremath{\mathbb{D}}}_R\circ {\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{E}}}^\vee\otimes {\ensuremath{\mathcal{N}}}).\end{aligned}$$ by (\[eqn:AR\]). Then the Grothendieck duality for $g$ implies that $\phi$ is isomorphic.
Put $$D^{\dag}(X)= \left\{
{\ensuremath{\mathcal{K}}}\in D(X) \bigm|
\Phi({\ensuremath{\mathcal{K}}})\in D^b(A) \right\}.$$
\[lem:adjoint\] The functor $\Phi:D^{\dag}(X)\to D^b(A)$ has a right adjoint functor.
Indeed, using ${\ensuremath{\mathbb{D}}}_A\circ \Phi\cong \Phi\circ {\ensuremath{\mathbb{D}}}_X$, we can readily check that ${\ensuremath{\mathbb{D}}}_X\circ\Psi \circ {\ensuremath{\mathbb{D}}}_A$ is a right adjoint functor of $\Phi$.
Non-commutative crepant resolution {#section:appendix}
==================================
First, let us recall the definition of non-commutative crepant resolutions introduced by Van den Bergh [@nonc].
Let $k$ be a field, $R$ a normal Gorenstein finitely generated $k$-domain. Furthermore we denote by $A$ an $R$-algebra that is finitely generated as an $R$-module. $A$ is called a non-commutative crepant resolution of $R$ if the following conditions hold:
1. There is a reflexive $R$-module $E$ such that $A={\mathop{\mathrm{End}}\nolimits}_R(E)$.
2. The global dimension of $A$ is finite.
3. $A$ is a Cohen-Macaulay $R$-module.
The next assertion is essentially shown in [@nonc].
Let $Y={\operatorname{Spec}}R$ be an affine normal Gorenstein variety and assume that there is a crepant resolution $f\colon X\to Y$: that is, $f$ is a birational projective morphism from a smooth variety $X$ and $f^*\omega_Y=\omega_X$. If we have a tilting generator ${\ensuremath{\mathcal{E}}}$ of $D^-(X)$ such that $${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{O}}}_X)={\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{O}}}_X,{\ensuremath{\mathcal{E}}})=0$$ for $i\ne0$, then $R$ has a non-commutative crepant resolution.
When $\dim R\le 1$, then $R$ is itself a non-commutative resolution of $R$. Thus, we assume that $\dim R\ge 2$ in what follows. We define as $$\begin{aligned}
E&={\ensuremath{\mathbb{R}}}\Gamma({\ensuremath{\mathcal{E}}})(\cong{\ensuremath{\mathbb{R}}}^0\Gamma({\ensuremath{\mathcal{E}}})),
\quad {\ensuremath{\mathcal{A}}}={\mathop{{\ensuremath{\mathbb{R}}}\mathcal{H}om}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}}),\\A&={\mathop{{\ensuremath{\mathbb{R}}}\Gamma}\nolimits}({\ensuremath{\mathcal{A}}})(\cong{\mathop{{\ensuremath{\mathbb{R}}}\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}})\cong {\mathop{\mathrm{Hom}}\nolimits}_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}})).\end{aligned}$$ By $f^*\omega_Y=\omega_X$, we have $f^!{\ensuremath{\mathcal{O}}}_Y={\ensuremath{\mathcal{O}}}_X$. Then the Grothendieck duality and our assumptions imply that $$\begin{aligned}
{\mathop{\mathrm{Hom}}\nolimits}^i_R(E,R)&\cong{\mathop{\mathrm{Hom}}\nolimits}^i_X({\ensuremath{\mathcal{E}}},f^!{\ensuremath{\mathcal{O}}}_Y)\\
&\cong{\mathop{\mathrm{Hom}}\nolimits}^i_X({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{O}}}_X)\\
&=0\end{aligned}$$ for any $i\ne 0$, which implies that $E$ is Cohen-Macaulay. We can show similarly that $A$ is Cohen-Macaulay, since $$\begin{aligned}
{\mathop{\mathrm{Hom}}\nolimits}^i_X({\ensuremath{\mathcal{A}}},{\ensuremath{\mathcal{O}}}_X)
&\cong{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{O}}}_X,{\ensuremath{\mathcal{A}}})\\
&\cong{\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}})=0\end{aligned}$$ for any $i\ne 0$. Note that ${\mathop{\mathrm{End}}\nolimits}_R(E)$ and $A$ are reflexive, since they are Cohen-Macaulay and $\dim R\ge 2$. Then the natural homomorphism $A\to {\mathop{\mathrm{End}}\nolimits}_R(E)$ is isomorphic in codimension one, as well as everywhere else. Moreover, $D^b(A)$ and $D^b(X)$ are derived equivalent, and therefore the global dimension of $A$ is finite.
Let $Y={\operatorname{Spec}}R$ be an affine normal Gorenstein variety defined over $\mathbb{C}$, and suppose that there is a crepant resolution $f\colon X\to Y$ with at most two-dimensional fibers. Further assume that we have a globally generated, ample line bundle ${\ensuremath{\mathcal{L}}}$ on $X$ which satisfies ${\ensuremath{\mathbb{R}}}^2 f_{\ast}{\ensuremath{\mathcal{L}}}^{-1}=0$. Then $R$ has a non-commutative crepant resolution.
Note that ${\ensuremath{\mathbb{R}}}f_*{\ensuremath{\mathcal{O}}}_X\cong{\ensuremath{\mathcal{O}}}_Y$ by the vanishing theorem. Because ${\ensuremath{\mathcal{O}}}_X$ is a direct summand of the tilting generator ${\ensuremath{\mathcal{E}}}$ constructed in Theorem \[thm:rel.dim2\] we obtain $${\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{O}}}_X)={\mathop{\mathrm{Hom}}\nolimits}_X^i({\ensuremath{\mathcal{O}}}_X,{\ensuremath{\mathcal{E}}})=0$$ for $i\ne0$. We can apply the above proposition.
[10]{}
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Yukinobu Toda
Institute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo, Kashiwano-ha 5-1-5, Kashiwa City, Chiba, 277-8582, Japan
[*e-mail address*]{} : toda@ms.u-tokyo.ac.jp
Hokuto Uehara
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minamiohsawa, Hachioji-shi, Tokyo, 192-0397, Japan
[*e-mail address*]{} : hokuto@tmu.ac.jp
|
---
abstract: 'I provide a complete analysis of the motivic Adams spectral sequences converging to the bigraded coefficients of the 2-completions of the motivic spectra $\BPGL$ and $\kgl$ over $p$-adic fields, $p>2$. The former spectrum is the algebraic Brown-Peterson spectrum at the prime 2 (and hence is part of the study of algebraic cobordism), and the latter is a certain $\BPGL$-module that plays the role of a “connective" motivic algebraic $K$-theory spectrum. This is the first in a series of two papers investigating motivic invariants of $p$-adic fields, and it lays the groundwork for an understanding of the motivic Adams-Novikov spectral sequence over such base fields.'
author:
- 'Kyle M. Ormsby'
bibliography:
- 'biblio.bib'
title: 'Motivic invariants of $p$-adic fields'
---
Introduction {#sec:intro}
============
Arithmetic input from $p$-adic fields {#sec:pAdic}
=====================================
Comodoules over the dual motivic Steenrod algebra {#sec:comod}
=================================================
Motivic $\Ext$-algebras {#sec:Ext}
=======================
$(\BPGL\comp)_*$ and $(\kgl\comp)_*$ via the motivic Adams spectral sequence {#sec:motASS}
============================================================================
|
---
abstract: 'We examine the current directions in the search for spin-dependent dark matter. We discover that, with few exceptions, the search activity is concentrated towards constraints on the WIMP-neutron spin coupling, with significantly less impact in the WIMP-proton sector. We review the situation of those experiments with WIMP-proton spin sensitivity, toward identifying those capable of reestablishing the balance.'
author:
- TA Girard
- 'F. Giuliani'
title: 'On the Direct Search for Spin-dependent WIMP Interactions'
---
INTRODUCTION
============
The direct search for weakly interacting massive particle (WIMP) dark matter continues among the forefront efforts of experimental physics. The search is largely motivated by the continuing absence of a second positive signal with annual modulation confirming the result of the DAMA/$NaI$ experiment despite significantly improved detectors, and especially following the several recent reports of possible indirect observation of dark matter annihilation [@egret; @hess; @heat].
Direct search efforts, based on the detection of nuclear recoils in WIMP-nucleon interactions, have been traditionally classified as to whether sensitive for the spin-independent or spin-dependent WIMP channel [@Lewin]. While the former are generally constructed simply on the basis of heavy target nuclei (since the interaction cross section varies as $A^{2}$), the latter require consideration of the spin structure of the detector nuclei and are customarily defined by whether the primary experiment sensitivity is to WIMP-proton or WIMP-neutron spin coupling. The main efforts to date have been in spin-independent searches, generally because the anticipated cross sections are larger owing to a coherent interaction across the nucleus. As recently noted by Bednyakov and $\check{S}$imkovic [@zero], however, the importance of the spin-dependent sector cannot however be ignored: such searches provide twice stronger constraints on SUSY parameter space, permit the detection of large nuclei recoil energy due to nuclear structure effects in the case of heavy target nuclei, and prevent missing a dark matter signal which might be suppressed in the spin-independent sector. If the neutralino is predominantly gaugino or higgsino states, the coupling is only spin-dependent [@gaitskell].
The above distinction in search efforts has become somewhat blurred, since many detector heavy isotopes also possess spin and even a small natural isotopic abundance can produce significant constraints. In fact, a single detector can simultaneously provide restrictions on both channels of the WIMP-nucleon interaction, but with sensitivity in each channel dependent on the nature of the detector material. The most stringent limits in the spin-dependent sector are currently provided by experiments traditionally considered spin-independent.
The future thrust of direct searches for WIMP dark matter is defined by a number of project upgrades and several new high profile activities. Basically designed for deeply probing the spin-independent phase space, these experiments project eventual sensitivities beginning well below the controversial DAMA/$NaI$ result and extending to cross sections as small as $10^{-10}$ pb in the WIMP-nucleon interaction. We here consider the impact of this activity thrust on the spin-dependent sector, finding that of those experiments with spin-sensitivity, most all will provide increasingly improved restrictions predominantly on the possible WIMP-neutron spin coupling, with significantly less impact on the WIMP-proton coupling. This latter sector is in fact observed to have previously received comparatively little direct experiment attention, with the current restrictions derived from $NaI$ experiments already surpassed by two orders of magnitude in sensitivity by the indirect searches [@superK; @baksan]. Given that there however remain important theoretical questions regarding the extraction of the indirect results, we examine the situation of direct experiments with predominantly WIMP-proton spin sensitivity, towards identifying those with capacity to provide similar restrictions.
Sec. II reviews the current experimental situation and thrust of new initiatives in the search effort. The impact of these is discussed in Sec. III, and summarized in Sec. IV.
EXPERIMENTAL SITUATION
======================
The situation for spin-dependent (SD) activity is shown in Fig. \[SDexc\] at $90\%$ C.L. for a WIMP mass (M$_{W}) = 50$ GeV/c$^{2}$, obtained in a model-independent, zero momentum transfer approximation [@FGprl; @Tovey]. The $a_{p},a_{n}$ are the WIMP-proton (neutron) coupling strengths in the spin-dependent WIMP-nucleus cross section
$$\label{cross2}
{\sigma_{A}^{(SD)} = \frac{32}{\pi}G_{F}^{2}\mu_{A}^{2}
(a_{p}\langle S_{p}\rangle +a_{n}\langle S_{n}\rangle
)^{2}\frac{J+1}{J}} ,$$
where $\mu_{p,n,A}$ = $\frac{M_{W}m_{p,n,A}}{M_{W}+m_{p,n,A}}$ is the WIMP-proton (-neutron,-nucleus) reduced mass, $\langle S_{p,n} \rangle $ is the expectation value of the proton (neutron) group spin, $G_{F}$ is the Fermi coupling constant, and J is the total nuclear spin. The figure is constructed from the published results of the respective experiments, using
$$\label{elli}
{\sum_{A}\left( \frac{a_{p}}{\sqrt{\sigma_{p}^{lim(A)}}}\pm
\frac{a_{n}}{\sqrt{\sigma_{n}^{lim(A)}}}\right)^{2} \leq
\frac{\pi}{24G_{F}^{2}\mu _{p}^{2}} } ,$$
with $\sigma_{p,n}^{lim(A)}$ the proton and neutron cross section limits defined by
$$\label{cross} {\sigma_{p,n}^{lim(A)} =
\frac{3}{4}\frac{J}{J+1}\frac{\mu_{p,n}^{2}}{\mu_{A}^{2}}\frac{\sigma_{A}^{lim}}{\langle
S_{p,n}\rangle^{2}} } ,$$
where $ \sigma_{A}^{lim}$ is the upper limit on $ \sigma
_{A}^{(SD)}$ obtained from experimental data, and the small difference between $m_{p}$ and $m_{n}$ is neglected. The sum in Eq. (\[elli\]) is over each of the detector nuclear species, with the sign of the addition in parenthesis being that of $\langle
S_{p}\rangle/\langle S_{n}\rangle$, and is an ellipse except in the case of single-nuclei experiments for which the ellipse degenerates into a band. When $\sigma_{p,n}^{lim(A)}$ were not available, they have been obtained from published cross section limits as described in Ref. [@FGprl; @FGprd].
At this magnification, with the exception of CDMS/$Ge$, the exclusion plot is seen to consist of essentially horizontal ($a_{p}$-sensitive), vertical ($a_{n}$-sensitive) and diagonal bands, within which lies the allowed area, the exterior being excluded.
isotope Z J$^{\pi}$ abundance (%) experiment
------------ ---- ----------- --------------- -----------------------------------------------------------------------------
$^{3}$He 2 1/2$^{+}$ $<<$1 MIMAC [@mimac]
$^{7}$Li 3 3/2$^{-}$ 93 Kamioka/$LiF$ [@lif]
$^{13}$C 6 1/2$^{-}$ 1.1 PICASSO [@newpicasso], SIMPLE [@plb2], COUPP [@coupp]
$^{17}$O 8 5/2$^{+}$ $<<$1 ROSEBUD [@rosebudII], CRESST [@cresst]
$^{19}$F 9 1/2$^{+}$ 100 SIMPLE [@plb2], PICASSO [@newpicasso], Kamioka [@lif; @naf], COUPP [@coupp]
$^{21}$Ne 10 3/2$^{+}$ $<<$1 CLEAN [@clean]
$^{23}$Na 11 3/2$^{+}$ 100 DAMA [@danai], NAIAD [@naiad2], ANAIS [@anais], LIBRA [@libra]
Kamioka/$NaF$ [@naf]
$^{27}$Al 13 5/2$^{+}$ 100 ROSEBUD [@rosebudII]
$^{29}$Si 14 7/2$^{+}$ 4.7 CDMS [@cdms06SD]
$^{35}$Cl 17 3/2$^{+}$ 76 SIMPLE [@plb2]
$^{37}$Cl 17 3/2$^{+}$ 24 SIMPLE [@plb2]
$^{43}$Ca 20 7/2$^{-}$ $<<$1 CRESST-II [@papcresstII], Kamioka/$CaF_{2}$ [@CaF]
$^{67}$Zn 30 5/2$^{-}$ 4.1 CRESST-II [@papcresstII]
$^{73}$Ge 32 9/2$^{+}$ 7.8 HDMS [@hdms], CDMS [@cdms06SD], GENIUS [@genius2], EDELWEISS [@eureka]
$^{127}$I 53 5/2$^{+}$ 100 DAMA [@danai], NAIAD [@naiad2], KIMS [@kims], ANAIS [@anais]
LIBRA [@libra], COUPP [@coupp]
$^{129}$Xe 54 1/2$^{+}$ 26 ZEPLIN [@zepmax], XENON [@xenon], XMASS [@xmass2], DRIFT [@drift]
$^{131}$Xe 54 3/2$^{+}$ 21 ZEPLIN [@zepmax], XENON [@xenon], XMASS [@xmass2], DRIFT [@drift]
$^{133}$Cs 55 7/2$^{+}$ 100 KIMS [@kims]
$^{183}$W 74 1/2$^{-}$ 14 CRESST-II [@papcresstII]
$^{209}$Bi 83 9/2$^{-}$ 100 ROSEBUD [@rosebudII]
The CDMS results derive from the use of the nonzero momentum transfer analysis of Ref. [@Savage] (the CDMS/$Si$ result, not shown, constitutes a near-vertical band at $|a_{n}| \sim 1.5$, overlapping to some extent the results of EDELWEISS and DAMA/Xe-2). Note that the zero momentum transfer approximation does not simply set the nuclear structure form factor appearing in the differential WIMP-nucleus scattering rate to 1: the calculation of $
\sigma_{A}^{lim}$ involves dividing the experimental upper limit on the WIMP rate by the convolution over the detector recoil energy range of the form factor with the average inverse WIMP velocity. Some experiments use a form factor independent of $a_{p,n}$ as suggested in Ref. [@Lewin], such that $\sigma_{A}^{lim}$ is also independent of $a_{p,n}$. Other experiments (such as [@naiad2; @damasumma]) employ form factors dependent on $a_{p,n}$ (*e.g.*, those of Ref. [@Ressell]) and the $
\sigma_{A}^{lim}$ in Eq. (\[cross\]) is not the same for $\sigma_{p}^{lim(A)}$ as for $\sigma_{n}^{lim(A)}$. When $\sigma_{p,n}^{lim(A)}$ have been published [@naiad; @naiad2; @ZEPLINI], it is straightforward to use them in Eq. (\[elli\]) to obtain zero momentum transfer exclusions. Provided that the form factor is not changed, changing to a nonzero momentum transfer analysis of the same data leaves the ($a_{p}$,0) and ($0,a_{n}$) points fixed, while rotating the major axis of the ellipse, generally towards the nearest coordinate axis because the absolute value of the coefficient of $a_{p}a_{n}$ is lowered. In particular, for a single sensitive nucleus the coefficient of $a_{p}a_{n}$ is generally less than twice the geometric average of the coefficients of $a_{p,n}^{2}$. This removes the degeneration of the ellipses to infinite bands predicted by the zero momentum transfer framework for single nuclei experiments.
![spin-dependent exclusions for various direct search activities, the region permitted by each experiment lying inside the respective band: DAMA/$Xe$-2 (small dot), EDELWEISS/$Ge$ (long dash), ZEPLIN-1/$Xe$ (big dot), CDMS/$Ge$ (solid), NAIAD/$NaI$ (solid), PICASSO/$C_{4}F_{10}$ (dash-dot), SIMPLE/$C_{2}ClF_{5}$ (short dash), Kamiokana/$CaF_{2}$ (short dash), CRESST-1/$Al_{2}O_{3}$ (short dash), KIMS/$CsI$ (dash-dot); the controversial positive result of DAMA/$NaI$ is shown as shaded. The unexcluded region, defined by the intersection of CDMS [@cdms06SD] and NAIAD [@naiad2], is shown as crosshatched.[]{data-label="SDexc"}](fig1.eps "fig:"){width="8"}\
Fig. \[SDexc\] includes the results from CRESST-I/$Al_{2}O_{3}$ [@cresst], several recently-reported fluorine-based experiments, and heavy nuclei searches normally considered spin-INdependent, such as EDELWEISS [@edel], ZEPLIN-I [@ZEPLINI] and CDMS [@cdms06SD]; 50 GeV/c$^{2}$ is chosen since it lies near the maximum sensitivities of the various experiments: for larger or smaller $M_{W}$, all results are generally less restrictive, and vary differentially. Each experiment is identified with the full detector exposure in achieving the limit, rather than the normally-quoted effective exposure (spin-sensitive detector mass $\times$ measurement time), in order to make clear the difference between detectors with 100% spin sensitivity material and those with less. Also note that the results of the indirect searches [@superK; @baksan], which have been recently used to set very restrictive limits on the WIMP-proton coupling [@Savage; @ulka], are not included. The 3$\sigma$ C.L. observation of the DAMA/$NaI$ annulus, appearing as two shaded bands, is taken from Ref. [@danai], which uses the standard halo model and Nijmegen form factor, spin matrix elements [@Ressell]. Although this report is from only a 159 kgy exposure, the most recent DAMA/$NaI$ [@damasumma] result confirms the same amplitude and phase of the annual modulation, simply refining the error bars; as a consequence, the shell decreases in thickness without shrinking.
As evident from Fig. \[SDexc\], the intersection of any two search results, one of which is predominantly $a_{p}$-sensitive and the other $a_{n}$, yields more restrictive limits than either of the two alone. Clearly, the spin-independent group of experiments is efficient in reducing the allowed spin-dependent parameter space, despite the small (7.8%) component of spin-sensitive $^{73}Ge$ isotope in the case of CDMS and EDELWEISS (see Table I). In Fig. \[SDexc\], the predominantly WIMP-neutron sensitivity of CDMS/$Ge$ is seen to reduce the range of $|a_{n}|$ allowed by NAIAD by more than a factor 30 (with a small reduction in $|a_{p}|$), corresponding to the cross section limits of $\sigma _{p}\leq 0.320$ pb; $\sigma _{n} \leq 0.166$ pb obtained via Eq. (\[cross2\]) rewritten for a single nucleon.
The future thrust of direct search activity is defined by a number of traditionally-classified spin-independent project upgrades, including ZEPLIN-MAX/$Xe$ [@zepmax], CRESST-II/$CaWO_{4}$ [@papcresstII], LIBRA/$NaI$ [@libra], EDELWEISS-II/$Ge$ [@edelII], GENIUS/$Ge$ [@genius2], superCDMS/$Ge$ [@supercdms], HDMS/$^{73}Ge$ [@hdms], KIMS/$CsI$ [@kims], WARP/$Ar$ [@warp] and ELEGANT VI/**$CaF_{2}$** [@libra]. New high profile projected activity includes XENON/$Xe$ [@xenon], XMASS/$Xe$ [@xmass2], EUREKA (CRESST-II+EDELWEISS-II) [@eureka], COUPP/$CF_{3}I$ [@coupp], CLEAN/$Ne$ [@clean], DRIFT/$CS_{2}$ [@drift], ArDM/$Ar$ [@ardm], DEAP/$Ar$ [@deap] and MIMAC/$He$ [@mimac]. It is not our point to review in detail these efforts: descriptions exist as indicated and elsewhere [@libra; @gaitskell]. Suffice it to mention, with the exception of the light noble liquid projects, all are “heavy” in the sense of A. Most all of the cryogenic activities envision eventual detector masses of up to 500 kg; the noble liquid activities, 1-10 ton. As evident, the new activity emphasis appears to have shifted from cryogenic searches to scintillators employing noble liquids, reflecting a shift from phonon+ionization to ionization+scintillation discrimination techniques in identifying and rejecting backgrounds, as well as providing directional sensitivity. The current background levels are ***$\sim
10^{-1}$*** evt/kgd; projections for the new devices range to $\sim$ $10^{-2}$ evt/kgy. Generally, the bolometers have a few-keV recoil threshold capacity, in contrast to the $>$ 10 keV thresholds of the noble liquid experiments; since the Na presence permits DAMA/$NaI$ to observe a signal below the Ge recoil thresholds of CDMS and EDELWEISS, the strong reduction of the spin-independent parameter space still compatible with the DAMA/$NaI$ signal disappears at masses below $\sim 20$ GeV/c$^{2}$. The recent CDMS Si-based measurement [@cdms06SI] further reduces this region by more than a factor two, but does not yet eliminate it.
DISCUSSION
==========
The spin-dependent nuclei of the above experiments are shown in Table I. As evident, not all of the above experiments will contribute to further constraining this sector, in particular those based on argon which lacks spin-sensitive isotopes.
The problem with the above activity thrust for spin-dependent investigations is shown in Fig. \[projSDexc\], with the projected experimental limits of the $|a_{n}|$ sensitive experiments (ZEPLIN, CRESST-II, EDELWEISS-II, superCDMS, HDMS, XENON, XMASS) subsumed under the label “future” suggested by a superCDMS projection [@supercdms], and intended only to serve as an indication of the general impact to be anticipated from the above activity (note the change in the $a_{n}$ scale). The currently allowed area in the parameter space of Fig. \[SDexc\] is indicated by the shaded area, and suggests the reduction in the allowed $a_{p}$ - $a_{n}$ space to be achieved with the “future” thrust: generally, the limiting ellipses of all will shrink in both parameters, but with the bounds on $|a_{p}|$ still an order of magnitude larger than $|a_{n}|$.
![general projections (at $M_{W}$ = 50 GeV/$c^{2}$) of results to be expected from the current experimental direct search activity, in comparison with the current $a_{p}$-sensitive experimental results. The near-vertical ellipse denoted by “future” and suggested by phase A of superCDMS [@supercdms] indicates the general improvement to be achieved by the $a_{n}$-sensitive experiments discussed in the text. The near-horizontal ellipse indicates a similar projection for the fluorine-based experiments (obtained from a 200 kgd projection of the current SIMPLE result), with the croshatched area indicating the intersection of the two; the crosshatched region of Fig. \[SDexc\] is now shown as shaded.[]{data-label="projSDexc"}](fig2.eps "fig:"){width="8"}\
The point of any search experiment is however discovery, which will be exacerbated should one or more of the spin-independent “future” experiments obtain a positive signal. This is illustrated schematically in Fig. \[posSDexc\] for the case of two “discoveries”, NaI and “future”. The four allowed $a_{p} - a_{n}$ regions (shaded), defined by the intersection of the hypothetical new positive result from the “future” experiments with that of $NaI$ (corresponding to two regions of $\sigma_{p} - \sigma_{n}$), will require at least one additional and different detector experiment of sufficient sensitivity to further reduce the parameter space to two allowed areas corresponding to a single pair of cross sections.
![schematic of a positive result in one of the “future” experiments (shaded), intersecting (hatched) that of a positive $NaI$ result. The parameter space shown as crosshatched represents the area allowed by the intersection of $NaI$, “future” and a similar positive result from one of the fluorine-based experiments. []{data-label="posSDexc"}](fig3.eps "fig:"){width="8"}\
Both NAIAD [@naiad2] and DAMA/$NaI$ are ended. As evident, without some additional effort, the direct search restrictions on $a_{p}$ would remain essentially unchanged from those provided by these measurements. The DAMA/$NaI$ experiment has been replaced by DAMA/LIBRA [@libra], an upgrade of the $NaI$ experiment to 250 kg with improved radiopurity, running since 2003. The mass increase however provides only a factor 2.5 decrease in the exposure necessary to confirm the original DAMA/$NaI$ signal, with further improvement in Fig. \[projSDexc\] scaling as $\sqrt[4]{\textrm{exposure}}$; R&D is in progress for a mass upgrade to 1 ton. A second $NaI$ experiment, ANAIS, reports an exposure of $5.7$ kgy with a $10.7$ kg prototype [@anais], and will be eventually upgraded to 100 kg. It is however only projected to repeat the DAMA/$NaI$ measurements for confirmation of the annual modulation.
All of these experiments rely on pulse-shape analyses for discrimination of backgrounds. All appear to require, relative to the leading $a_{n}$-sensitive experiments, exceedingly large exposures, despite active masses significantly larger than the bolometer experiments of the a$_{n}$ sector. To further limit $a_{p}$ via direct observation, one or more experiments with improved WIMP-proton sensitivity is required.
Other $a_{p}$-sensitive measurements
------------------------------------
The KIMS experiment, based on $CsI$, has recently reported competitive spin-independent limits with a 247 kgd exposure. Since its sensitivity is similar to $NaI$, we show in Fig. \[SDexc\] the corresponding spin-dependent constraints as recalculated from the raw data in Ref. [@kims], after first reproducing the reported spin-independent exclusion. Lacking any calculated $<S_{p,n}>$, the result is obtained using an odd group approximation (OGA), and the spin-dependent form factor of Ref. [@Lewin]. The small gyromagnetic ratio of $^{133}$Cs yields $<S_{p}>$ $\sim$ -0.2, near that of $^{23}Na$ (recall that the spins enter quadratically), although the higher J of Cs implies a factor 77% reduction in Eq. (\[cross2\]). To achieve the level of NAIAD, an exposure of 25.1 kgy would be required with the current background level and discrimination; the experiment is to be upgraded to 80 kg.
As seen in Fig. \[SDexc\], CRESST-I achieved [@cresst] a competitive result with only a 1.51 kgd exposure of a 262 g $Al_{2}O_{3}$ bolometer, and could achieve the level of NAIAD with a factor 50 less exposure ($\sim 300$ kgd). It has however been abandoned in favor of $CaWO_{4}$ (CRESST-II), which is primarily $a_{n}$ sensitive through its naturally abundant 14.3 $\%$ $^{183}W$, 0.14% $^{43}Ca$ and 0.038% $^{17}O$ isotopes [@cresstII]. The main spin-dependent sensitivity derives from the almost negligible $^{43}Ca+^{17}O$, the small gyromagnetic ratio of $^{183}$W pointing to a negligible spin-dependent OGA sensitivity ($\langle S_{n} \rangle \sim$ 0.031); the current CRESST-II result lies near $|a_{n}| \leq$ 20, well outside Fig. \[SDexc\].
Several activities based on new prototype devices have been recently reported. ROSEBUD includes an $Al_{2}O_{3}$ bolometer, but the device is only $50$ g [@rosebudII] and assuming the same sensitivity as CRESST-I would require almost 16 years exposure to achieve the current NAIAD limits. A mass increase to 1 kg would enable limits on $a_{p}$ similar to, and more restrictive than, the current NAIAD result with a relatively short time exposure of $\sim$ 0.8 y. Like CRESST-II [@cresstII], ROSEBUD however pursues scintillating bolometers to further reject backgrounds, which if successful could yield restrictions on $a_{p}$ equivalent to those of NAIAD with as little as a 2 kgd exposure.
ROSEBUD also pursues measurements in BGO (= $Bi_{4}Ge_{3}O_{12}$). As seen in Table 1, although $Ge$ and $O$ are both neutron-sensitive, $Bi$ is proton-sensitive. As with $^{183}W$ however, a small $^{209}Bi$ gyromagnetic ratio yields an OGA estimate of $<S_{p}>$ $\sim$ -0.085. Successful scintillation discrimination in this case could also yield results equivalent to those of NAIAD (although a 200 kgd exposure would be required).
The Kamioka/$CaF_{2}$ scintillator experiment of Fig. \[SDexc\] reports a new, very competitive limit with a total 14 kgd exposure of a 310 g device [@CaF], realized via careful attention to component intrinsic radioactive backgrounds. It surpasses both bolometer-based Kamioka/$NaF$ [@naf] and Kamioka/$LiF$ [@lif]. The background rate is however still roughly a factor 10 higher than those of the $NaI$ experiments, and may limit the future performance of the detector. The recently ended ELEGANT VI is being replaced by CANDLES III, but both are primarily focused on $\beta\beta$ decay and have yet to provide a WIMP exclusion.
The Kamioka/$CaF_{2}$ result is essentially equivalent to recent results reported by the two superheated droplet detector (SDD) experiments (SIMPLE/$C_{2}ClF_{5}$ [@plb2], PICASSO/$C_{4}F_{10}$ [@newpicasso]), with significantly less active mass and exposure owing to inherent SDD background insensitivity. These have so far received little attention, most likely because of only prototype results having so far been reported, with an unfamiliar technique. Nevertheless, given their current results, they offer significant room for rapid improvement in parameter space restrictions. This is shown in Fig. \[projSDexc\] by “fluorine” for a 10 kgd exposure with background level of 1 evt/kgd (the Kamioka/$CaF_{2}$ experiment would require 34.5 kgy exposure with current sensitivity to achieve the same limit, requiring either significant detector mass increase and/or improved background discrimination to remain competitive). Being also comparatively inexpensive and simple in construct, large volume SDD efforts may easily be envisioned (a 2.6 kg, 336 kgd exposure PICASSO effort is in progress, which if successful will further reduce the crosshatched area of Fig. \[projSDexc\]).
The simplicity argument is similarly true for COUPP [@coupp], which is based on a 2 kg $CF_{3}I$ bubble chamber with the background insensitivity of the SDDs. In this case, the 10 kgd exposure could be achieved more quickly since the $CF_{3}I$-loading of a SDD is only 1% in volume [@tomo]. The COUPP technique however requires a significant extension of the metastability lifetime of the refrigerant beyond previous bubble chamber technology. This has apparently been addressed with some success [@coupp], but a first result is still lacking.
In either case, given sufficient exposure, the fluorine experiments combined with current CDMS results have the ability to severely constrain the currently allowed parameter space of Fig. \[SDexc\].
Spin Sensitivities
------------------
The above complementarity of various experiments of differing orientation in the parameter space is strongly governed by the spin matrix elements of the involved nuclei. Unfortunately, many of the new projected experiments lack specific spin matrix element calculations; in their absence, several of the results are obtained from an odd group approximation (OGA). This approximation strictly allows only a WIMP-proton or WIMP-neutron sensitivity, by assuming the even group to be an inert spectator so that the WIMP interacts with only the odd group of detector nucleons. This is reflected in the traditional spin-dependent exclusion plots, in which for $<S_{p}>$= 0, only $a_{n}$ is constrained.
In the OGA, an experiment using only odd Z isotopes cannot constrain the WIMP-neutron coupling. Nuclear structure calculations however show that the even group of nucleons has a non-negligible (though subdominant) spin. An example is $^{39}$K, which surprisingly possesses $<S_{n}>$ = 0.05 [@ERTO] in spite of having a magic number of neutrons (closed neutron shell). This can be understood because the gyromagnetic ratio of the nucleus is low compared to that of the nucleon, indicating a dominant contribution from the orbital angular momentum of the proton structure; when Z $\approx$ N and large, if the protons have a high angular momentum, so also will the neutrons in general.
![exclusion comparison for $C_{4}F_{10}$ for each of two sets of fluorine ($^{19}F$) spin matrix elements [@Pacheco; @Divari], for a zero event 30 kgd exposure.[]{data-label="posSDex"}](fig4.eps "fig:"){width="8"}\
For A $<$ 50, the OGA has been extended by including additional information regarding the $\beta$ decay $ft$ values and measured magnetic moments of mirror pairs for nuclear systems [@Engel2], which provides nonzero estimates of the spin matrix element for the odd group, with seemingly small variations in the odd group spin matrix element.
Generally, the OGA yields $<S_{p,n}>$ significantly different from the model calculations. The refined $<S_{p,n}>$ of nuclear structure calculations are however not measured, but obtained from various nuclear models which reproduce known nuclear data, so that different sets of results may exist for the same nuclide. In some cases ($^{23}$Na, $^{35}$Cl), there is even a sign reversal. Some indication of the impact of the model difference on the contour orientation is seen in Fig. \[posSDex\], for an otherwise identical 30 kgd projection with $C_{4}F_{10}$ assuming full discrimination.
For heavy nuclei, and/or heavy WIMPs, the zero momentum transfer approximation breaks down and the finite momentum transfer must be taken into account, as discussed extensively in Ref. [@newnonzero]. In general this involves consideration of the nuclear form factor ($F$) in the interaction scattering rate
$$\frac{dN}{dE_{r}} \sim
\mu^{-2}M_{W}^{-1} \sigma^{(SD)}_{A} F^{2}(q)\int^{v_{max}}_{v_{min}}
\frac{f(v)}{v}dv ,$$
where $f(v)$ is related to the velocity distribution of halo WIMPS, $v_{min}$ is the minimum incident WIMP speed required to cause a recoil of energy $E_{r}$, $v_{max}$ is the maximum incident WIMP speed, and $F^{2}(q) = \frac{S^{A}(q)}{S^{A}(0)}$ with the $S^{A}$ related to the $a_{p},a_{n}$ by $S^{A}(q) =
(a_{p}+a_{n})^{2}S_{00}(q) + (a_{p}-a_{n})^{2}S_{11}(q) +
(a_{p}+a_{n})(a_{p}-a_{n})S_{01}(q)$. Calculations of the structure functions $S_{jk}$ so far have included only $^{19}F$, $^{23}Na$, $^{27}Al$, $^{29}Si$, $^{73}Ge$, $^{127}I$, and $^{129,131}Xe$, and the results for the same isotope differ significantly among calculations, depending on the nuclear potential employed [@newnonzero].
CONCLUSIONS
===========
The future search for “spin-independent” WIMP dark matter is particularly well-motivated and directed towards improvements of several orders of magnitude in probing the phase space; due to the spin sensitivity of several of the new detector isotopes, it will also provide significant impact in the $a_{n}$ sector of the “spin-dependent” phase space.
In contrast, the direct search in the $a_{p}$ sector is somewhat neglected. This seems particularly strange given that the spin-sensitivity of fluorine is well-known and that several fluorine-based prototype experiments ($LiF, NaF, CaF_{2}$) have been reported over the years. At present, new experiments based on $Al_{2}O_{3}$ and fluorine are seen as possibly capable of providing restrictions on $a_{p}$ surpassing those from $NaI$ and complementary to those to be obtained on $a_{n}$. Of these, the SDD and bubble chamber experiments appear to offer the greatest possibility of achieving significantly improved restrictions with least exposure, given their intrinsic insensitivity to most common backgrounds; being also relatively simple in construct and less expensive by at least an order of magnitude, large volume efforts are readily possible. None of these experiments however seem receiving of attention comparable to those of the $a_{n}$ activity, which will prove problematic should any of the latter in fact observe a positive signal in the near future.
The projected impact of several of the new, possibly interesting spin-dependent projects, such as $CaWO_{4}$, $BGO$ and $CsI$, suffer from the availability of only OGA estimates of their spin values, which constrains *a priori* their orientation in the parameter space, and could profit from more detailed nuclear structure calculations.
F. Giuliani is supported by grant SFRH/BPD/13995/2003 of the Foundation for Science and Technology (FCT) of Portugal. This work was supported in part by POCI grants FIS/57834/2004 and FIS/56369/2004 of the National Science & Technology Foundation of Portugal, co-financed by FEDER.
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abstract: 'We theoretically investigate magnetic properties of a trapped ultracold Fermi gas. Including pairing fluctuations within the framework of an extended $T$-matrix approximation (ETMA), as well as effects of a harmonic trap in the local density approximation (LDA), we calculate the local spin susceptibility $\chi_{\rm t}(r,T)$ in the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover region. We show that pairing fluctuations cause non-monotonic temperature dependence of $\chi_{\rm t}(r,T)$. Although this behavior looks similar to the spin-gap phenomenon associated with pairing fluctuations in a [*uniform*]{} Fermi gas, the trapped case is found to also be influenced by the temperature-dependent density profile, in addition to pairing fluctuations. We demonstrate how to remove this extrinsic effect from $\chi_{\rm t}(r,T)$, to study the interesting spin-gap phenomenon purely originating from pairing fluctuations. Since experiments in cold atom physics are always done in a trap, our results would be useful for the assessment of preformed pair scenario, from the viewpoint of spin-gap phenomenon.'
author:
- 'Hiroyuki Tajima$^1$, Ryo Hanai$^2$, and Yoji Ohashi$^3$'
title: 'Spin susceptibility and effects of a harmonic trap in the BCS-BEC crossover regime of an ultracold Fermi gas'
---
Introduction
============
Since the realization of superfluid $^{40}$K [@Regal] and $^6$Li [@Zwierlein; @Kinast; @Bartenstein] Fermi gases, strong-coupling properties in the BCS (Bardeen-Cooper-Schrieffer)-BEC (Bose-Einstein condensation) crossover region have attracted much attention in this field [@Holland; @Ohashi2; @Levin2005; @Giorgini; @Bloch; @Chin; @Zwerger]. In this regime, the system properties are dominated by strong pairing fluctuations, that are physically described as repeating the formation and dissociation of preformed (Cooper) pairs[@Nozieres; @SadeMelo; @Randeria; @Randeria1992]. Thus, ultracold Fermi gases are expected to provide a useful testing ground for the assessment of the so-called preformed pair scenario, which has been proposed as a possible mechanism of the pseudogap observed in the underdoped regime of high-$T_{\rm c}$ cuprates [@Randeria1992; @Singer; @Janko; @Renner; @Yanase; @Rohe; @Perali; @Lee2; @Fischer; @Varma] (where a gap-like structure appears in the density of states (DOS) even above the superconducting phase transition temperature $T_{\rm c}$). Although the origin of this anomaly is still unclear in high-$T_{\rm c}$ cuprates because of the complexity of this electron system [@Randeria1992; @Singer; @Janko; @Renner; @Yanase; @Rohe; @Perali; @Lee2; @Fischer; @Varma; @Pines; @Kampf; @Chackravarty], if it is observed in an ultracold Fermi gas, the origin must be strong pairing fluctuations, or the formation of preformed Cooper pairs [@Tsuchiya; @Watanabe; @Mueller; @Magierski; @Su]. Although this observation would not immediately clarify the pseudogap phenomenon in high-$T_{\rm c}$ cuprates, one may regard it as an evidence for the validity of the preformed pair scenario, at least, in the presence of strong pairing fluctuations.
At present, the pseudogap has not been observed in an ultracold Fermi gas yet, because of the difficulty of the direct observation of DOS in this field. Although a photoemission-type experiment supports the preformed pair scenario[@Stewart; @Gaebler; @Sagi2], thermodynamic measurements[@Nascimbene2; @Nascimbene3; @Nascimbene] report the Fermi liquid-like behavior of the system with no pseudogap. Thus, further studies are necessary to resolve this controversial situation.
Recently, we have theoretically pointed out [@Tajima; @Tajima2] that the spin-gap may be an alternative key phenomenon to assess the preformed pair scenario in an ultracold Fermi gas. This magnetic phenomenon is characterized by the anomalous suppression of spin susceptibility in the normal state near the superfluid phase transition temperature. This many-body phenomenon has been observed in high-$T_{\rm c}$ cuprates [@Takigawa; @Yoshinari; @Takigawa2; @Bobroff; @Ogata], although the origin is still in controversial. In the preformed pair scenario, the pseudogap and spin-gap are understood as different aspects of the same pairing phenomenon. That is, while the former is explained from the viewpoint of “binding energy" of preformed pairs, the latter is understood as a result of the formation of [*spin-singlet*]{} preformed pairs. Indeed, it has theoretically been shown [@Tajima] that the pseudogap temperature (below which a dip structure appears in DOS) is very close to the spin-gap temperature (below which the spin susceptibility is anomalously suppressed) in the BCS-unitary regime. Recently, the spin susceptibility has become observable in cold Fermi gas physics [@Sanner; @Sommer; @Meineke; @Lee], and theoretical analyses on the observed spin susceptibility have been started [@Tajima; @Tajima2; @Palestini; @Mink; @Enss]. Thus, this alternative approach seems promising in the current stage of cold Fermi gas physics.
In this paper, we extend our previous work [@Tajima; @Tajima2] for the spin susceptibility in a uniform Fermi gas to include effects of a harmonic trap. This extension is really important, because experiments are always done in a trap potential. Thus, it is a crucial issue how spatially inhomogeneous pairing fluctuations affect spin susceptibility. In addition, to assess the preformed pair scenario without any ambiguity, we need to know spin susceptibility in a [*uniform*]{} Fermi gas, from observed data in a [*trapped*]{} Fermi gas. Regarding this, the pseudogap case is simpler, because, once the local density of states $\rho({\bm r},\omega)$ becomes observable in the future, the observed dip structure in $\rho({\bm r},\omega)$ around $\omega=0$ can immediately be interpreted as the pseudogap in the uniform case with the uniform density $n$ being equal to the local density $n({\bm r},T)$ at the observed spatial position ${\bm r}$. On the other hand, since the spin-gap appears in the [*temperature dependence*]{} of the spin susceptibility, it is sensitive to the temperature-dependence of the density profile $n({\bm r},T)$. To examine the preformed pair scenario proposed in the uniform system, we need to remove the latter extrinsic effect from the observed temperature dependence of the spin susceptibility in a trap.
For our purpose, we include effects of a harmonic trap in the local density approximation (LDA) [@Ohashi6; @Perali3; @Haussmann3; @Tsuchiya2; @Tsuchiya3; @Watanabe]. Pairing fluctuations are taken into account within the framework of an extended $T$-matrix approximation (ETMA) [@Tajima; @Tajima2; @Kashimura; @Hanai; @Kashimura2; @Tajima3]. Using this combined theory, we calculate the local spin susceptibility $\chi_{\rm t}(r,T)$, as well as the spatially averaged one $X_{\rm t}(T)$, in the whole BCS-BEC crossover region. We demonstrate how we can map $\chi_{\rm t}(r,T)$ onto the spin susceptibility $\chi_{\rm u}(T)$ in a [*uniform*]{} Fermi gas. We also compare our results with the recent experiment in a $^6$Li Fermi gas [@Sanner].
This paper is organized as follows. In Sec. II, we present our combined extended $T$-matrix approximation (ETMA) with the local density approximation (LDA). In Sec. III, we show our numerical results on the local spin susceptibility $\chi_{\rm t}(r,T)$ in the BCS-BEC crossover region of a trapped Fermi gas. Here, we also explain how to relate $\chi_{\rm t}(r,T)$ to $\chi_{\rm u}(T)$ in a uniform Fermi gas, to examine the spin-gap phenomenon purely originating from pairing fluctuations. In Sec. IV, we consider the trap-averaged spin susceptibility $X_{\rm t}(T)$, to compare our results with the recent experiment on a $^6$Li Fermi gas [@Sanner]. Throughout this paper, we set $\hbar=k_{\rm B}=1$, for simplicity.
Formulation
===========
To explain our formalism, we start from a uniform superfluid Fermi gas. Effects of a harmonic trap will be included later. In the two-component Nambu representation [@Fukushima; @Schrieffer; @Ohashi4; @Watanabe], our model Hamiltonian is given by $$H=\sum_{\bm p}
{\hat \Psi}_{\bm p}^\dagger
\left[
\xi_{\bm p}\tau_3-h-\Delta\tau_1
\right]
{\hat \Psi}_{\bm p }
-{U \over 4}\sum_{\bm q}
\left[
\rho_{1,{\bm q}}\rho_{1,-{\bm q}}+
\rho_{2,{\bm q}}\rho_{2,-{\bm q}}
\right].
\label{eq1}$$ Here, $$\begin{aligned}
{\hat \Psi}_{\bm p}=
\left(
\begin{array}{c}
c_{{\bm p},\uparrow} \\
c_{-{\bm p},\downarrow}^\dagger
\end{array}
\right)
\label{eq1b}\end{aligned}$$ is the two-component Nambu field, where $c_{{\bm p},\sigma}^\dagger$ is the creation operator of a Fermi atom with pseudospin $\sigma=\uparrow,\downarrow$, describing two atomic hyperfine states. $\xi_{\bm p}=\varepsilon_{\bm p}-\mu={\bm p}^2/(2m)-\mu$ is the kinetic energy of a Fermi atom, measured from the Fermi chemical potential $\mu$, where $m$ is an atomic mass. Although we consider the population-balanced case, the model Hamiltonian in Eq. (\[eq1\]) involves an infinitesimally small fictious magnetic field $h$, in order to calculate the spin susceptibility later. $\tau_i$ ($i=1,2,3$) are Pauli matrices acting on particle-hole space. In Eq. (\[eq1\]), the superfluid order parameter $\Delta$ is taken to be parallel to the $\tau_1$-component, without loss of generality. In this choice, the generalized density operators, $$\rho_{i,{\bm q}}=
\sum_{\bm p}{\hat \Psi}_{{\bm p}+{\bm q}/2}^\dagger\tau_i
{\hat \Psi}_{{\bm p}-{\bm q}/2}~~(i=1,2),
\label{eq1c}$$ physically describe amplitude fluctuations $(i=1)$ and phase fluctuations $(i=2)$ of the superfluid order parameter $\Delta$ [@Fukushima; @Ohashi4; @Watanabe]. We briefly note that the ordinary contact-type $s$-wave pairing interaction is described by the sum of amplitude-amplitude ($\rho_1\rho_1$) and phase-phase ($\rho_2\rho_2$) interactions in Eq. (\[eq1\]).
As usual, we measure the interaction strength in terms of the $s$-wave scattering length $a_s$, which is related to the bare coupling constant $-U~(<0)$ as, $${4\pi a_s \over m}=-
{U \over \displaystyle 1-U\sum_{\bm p}{1 \over 2\varepsilon_{\bm p}}}.
\label{eq2}$$ In this scale, the weak-coupling BCS regime and strong-coupling BEC regime are conveniently characterized as $(k_{\rm F}a_s)^{-1}\lesssim -1$ and $(k_{\rm F}a_s)^{-1}\gesim 1$, respectively. The region between the two is called the BCS-BEC crossover region.
![(a) Self-energy correction ${\hat \Sigma}$ in combined ETMA with LDA (LDA-ETMA). The $2\times 2$-matrix particle-particle scattering vertex ${\hat \Gamma}=\{\Gamma\}^{j,j'}$ is given in (b). In this figure, the double and single solid lines denote the LDA-ETMA dressed Green’s function ${\hat G}$ in Eq. (\[eq4\]), and the bare one ${\hat G}^{0}$ in Eq. (\[eq4b\]), respectively. The filled circle is a pairing interaction $-U$.[]{data-label="fig1"}](fig1.eps){width="8cm"}
Now, we include effects of a harmonic trap. In the local density approximation (LDA)[@Ohashi6; @Perali3; @Haussmann3; @Tsuchiya2; @Tsuchiya3; @Watanabe], this extension is achieved by simply replacing the Fermi chemical potential $\mu$ by the LDA one, $\mu(r)=\mu-V(r)$, where $$V(r)={1 \over 2}m\Omega_{\rm tr}^2 r^2
\label{eq3}$$ is a harmonic potential with a trap frequency $\Omega_{\rm tr}$, with $r$ being the radial position, measured from the trap center. The $2\times 2$-matrix single-particle thermal Green’s function in LDA has the form, $${\hat G}_{\bm p}(i\omega_n,r)=
{1 \over
(i\omega_n+h)-\xi_{\bm p}(r)\tau_3+\Delta(r) \tau_1-\hat{\Sigma}_{\bm p}
(i\omega_n,r)},
\label{eq4}$$ where $\omega_n$ is the fermion Matsubara frequency, $\xi_{\bm p}(r)=\varepsilon_{\bm p}-\mu(r)$, and $\Delta(r)$ is the LDA position-dependent superfluid order parameter. The $2\times 2$-matrix self-energy ${\hat \Sigma}_{\bm p}(i\omega_n,r)$ describes strong-coupling corrections to single-particle excitations. Within the framework of the combined extended $T$-matrix approximation (ETMA) with LDA (LDA-ETMA), it is diagrammatically described as Fig. \[fig1\], which gives $${\hat \Sigma}_{\bm p}(i\omega_n,r)=-T
\sum_{{\bm q},\nu_n}\sum_{j,j'=\pm}\Gamma^{j,j'}_{\bm q}(i\nu_n,r)
\tau_j {\hat G}_{{\bm p}+{\bm q}}(i\omega_n+i\nu_n,r)\tau_{j'}.
\label{eq5}$$ Here, $\nu_n$ is the boson Matsubara frequency, $\tau_{\pm}=(\tau_1+i\tau_2)/2$, and $$\begin{aligned}
\left(\begin{array}{cc}
\Gamma^{-+}_{\bm{q}}(i\nu_n,r) & \Gamma^{--}_{\bm{q}}(i\nu_n,r) \\
\Gamma^{++}_{\bm{q}}(i\nu_n,r) & \Gamma^{+-}_{\bm{q}}(i\nu_n,r)
\end{array}\right)
=-U\left[1+U
\left
(\begin{array}{cc}
\Pi^{-+}_{\bm{q}}(i\nu_n,r) & \Pi^{--}_{\bm{q}}(i\nu_n,r) \\
\Pi^{++}_{\bm{q}}(i\nu_n,r) & \Pi^{+-}_{\bm{q}}(i\nu_n,r)
\end{array}
\right)\right]^{-1}
\label{eq6}\end{aligned}$$ is the $2\times 2$-matrix particle-particle scattering vertex, describing fluctuations in the Cooper channel. In Eq. (\[eq6\]), $$\Pi^{j,j'}_{\bm q}(i\nu_n,r)=
T\sum_{{\bm p},i\omega_n}{\rm Tr}
\left[
\tau_j\hat{G}^0_{{\bm p}+{\bm q}}(i\omega_n+i\nu_n,r)
\tau_{j'}\hat{G}^0_{\bm p}(i\omega_n,r)
\right],
\label{eq7}$$ is the lowest-order pair correlation function, where $$\hat{G}^{0}_{\bm p}(i\omega_n,r)
={1 \over i\omega_n-\xi_{\bm p}(r)\tau_3+\Delta(r)\tau_1}
\label{eq4b}$$ is the $2\times 2$-matrix mean-field BCS single-particle thermal Green’s function.
An advantage of ETMA is that one can obtain the expected [*positive*]{} spin susceptibility in the whole BCS-BEC crossover region[@Kashimura]. The ordinary (non-selfconsistent) $T$-matrix approximation (TMA), as well as the strong-coupling theory developed by Nozières and Schmitt-Rink (NSR), are known to unphysically give [*negative*]{} spin susceptibility in the crossover region, because of unsatisfactory treatment of strong-coupling corrections to spin-vertex and single-particle density of states, respectively [@Kashimura].
We calculate the local spin susceptibility $\chi_{\rm t}(r,T)$ from $$\chi_{\rm t}(r,T)=
\lim_{h\rightarrow 0}{n_\uparrow(r,T)-n_\downarrow(r,T) \over h}.
\label{eq11}$$ Here, $n_\sigma(r,T)$ is the density profile of $\sigma$-spin atoms, which is calculated from LDA-ETMA dressed Green’s function in Eq. (\[eq4\]) as, $$\begin{aligned}
\begin{array}{l}
\displaystyle
n_{\up}(r,T)=T\sum_{{\bm p},i\omega_n}G^{11}_{\bm p }(i\omega_n,r),
\\
\displaystyle
n_{\dwn}(r,T)=\sum_{\bm{p}}1-
T\sum_{{\bm p},i\omega_n}G^{22}_{\bm p}(i\omega_n,r).
\end{array}
\label{eq10}\end{aligned}$$ In this paper, we numerically evaluate Eq. (\[eq11\]), by setting $h/\varepsilon_{\rm F}^{\rm t}=0.01$, where $\varepsilon_{\rm F}^{\rm t}$ is the Fermi energy of a trapped free Fermi gas. We have numerically confirmed that the difference $n_\uparrow(r)-n_\downarrow(r)$ is proportional to $h$, when $h/\varepsilon_{\rm F}^{\rm t}=O(10^{-2})$.
In this paper, we also consider the spatially averaged (or total) spin susceptibility, $$X_{\rm t}(T)=
\int d{\bm r}\chi_{\rm t}(r,T)=\lim_{h\rightarrow0}\frac{N_{\up}-N_{\dwn}}{h},
\label{eq12}$$ where $N_\sigma$ is the number of $\sigma$-spin atoms.
In calculating Eqs. (\[eq11\]) and (\[eq12\]), we note that effects of fictitious field $h$ on the superfluid order parameter $\Delta(r)$ and the chemical potential $\mu$ are $O(h^2)$. For example, the gap equation, which is obtained from the condition for the gapless Goldstone mode (${\rm det}[{\hat \Gamma}_{{\bm q}=0}(\nu_m=0,r)]=0$, where ${\hat \Gamma}=\{\Gamma^{j,j'}\}$), has the form, in the presence of $h$, $$1=-{4\pi a_s \over m}
\sum_{\bm p}
\left[
{1 \over 4E_{\bm p}(r)}
\left[
\tanh{E_{\bm p}(r)+h \over 2T}
+
\tanh{E_{\bm p}(r)-h \over 2T}
\right]
-{1 \over 2\varepsilon_{\bm p}}
\right],
\label{eq11b}$$ where $E_{\bm p}(r)=\sqrt{\xi_{\bm p}^2(r)+\Delta^2(r)}$ describes local Bogoliubov single-particle excitations in LDA. The right-hand side of Eq. (\[eq11b\]) is clearly an even function of $h$, indicating the even function of $\Delta(r)$ in terms of $h$. Because of this, we can safely ignore $h$ in determining $\Delta(r)$ and $\mu$ for our purpose. The gap equation (\[eq11b\]) is then simplified as ($h=0$), $$1=-{4\pi a_s \over m}
\sum_{\bm p}
\left[{1 \over 2E_{\bm p}(r)}\tanh{E_{\bm p}(r) \over 2T}-
{1 \over 2\varepsilon_{\bm p}}
\right].
\label{eq8}$$ We solve Eq. (\[eq8\]), together with the equation for the total number $N$ of Fermi atoms, $$N=\sum_\sigma N_\sigma=\sum_\sigma\int d^3{\bm r}n_\sigma(r,T)_{h=0},
\label{eq9}$$ to self-consistently determine $\Delta(r)$ and $\mu$.
Although the LDA gap equation (\[eq8\]) gives position-dependent superfluid phase transition temperature $T^{\rm t}_{\rm c}(r)$, it is an artifact of this approximation. The superfluid order parameter should become finite everywhere in a gas cloud below the superfluid phase transition $T^{\rm t}_{\rm c}$ of the system. In this sense, $T^{\rm t}_{\rm c}(r)$ should physically be regarded as a characteristic temperature below which the superfluid order parameter at $r$ becomes large. In LDA, the superfluid phase transition temperature $T_{\rm c}^{\rm t}$ is determined from the $T_{\rm c}^{\rm t}(r)$-equation at $r=0$, $$1=-{4\pi a_s \over m}
\sum_{\bm p}
\left[{1 \over 2\xi_{\bm p}}\tanh{\xi_{\bm p} \over 2T^{\rm t}_{\rm c}}-
{1 \over 2\varepsilon_{\bm p}}
\right].
\label{eq8b}$$ Above $T^{\rm t}_{\rm c}$, as well as in the spatial region with vanishing superfluid order parameter $\Delta(r)=0$ even below $T^{\rm t}_{\rm c}$ (Note that $T_{\rm c}^{\rm t}(r)\le T_{\rm c}^{\rm t}$.), we only solve the number equation (\[eq9\]), to determine $\mu$.
![(a) Calculated local spin susceptibility $\chi_{\rm t}(r,T)$ at $r=0.4R_{\rm F}$, as a function of temperature. $R_{\rm F}=\sqrt{2\varepsilon_{\rm F}^{\rm t}/(m\Omega_{\rm tr}^2)}$ is the Thomas-Fermi radius, where $\varepsilon_{\rm F}^{\rm t}={k_{\rm F}^{\rm t}}^2/(2m)=(3N)^{1/3}\Omega_{\rm tr}$ is the LDA Fermi energy in a trap (which equals the LDA Fermi temperature $T_{\rm F}^{\rm t}$). $\chi_{\rm t}^0(r,T)=3m n(r,T)^{1/3}/(3\pi^2)^{2/3}$ is the expression for the spin susceptibility in a free Fermi gas at $T=0$ where the number density is replaced by the LDA-ETMA local density $n(r,T)=n_\uparrow(r,T)+n_\downarrow(r,T)$ at $r=0.4R_{\rm F}$. At each line, the short vertical line shows $T^{\rm t}_{\rm c}$, and the open circle represents the peak position of $\chi_{\rm t}(r,T)$ in the normal state. The arrow shows $T^{\rm t}_{\rm c}(r)$, below which the LDA superfluid order parameter $\Delta(r=0.4R_{\rm F})$ becomes non-zero. (b) Density profile $n(r)=n_\uparrow(r)+n_\downarrow(r)$, when $(k^{\rm t}_{\rm F}a_s)^{-1}=-0.6$. []{data-label="fig2"}](fig2.eps){width="8cm"}
Local spin susceptibility and spin-gap phenomenon in a trapped Fermi gas
========================================================================
Figure \[fig2\](a) shows the local spin susceptibility $\chi_{\rm t}(r,T)$ at $r=0.4R_{\rm F}$ (where $R_{\rm F}$ is the Thomas-Fermi radius). In this figure, $\chi_{\rm t}(r,T)$ is found to exhibit a peak structure at a certain temperature ($\equiv T^{\rm t}_{\rm p}(r)$) in the normal state, and is suppressed below this. Since the local superfluid order parameter $\Delta(r)$ only becomes non-zero below the temperature at the arrow in Fig. \[fig2\](a), this anomaly is found to occur in the absence of $\Delta(r=0.4R_{\rm F})$.
At a glance, the non-monotonic behavior of $\chi_{\rm t}(r,T)$ around $T_{\rm p}^{\rm t}(r)$ looks similar to the spin-gap phenomenon discussed in the BCS-BEC crossover regime of a [*uniform*]{} Fermi gas [@Tajima], where this anomaly originates from the formation of spin-singlet preformed (Cooper) pairs. In this magnetic phenomenon, the spin-gap temperature $T_{\rm SG}^{\rm u}$ is defined as the temperature at which the uniform spin susceptibility $\chi_{\rm u}(T)$ takes a maximum value. Regarding this, if the density profile were $T$-independent, each result in Fig. \[fig2\](a) could be immediately regarded as the spin susceptibility $\chi_{\rm u}(T)$ in an assumed [*uniform*]{} Fermi gas with the uniform density $n=n(r=0.4R_{\rm F})=\sum_{\sigma}n_\sigma(r=0.4R_{\rm F})$. However, Fig. \[fig2\](b) shows that the density profile $n(r,T)$ actually depends on $T$. Thus, $\chi_{\rm t}(r,T)$ in Fig. \[fig2\](a) is also affected by this $T$-dependent density profile, in addition to pairing fluctuations. Since the former effect does not exist in the uniform case, the peak temperature $T_{\rm p}^{\rm t}(r)$ in $\chi_{\rm t}(r,T)$ cannot be immediately identified as the spin-gap temperature $T_{\rm SG}^{\rm u}$ in the uniform case. To examine the spin-gap phenomenon purely originating from pairing fluctuations, we need to remove effects of the $T$-dependent density profile from $\chi_{\rm t}(r,T)$. This would be particularly important, when the local spin susceptibility in a trapped Fermi gas becomes experimentally accessible in the future.
.[]{data-label="fig3"}](fig3.eps){width="7cm"}
![Mapping of local spin susceptibility $\chi_{\rm t}(r,T)$ in a trapped Fermi gas onto spin susceptibility $\chi_{\rm u}(T)$ in a uniform Fermi gas. (a) Each solid line with a filled circle (“a"-“i") is the scaled temperature $T/T^{\rm u}_{\rm F}(r)$ as a function of the scaled interaction strength $(k_{\rm F}^{\rm u}a_s)^{-1}$, which is obtained from the local density $n(r,T)$ in Fig. \[fig3\] at the same label (“a"-“i"). (However, since the solid lines in the cases of “d"-“f" are the same vertical line at $(k_{\rm F}^{\rm u}a_s)^{-1}=0$, we do not draw them in the figure.) $k_{\rm F}^{\rm u}$, $T_{\rm F}^{\rm u}$, $T_{\rm c}^{\rm u}$, and $T^{\rm u}_{\rm SG}$, are the Fermi momentum, Fermi temperature, the superfluid phase transition temperature, and the spin-gap temperature, in a uniform Fermi gas, respectively. (b) Spin susceptibility $\chi_{\rm u}(T)$ in a uniform Fermi gas [@Tajima]. The filled circles “a"-“i" are the values of the local spin susceptibility $\chi_{\rm t}(r,T)$ at the same labels in Fig. \[fig3\]. $\chi_{\rm u}^0(0)=mk_{\rm F}^{\rm u}/\pi^2$ is the uniform spin susceptibility in a free Fermi gas at $T=0$. []{data-label="fig4"}](fig4.eps){width="8cm"}
We demonstrate how to extract information about the spin-gap phenomenon in a uniform Fermi gas from the local spin susceptibility $\chi_{\rm t}(r,T)$ in a trapped one. For this purpose, we recall that LDA treats a gas at each spatial position ${\bm r}$ as a [*uniform*]{} one with the “(effective) local Fermi momentum", $$k^{\rm t}_{\rm F}(r,T)=[3\pi^2n(r,T)]^{1/3}.
\label{eq.80aa}$$ For example, “a" in Fig. \[fig3\](a) is regarded as a uniform Fermi gas with the Fermi momentum, $$k^{\rm t}_{\rm F}(r,T)=
\left[
3\pi^2\times{8 \over \pi^2}{N \over R_{\rm F}^3}
\right]^{1/3}=k^{\rm t}_{\rm F}.
\label{eq.80a}$$ Here, we have used the LDA relation, $R_{\rm F}=\sqrt{2\varepsilon^{\rm t}_{\rm F}/(m\Omega_{\rm tr}^2)}$, where $\varepsilon^{\rm t}_{\rm F}={k_{\rm F}^{\rm t}}^2/(2m)=(3N)^{1/3}\Omega_{\rm tr}$ is the LDA Fermi energy in a trapped Fermi gas, with $k_{\rm F}^{\rm t}$ being the LDA Fermi momentum [@Ohashi6; @note80a]. The local spin susceptibility $\chi_{\rm t}(r,T)$ at “a" in Fig. \[fig3\](a) can then be regarded as the susceptibility $\chi_{\rm u}(T)$ in a uniform Fermi gas at the scaled temperature $T/T_{\rm F}^{\rm u}=T/T^{\rm t}_{\rm F}(r)=T/T_{\rm F}^{\rm t}=0.16$, and the scaled interaction strength $(k_{\rm F}^{\rm u} a_s)^{-1}=(k_{\rm F}^{\rm t}(r,T)a_s)^{-1}=(k_{\rm F}^{\rm t}a_s)^{-1}=-0.6$ (“a" in Fig. \[fig4\](a)). Here, $T_{\rm F}^{\rm t}={k_{\rm F}^{\rm t}}^2/(2m)$ is the LDA Fermi temperature in a [*trapped*]{} Fermi gas, and $k_{\rm F}^{\rm u}$ and $T_{\rm F}^{\rm u}={k_{\rm F}^{\rm u}}^2/(2m)$ are the Fermi momentum and Fermi temperature in a [*uniform*]{} Fermi gas, respectively. In the same manner, the spatial position “b" and “c" in Fig. \[fig3\](a) are mapped onto the uniform system with the same scaled interaction strength $(k_{\rm F}^{\rm u} a_s)^{-1}=-0.6$, but at $T/T_{\rm F}^{\rm u}=0.2$ and 0.25, respectively (see Fig. \[fig4\](a)). As shown in Fig. \[fig4\](b), the values of $\chi_{\rm t}(r,T)$ at “a"-“c" in Fig. \[fig3\](a) coincide with the previous ETMA result for a uniform Fermi gas at $(k_{\rm F}^{\rm u}a_s)^{-1}=-0.6$ [@Tajima], as expected.
The above prescription is also valid for stronger coupling cases. Indeed, the positions “d"-“i" in Figs. \[fig3\](b) and (c) are mapped onto the uniform case at the same labels in Fig. \[fig4\], respectively.
We note that this mapping can be simplified to some extent at the unitarity, because $\chi_{\rm t}(r,T)$ in this special case is always mapped onto $\chi_{\rm u}(T)$ in a uniform unitary Fermi gas. (Note that the scaled interaction $(k_{\rm F}^{\rm u}a_s)^{-1}$ identically vanishes when $a_s^{-1}=0$, irrespective of the value of the Fermi momentum $k_{\rm F}^{\rm u}$.) Using this, we can construct the temperature dependence of $\chi_{\rm u}(T)$ at the unitarity only from the temperature dependence of $\chi_{\rm t}(r,T)$ at a fixed position $r$. The maximum $\chi_{\rm t}(r,T)$ is mapped onto the maximum $\chi_{\rm u}(T)$ in this case, so that one can exceptionally relate the peak temperature $T_{\rm p}^{\rm t}(r)$ in the trapped case (open circle in Fig. \[fig2\](a)) to the spin-gap temperature $T_{\rm SG}^{\rm u}$ in the uniform case as, $${T_{\rm SG}^{\rm u} \over T_{\rm F}^{\rm u}}=
\left(
{8 \over \alpha(r)\pi^2}
\right)^{2/3}
{T_{\rm p}^{\rm t}(r) \over T_{\rm F}^{\rm t}},
\label{eq.80b}$$ where $\alpha(r)=(R_{\rm F}^3/N)n(r,T_{\rm p}^{\rm t}(r))$.
![$r-T$ phase diagram of a trapped Fermi gas. (a) $(k^{\rm t}_{\rm F}a_s)^{-1}=-0.6$ (weak-coupling BCS side). (b) $(k^{\rm t}_{\rm F}a_s)^{-1}=0$ (unitarity limit). (c) $(k^{\rm t}_{\rm F}a_s)^{-1}=0.6$ (strong-coupling BEC side). The LDA superfluid order parameter $\Delta(r)$ becomes non-zero when $r\le r_{\rm c}(T)$ (SF). $T_{\rm p}^{\rm t}(r)$ is the temperature at which $\chi_{\rm t}(r,T)$ takes a maximum value, when $r$ is fixed. $r_{\rm p}(T)$ is the spatial position at which $\chi_{\rm t}(r,T)$ takes a maximum value, when $T$ is fixed. $\chi_{\rm t}(r_{\rm SG}(T),T)$ is mapped onto $\chi_{\rm u}(T_{\rm SG}^{\rm u})$ in a uniform Fermi gas. The region $r_{\rm c}(T)\le r\le r_{\rm SG}(T)$ (SG) is mapped onto the spin-gap regime in a uniform Fermi gas, where $\chi_{\rm u}(T)$ is suppressed by pairing fluctuations. The region $r>r_{\rm SG}(T)$ (NF) is mapped onto the normal Fermi gas regime in the uniform case, where $\chi_{\rm u}(T)$ monotonically increases with decreasing the temperature. The open and filled circles represent $T_{\rm p}^{\rm t}(r)$ and $r_{\rm p}(T)$ obtained from Figs. \[fig2\] and \[fig6\], respectively. Because of computational problems at low temperatures ($T\lesssim 0.02T^{\rm t}_{\rm F}$), we only draw eye-guide (thin dashed line) for each line there. []{data-label="fig5"}](fig5.eps){width="7cm"}
Figure \[fig5\] shows the phase diagram of a trapped Fermi gas with respect to the spatial position $r$ (measured from the trap center) and the temperature $T$ in LDA-ETMA. In each panel, $r_{\rm SG}(T)$ is the spatial position which is mapped onto the spin-gap temperature $T_{\rm SG}^{\rm u}$ in a uniform Fermi gas with the uniform density $n=n(r_{\rm SG}(T),T)$ and the interaction strength $(k_{\rm F}^{\rm u}a_s)^{-1}=(k_{\rm F}^{\rm t}(r_{\rm SG}(T))a_s)^{-1}$, for a given interaction strength $(k_{\rm F}^{\rm t}a_s)^{-1}$. As expected, one sees in Fig. \[fig5\](b) that the peak temperature $T_{\rm p}^{\rm t}(r)$ coincides with the “spin-gap line" $r_{\rm SG}(T)$ in the unitarity limit, except for the outer region of the gas cloud, $r\gesim 0.8R_{\rm F}$ (which will be separately discussed later).
Although this coincidence is only guaranteed at the unitarity, Fig. \[fig5\](a) shows that $T_{\rm p}^{\rm t}(r)$ is still close to $r_{\rm SG}(T)$ in the weak-coupling BCS side (as far as we consider the region $r\lesssim 0.8R_{\rm F}$). Thus, the peak-temperature $T_{\rm p}^{\rm t}(r)$ in the trapped case is still useful for [*roughly*]{} estimating the spin-gap temperature $T_{\rm SG}^{\rm u}$ in the BCS side of a uniform Fermi gas. On the other hand, we see in Fig. \[fig5\](c) that $T_{\rm p}^{\rm t}(r)$ is very different from $r_{\rm SG}(T)$ in the BEC side, indicating that we need to faithfully fulfil the above-mentioned mapping, in order to examine the spin-gap there.
The LDA superfluid order parameter $\Delta(r)$ only becomes non-zero when $T\le T_{\rm c}^{\rm t}(r)~(\le T_{\rm c}^{\rm t})$, which leads to the shell structure of the system below $T_{\rm c}^{\rm t}$, being composed of the superfluid core region ($\Delta(r\le r_{\rm c}(T))\ne 0$) which is surrounded by the normal-fluid region ($\Delta(r>r_{\rm c}(T))=0$). In this case, the region “SG" in Fig. \[fig5\] ($r_{\rm c}(T)\le r\le r_{\rm SG}(T)$) is mapped onto the spin-gap regime ($T_{\rm c}^{\rm u} \le T\le T_{\rm SG}^{\rm u}$) of an uniform Fermi gas, where $\chi_{\rm u}(T)$ is suppressed by pairing fluctuations (where $T_{\rm c}^{\rm u}$ is the superfluid phase transition temperature in the uniform case). The region “NF" and “SF" in Fig. \[fig5\], are, respectively, mapped onto the normal Fermi gas regime (where $\chi_{\rm u}(T)$ monotonically increases as the temperature decreases), and the superfluid regime (where $\chi_{\rm u}(T)$ is suppressed by the superfluid order) of a uniform Fermi gas, respectively.
![(a) Temperature dependence of local density $n(r,T)$ in the outer region of the gas cloud at $r=0.9R_{\rm F}$. (b) Scaled local temperature $T/T_{\rm F}^{\rm t}(r=0.9R_{\rm F})$, as a function of $T/T_{\rm F}^{\rm t}$. []{data-label="fig6"}](fig6.eps){width="7cm"}
Of course, the above-mentioned shell structure is, strictly speaking, an artifact of LDA. The superfluid order parameter $\Delta(r)$ should actually become non-zero everywhere in a gas below $T_{\rm c}^{\rm t}$. Thus, when we experimentally examine the spin-gap phenomenon purely caused by [*normal-state*]{} pairing fluctuations, we should examine the region surrounded by the vertical $T_{\rm c}^{\rm t}$-line and the spin-gap line $r_{\rm SG}(T)$ in Fig. \[fig5\].
As briefly mentioned previously, in the unitarity limit shown in Fig. \[fig5\](b), while the peak temperature $T_{\rm p}^{\rm t}(r)$ coincides with the spin-gap line $r_{\rm SG}(T)$ in the central region of the gas cloud ($r\lesssim 0.8R_{\rm F}$), such coincidence is not obtained in the outer region, $r\gesim 0.8R_{\rm F}$, implying that $T_{\rm p}^{\rm t}(r\gesim 0.8R_{\rm F})$ comes from a different origin from the spin-gap phenomenon. To understand the origin of this peak temperature, the key is that, when one increases the temperature from $T=0$, the local density $n(r\gesim 0.8R_{\rm F})$ first increases because of the thermal expansion of the gas cloud, as shown in Fig. \[fig6\](a). As a result, the scaled local temperature $T/T_{\rm F}^{\rm t}(r\gesim 0.8R_{\rm F})$ exhibits a non-monotonic temperature dependence, as shown in Fig. \[fig6\](b). In the case of Fig. \[fig6\] ($r=0.9R_{\rm F}$), denoting the dip temperature in Fig. \[fig6\](b) as $T_{\rm dip}$, one finds that the increase of $T/T_{\rm F}^{\rm t}$ in the low temperature region of a trapped Fermi gas ($T\le T_{\rm dip}$) corresponds to the [*decrease*]{} of $T/T^{\rm u}_{\rm F}=T/T_{\rm F}^{\rm t}(r=0.9R_{\rm F})$ in the high-temperature region of a [*uniform*]{} Fermi gas. Thus, reflecting the increasing of $\chi_{\rm u}(T)$ with decreasing $T/T^{\rm u}_{\rm F}$ in the high-temperature region, the corresponding $\chi_{\rm t}(r=0.9R_{\rm F},T)$ [*increases*]{} with increasing $T/T_{\rm F}^{\rm t}$ when $T\le T_{\rm dip}$. On the other hand, when $T\ge T_{\rm dip}$, the increase of $T/T_{\rm F}^{\rm t}$ corresponds to the [*increase*]{} of $T/T^{\rm u}_{\rm F}$. Thus, the decrease of $\chi_{\rm u}(T)$ with increasing $T/T_{\rm F}^{\rm t}$ leads to the [*decrease*]{} of $\chi_{\rm t}(r=0.9R_{\rm F},T)$ with increasing $T/T_{\rm F}^{\rm t}$ when $T\ge T_{\rm dip}$.
To conclude, although the resulting $\chi_{\rm t}(r\gesim 0.8R_{\rm F})$ takes a maximum value at $T_{\rm dip}$, it is clearly not due to pairing fluctuations, but simply originates from the temperature dependence of the density profile around the edge of the gas cloud. Since the non-monotonic behavior of $T/T_{\rm F}^{\rm t}(r=0.9R_{\rm F})$ is also seen in the other two cases shown in Fig. \[fig6\], $T_{\rm p}^{\rm t}(r\gesim 0.8R_{\rm F})$ in Fig. \[fig5\](a), as well as that in Fig. \[fig5\](c), are also nothing to do with the spin-gap phenomenon.
Regarding the above-mentioned effects of $T$-dependent density profile, we briefly note that, while the thermal expansion of the trapped gas increases the density $n(r,T)$ in the outer region of the gas cloud at low temperatures, it decreases $n(r)$ in the central region, as seen in Fig. \[fig3\]. Because of this, the scaled local temperature $T/T_{\rm F}^{\rm t}(r)$ in the trap center monotonically increases with increasing $T/T_{\rm F}^{\rm t}$. Thus, the increase of $T/T_{\rm F}^{\rm t}$ in the trapped case can simply be related to the increase of $T/T_{\rm F}^{\rm u}$ in the uniform case there.
![Local spin susceptibility $\chi_{\rm t}(r,T)$, as a function of the spatial position $r$, measured from the trap center. We also plot the superfluid order parameter $\Delta(r)$. []{data-label="fig7"}](fig7.eps){width="8cm"}
Figure \[fig7\] shows the spatial variation of $\chi_{\rm t}(r,T)$ in a trapped Fermi gas. In addition to the well-known suppression of spin susceptibility in the superfluid phase ($\Delta(r,T)\ne 0$), $\chi_{\rm t}(r,T)$ is found to be suppressed in the trap center ($r\sim 0$), even in the normal state. Conveniently defining the peak radius $r_{\rm p}(T)$ as the position at which the spatial variation of $\chi_{\rm t}(r,T)$ takes a maximum value above $T_{\rm c}^{\rm t}$, we find that it agrees with the spin-gap radius $r_{\rm SG}(T)$ in the unitarity limit (see Fig. \[fig6\](b)). This is simply because the scaled local interaction strength $(k_{\rm F}^{\rm t}(r)a_s)^{-1}$ always vanishes at the unitarity ($a_s^{-1}=0$), irrespective of the value of $k_{\rm F}^{\rm t}(r)$, so that $\chi_{\rm t}(r,T)$ in the unitarity limit is always mapped onto $\chi_{\rm u}(T)$ in a uniform unitary Fermi gas. This means that we can evaluate the spin-gap temperature without measuring the temperature dependence of $\chi_{\rm t}(r,T)$ at the unitarity.
Of course, the peak radius $r_{\rm p}(T)$ does not coincide with the spin-gap line $r_{\rm SG}(T)$ for $(k_{\rm F}^{\rm t}a_s)^{-1}\ne 0$ (see Figs. \[fig5\](a) and (c)), because of the position dependent $T/T_{\rm F}^{\rm t}(r)$, and $(k_{\rm F}^{\rm t}(r)a_s)^{-1}$.
![Calculated trap-averaged spin susceptibility $X_{\rm t}(T)$ in Eq. (\[eq12\]), normalized by the value $X_{\rm t}^0(0)=3N/\varepsilon_{\rm F}^{\rm t}$ in a trapped free Fermi gas at $T=0$. (a) $(k^{\rm t}_{\rm F}a_s)^{-1}=-0.8$. (b) $(k^{\rm t}_{\rm F}a_s)^{-1}=0$. (c) $(k^{\rm t}_{\rm F}a_s)^{-1}=-0.8$. For comparison, we also plot the ETMA spin susceptibility $\chi_{\rm u}(T)$ in a uniform Fermi gas[@Tajima2], normalized by the value $\chi_{\rm u}^0(0)=mk_{\rm F}^{\rm u}/\pi^2$ in a free Fermi gas at $T=0$. In each result, the filled circle shows the temperature at which the spin susceptibility takes a maximum value. In the uniform case, it gives the spin-gap temperature $T_{\rm SG}^{\rm u}$. In the trapped case, it gives ${\tilde T}_{\rm p}^{\rm t}$. The open circles are the recent experimental data on a $^6$Li Fermi gas [@Sanner]. Because of computational problems, our LDA-ETMA results end at $T\simeq 0.02T^{\rm t}_{\rm F}$; the thin dashes lines at lower temperatures in panels (a) and (b) are eye-guide. []{data-label="fig8"}](fig8.eps){width="8cm"}
Trap-averaged spin susceptibility in the BCS-BEC crossover region
=================================================================
Figure \[fig8\] compares the trap-averaged spin susceptibility $X(T)$ in Eq. (\[eq12\]) with the spin susceptibility $\chi_{\rm u}(T)$ in a uniform Fermi gas. We find that the behavior of $X(T)$ is relatively close to that of $\chi_{\rm u}(T)$, in spite of the fact that $X_{\rm t}(T)$ is affected by $T$-dependent density profile $n(r,T)$. Figure \[fig8\] also shows that the both $X(T)$ and $\chi_{\rm u}(T)$ agree with the recent experiment on a $^6$Li Fermi gas[@Sanner] in the weak-coupling regime, as well as in the unitarity limit. Although the spatial resolution of this experiment[@Sanner] is unclear, our results indicate that the spatial inhomogeneity is not so crucial for the observed spin susceptibility, at least in the cases of Figs. \[fig8\](a) and (b).
In our previous paper [@Tajima2], we pointed out the the observed spin susceptibility in the strong-coupling BEC side ($(k_{\rm F}^{\rm u}a_s)^{-1}=0.8>0$) cannot be explained by ETMA spin susceptibility in a uniform Fermi gas. In this regard, Fig. \[fig8\](c) shows that this problem still remains in the trapped case, because $X_{\rm t}(T)$ is still much smaller than the observed value. In order to reproduce the experimental result in the strong-coupling regime [@Sanner] within the current LDA-ETMA formalism, we need to raise the temperature to $T\simeq 0.6T_{\rm F}^{\rm t}$. At preset, we have no idea to fill up this discrepancy, which remains our future problem.
![Peak temperature ${\tilde T}_{\rm p}^{\rm t}$ of $X_{\rm t}(T)$. For comparison, we also plot the spin-gap temperature $T_{\rm SG}^{\rm u}$ [@Tajima]. ${\tilde T}_{\rm p}^{\rm t:BEC}$ is the solution of Eq. (\[eq30\]). []{data-label="fig9"}](fig9.eps){width="8cm"}
Figure \[fig9\] shows the peak temperature ${\tilde T}_{\rm p}^{\rm t}$ at which the averaged spin susceptibility $X_{\rm t}(T)$ takes a maximum value. As expected from the similarity between $X_{\rm t}(T)$ and $\chi_{\rm u}(T)$ in Fig. \[fig8\], ${\tilde T}_{\rm p}^{\rm t}$ is relatively close to the spin-gap temperature $T_{\rm SG}^{\rm u}$ in a uniform Fermi gas, although the former also involves effects of the $T$-dependent density profile. Indeed, when we ignore pairing fluctuations by replacing the ETMA self-energy in Eq. (\[eq5\]) with that in the mean-field approximation [@FW; @noteMF],
$$\hat{\Sigma}^{\rm MF}(r,T)=\frac{4\pi a_s}{m}\left[n_{\dwn}(r,T)\frac{(1+\tau_3)}{4}-n_{\up}(r,T)\frac{(1-\tau_3)}{4}\right],
\label{eq5HF}$$
the resulting averaged spin susceptibility ($\equiv X_{\rm t}^{\rm MF}(T)$) exhibits “spin-gap" like temperature dependence, as shown in Fig. \[fig10\]. Since the averaged spin susceptibility does not exhibit such a non-monotonic behavior when the density profile is $T$-independent, it purely comes from the $T$-dependent $n_\sigma(r,T)$. The peak temperature ${\tilde T}_{\rm p}^{\rm t}$ is considered to also involve this effect, in addition to spin-gap effects associated with pairing fluctuations.
![Averaged spin susceptibility $X_{\rm t}^{\rm MF}(T)$ in the mean-field approximation, where the Hartree-Fock mean-field self-energy $\hat{\Sigma}^{\rm MF}(r,T)$ in Eq. (\[eq5HF\]) is used for the ETMA one in Eq. (\[eq5\]).[]{data-label="fig10"}](fig10.eps){width="8cm"}
In Ref. [@Tajima], we showed that the spin-gap temperature $T_{\rm SG}^{\rm u}$ in the strong coupling regime of a uniform Fermi gas can be explained by a classical gas mixture, consisting of two kinds of atoms with active spins $\sigma=\uparrow,\downarrow$ and one-component spinless molecules [@Saha]. When we simply extend this to the present trapped case, the equation for the peak temperature ${\tilde T}_{\rm p}^{\rm t:BEC}$ of $X_{\rm t}(T)$ in this classical gas mixture is obtained as, $${(2mR_{\rm F}^2{\tilde T}_{\rm p}^{\rm t:BEC})^3 \over 108N^2
}\exp \left(-\frac{E_{\rm b}}{T}\right)=
\frac{\left[\left(\frac{E_{\rm b}+3{\tilde T}_{\rm p}^{\rm t:BEC}}{E_{\rm b}+2{\tilde T}_{\rm p}^{\rm t:BEC}}\right)-2\right]^2}{\frac{E_{\rm b}+3{\tilde T}_{\rm p}^{\rm t:BEC}}{E_{\rm b}+2{\tilde T}_{\rm p}^{\rm t:BEC}}-1},
\label{eq30}$$ where $E_{\rm b}=1/(ma_s^2)$ is the molecular binding energy. (For the derivation of Eq. (\[eq30\]), see the Appendix.) The calculated ${\tilde T}_{\rm p}^{\rm t:BEC}$ well reproduces ${\tilde T}_{\rm p}^{\rm t}$ in the strong-coupling regime (see Fig. \[fig10\]), indicating that the simple classical gas mixture is also valid in considering $X_{\rm t}(T)$ in a trapped Fermi gas when $(k_{\rm F}^{\rm t}a_s)\gesim 0.5$.
Summary
=======
To summarize, we have discussed magnetic properties of a trapped ultracold Fermi gas. Including effects of strong pairing fluctuations within in the framework of an extended $T$-matrix approximation (ETMA), as well as effects of a harmonic trap in the local density approximation (LDA), we have calculated local spin susceptibility $\chi_{\rm t}(r,T)$, as well as the spatially averaged one in the whole BCS-BEC crossover region.
We showed that the local spin susceptibility $\chi_{\rm t}(r,T)$ in the BCS-BEC crossover region exhibits a non-monotonic temperature dependence, taking a maximum value at a certain temperature $T^{\rm t}_{\rm p}(r)$. At a glance, it looks similar to the spin-gap behavior of the spin susceptibility $\chi_{\rm u}(T)$ in a uniform Fermi gas. However, the former peak temperature $T^{\rm t}_{\rm p}(r)$ cannot actually be simply related to the latter spin-gap temperature $T^{\rm u}_{\rm SG}$ (except at the unitarity), because the former also involves effects of [*temperature-dependent*]{} density profile, in addition to effects of pairing fluctuations. We explained how to evaluate $T^{\rm u}_{\rm SG}$, by properly mapping $\chi_{\rm t}(r,T)$ onto $\chi_{\rm u}(T)$. Using this, we also identified the region which is mapped onto the spin-gap regime ($T_{\rm c}^{\rm u}\le T\le T^{\rm u}_{\rm SG}$) of a uniform Fermi gas, in the phase digram of a trapped Fermi gas with respect to the spatial position $r$ measured from the trap center and the temperature.
We pointed out that this mapping can be simplified to some extent in the unitarity limit, because the local spin susceptibility $\chi_{\rm t}(r,T)$ in a trapped [*unitary*]{} Fermi gas is always mapped onto $\chi_{\rm u}(T)$ in a uniform [*unitary*]{} Fermi gas. Using this advantage, we can immediately relate the peak temperature $T_{\rm p}^{\rm t}$ to the spin-gap temperature $T^{\rm u}_{\rm SG}$, by way of the simple relation in Eq. (\[eq.80b\]). We pointed out that this advantage also enables us to evaluate $T_{\rm SG}^{\rm u}$ from the spatial variation of $\chi_{\rm t}(r,T)$ for a fixed temperature.
Besides the local spin susceptibility, we also examined the spatially averaged spin susceptibility $X_{\rm t}(T)$. The calculated $X_{\rm t}(T)$ was shown to agree with the recent experiment on a $^6$Li Fermi gas in the weak-coupling regime, as well as in the unitarity limit. However, in the strong-coupling BEC regime, our result was found to be much smaller than the observed value. In this regard, our previous work for a uniform Fermi gas has already faced the same discrepancy in the strong-coupling regime [@Tajima]. Thus, our result in this paper indicates that this problem is nothing to do with effects of a harmonic trap. Explaining theoretically the observed large spin susceptibility in the strong-coupling regime remains as our future problem.
Even when the local measurement of spin susceptibility in an ultracold Fermi gas becomes possible in the future, experimental data would more or less involve effects of finite spatial resolution. In this regard, this paper has only dealt with the two extreme cases, that is, the local spin susceptibility $\chi_{\rm t}(r,T)$ and the fully averaged one $X_{\rm t}(T)$. Thus, as a future challenge, it would be interesting to theoretically clarify the minimal spatial resolution which is necessary to examine the spin-gap phenomenon, by using the observed spin susceptibility in a trapped Fermi gas. We briefly note that this kind of theoretical estimation has recently been done [@Ota] for the local photoemission-type experiment developed by JILA group [@Sagi2]. At present, because cold atom physics has no experimental technique to directly observe the pseudogapped density of states, our results would be useful for the assessment of preformed pair scenario from the viewpoint of spin-gap phenomenon in a trapped ultracold Fermi gas.
We thank T. Kashimura, R. Watanabe, D. Inotani, and M. Ota for useful discussions. This work was supported by KiPAS project at Keio University. H.T. and R.H. were supported by a Grant-in-Aid for JSPS fellows (No.17J03975, No.17J01238). Y.O. was also supported by Grant-in-Aid for Scientific research from MEXT and JSPS in Japan (No.16K05503, No.15K00178, No.15H00840).
Derivation of Eq. (\[eq30\]) {#apA}
============================
We consider a non-interacting classical gas mixture, consisting of two-component atoms with active spins $\sigma=\uparrow,\downarrow$ (with the number density $n^0_\sigma(r,T)$) and one-component spinless molecules (with the molecular density $n_{\rm M}(r,T)$), in a harmonic trap potential. In the BEC regime of an ultracold Fermi gas, the former two and the latter correspond to unpaired Fermi atoms and tightly bound molecular bosons, respectively. The total atomic number density $n(r,T)$ is given by $$n(r,T)=n^0_\uparrow(r,T)+n^0_\downarrow(r,T)+2n_{\rm M}(r,T),
\label{eq25}$$ where $$\begin{aligned}
\label{eq26}
n^0_\sigma(r,T)
=\sum_{\bm{p}}\exp\left[-\frac{\xi_{\bm{p}}(r)-\s h}{T}\right]
=\frac{3\sqrt{\pi}}{8}\left(\frac{T}{T_{\rm F}^{\rm t}}\right)^{\frac{3}{2}}
\lambda\exp\left(\frac{\s h - m\Omega_{\rm tr}^2r^2/2}{T}\right),\end{aligned}$$ $$\begin{aligned}
\label{eq27}
n_{\rm M}(r)=\sum_{\bm{q}}\exp\left[-\frac{\varepsilon_{\bm{q}}^{\rm M}-2\mu(r)-E_{\rm b}}{T}\right]
=\frac{3\sqrt{2\pi}}{4}\lambda^2
\exp\left(\frac{E_{\rm b}-m\Omega_{\rm tr}^2r^2}{T}\right).\end{aligned}$$ Here, $\varepsilon_{\bm q }^{\rm M}=\bm{q}^2/(4m)$ is the molecular kinetic energy and $\lambda=\exp(\mu/T)$ is the fugacity. Solving the total number equation,$$N=\int d{\bm r}n(r,T),
\label{eq27a}$$ in terms of the fugacity $\lambda$, one obtains, $$\label{eq28}
\lambda=
{1 \over 2}
\exp\left(-\frac{E_{\rm b}}{T}\right)
\left[\sqrt{1+{2 \over 3}\left(\frac{T^{\rm t}_{\rm F}}{T}\right)^3
\exp\left(\frac{E_{\rm b}}{T}\right)}-1\right].$$
Noting that the averaged spin susceptibility ($\equiv X_{\rm t}^{\rm cl}(T)$) in the present model classical gas is obtained from spin-active atoms. Therefore, we reach, $$\begin{aligned}
\label{eq29}
X_{\rm t}^{\rm cl}(T)=
\lim_{h\rightarrow 0}\int d\bm{r}
{n_{\up}^0(r)-n_{\dwn}^0(r) \over h}
= 2\left({T \over T^{\rm t}_{\rm F}}\right)^2\lambda X_{\rm t}^0(0).\end{aligned}$$ Equation (\[eq30\]) is straightforwardly obtained from the extremum condition $(\partial X_{\rm t}^{\rm cl}/\partial T)=0$.
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|
---
author:
- Samriddhi Sankar Ray
- Dario Vincenzi
title: Elastic turbulence in a shell model of polymer solution
---
Introduction
============
Elastic turbulence is a chaotic regime that develops in low-inertia viscoelastic fluids when the elasticity of the fluid exceeds a critical value [@GS00]. It is characterised by power-law velocity spectra (both in time and in space) and by a strong increase of the flow resistance compared to the laminar regime. Elastic turbulence differs from hydrodynamic turbulence in that inertial nonlinearities are irrelevant and the chaotic behaviour of the flow is entirely generated by elastic instabilities. In addition, the spatial spectrum of the velocity decays faster than in hydrodynamics turbulence; thus, the velocity field is smooth in space. Elastic turbulence has important applications, since the possibility of inducing instabilities at low Reynolds numbers allows the generation of mixing flows in microfluidics devices [@GS01; @BSBGS04]. This phenomenon has been used, for instance, to study the deformation of DNA molecules in chaotic flows [@LS14]. Furthermore, elastic turbulence provides a possible explanation of the improvement in oil-displacement efficiency that is observed when polymer solutions are used to flood reservoir rocks [@MLHC16].
The first experiments on elastic turbulence have used confined flows with curved stream lines [@GS04]. Nonetheless, purely elastic instabilities have been shown to develop also in a viscoelastic version of the Kolmogorov flow, which is periodic and parallel [@BCMPV05; @BBCMPV07]. Indeed, elastic turbulence in the viscoelastic Kolmogorov flow exhibits a phenomenology qualitatively similar to that observed in experiments [@BBBCM08; @BB10]. Low-Reynolds-number elastic instabilities have also been predicted for the Poiseuille flow [@MSMS04] and for the planar Couette flow [@MS05] of a polymer solution at large elasticity. More recently, elastic turbulence has been observed experimentally in a straight microchannel [@PMWA13; @BBMMKC15] and numerically in a periodic square [@G14]. These findings indicate that elastic turbulence also develops in simplified flow configurations and that the specific geometry of the system may not play a crucial role in this phenomenon. In this Letter, we take a step further in this direction and study elastic turbulence in a shell model of polymer solution.
Hydrodynamical shell models are low-dimensional models that preserve the essential shell-to-shell energy transfer feature of the original partial differential equations in Fourier space. Despite the fact that they are not derived from the principle hydrodynamic equations in any rigorous way, they have played a fundamental role in the study of fluid turbulence since they are numerically tractable [@F95; @BJPV98; @Biferale; @PPR09]. Shell models have also achieved remarkable success in problems related to passive-scalar turbulence [@JPV92; @WB96; @MP05; @RMP08], magnetohydrodynamic turbulence [@PSF13], rotating turbulence [@Sagar], binary fluids [@JO98; @RB11], and fluids with polymer additives [@BdAGP03; @KGP05]. Furthermore, the mathematical study of shell models has yielded several rigorous results, whose analogs are still lacking for the three-dimensional Navier–Stokes equations (e.g., Refs. [@CLT06; @F11]).
A shell model of polymer solution can be obtained by coupling the evolution of the velocity variables with the evolution of an additional set of variables representing the polymer end-to-end separation field. Shell models of polymer solutions have been successfully applied to the study of drag reduction in forced [@BdAGP03; @BCHP04; @BCP04] and decaying [@KGP05] turbulence, two-dimensional turbulence with polymer additives [@BHP04], and turbulent thermal convection in viscoelastic fluids [@BCdA10; @BCW14]. Here, we study a shell model of polymer solution in the regime of low inertia and high elasticity. We show that this shell model undergoes a transition froam a laminar to a chaotic regime with properties remarkably similar to those of elastic turbulence. Moreover, the use of a low-dimensional model allows us to explore the properties of elastic turbulence over a wide range both in polymer concentration and in Weissenberg number, which would be difficult to cover with direct numerical simulations of constitutive models of viscoelastic fluids.
Shell model of polymer solution
===============================
We consider the shell model of polymer solution introduced by Kalelkar *et al.* [@KGP05], which is based on a shell model initially proposed for three-dimensional magnetohydrodynamics [@FS98; @BSDR98] and reduces to the GOY model [@G73; @YO87] when polymers are absent. The shell model by Kalelkar *et al.* [@KGP05] can be regarded as a reduced, low-dimensional version of the FENE model [@BCAH87]. It describes the temporal evolution of a set of complex scalar variables $v_n$ and $b_n$ representing the velocity field and the polymer end-to-end separation field, respectively. The variables $v_n$ and $b_n$ evolve according to the following equations [@KGP05]:
$$\begin{aligned}
\label{eq:v}
\frac{{\mathrm{d}}v_n}{{\mathrm{d}}t}&=&\Phi_{n,vv}-\nu_s k^2_n
v_n+\frac{\nu_p}{\tau_p} P(b)\Phi_{n,bb}+f_n, \\
\label{eq:b}
\frac{{\mathrm{d}}b_n}{{\mathrm{d}}t}&=&\Phi_{n,vb}+\Phi_{n,bv} -\frac{1}{\tau_p}P(b)b_n-\nu_b k_n^2 b_n.\end{aligned}$$
where $n=1,\dots,N$, $k_n=k_0 2^n$, $P(b)=1/(1-\sum_n\vert b_n\vert^2)$ and $\Phi_{n,vv}=i(a_1k_nv_{n+1}v_{n+2}+a_2k_{n-1}v_{n+1}v_{n-1}+a_3k_{n-2}v_{n-1}
v_{n-2})^*$, $\Phi_{n,bb}=-i(a_1 k_n b_{n+1}b_{n+2}+a_2 k_{n-1}
b_{n+1}b_{n-1}+a_3 k_{n-2} b_{n-1}b_{n-2})^*$, $\Phi_{n,vb}=i(a_4 k_n
v_{n+1}b_{n+2}+a_5 k_{n-1}v_{n-1}b_{n+1}+a_6 k_{n-2} v_{n-1} b_{n-2})^*$, and $\Phi_{n,bv}=-i(a_4 k_n b_{n+1} v_{n+2}+a_5 k_{n-1} b_{n-1}v_{n+1}+a_6 k_{n-2}
b_{n-1} v_{n-2})^*$ with $k_0=1/16$, $a_1=1$, $a_2=-\epsilon$, $a_3=-(1-\epsilon)$, $a_4=1/6$, $a_5=1/3$, $a_6=-2/3$, and the single free parameter $\epsilon$ determines whether or not, in the absence of polymers, the behaviour of the shell model is chaotic. The GOY shell model for fluids indeed shows a chaotic behaviour for $0.33 \lesssim \epsilon
\lesssim 0.9$ and a non-chaotic behaviour for $\epsilon\lesssim 0.33$ [@BLLP95; @KOJ98]; the standard choice for hydrodynamic turbulence is $\epsilon=0.5$ [@F95; @BJPV98; @Biferale; @PPR09]. As we shall see later, it is useful to study elastic turbulence in both these regimes.
![Log-log plots of the kinetic energy spectrum $E(k)$ vs the wavenumber $k$ for a highly viscous flow with (red, filled squares) and without (blue, filled circles) polymer additives (see text). The curve without the addition of polymers do not show any algebraic scaling. However, the addition of polymers leads to the development of a power-law scaling $k^{-4}$ in the energy spectrum (as indicated by the thick black line).[]{data-label="fig:spectra"}](spectra_showing_power_law.png){width="\columnwidth"}
![The kinetic energy vs time for various values of $c$ and $Wi$. From the uppermost to the lowermost curve the curves correspond to $c = 1.0$, $Wi \approx 0.25$; $c =
4.0$, $Wi \approx 0.25$; $c = 1.0$, $Wi \approx 25$; $c = 4.0$, $Wi \approx 25$; and $c = 20.0$, $Wi \approx 25$. The top two curves show non-chaotic, laminar behaviour with a transition to periodic dynamics in the middle curve and then fully elastic turbulence in the bottom two.[]{data-label="fig:all_energy"}](all_energies.png){width="\columnwidth"}
{width="0.5\linewidth"}{width="0.5\linewidth"}
The number of shells that are used is given by $N$, the coefficient of kinematic viscosity by $\nu_s$, the polymer relaxation time by $\tau$, $\nu_p$ is the polymer viscosity parameter, $\nu_b=10^{-13}\nu_s$ is a damping coefficient to allow for the dissipation term $-\nu_b k_n^2 b_n$ to be added to Eq. in order to improve numerical stability [@BCHP04; @BCP04; @KGP05], and the forcing $f_n$ drives the system to a non-equilibrium statistically stationary state. In particular we use either a deterministic forcing $f_n = f_0(1 + \imath)\delta_{n,2}$ or a white-in-time Gaussian stochastic forcing with amplitude $f_0$ acting on the $n=2$ shell. We choose initial conditions of the form $v_n^0 = k_n^{1/2}e^{\imath \phi_n}$ for $n = 1, 2$ and $v_n^0 = k_n^{1/2}e^{-k_n^2}e^{\imath \phi_n}$ for $3 \le n \le
N$ and, for the polymer field, $b_n^0 = k_n^{1/2}e^{\imath \theta_n}$ for $1
\le n \le N$. Here $\phi_n$ and $\theta_n$ are random phases uniformly distributed between 0 and $2\pi$. Equations and are solved numerically through a second-order Adams–Bashforth method with a time step $\delta t$ for all our simulations. The numerical values for the various parameters of our simulations are given in Table 1.
By analogy with continuum models of polymer solutions, we interpret the ratio $c=\nu_p/\nu_s$ as the polymer concentration [@BCHP04; @BCP04; @KGP05]. We define the mean dissipation rate of the flow as $\varepsilon=\langle\nu_s\sum_n k_n^2 v_n^2\rangle$; thence we extract the large-scale time $T=(k_1^2\varepsilon)^{-1/3}$, which allows us to define the Weissenberg number as $\mathrm{Wi}=\tau/T$. In our simulations we choose eight different values both of $c$ and of $\mathrm{Wi}$ such that $0 \le c \le 20$ and $0 \le \mathrm{Wi} \lesssim 25$. The Weissenberg number is varied by varying $\tau$, so that the inertia of the system remains constant and negligible for all Wi.
Elastic-turbulence regime
=========================
In order to understand whether the shell model defined via Eqs. and indeed shows the typical features of elastic turbulence, we perform numerical simulations of the shell model with $\epsilon=0.5$ and a stochastic forcing. The parameters (see Table \[dec\_para\]) are such that for $c=0$ the shell model is not turbulent. The time-averaged (in the steady state) kinetic-energy spectrum $E(k_n)=\vert v_n\vert^2/k_n$ indeed decreases sharply with the wavenumber $k_n$ without any apparent power-law scaling (see the blue line with filled circles in Fig. \[fig:spectra\]).
A typical signature for elastic turbulence is the development of a power-law energy spectrum with an exponent smaller than $-3$ as the Weissenberg number is increased at fixed Reynolds number much smaller than 1 [@GS04; @BSS07]. We therefore turn on the polymer field in the shell model ($c \neq 0$), and for sufficiently large $c$ and $\mathrm{Wi}$ a power-law scaling emerges. In Fig. \[fig:spectra\], the energy spectrum for $c = 20$ and $\mathrm{Wi} =
25$ is shown (red squares). We see a clear power-law behaviour, namely $E(k_n)
\sim k_n^{-4}$, as is indicated by the thick black line. The value of the exponent is close to that found in experiments [@GS04; @BSS07] and in numerical simulations [@BBBCM08; @BB10; @WG13; @WG14] and is consistent with the theoretical predictions based on the Oldroyd-B model [@FL03]. The spectrum of the polymer end-to-end separation field does not show a power-law behaviour and is concentrated around small wave numbers, in agreement with direct numerical simulations of elastic turbulence [@G14]. An analogous behaviour is found with a deterministic forcing. This is accompanied by a corresponding increase of the largest Lyapunov exponent as discussed in detail later. The spectrum of the polymer end-to-end separation field does not show a power-law behaviour and is concentrated around small wave numbers, in agreement with direct numerical simulations of elastic turbulence [@G14]. Thus, the shell model reproduces the most obvious signature of elastic turbulence, namely, the emergence of a large-scale chaotic dynamics in a [*laminar*]{} flow with the addition of polymers.
{width="0.33\linewidth"}{width="0.33\linewidth"}{width="0.33\linewidth"}
{width="0.45\linewidth"} {width="0.45\linewidth"}
A global quantity like the total kinetic energy $K(t)=\sum_n v_n^2(t)$ provides further insight into the transition to elastic turbulence; its temporal behaviour with varying Wi and $c$ indeed is an indicator of the changes of dynamical regime which happen in the system [@BB10]. In Fig. \[fig:all\_energy\] we show time series of $K(t)$ for various combinations of $c$ and $\mathrm{Wi}$. For cases with very small values of $\mathrm{Wi}$—and independent of the value of $c$—the total energy quickly saturates to an asymptotic value with no noticeable fluctuations, as is typical for laminar flows (Fig. \[fig:all\_energy\], top two panels). However, as $\mathrm{Wi}$ increases, even for a small enough value of $c = 1.0$, tiny but regular oscillations are seen in the temporal dynamics of $K(t)$ vs $t$ (Fig. \[fig:all\_energy\], middle panel). This behaviour is shown clearly in a zoomed plot in Fig. \[fig:zoomed\_energy\](a). Keeping the Weissenberg number fixed, we now increase the concentration (Fig. \[fig:all\_energy\], bottom two panels) and see that the total kinetic energy vs time shows increasingly chaotic dynamics with large irregular fluctuations. This behaviour is highlighted in the zoomed-in Fig. \[fig:zoomed\_energy\](b). Figures \[fig:all\_energy\] and \[fig:zoomed\_energy\] show that the shell model (which for $c=0$ is laminar because of our choice of parameters), with increasing effect of polymers characterised by the concentration or the Weissenberg number, undergoes a transition from a laminar phase to one with strong fluctuations through a series of intermediate periodic phases for moderate values of $c$ and $\mathrm{Wi}$. This phenomenon was first observed as a function of Wi in direct numerical simulations of the Oldroyd-B model [@BCAH87] with periodic Kolmogorov forcing [@BB10] and of the FENE-P model [@BCAH87] in a cellular flow [@G14]. The shell model considered here not only reproduces such a transition to chaos through periodic states as the Weissenberg number is increased, but also shows that an analogous transition occurs as a function of polymer concentration. Following Ref. [@KOJ98], in Fig. \[fig:map\] we also show the map $\vert
v_{n+1}\vert$ vs $\vert v_n\vert$, for $n = 2$. The structure of this map for increasing values of $c$ further shows that the elastic-turbulence regime emerges through period doubling.
The above results confirm that the shell model with polymer additives replicates the global features of elastic turbulence. Given the relative numerical simplicity of shell models, it now behooves us to study in detail the effects of concentration on the small-scale mixing properties of elastic turbulence, which determine the importance of this phenomenon for practical applications. We quantify mixing in elastic turbulence and its dependence on $c$ and $\mathrm{Wi}$ by calculating the largest Lyapunov exponent $\lambda$ of the projection of the shell model on the $v_n$ variables. We recall that for the fluid GOY shell model such calculations show the chaotic–non-chaotic transitions as a function of the single parameter $\epsilon$ [@BLLP95; @KOJ98]. For this set of calculations we would like to ensure that, in the absence of polymers, the flow is non-chaotic in order to reveal the transition to chaos more clearly. Thus we now study the shell model with $\epsilon = 0.3$ (see Table \[dec\_para\]), for which $\lambda = 0$ when $c = 0$. In order to check the generality of our conclusions, we use both a deterministic and a stochastic forcing and find our results insensitive to the precise nature of forcing.
Before we turn our attention to a quantitative measure of the transition to elastic turbulence below, we immediately note, as seen in Fig \[fig:eps0p3\_shellmap\], that the basic feature of transition to chaos, with increasing concentration for a fixed $\mathrm{Wi}$, persists even for the case of $\epsilon = 0.3$.
In Figure \[fig:lambda\_vs\_Wi\] we show the Lyapunov exponent rescaled with the polymer relaxation time, $\lambda\tau$, as a function of $\mathrm{Wi}$ for different values of $c$ both for deterministic and for stochastic forcing (inset). For small values of $c \lesssim 5$, the rescaled Lyapunov exponent remains close to 0, and hence the system is non-chaotic or [*laminar*]{}. For sufficiently large values of $c \gtrsim 5$, we find that beyond a threshold value of the Weissenberg number ($\mathrm{Wi} \approx 5$) the rescaled Lyapunov exponent increases approximately linearly and for the largest value of $c$, at $\mathrm{Wi} \gtrsim 25$, we find $\lambda \approx 1/\tau$. Our findings are in agreement with analogous calculations made in the viscoleastic Kolmogorov flow for a single, fixed value of the elasticity of the flow [@BBBCM08]. It is important to note, though, that the results for the Kolmogorov flow [@BBBCM08] indicates a slightly more dramatic increase of $\lambda\tau$ as a function of the Weissenberg number than seen in the shell model.
We now turn to the behaviour of $\lambda\tau$ as a function of $c$ for different values of $\mathrm{Wi}$, as shown in Fig. \[fig:lambda\_vs\_c\]. As before we find that for low Weissenberg numbers, the flow remains non-chaotic even when the polymer concentration increases. Beyond a threshold value of $\mathrm{Wi}$, there is a sharp increase in $\lambda\tau$ when the concentration becomes greater than 5. Thus, provided Wi is sufficiently large, increasing the concentration has a destabilizing effect comparable to that observed when the Weissenberg number is increased.
![$\lambda\tau$ vs the Weissenberg number for $\epsilon = 0.3$ for various values of the concentration $c$ and for deterministic and stochastic (inset) forcing. The symbols sizes are proportional to the error bars in our calculations. The concentration varies between 0 and $20$ from bottom to top. []{data-label="fig:lambda_vs_Wi"}](lambda_with_inset.pdf){width="\columnwidth"}
![$\lambda\tau$ vs the concentration for $\epsilon = 0.3$ for various values of the Weissenberg number Wi and for deterministic and stochastic (inset) forcing. The symbol sizes are proportional to the error bars in our calculations. The Weissenberg number varies from 0 to 25 from bottom to top. []{data-label="fig:lambda_vs_c"}](lambda_vs_conc_with_inset.pdf){width="\columnwidth"}
Finally, the interaction between the polymers and the flow is described by the energy exchange [@BCP04]: $$\Pi=-\dfrac{\nu_p
P(b)}{\tau}\,\mathrm{Re}\Big(\sum_n v_n^*\Phi_{n,bb}\Big).$$ Negative values of $\Pi$ indicate that energy flows from the velocity variables towards the polymers; positive values of $\Pi$ correspond to energy transfers in the opposite direction. In turbulent drag reduction, the timeseries of $\Pi$ is predominantly negative [@BdAGP03; @BCP04]. This fact signals that polymers drain energy from the flow and justifies the description of their effect as a scale-dependent effective viscosity [@BCP04]. In Fig. \[fig:Pi\], we show the probability density function of $\Pi$ in the elastic-turbulence regime of the shell model with $\epsilon=0.3$ at different concentrations for a fixed value $\mathrm{Wi}$. We find that $\Pi$ takes positive and negative values with comparable probabilities, i.e. in elastic turbulence there are continuous energy transfers between the flow and the polymers without a definite preferential direction (see also inset of Fig. \[fig:Pi\]. This result is consistent with the behaviour of the energy-exchange rate in decaying isotropic turbulence with polymer additives, the long-time stage of which has properties in common with elastic turbulence [@WG14].
![Probability density function of $\Pi$ for $\epsilon = 0.3$ at $\mathrm{Wi} = 5$ for concentration values $c = 5.0$ (red triangles), 10.0 (blue squares), and 20.0 (green circles). The inset shows a typical timeseries of $\Pi$ for $c = 10.0$.[]{data-label="fig:Pi"}](flux_pdf.png){width="\columnwidth"}
Conclusions
===========
We have considered a shell model of viscoelastic fluid that describes the coupled dynamics of the velocity and polymer fields in the flow of a polymer solution. In the regime of large inertia and large elasticity, this model was previously shown to reproduce the main features of turbulent drag reduction. We have studied the regime in which inertial nonlinearities are negligible and have shown that, when the Weissenberg number becomes sufficiently high, the system shows a transition to a chaotic state. A detailed analysis of this chaotic state indicates that the shell model under consideration qualitatively reproduces the transition to elastic turbulence observed in experiments and in numerical simulations. The simplicity of the shell model also allows us to study the elastic-turbulence regime over a wide range of values not only of the Weissenberg number but also of polymer concentration. In particular, we find that, when the concentration is increased while the Weissenberg number is fixed, the emergence of the chaotic regime follows a dynamics analogous to that observed when the Weissenberg number is increased at fixed polymer concentration. Thus the transition to elastic turbulence shows similar features as a function of Wi and of $c$.
This study enhances our understanding of the transition to elastic turbulence in polymer solutions. The shell model that we have studied mimicks the interactions between the Fourier modes of the velocity field and of the polymer end-to-end separation field in a viscoelastic fluid, but it contains no information on the spatial structure of these fields. The fact that such a model can replicate the main features of elastic turbulence shows that the specific geometrical configuration of the system does not play an essential role in the transition to elastic turbulence and that the physical mechanisms leading to elastic turbulence do not rely on the boundary conditions or on the mean flow.
Acknowledgments
===============
The authors are grateful to <span style="font-variant:small-caps;">Chirag Kalelkar</span> and <span style="font-variant:small-caps;">Stefano Musacchio</span> for useful discussions. This work was supported in part by the Indo–French Centre for Applied Mathematics (IFCAM) and by the EU COST Action MP 1305 “Flowing Matter”. SSR acknowledges support from the AIRBUS Group Corporate Foundation in Mathematics of Complex Systems established at ICTS and the hospitality of the Observatoire de la Côte d’Azur, Nice, France. DV acknowledges the hospitality of ICTS-TIFR, Bangalore, India.
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abstract: 'A van der Waals heterostructure combined with intrinsic magnetism and topological orders have recently paved attractive avenues to realize quantum anomalous Hall effects. In this work, using first-principles calculations and effective model analysis, we propose that the robust quantum anomalous Hall states with sizable band gaps emerge in the van der Waals heterostructure of germanene/Cr$_2$Ge$_2$Te$_6$. This heterostructure possesses high thermodynamic stability, thus facilitating its experimental fabrication. Furthermore, we uncover that the proximity effect enhances the coupling between the germanene and Cr$_2$Ge$_2$Te$_6$ layers, inducing the nontrivial band gaps in a wide range from 29 meV to 72 meV. The chiral edge states inside the band gap, leading to Hall conductance quantized to $-e^2/h$, are clearly visible. This findings provide an ideal candidate to detect the quantum anomalous Hall states and realize further applications to nontrivial quantum transport at a high temperature.'
author:
- Runling Zou
- Fangyang Zhan
- Baobing Zheng
- Xiaozhi Wu
- Jing Fan
- Rui Wang
title: 'Intrinsic quantum anomalous Hall phase induced by proximity in germanene/Cr$_2$Ge$_2$Te$_6$ van der Waals heterostructure'
---
Due to the coexistence of electronic band topology and magnetic orders, the design and exploration of exotic topological quantum states in magnetic materials attract intensive interest in condensed matter physics as well as materials science. Among these topological phases of matter, the quantum anomalous Hall (QAH) effect is one of the most fascinating topics. The QAH insulators (i.e., Chern insulators) possess dissipationless chiral edges states characterized by integer Chern numbers $\mathcal{C}$, resulting in quantized Hall conductance in units of $e^2/h$ [@PhysRevLett.49.405; @Haldane1988]. Topological properties in QAH insulators, induced by spontaneous magnetization without external magnetic fields, provide a reliable avenue to realize next-generation electronic devices with ultralow power dissipation. Therefore, extensive studies, including theoretical and experimental proposals, have always been carried out to explore candidates with QAH effects [@Yu2010; @PhysRevB.85.045445; @Fang2014; @PhysRevLett.115.186802; @PhysRevLett.119.026402; @PhysRevB.92.165418; @PhysRevB.97.085401; @PhysRevLett.110.116802; @Chang2013; @PhysRevLett.114.187201; @PhysRevLett.113.137201; @Natphys2014]. The QAH effect was firstly realized in magnetically doped topological insulators (TIs), i.e., Cr-doped (Bi,Sb)$_2$Te$_3$ thin films [@Chang2013; @PhysRevLett.114.187201; @PhysRevLett.113.137201; @Natphys2014]. But, unfortunately, disorders of random magnetic dopants in doped magnetic systems do not prefer to form ordered magnetic structures. Therefore, the laboratory synthesis of these doped magnetic TIs is quite challenging, leading to important quantum phenomena only observed at ultralow temperatures. In comparison with the doped magnetic systems, a pristine crystal is in favor of forming long-rang magnetic orders. Thus, the realization of intrinsic QAH effects facilitates their realistic applications at high temperatures. However, due to the lack of suitable candidates, the experimental observation of QAH effects in intrinsic magnetic materials is relatively slow even though there are lots of previously predicted QAH systems [@PhysRevB.85.045445; @PhysRevLett.115.186802; @PhysRevLett.119.026402; @PhysRevB.92.165418; @PhysRevB.97.085401; @PhysRevLett.110.196801; @PhysRevLett.116.096601; @PhysRevB.98.245127].
Very recently, breakthroughs of intrinsic QAH effects have been achieved in a van der Waals (vdW) layered magnetic TI MnBi$_2$Te$_4$ [@PhysRevLett.123.096401; @PhysRevX.9.041038; @Lieaaw5685; @NC2020; @Natmater2020; @Dengeaax8156]. The compound MnBi$_2$Te$_4$ is stacked in a configuration of Te-Bi-Te-Mn-Te-Bi-Te septuple layers. Significantly, the evidence of zero-field QAH effects has been observed in a five-septuple-layer of MnBi$_2$Te$_4$ up to 1.4 K by Deng *et.al* [@Dengeaax8156]. Beyond MnBi$_2$Te$_4$, other vdW superlattice-like Mn-Bi-Te structures with tunable chemical formulas as (MnBi$_2$Te$_4$)$_m$(Bi$_2$Te$_3$)$_n$ (where $m$ and $n$ are integer) have also been proposed [@PhysRevLett.123.096401; @Wueaax9989; @PhysRevX.9.041039; @PhysRevB.100.155144]. Through tuning numbers of layers, stacking configurations, or compositions, these vdW heterostructures have been further proposed to investigate rich topological quantum phenomena between magnetism and topology, such as QAH phases [@PhysRevLett.123.096401; @Lieaaw5685; @Dengeaax8156], axion insulator states [@Natmater2020], antiferromagnetic TIs [@NC2020; @Natmater2020], and their phase transitions [@PhysRevLett.123.096401].
These advances mentioned above forward a strategy for investigating magnetic topological phases and especially QAH effects in vdW layered materials. Inspired by the significant achievement in vdW Mn-Bi-Te systems, it is desirable to find other two-dimensional (2D) vdW heterostructures with intrinsic magnetism incorporating topological orders. However, as is well known, the long-rang magnetic order in 2D systems is often strongly suppressed by thermal fluctuations [@PhysRevLett.17.1133], and thus 2D magnets are rarely reported in literatures. Recently, the growth technology for 2D intrinsic ferromagnetic (FM) semiconductors has achieved important progress, and atomic layers of CrI$_3$ [@Nature2017cri3] and Cr$_2$Ge$_2$Te$_6$ [@Naturecrgete] were realized by mechanical exfoliation. These 2D ferromagnets can potentially act as ideal basic units for designing 2D vdW heterostructures with FM orders driven by proximity effects [@PhysRevLett.123.016804; @Nano4567; @Houeaaw1874]. With this guidance, the QAH effects in graphene-based heterostructures were predicted [@PhysRevB.92.165418; @PhysRevB.97.085401], but nontrivial band gaps of these heterostructures are very small due to the extremely weak spin-orbital coupling (SOC) of graphene. This seriously hinders to observe QAH effects at high temperatures. Therefore, the design of promising vdW heterostructures with large nontrivial band gaps, which can also easily be fabricated in the laboratory, is highly desired.
In this work, we demonstrate that the vdW heterostructure germanene/Cr$_2$Ge$_2$Te$_6$ can satisfy the above criteria. Germanene is a well-known time-reversal ($\mathcal{T}$)-preserved QSH insulator, which is a monolayer composed of Ge atoms in a 2D honeycomb lattice with a low-buckled geometry [@PhysRevLett.102.236804; @PhysRevLett.107.076802]. Based on first-principles calculations, we show that this heterostructure with the out-of-plane FM order hosts high thermodynamic stability that benefits its growth in experiments. In the absence of SOC, the effective model and symmetry analysis reveal that germanene/Cr$_2$Ge$_2$Te$_6$ is a spin-polarized semimetal with a quadratic nodal point. When SOC is considered, the quadratic nodal point shrinks to a sizable band gap of 29 meV. Due to weak vdW interactions, the nontrivial band gap can be tunable up to 72 meV by slightly applying an external pressure. Therefore, our work provides a reliable platform to realize intrinsic QAH effects at high temperatures.
To depict electronic and topological properties of the vdW heterostructure germanene/Cr$_2$Ge$_2$Te$_6$, we carried out first-principles calculations based on the density functional theory (DFT) [@Hohenberg; @Kohn] as implemented in the Vienna Ab initio Simulation Package (VASP) [@Kressecom; @Kresse2] (see details in the Supplemental Materials (SM) [@SM]). As illustrated in Figs. \[figure1\](a) and \[figure1\](b), we respectively show the side and top views of the germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure, i.e., germanene on top of a single layer of Cr$_2$Ge$_2$Te$_6$. The unit cell of this heterostructure is composed by a $p$($\sqrt{3}\times \sqrt{3}$) unit cells of germanene and a $p$($1\times 1$) unit cell of Cr$_2$Ge$_2$Te$_6$. The lattice mismatch between the germanene and Cr$_2$Ge$_2$Te$_6$ layers is only 1%. This low mismatch indicates small strain, which will facilitate the growth in experiments. To obtain the equilibrium geometry of germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure, we optimize the compositive structure starting from several typical stacking configurations. The optimized equilibrium distance is $d_{0}=3.48$ [Å]{}, i.e., the interaction between the germanene and Cr$_2$Ge$_2$Te$_6$ layers is a typical vdW bond. From the lowest energy structure that we have obtained, the germanene layer is further moved along the representative $\mathbf{a}_{1}$ and $\mathbf{a}_{1}+\mathbf{a}_{2}$ directions \[see Fig. \[figure1\](b)\] with respect to the Cr$_2$Ge$_2$Te$_6$ layer to make sure that this configuration is really minimum in energy. Moreover, we find that the energy differences among different stacking configurations are quite small; that is, there may be the possibility that the germanene layer is trapped in a local minimum rather than the groundstate while the growth process. Therefore, we calculated electronic properties of several stacking configurations in comparison with that of the equilibrium structure. The results are insensitive to different stacking formalisms, implying that topological features of the germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure are robust \[see Fig. S1 in the SM [@SM]\]. Given the equilibrium structure, the binding energy per Ge atom is calculated by $E_{\mathrm{b}}=(E_{\mathrm{Ge}}+E_{\mathrm{Cr}{_2}\mathrm{Ge}_{2}\mathrm{Te}{_6}}-E_{\mathrm{total}})/N$, where $E_{\mathrm{Ge}}$, $E_{\mathrm{Cr}{_2}\mathrm{Ge}_{2}\mathrm{Te}{_6}}$, and $E_{\mathrm{total}}$ are the total energies of the isolated germanene layer, isolated Cr$_2$Ge$_2$Te$_6$ layer, and germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure, respectively; and $N=6$ is the number of Ge atoms in one unit cell. We obtain that the binding energy $E_{\mathrm{b}}$ is about 133 meV, indicating that the formation of this heterostructure further improves the thermodynamic stability.
![(a) Side and (b) top views of the vdW heterostructure germanene/Cr$_2$Ge$_2$Te$_6$ with the interlayer distance $d$. The hexagonal unit cell is marked by red-lines in (b), which is composed by $p$($\sqrt{3}\times \sqrt{3}$) unit cells of germanene and a $p$($1\times 1$) unit cell of Cr$_2$Ge$_2$Te$_6$. \[figure1\]](figure1.pdf)
![(a) The electronic band structure of $p$($\sqrt{3}\times \sqrt{3}$) germanene. The upper and lower insets are the enlarged views without and with SOC at the $\Gamma$ point, respectively. (b) The spin-polarized electronic band structure of the Cr$_2$Ge$_2$Te$_6$ monolayer in the absence of SOC. (c) The spin-polarized electronic band structure of the vdW germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure in the absence of SOC. The inset exhibits a quadratic nodal point at the $\Gamma$ point. In (b) and (c), the red and blue lines respectively represent the majority and minority spin channels. (d) The spin-resolved partial density of states in the absence of SOC. Ge1 denotes the Ge atoms in the germanene layer, and Ge2 denotes the Ge atoms in the Cr$_2$Ge$_2$Te$_6$ layer. \[figure2\]](figure2.pdf)
The electronic band structure of the pristine $p$($\sqrt{3}\times \sqrt{3}$) germanene is shown in Fig. \[figure2\](a). Due to the band folding, the bands at the K point in the case of a germanene unit cell are mapped to the $\Gamma$ point, resulting in a linear Dirac cone at the $\Gamma$ point in the absence of SOC. When the SOC effect is present, germanene is a $\mathbb{Z}_2$ QSH insulator with a nontrivial gap $\sim$23 meV, which agrees well with the previous result [@PhysRevLett.107.076802]. The spin-polarized electronic band structure of the Cr$_2$Ge$_2$Te$_6$ monolayer is depicted in Fig. \[figure2\](b). We can see that the valence band maximum and conduction band minimum are both contributed by the majority-spin channel, exhibiting a indirect gap. As a result, the Cr$_2$Ge$_2$Te$_6$ monolayer is a FM semiconductor with a saturation magnetic moment of $\sim$ 6 $\mu_{B}$ per unit cell [@Naturecrgete].
Next, we focus on the electronic properties of the vdW germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure. In Fig. \[figure2\](c), we show the spin-polarized electronic band structure along high-symmetry directions. The bands in the absence of SOC reveal that the Cr$_2$Ge$_2$Te$_6$ layer completely magnetizes the germanene layer. In this case, this heterostructure is a half-metal; that is, there is a semiconducting feature with a band gap $\sim$163 meV in the minority-spin channel and a semimetallic feature in the majority-spin channel. The one unit cell of the vdW heterostructure has a magnetic moment of $\sim$ 6.04 $\mu_{B}$, which is slightly larger than that $\sim$6 $\mu_{B}$ of the isolated Cr$_2$Ge$_2$Te$_6$ layer. In Fig. \[figure2\](d), we illustrate the spin-resolved partial density of states in the absence of SOC. It is found that the states around the Fermi level are mainly contributed by the majority-spin channel of Ge1 $p$ orbitals of germanene. Besides, there are also a small amounts of Cr $d$ and Ge2 $p$ orbitals near the Fermi level, implying the presence of hybridization and charge transfer between the germanene and Cr$_2$Ge$_2$Te$_6$ layers. Therefore, the electronic properties of germanene are changed due to the formation of this vdW heterostructure. To determine the behavior of low-energy excitations, we plot the enlarged view of band structures around the $\Gamma$ point \[see the inset of Fig. \[figure2\](c)\]. It is found that there is a nodal point at the $\Gamma$ point. This nodal point is about $\sim$ 7 meV above the Fermi level. That is to say, the proximity makes lightly hole doped in germanene. Furthermore, the dispersion around this nodal point are quadratic, which is different from the linear Dirac Cone of the isolated germanene.
In order to insightfully understand the quadratic dispersions around the $\Gamma$ point, we construct a $2\times 2$ $k\cdot p$ effective Hamiltonian to describe the two crossing bands as $$\label{eqH}
\mathcal{H}(\mathbf{k})=f(\mathbf{k})\sigma_{+}+f(\mathbf{k})^{*}\sigma_{-},$$ where $\mathbf{k}$ is the wave vector referenced to the $\Gamma$ point, $f(\mathbf{q})$ is a complex function, $\sigma_{\pm}= \sigma_x \pm i\sigma_y$, and $\sigma_{i}$ ($i=x$, $y$) are the Pauli matrices. Here, we ignore the the kinetic term in Eq. (\[eqH\]). In the absence of SOC, there are only majority-spin states across the Fermi level. The decoupling between the spin and orbital species indicates that the vdW germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure can be considered as a spinless ferromagnet. In this case, all crystalline symmetries as well as the $\mathcal{T}$-symmetry are preserved for each spin channel. At the $\mathcal{T}$-invariant $\Gamma$ point, the $k\cdot p$ Hamiltonian is subjected to the product of $C_{3}$ little group and $\mathcal{T}$, which constrains Eq. (\[eqH\]) as $$\label{HconT}
{C}_{3}\mathcal{T}\mathcal{H}(\mathbf{k})\mathcal{T}^{-1}{C}_{3}^{-1}=\mathcal{H}[(\mathbf{R}_{3}\mathcal{T})\mathbf{k}],$$ where $\mathbf{R}_{3}$ is a $2\times 2$ rotational matrix of $C_3$. In the spinless-like system, we have $\mathcal{T}^2=1$ and thus the $\mathcal{T}$-operator can be represented as $\mathcal{T}=K$ with a complex conjugate operator $K$. As a result, the constraint of Eq. (\[HconT\]) gives (see the SM [@SM]) $$\label{dft}
e^{i2\pi/3}d(k_{+}, k_{-})=f(k_{+}e^{-i\pi/3}, k_{-}e^{i\pi/3}),$$ where $k_{\pm}=k_x\pm ik_{y}$. Then, we can expand Eq. (\[dft\]) and obtain the symmetry-allowed expressions to the lowest orders as $$\label{AG}
f(\mathbf{k})=a_{+}k_+^{2}+a_{-}k_-^{2},$$ which uncovers that low-energy excitations around the $\Gamma$ point are indeed quadratic in the $k_x$-$k_y$ plane.
In the presence of SOC, the coupling between the spin and orbital species causes spontaneous breaking of spin rotational symmetry, and thus the $\mathcal{T}$-symmetry is also removed. In this case, the magnetic space group strongly influences the topological features. To determine the magnetization direction, we have carried out total energy calculations with different magnetic configurations, including magnetization along out-of-plane and in-plane directions. Our results confirm that the easy axis is perpendicular to the 2D plane, which agrees well with the magnetic groundstate properties of Cr$_2$Ge$_2$Te$_6$ in experiments [@Naturecrgete]. The electronic band structures with magnetization perpendicular to the 2D plane are shown in Figs. \[figure3\](a) and \[figure3\](b). The figures show that the vdW germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure converts into a insulator with a sizable band gap of 29 meV. It is worth noting that this band gap is larger than that of germanene. Since the Cr $3d$ and Te $5p$ orbitals host the greater SOC strength than the Ge $4p$ states, and thus the interlayer coupling and hybridization can enlarge the band gap. To confirm whether this band gap opened by SOC is nontrivial, we calculate the Berry curvature as implemented in the Wannier90 code [@Mostofi2008]. In Fig. \[figure3\](b), the calculated Berry curvature, illustrated as the red-dots, shows a peak near the $\Gamma$ point. We integrate the Berry curvature over the states below the Fermi level, and obtain the nonzero Chern number $\mathcal{C=}-1$. This implies that the QAH state is present in the vdW germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure. We also calculate the evolution of Wannier charge centers (WCCs) to further verify the nontrivial properties. The WCCs are calculated using Z2PACK package[@PhysRevB.95.075146], in which the Wilson loop method is employed [@Yu2011]. As shown in Fig. \[figure3\](c), we can see that the referenced line can always cross the WCC line once, confirming that this system is indeed a nontrivial QAH insulator. In addition, the energy differences between out-of-plane and in-plane magnetic configurations are smaller than $\sim$1 meV. This implies that there may be the possibility of the topological phase transition between the QAH and 2D Weyl semimetallic states by applying an external field [@PhysRevB.100.064408].
Due to the weak interaction of vdW heterostructures, the interlayer distance can be easily reduced by a vertical external pressure. The applied external pressure at the interlayer distance $d$ is calculated by $P=\delta E/(S \delta d)$, where $\delta E=E(d)-E(d_{0})$ represents the energy difference between the compressed $E(d)$ and equilibrium $E(d_{0})$ structures, $\delta d = |d-d_0|$, and $S$ is the area per unit cell of the heterostructure. The reduction of the interlayer distance can enhance the hybridization between the germanene layer and the Cr$_2$Ge$_2$Te$_6$ layer, which may further enlarge the nontrivial band gap. The corresponding electronic band structures without and with SOC are given in the SM [@SM]. As expected, we can see that the nontrivial band gaps are monofonically increased with reducing the interlayer distance in Fig. \[figure3\](e). When the vertical external pressure is 1.8 GPa (i.e., $\delta d= 0.6$ [Å]{}), the nontrivial band gap can be considerably up to $\sim$72 meV \[see Figs. \[figure3\](d) and \[figure3\](e)\]. Besides, we also find that the germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure will transform to a trivial phase when $\delta d>0.7$ [Å]{}.
![(a) The electronic band structure of the vdW germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure with magnetization perpendicular to the 2D plane. (b) The enlarged view of (a) near the $\Gamma$ point, showing a nontrivial band gap of $\sim$29 meV. The calculated Berry curvature illustrated as the red-circles. (c) Evolution of WCCs as a function of $k_y$. (d) Nontrivial band gap as a function of reduction of interlayer distance. (e) External pressure as a function of reduction of interlayer distance. \[figure3\]](figure3.pdf)
The QAH insulator with the nonzero Chern number $\mathcal{C}=-1$ possesses the chiral edge states and quantized Hall conductance. To illustrate these topologically protected features, we construct a tight-binding (TB) Hamiltonian based on maximally localized Wannier functions methods [@Mostofi2008]. The local density of states (LDOS) are calculated by the iterative Green’s method [@Sancho1984; @WU2017], and the anomalous Hall conductance are calculated from the Kubo formula with the TB Hamiltonian. The LDOS of a semi-infinite ribbon of the germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure and Hall conductance are respectively shown in Fig. \[figure4\](a) and \[figure4\](b). As expected, the chiral edge states connecting the valence and conduction bands are visible. The intrinsic Hall conductance $\sigma_{xy}$ is exactly quantized to $-e^2/h$ inside the nontrivial band gap.
![(a) The calculated LDOS of a semi-infinite ribbon of the germanene/Cr$_2$Ge$_2$Te$_6$ heterostructure. (b) The intrinsic Hall conductance inside the nontrivial band gap exactly quantized to $-e^2/h$. \[figure4\]](figure4.pdf)
In conclusion, we demonstrate by using first-principles calculations and effective model analysis that the robustly intrinsic QAH effect can be realized in the vdW heterostructure germanene/Cr$_2$Ge$_2$Te$_6$. We show that hybridization and charge transfer occur between the germanene and Cr$_2$Ge$_2$Te$_6$ layers. The proximity effect gives rise to a quadratic nodal point in band structures in the absence of SOC. When SOC is considered and the magnetization is aligned along the easy axis, the quadratic nodal point is opened, driving this heterostructure to be an QAH insulator with a sizable band gap of 29 meV. In addition, we also find that the nontrivial band gap can be tunable up to 72 meV by applying an external pressure. The chiral edge states and quantized Hall conductance have been investigated, which can be easily detectable. Considering that Cr$_2$Ge$_2$Te$_6$ is a nearly ideal 2D ferromagnet, the vdW heterostructure germanene/Cr$_2$Ge$_2$Te$_6$ with high thermodynamic stability is expected to its synthesis in the laboratory. Therefore, our work provides a reliable platform to observe intrinsic QAH effects and design topological devices in futures.
This work was supported by the National Natural Science Foundation of China (NSFC, Grants No. 11974062, No. 11704177, and 11947406), the Chongqing National Natural Science Foundation (Grants No. cstc2019jcyj-msxmX0563), and the Fundamental Research Funds for the Central Universities of China (Grant No. 2019CDXYWL0029)\
\
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---
abstract: |
We investigate and compare experimental and numerical excitonic spectra of the yellow series in cuprous oxide Cu$_2$O in the Voigt configuration and thus partially extend the results from Schweiner *et al.* \[Phys. Rev. B **95**, 035202 (2017)\], who only considered the Faraday configuration. The main difference between the configurations is given by an additional effective electric field in the Voigt configuration, caused by the motion of the exciton through the magnetic field. This Magneto-Stark effect was already postulated by Gross *et al.* and Thomas *et al.* in 1961 \[Sov. Phys. Solid State **3**, 221 (1961); Phys. Rev. **124**, 657 (1961)\]. Group theoretical considerations show that the field most of all significantly increases the number of allowed lines by decreasing the symmetry of the system. This conclusion is supported by both the experimental and numerical data.\
author:
- Patric Rommel
- Frank Schweiner
- Jörg Main
- Julian Heckötter
- Marcel Freitag
- Dietmar Fröhlich
- Kevin Lehninger
- 'Marc A[ß]{}mann'
- Manfred Bayer
title: 'Magneto-Stark-effect of yellow excitons in cuprous oxide'
---
[^1]
Introduction
============
Application of electric or magnetic fields, both representing well-controlled external perturbations, has offered detailed insight into quantum mechanical systems by inducing characteristic shifts and lifting degeneracies of energy levels. This potential became apparent already for the simplest quantum mechanical system in nature, the hydrogen atom, for which an external field reduces the symmetry from spherical to cylindrical [@Fri89; @Has89; @ruder1994atoms].
In condensed matter, the hydrogen model is often applied as a simple, but nevertheless successful model for problems in which Coulomb interaction mediates the coupling between opposite charges of quasi-particles [@WannierExcitonen1937; @knox1963excitons]. The most prominent example is the exciton, the bound complex of a negatively charged electron and a positively charged hole. In particular, for crystals of high symmetry such as cubic systems, the phenomenology of energy levels in an external field often resembles the corresponding hydrogen spectra after rescaling the energy axis with the corresponding material parameters that enter the Rydberg energy such as the effective mass and the dielectric constant which in these systems are isotropic.
Recent high resolution spectroscopy of excited exciton states in Cu$_2$O has allowed one to reveal features which result from the crystal environment having discrete instead of continuous symmetry, leading to deviations from a simple hydrogen description [@GiantRydbergExcitons; @ObservationHighAngularMomentumExcitons; @Heckoetter2017]. Theoretical investigations of excitons in external fields [@Sylwia2016; @Sylwia2017; @Kurz2017] also show differences to a simple hydrogen model. For example, the level degeneracy is lifted already at zero field. A detailed analysis shows that this splitting arises mostly from the complex valence band structure [@SchoeneLuttinger; @frankmagnetoexcitonscuprousoxide; @ImpactValence; @Bauer1988]. In effect, angular momentum is not a good quantum number anymore, but states of different angular momenta having the same parity are mixed. As the exciton size is large compared to that of the crystal unit cell, the mixing is weak, however, so that angular momentum may still be used approximately as quantum number. Simply speaking, this can be understood in the following way: The exciton wave function “averages” over the crystal lattice, assembled by the cubic unit cells. With increasing wave function extension a state becomes less and less sensitive to the arrangement of the atoms in a crystal unit cell and thus to the deviations from spherical symmetry.
For example, the size of the cubic unit cell in Cu$_2$O is about 0.4 nm [@korzhavyi2011literature], while the Bohr radius is [$a_{\mathrm{exc}} =$]{} $ 1.1\,\mathrm{nm}$ for the dipole-active $P$-type excitons dominating the absorption [@KavoulakisBohrRadius]. [Using the formula $\langle r \rangle_{n,L} = \frac{1}{2} a_{\mathrm{exc}} (3 n^2 - L(L+1))$ [@GiantRydbergExcitons],]{} the size of the lowest exciton $n=4$, for which the mixing becomes optically accessible, is [$\langle r\rangle_{4,1} = 25.4\,\mathrm{nm}$]{}, covering thousands of unit cells. As a consequence of the weak mixing, the state splitting is small compared to the separation between levels of different $n$. Generally, the average radius of the wavefunction increases as $n^2$, so that the zero-field state splitting decreases with increasing principal quantum number, leading to the levels becoming quasi-degenerate again for $n$ larger than about 10, as demonstrated in Ref. [@ObservationHighAngularMomentumExcitons]. For smaller principal quantum numbers, the splitting can be well resolved, in particular, because the mixing of levels also causes a redistribution of oscillator strength, so that not only the $P$-states but also higher lying odd states such as $F, H, \ldots$ within a multiplett can be resolved in single photon absorption.
Applying a magnetic field leads to a squeezing of the wave function mostly normal to the field, so that the influence of the crystal is enhanced. Even when assuming spatial isotropy, the symmetry is reduced to rotational invariance about the field, leaving only the magnetic quantum number $m$ as conserved quantity. As a consequence, states of the same $m$ but different orbital angular momentum become mixed, activating further lines optically. In combination with the Zeeman splitting of the levels this leads to a rich appearance of the absorption spectra, in particular, as for excitons due to their renormalized Rydberg energy being much smaller than in the atomic case, also resonances between states of different principal quantum numbers can be induced by the field application, at least for excited excitons.
As indicated above, a key ingredient for the renormalized Rydberg energy are the smaller masses of the involved particles forming an exciton, in particular of the hole compared to the nucleus. In Cu$_2$O, the electron mass is almost that of the free electron ($m_{\mathrm{e}} = 0.99 m_0$ [@HodbyEffectiveMasses]), while the hole represents the lighter particle ($m_{\mathrm{h}} = 0.58 m_0$ [@HodbyEffectiveMasses]), in contrast to most other semiconductors. This makes the reduced mass entering the relative motion of electron and hole only a factor of less than 3 smaller than in hydrogen. The mass of the exciton center of mass motion, on the other hand, is clearly more than three orders of magnitude smaller. This raises the question whether this mass [disparity]{} causes a difference of the optical spectra in magnetic field, as the excitons are generated with a finite wavevector identical to the wavevector of the exciting laser.
In that respect two different field configurations need to be distinguished, namely the Faraday and the Voigt configuration. In the first case the magnetic field is applied along the optical axis, while in the second case the field is oriented normal to the optical axis. [One would not expect a difference in the spectra between the two cases when probing atomic systems using a fixed polarization. ]{}
Here we have performed corresponding experiments which demonstrate that in contrast to the atomic case the exciton spectra differ significantly for the two field configurations. These findings are in good agreement with detailed theoretical calculations. From the comparison we trace the difference to the Magneto-Stark-effect [@Gross61; @Hopfield61; @Thomas61] which is acting only in the Voigt configuration, where the excited exciton is moving normal to the magnetic field, so that its two constituents are subject to the Lorentz force acting in opposite direction for electron and hole and therefore trying to move them apart, similar to the action of an electric field of the form $$\begin{aligned}
\mathcal{F}_{\mathrm{MSE}} = \frac{\hbar}{M} \left( \boldsymbol{K} \times \boldsymbol{B} \right),
\label{eq:MotionalStarkField}\end{aligned}$$ where $M$ is the exciton center-of-mass, $\boldsymbol{K}$ is the exciton wavevector, and $\boldsymbol{B}$ is the magnetic field. The action of this field is obviously absent in the Faraday configuration.
The main novelty in this work comes from the comparison of the spectra for atoms and for excitons. One would not expect any difference in the spectra for atoms when the magnetic field is aligned parallel or normal to the magnetic field. For excitons, on the other hand, it does make a difference as we demonstrate here. Further, we identify the origin of this difference as the Magneto-Stark-Effect (MSE), which does not show up prominently for atoms due to their heavy mass and small size. The MSE has been introduced theoretically already in the 1960s (Refs. [@Gross61; @Hopfield61; @Thomas61; @Zhilich1969]) but clear demonstrations of its effect are still scarce [@Lafrentz2013].
The symmetry reduction by a longitudinal field (electric or magnetic) was demonstrated also in previous work. Still, in atomic physics, but also in semiconductor physics, it is common believe that for a bulk crystal there should not be a difference between the two configurations as the symmetry reduction by the field is the same, considering only the impact of the magnetic field. The additional symmetry breaking here comes from the optical excitation, which in Faraday-configuration is parallel to the field, leading to no further symmetry reduction (as studied previously), while in Voigt the optical axis is normal to it. This lifts virtually all symmetries, as evidenced by the observation of basically all possible optical transitions which therefore represents also a point of novelty.
In this respect we want to emphasize that the difference between the two field configurations that has been reported for confined semiconductor quantum structures has a different origin, as the geometric confinement breaks the spatial isotropy already at zero field. When applying a field, the energy spectrum of the free particles depends strongly on the field orientation [@landwehr1991landau]: For example, in a quantum well application of the field normal to the quantum well leads to a full discretization of the energy levels [@Maan1984; @Bauer1988; @Bauer1988Magnetoexcitons], while application in the well plane leaves the carrier motion along the field free (see Ref. [@Bayer1995] and further references therein). Here one has a competition between the magnetic and the geometric confinement normal to the field, while for normal orientation there is only the magnetic confinement normal to the field. This causes an intrinsic difference of the magneto-optical spectra in terms of state energies. The same is true to other quantum structures like quantum dots for which the geometric confinement in most cases is spatially anisotropic, while for the bulk the energy spectrum is independent of the field orientation [@Wang1996]. We also note that in confined semiconductor quantum structures like quantum wells the difference in the spectra of the Faraday and the Voigt configuration depends on the difference in the number of degrees of freedoms involved in the interaction between the excitons and the magnetic field, i.e., two for Faraday and one for Voigt configuration [@Czajkowski2003].
However, also for quantum structures one can indeed use the different field orientations to vary the number of observed optical transitions, somewhat similar to the case studied here. For example, in quantum wells or flat quantum dots the field orientation normal to the structure leaves the rotational in-plane symmetry about the field unchanged, while this symmetry is broken for application in the structural plane, so that one can mix excitons of different angular moments and make dark excitons visible [@Bayer2000].
Hamiltonian \[sec:Hamiltonian\]
===============================
Our theoretical description of excitonic spectra in $\mathrm{Cu_{2}O}$ with an external magnetic field builds upon Schweiner *et al.*’s treatment in Ref. [@frankmagnetoexcitonscuprousoxide], where only the Faraday configuration was considered. The Hamiltonian without magnetic field is given by $$\begin{aligned}
H = E_{\mathrm{g}} + H_{\mathrm{e}}\left(\boldsymbol{p}_{\mathrm{e}}\right)+H_{\mathrm{h}}\left(\boldsymbol{p}_h\right) + V\left(\boldsymbol{r}_{\mathrm{e}} - \boldsymbol{r}_{\mathrm{h}}\right),
\label{eq:Hamiltonian}\end{aligned}$$ where $H_{\mathrm{e}}$ and $H_{\mathrm{h}}$ are the kinetic energies of the electron and hole, respectively. They are given by $$H_{\mathrm{e}}\left(\boldsymbol{p}_{\mathrm{e}}\right)=\frac{\boldsymbol{p}_{\mathrm{e}}^{2}}{2m_{\mathrm{e}}}
\label{eq:ElectronKinetic}$$ and $$\begin{aligned}
H_{\mathrm{h}}\left(\boldsymbol{p}_{\mathrm{h}}\right) & = & H_{\mathrm{so}}+\left(1/2\hbar^{2}m_{0}\right)\left\{ \hbar^{2}\left(\gamma_{1}+4\gamma_{2}\right)\boldsymbol{p}_{\mathrm{h}}^{2}\right.\phantom{\frac{1}{1}}\nonumber \\
& + & 2\left(\eta_{1}+2\eta_{2}\right)\boldsymbol{p}_{\mathrm{h}}^{2}\left(\boldsymbol{I}\cdot\boldsymbol{S}_{\mathrm{h}}\right)\phantom{\frac{1}{1}}\nonumber \\
& - & 6\gamma_{2}\left(p_{\mathrm{h}1}^{2}\boldsymbol{I}_{1}^{2}+\mathrm{c.p.}\right)-12\eta_{2}\left(p_{\mathrm{h}1}^{2}\boldsymbol{I}_{1}\boldsymbol{S}_{\mathrm{h}1}+\mathrm{c.p.}\right)\phantom{\frac{1}{1}}\nonumber \\
& - & 12\gamma_{3}\left(\left\{ p_{\mathrm{h}1},p_{\mathrm{h}2}\right\} \left\{ \boldsymbol{I}_{1},\boldsymbol{I}_{2}\right\} +\mathrm{c.p.}\right)\phantom{\frac{1}{1}}\nonumber \\
& - & \left.12\eta_{3}\left(\left\{ p_{\mathrm{h}1},p_{\mathrm{h}2}\right\} \left(\boldsymbol{I}_{1}\boldsymbol{S}_{\mathrm{h}2}+\boldsymbol{I}_{2}\boldsymbol{S}_{\mathrm{h}1}\right)+\mathrm{c.p.}\right)\right\} \phantom{\frac{1}{1}}
\label{eq:HoleKinetic}\end{aligned}$$ with the spin-orbit interaction $$H_{\mathrm{so}}=\frac{2}{3}\Delta\left(1+\frac{1}{\hbar^{2}}\boldsymbol{I}\cdot\boldsymbol{S}_{\mathrm{h}}\right).$$ Here, $\boldsymbol{I}$ is the quasi-spin and $\boldsymbol{S}_{\mathrm{h}}$ the spin $S_{\mathrm{h}}=\frac{1}{2}$ of the hole and c.p. denotes cyclic permutation. Electron and hole interact via the screened Coulomb potential $$V\left(\boldsymbol{r}_{\mathrm{e}} - \boldsymbol{r}_{\mathrm{h}}\right) =
-\frac{e^2}{4\pi\varepsilon_0 \varepsilon \left|\boldsymbol{r}_{\mathrm{e}} - \boldsymbol{r}_{\mathrm{h}} \right|} ,$$ with the dielectric constant $\varepsilon$. To account for the magnetic field $\boldsymbol{B}$, we use the minimal substitution $\boldsymbol{p}_{\mathrm{e}} \rightarrow \boldsymbol{p_{\mathrm{e}}} + e \boldsymbol{A}(\boldsymbol{r}_e)$ and $\boldsymbol{p}_{\mathrm{h}} \rightarrow \boldsymbol{p}_{\mathrm{h}} - e \boldsymbol{A}(\boldsymbol{r}_h)$ with the vector potential for a homogenous field $\boldsymbol{A}(\boldsymbol{r}_{\mathrm{e,h}}) = \left(\boldsymbol{B} \times \boldsymbol{r}_{\mathrm{e,h}} \right)/2$. The energy gained by the electron and hole spin in the external magnetic field is described by $$H_{B}=\mu_{\mathrm{B}}\left[g_{c}\boldsymbol{S}_{\mathrm{e}}+\left(3\kappa+g_{s}/2\right)\boldsymbol{I}-g_{s}\boldsymbol{S}_{\mathrm{h}}\right]\cdot\boldsymbol{B}/\hbar.$$ with the Bohr magneton $\upmu_B$ and the g-factor of the hole spin $g_s \approx 2$. We finally switch into the center of mass reference frame [@Schmelcher1992]: $$\begin{aligned}
\boldsymbol{r} &= \boldsymbol{r}_{\mathrm{e}} - \boldsymbol{r}_{\mathrm{h}},\nonumber\\
\boldsymbol{R} &= \frac{m_{\mathrm{e}}}{m_{\mathrm{e}} + m_{\mathrm{h}}} \boldsymbol{r}_{\mathrm{e}} + \frac{m_{\mathrm{h}}}{m_{\mathrm{e}} + m_{\mathrm{h}}} \boldsymbol{r}_{\mathrm{h}},\nonumber\\
\boldsymbol{p} &= \hbar \boldsymbol{k} - \frac{e}{2} \boldsymbol{B} \times \boldsymbol{R} = \frac{m_h}{m_{\mathrm{e}} + m_{\mathrm{h}}} \boldsymbol{p}_{\mathrm{e}} - \frac{m_{\mathrm{e}}}{m_{\mathrm{e}} + m_{\mathrm{h}}} \boldsymbol{p}_{\mathrm{h}}, \nonumber\\
\boldsymbol{P} &= \hbar \boldsymbol{K} + \frac{e}{2} \boldsymbol{B} \times \boldsymbol{r} = \boldsymbol{p}_{\mathrm{e}} + \boldsymbol{p}_{\mathrm{h}},
\label{eq:CentorOfMass}\end{aligned}$$ and set $\boldsymbol{R} = 0$. More details can be found in Refs. [@frankmagnetoexcitonscuprousoxide; @franklinewidth; @frankevenexcitonseries; @Luttinger52CyclotronResonanceSemiconductors] and values of material parameters for Cu$_2$O used in Eqs. - are listed in Table \[tab:Constants\].
---------------------------- --------------------------------------- -------------------------------------------
band gap energy $E_{\mathrm{g}}=2.17208\,\mathrm{eV}$ [@GiantRydbergExcitons]
electron mass $m_{\mathrm{e}}=0.99\, m_{0}$ [@HodbyEffectiveMasses]
hole mass $m_{\mathrm{h}} = 0.58\, m_0$ [@HodbyEffectiveMasses]
dielectric constant $\varepsilon=7.5$ [@LandoltBornstein1998DielectricConstant]
spin-orbit coupling $\Delta=0.131\,\mathrm{eV}$ [@SchoeneLuttinger]
valence band parameters $\gamma_{1}=1.76$ [@SchoeneLuttinger]
$\gamma_{2}=0.7532$ [@SchoeneLuttinger]
$\gamma_{3}=-0.3668$ [@SchoeneLuttinger]
$\eta_{1}=-0.020$ [@SchoeneLuttinger]
$\eta_{2}=-0.0037$ [@SchoeneLuttinger]
$\eta_{3}=-0.0337$ [@SchoeneLuttinger]
fourth Luttinger parameter $\kappa = -0.5$ [@frankmagnetoexcitonscuprousoxide]
g-factor of cond. band $g_{\mathrm{c}}=2.1$ [@ArtyukhinGFactor]
\[tab:Constants\]
---------------------------- --------------------------------------- -------------------------------------------
: Material parameters used in Eqs. -.
Faraday and Voigt configuration, Magneto-Stark-effect
-----------------------------------------------------
We consider two different relative orientations of the magnetic field to the optical axis. In the Faraday configuration, both axes are aligned to be parallel, whereas in the Voigt configuration, they are orthogonal to each other. Generally, the exciting laser will transfer a finite momentum $\hbar
\boldsymbol{K}$ onto the exciton. This center of mass momentum would have to be added in the terms for the kinetic energies. Even without a magnetic field, this leads to quite complicated formulas (cf. the expressions for the Hamiltonian in the supplemental material of Ref. [@frankjanpolariton]) which are further complicated by the minimal substitution. Since the effect of many of the arising terms is presumably negligible due to the smallness of $K$, we simplify the problem and only consider the leading term [@frankmagnetoexcitonsbreak; @ruder1994atoms] $$H_{\mathrm{ms}} = \frac{\hbar e}{M} (\boldsymbol{K} \times \boldsymbol{B})\cdot \boldsymbol{r}
\label{eq:Magneto-Stark-term}$$ in our numerical calculations, which is the well-known motional Stark effect term of the hydrogen atom. This term has the same effect as an external electric field perpendicular to the plane spanned by the wavevector $\boldsymbol{K}$ and the magnetic field vector $\boldsymbol{B}$. Evidently then, the significance of this term depends on the used configuration.
For the Faraday configuration, the effective electrical field vanishes. A previous investigation of Schweiner *et al.* [@frankmagnetoexcitonscuprousoxide] was thus conducted under the approximation of vanishing cent[e]{}r of mass momentum. They report a complicated splitting pattern where the magnetic field lifts all degeneracies. For a magnetic field oriented along one of the high symmetry axes of the crystal, the symmetry of the exciton is reduced from $O_{\mathrm{h}}$ to $C_{4\mathrm{h}}$. Still, some selection rules remain, and not all lines become dipole-allowed. Parity remains a good quantum number and since only states with an admixture of P states have nonvanishing oscillator strengths, only states with odd values of L contribute to the exciton spectrum.
In the Voigt configuration on the other hand, the excitons have a nonvanishing momentum perpendicular to the magnetic field and the Magneto-Stark term has to be included. For our calculations, we therefore include an electric field, the size of which is given by the wavevector $K_0 = 2.79 \times 10^7 \frac{1}{\mathrm{m}}$ of the incident light and the magnetic field. This value is obtained by the condition $$\frac{\hbar c K_0}{\sqrt{\varepsilon_{b2}}} = E_g - \frac{R_{\mathrm{exc}}}{n^2}$$ for $n=5$ and with $\varepsilon_{b2} = 6.46$ [@LandoltBornstein1998DielectricConstant] and $R_{\mathrm{exc}} =
86 \, \mathrm{meV}$ [@SchoeneLuttinger], i.e., $\hbar K_0$ is the momentum of a photon that has the appropriate energy to create an exciton in the energy range we consider. [Note that, in contrast to Table \[tab:Constants\], we here use the dielectric constant in the high frequency limit to describe the refractive index of the incident light.]{} Since the total mass $M$ of the exciton is some three orders of magnitudes smaller than for a hydrogen atom, this term will have a significant effect on the spectra, even more so if we consider that the region of high fields is shifted to much lower values for the exciton [@frankmagnetoexcitonscuprousoxide]. The term breaks the inversion symmetry and parity ceases to be a good quantum number. While in the Faraday configuration only the dipole-allowed exciton states of odd angluar momentum have been important, now also the states of even angluar momentum need to be considered. Hence, we need to include the terms for the central cell corrections with the Haken potential as given in Refs. [@frankevenexcitonseries; @frankjanpolariton] in our treatment to correctly take the coupling to the low lying S states into account.
[In general, polariton effects have to be considered when the center of mass momentum $\boldsymbol{K}$ is nonzero. The experimental results in Refs. [@PhysRevLett.91.107401; @doi:10.1002/pssc.200460331; @PhysRevB.70.045206] on the other hand show, that the polariton effects for the $1S$ state are of the order of $10\, \upmu \mathrm{eV}$ and thus small in comparison with the effects considered in this paper. Furthermore, a recent discussion by Stolz *et al.* [@1367-2630-20-2-023019] concluded that polariton effects should only be observable in transmission experiments for $n \geq 28$. Hence, we will not include them in our discussion.]{}
Numerical approach
==================
Using the Hamiltonian with the additional terms for the central cell corrections $H_{\mathrm{CCC}}$ and the Magneto-Stark effect $H_{\mathrm{ms}}$ with a suitable set of basis vectors, the Schrödinger equation can be brought into the form of a generalized eigenvalue equation $$\boldsymbol{D} \boldsymbol{c} = E \boldsymbol{M} \boldsymbol{c}.
\label{eq:GeneralizedEigenvalueEquation}$$ We choose a basis consisting of Coulomb-Sturmian functions with an appropriate part for the various appearing spins and angular momenta. Due to the broken inversion symmetry, it is not sufficient to include only basis functions of odd parity as in reference [@frankmagnetoexcitonscuprousoxide]. Instead, basis functions of even symmetry have to be included as well. The resulting equation can then be solved using a suitable LAPACK routine [@lapackuserguide3]. For details we refer to the discussions in Refs. [@frankmagnetoexcitonscuprousoxide; @frankevenexcitonseries; @PaperMotionalStark].
Oscillator strengths
--------------------
The extraction of the dipole oscillator strengths is performed analogously to the calculation for the Faraday configuration [@frankmagnetoexcitonscuprousoxide]. For the relative oscillator strengths we use $$f_{\mathrm{rel}}\sim\left|\lim_{r\rightarrow0}\frac{\partial}{\partial r}\left\langle \pi_{x,z}\middle|\Psi\left(\boldsymbol{r}\right)\right\rangle\right|^2\label{eq:frel}$$ for light linearly polarized in $x$- or $z$-direction. The states $\left| \pi_{x,z} \right\rangle$ are given by
$$\begin{aligned}
\left|\pi_x\right\rangle= &\; \frac{i}{\sqrt{2}}\left[\left|2,\,-1\right\rangle_D+\left|2,\,1\right\rangle_D\right],\\
\left|\pi_z\right\rangle= &\; \frac{i}{\sqrt{2}}\left[\left|2,\,-2\right\rangle_D-\left|2,\,2\right\rangle_D\right],\end{aligned}$$
\[eq:Dxz\]
where $\left|F_t,\,M_{F_t}\right\rangle_D$ is an abbreviation [@frankmagnetoexcitonscuprousoxide] for $$\begin{aligned}
& & \left|\left(S_{\mathrm{e}},\,S_{\mathrm{h}}\right)\,S,\,I;\,I+S,\,L;\,F_t,\,M_{F_t}\right\rangle\nonumber\\
& = & \left|\left(1/2,\,1/2\right)\,0,\,1;\,1,\,1;\,F_t,\,M_{F_t}\right\rangle.\end{aligned}$$ In this state, the electron and hole spin $S_{\mathrm{e}}$ and $S_{\mathrm{h}}$ are coupled to the total spin $S$. This is combined first with the quasispin $I$ and then with the orbital angular momentum $L$ to obtain the total angular momentum $F_t$. $M_{F_t}$ is the projection onto the axis of quantization.
Experiment
==========
In the experiment, we investigated the absorption of thin Cu$_2$O crystal slabs. Three different samples with different orientations were available: In the first sample the \[001\] direction is normal to the crystal surface, in the other two samples the normal direction corresponds to the \[110\] and \[111\] orientation, respectively. The thicknesses of these samples differed slightly from 30 to $50 \, \upmu \mathrm{m}$ which is, however, of no relevance for the results described below. For application of a magnetic field, the samples were inserted at a temperature of $1.4\,\mathrm{K}$ in an optical cryostat with a superconducting split coil magnet. Magnetic fields with strengths up to $7\,\mathrm{T}$ could be applied with orientation either parallel to the optical axis (Faraday configuration) or normal to the optical axis (Voigt configuration).
The absorption was measured using a white light source which was filtered by a double monochromator such that only the range of energies in which the exciton states of interest are located was covered. A linear polarization of the exciting light, hitting the crystal normal to the slabs, was used. The transmitted light was dispersed by another double monochromator and detected by a liquid-nitrogen cooled charge coupled device camera, providing a spectral resolution of about $10\,\upmu$eV. Since the spectral width of the studied exciton resonances is significantly larger than this value, the setup provides sufficient resolution.
Results and discussion\[sub:field\]
===================================
![[Experimental transmission spectra in arbitrary units for $n=4$ to $n=7$ taken in Voigt configuration for polarization (a) orthogonal \[010\] and (b) parallel \[100\] to the magnetic field.]{} \[fig:Voigt\_Senkrecht\_Parallel\_Experiment\]](Fig1.pdf){width="1.02\columnwidth"}
![[Second derivative of experimental transmission spectra for $n=4$ to $n=7$ taken in (a) Voigt configuration and (b) Faraday configuration with polarization orthogonal \[010\] to the magnetic field. Data for the Faraday configuration were obtained by combining $\sigma^+$- and $\sigma^-$-polarized spectra from Ref. [@frankmagnetoexcitonscuprousoxide] in an appropriate linear combination. We use the second derivative for better visibility of weak lines.]{} \[fig:Voigt\_Faraday\_Experiment\]](Fig2.pdf){width="1.02\columnwidth"}
[Figure \[fig:Voigt\_Senkrecht\_Parallel\_Experiment\] shows experimental spectra for $n=4$ to $n=7$ in Voigt configuration with polarization orthogonal and parallel to the magnetic field respectively. The spectra for the two cases show clear differences due to the different selection rules for different polarizations. We will show in Sec. \[subsec:MagnetoStarkGroupTheory\] that all lines in principle become dipole allowed and can be excited by exactly one of the two polarizations shown here.]{}
![[Comparison between numerical and experimental line positions for the Voigt configuration with light polarized orthogonally \[010\] to the magnetic field. (a) Numerical data in greyscale with read out experimental line positions (blue triangles) and (b) experimental data using the second derivative to enhance visibility of weak lines. Note that the resolution of the numerical data is not uniform for all field strengths. \[fig:VoigtXTheoExpconv\]]{} ](Fig3.pdf){width="1.02\columnwidth"}
![[Comparison between numerical and experimental line positions for the Voigt configuration with light polarized parallely \[100\] to the magnetic field. (a) Numerical data in greyscale with read out experimental line positions (blue triangles) and (b) experimental data using the second derivative to enhance visibility of weak lines. We increase the visibility of the experimental 4D line by using a different filter width and higher contrast. Note that the resolution of the numerical data is not uniform for all field strengths. \[fig:VoigtZTheoExpconv\]]{} ](Fig4.pdf){width="1.02\columnwidth"}
[For the comparison between the Faraday and Voigt configuration we show in Fig. \[fig:Voigt\_Faraday\_Experiment\] experimental spectra taken (a) in Voigt configuration and (b) in Faraday configuration with polarization orthogonal to the magnetic field respectively. The polarizations are chosen in such a way that the same selection rules would apply to both spectra in Fig. \[fig:Voigt\_Faraday\_Experiment\] without the Magneto-Stark field. Thus, the differences between them must be due to the different geometries. S-lines are visible for both configurations. This can be attributed to quadrupole-allowed transitions in the case of the Faraday configuration [@frankmagnetoexcitonscuprousoxide]. For the Voigt configuration, these lines quickly fade away. This is a sign that the additional mixing from the electric field transfers quadrupole oscillator strength away from the S excitons. This effect is not reproduced in the numerical spectra since we only extracted dipole oscillator strengths. In general, the effective electric field lifts selection rules, revealing additional lines not visible in the Faraday configuration. This can for example clearly be seen for the $n=5$ states. ]{}
![Comparison of numerical and experimental spectra of the $n=4$ and $n=5$ excitons in an external magnetic field $\mathbf{B}$ $\parallel$ \[100\]. (a) [Faraday configuration with light polarized along the \[010\] direction. Data were obtained by combining $\sigma^+$- and $\sigma^-$-polarized spectra from Ref. [@frankmagnetoexcitonscuprousoxide] in an appropriate linear combination.]{} (b) and (c) Voigt configuration with a wavevector aligned with the \[001\] direction and the light polarized (b) orthogonally [\[010\]]{} and (c) parallely [\[100\]]{} to the magnetic field. Numerically calculated relative oscillator strengths are shown in grayscale in arbitrary units. Experimentally measured absorption coefficients $\alpha$ are superimposed in arbitrary units for a few selected values of B (blue solid lines). Note that the resolution of the numerical data is not uniform for all field strengths. We point out the theoretical visibility of S- and D-excitons as marked in (c). [See text for further information.]{} \[fig:VoigtSenkrechtParallelFaraday\]](Fig5.pdf){width="1.02\columnwidth"}
[In Figs. \[fig:VoigtXTheoExpconv\] and \[fig:VoigtZTheoExpconv\] we show a comparison between experimental and numerically obtained line positions for $n=4$ and $n=5$.]{} [To improve the presentation of areas with many densely lying lines that individually have very low oscillator strengths, numerical spectra are convoluted using a Gaussian function with a constant width of $13.6$ $\, \upmu
\mathrm{eV}$. This value is of the same order of magnitude as the width of the sharpest lines visible in the experiment.]{} [While the position of the P- and F-lines is reproduced very well, noticeable disagreement is observed for the S-lines and also the faint 4D-line visible in Fig. \[fig:VoigtZTheoExpconv\]. Since our model is not explicitely constructed on a lattice [@Alvermann2018], we have to include the centrel cell corrections as an approximation into our Hamiltonian. As the centrel cell corrections influence the even parity states much more strongly than the odd parity states, the error involved in this is more pronounced for the former than for the latter. A similar effect can also be seen in Ref. [@frankevenexcitonseries].]{} [To make additional comparision involving the oscillator strengths possible]{} we [also]{} present in Fig. \[fig:VoigtSenkrechtParallelFaraday\] (a) [data with light linearly polarized orthogonally \[010\] to the magnetic field]{} in Faraday configuration taken from Ref. [@frankmagnetoexcitonscuprousoxide] and in (b) and (c) spectra in the Voigt configuration with light polarized orthogonally [\[010\]]{} and parallely [\[100\]]{} to the magnetic field axis, respectively, for the principal quantum numbers $n=4$ and $n=5$. [ The experimental absorption coefficients do not fall to zero far away from the peaks due to phonon background. We lowered the values with a constant shift to counteract this effect.]{} Note that we investigate a parameter region where the effects of quantum chaos as discussed in Ref. [@AszmannGUE2016] are not important.
In general, a good agreement between the experimental and numerical data sets is obtained. In the Voigt configuration in [Figs. \[fig:VoigtXTheoExpconv\], \[fig:VoigtZTheoExpconv\] and \[fig:VoigtSenkrechtParallelFaraday\]]{}, a rich splitting is observed, especially of the F-states of the $n=5$ excitons. We see that light polarized orthogonally to the magnetic field probes complementary lines to the ones excited by light polarized in the direction of the field, a result that will also follow from our discussion below.
In experiment, we are not able to resolve the multiplicity of lines that the calculations reveal. This is related to the increased linewidth of the individual features arising from exciton relaxation by radiative decay and phonon scattering that are not included in the model. Still the field dependences of the main peaks with largest oscillator strength are nicely reproduced as are the broadenings of the multiplets due to level splitting.
Influence of the Magneto-Stark effect {#subsec:MagnetoStarkGroupTheory}
-------------------------------------
In this section we want to discuss the effects of the additional effective electric field on the line spectra. As we will see in the following group theoretical derivations, the most pronounced effect is a significant increase in the number of dipole-allowed lines due to the decreased symmetry with the electric field. Panels (a) and (b) in Fig. \[fig:VoigtSenkrechtParallelFaraday\] show this quite clearly, especially for the large number of additional F-lines and also G-lines for $n=5$ in the Voigt configuration. [This is most obvious for the theoretical spectra, but can also distinctly be seen in the experiment for $n=5$.]{} Note that without the Magneto-Stark effect the same selection rules would apply to the spectra in (a) and (b), but not in (c). In contrast to the Faraday configuration [@frankmagnetoexcitonscuprousoxide], we can not limit ourselves to the states with odd values for L, owing to the mixture of the even and odd series in the electric field. We discuss the case of a magnetic field aligned in \[001\] direction and will disregard the influence of the central cell corrections in this discussion.
We consider the reduction of the irreducible representations $\tilde{D}^{F \pm}$ of the full rotation group in the presence of the crystal as well as the magnetic and effective electric field, where $F = J + L = (I + S_{\mathrm{h}}) + L$ is the angular momentum without the electron spin. [Here, the quasispin $I$ and hole spin $S_{\mathrm{h}}$ are first coupled to the effective hole spin $J$ and then combined with the orbital angular momentum $L$ to form $F$.]{} With this information we will be able to deduce the splitting of the lines due to the reduced symmetry [@abragam2012electron]. Additionally we can compare the resulting irreducible representations with those that the dipole operator belongs to. This will tell us which lines are dipole-allowed and which are not. Note that the symmetry of the quasispin $I$ in $O_{\mathrm{h}}$ is given by $\Gamma^+_5 = \Gamma^+_4 \otimes \Gamma^+_2$ [@frankmagnetoexcitonscuprousoxide] and therefore all irreducible representations have to be multiplied by $\Gamma^+_2$ in comparison with the case of an ordinary spin. Keeping this in mind, we have [@koster1963properties]
$$\begin{aligned}
L=0:\hspace{1.7cm}\nonumber\\
\tilde{D}^{\frac{1}{2}+}=D^{\frac{1}{2}+}\otimes\Gamma_{2}^{+} &\: =\Gamma_{6}^{+}\otimes\Gamma_{2}^{+}=\Gamma_{7}^{+},\\
\displaybreak[1]
\nonumber \\
L=1:\hspace{1.7cm}\nonumber\\
\tilde{D}^{\frac{1}{2}-}=D^{\frac{1}{2}-}\otimes\Gamma_{2}^{+} &\: =\Gamma_{6}^{-}\otimes\Gamma_{2}^{+}=\Gamma_{7}^{-},\\
\displaybreak[1]
\nonumber \\
\tilde{D}^{\frac{3}{2}-}=D^{\frac{3}{2}-}\otimes\Gamma_{2}^{+} &\: =\Gamma_{8}^{-}\otimes\Gamma_{2}^{+}=\Gamma_{8}^{-},\\
\displaybreak[1]
\nonumber \\
L=2:\hspace{1.7cm}\nonumber\\
\tilde{D}^{\frac{3}{2}+}=D^{\frac{3}{2}+}\otimes\Gamma_{2}^{+} &\: =\Gamma_{8}^{+}\otimes\Gamma_{2}^{+}=\Gamma_{8}^{+},\\
\displaybreak[1]
\nonumber \\
\tilde{D}^{\frac{5}{2}+}=D^{\frac{5}{2}+}\otimes\Gamma_{2}^{+} &\: =\left(\Gamma_{7}^{+}\oplus\Gamma_{8}^{+}\right)\otimes\Gamma_{2}^{+}\nonumber \\
&\: =\Gamma_{6}^{+}\oplus\Gamma_{8}^{+},\\
\displaybreak[1]
\nonumber \\
L=3:\hspace{1.7cm}\nonumber\\
\tilde{D}^{\frac{5}{2}-}=D^{\frac{5}{2}-}\otimes\Gamma_{2}^{+} &\: =\left(\Gamma_{7}^{-}\oplus\Gamma_{8}^{-}\right)\otimes\Gamma_{2}^{+}\nonumber \\
&\: =\Gamma_{6}^{-}\oplus\Gamma_{8}^{-},\\
\displaybreak[1]
\nonumber \\
\tilde{D}^{\frac{7}{2}-}=D^{\frac{7}{2}-}\otimes\Gamma_{2}^{+} &\: =\left(\Gamma_{6}^{-}\oplus\Gamma_{7}^{-}\oplus\Gamma_{8}^{-}\right)\otimes\Gamma_{2}^{+}\nonumber \\
&\: =\Gamma_{7}^{-}\oplus\Gamma_{6}^{-}\oplus\Gamma_{8}^{-},\\
\displaybreak[1]
\nonumber \\
L=4:\hspace{1.7cm}\nonumber\\
\tilde{D}^{\frac{7}{2}+}=D^{\frac{7}{2}+}\otimes\Gamma_{2}^{+} &\: =\left(\Gamma_{6}^{+}\oplus\Gamma_{7}^{+}\oplus\Gamma_{8}^{+}\right)\otimes\Gamma_{2}^{+}\nonumber \\
&\: =\Gamma_{7}^{+}\oplus\Gamma_{6}^{+}\oplus\Gamma_{8}^{+},\\
\displaybreak[1]
\nonumber \\
\tilde{D}^{\frac{9}{2}+}=D^{\frac{9}{2}+}\otimes\Gamma_{2}^{+} &\: =\left(\Gamma_{6}^{+}\oplus\Gamma_{8}^{+}\oplus\Gamma_{8}^{+}\right)\otimes\Gamma_{2}^{+}\nonumber\\
&\: = \Gamma_{7}^{+}\oplus\Gamma_{8}^{+}\oplus\Gamma_{8}^{+}.\end{aligned}$$
We still need to include the spin of the electron which transforms according to $\Gamma^+_6$. For vanishing magnetic field strengths, the representations belonging to an irreducible representation without the spin are degenerate. Those will be written in brackets. The reduction [@koster1963properties] will only be specified for even parity, since the odd case only changes the sign. We obtain
$$\begin{aligned}
\tilde{D}^{\frac{1}{2}+} \otimes \Gamma^+_6 &= (\Gamma^+_2 \oplus \Gamma^+_5),\\
\displaybreak[1] \nonumber\\
\tilde{D}^{\frac{3}{2}+} \otimes \Gamma^+_6 &= (\Gamma^+_3 \oplus \Gamma^+_4 \oplus \Gamma^+_5),\\
\displaybreak[1] \nonumber\\
\tilde{D}^{\frac{5}{2}+} \otimes \Gamma^+_6 &= (\Gamma^+_1 \oplus \Gamma^+_4) \oplus (\Gamma^+_3 \oplus \Gamma^+_4 \oplus \Gamma^+_5),\\
\displaybreak[1] \nonumber\\
\tilde{D}^{\frac{7}{2}+} \otimes \Gamma^+_6 &= (\Gamma^+_2 \oplus \Gamma^+_5) \oplus (\Gamma^+_1 \oplus \Gamma^+_4)\\
&\phantom{=}\oplus (\Gamma^+_3 \oplus \Gamma^+_4 \oplus \Gamma^+_5),\nonumber\\
\displaybreak[1] \nonumber\\
\tilde{D}^{\frac{9}{2}+} \otimes \Gamma^+_6 &= (\Gamma^+_2 \oplus \Gamma^+_5) \oplus (\Gamma^+_3 \oplus \Gamma^+_4 \oplus \Gamma^+_5)\\
&\phantom{=}\oplus (\Gamma^+_3 \oplus \Gamma^+_4 \oplus \Gamma^+_5).\nonumber\end{aligned}$$
$\Gamma^+_1$ and $\Gamma^+_2$ are one-dimensional, $\Gamma^+_3$ is two-dimensional, and $\Gamma^+_4$ and $\Gamma^+_5$ are three-dimensional. So without the field, we have for example fourfold degenerate S-states and P-states that are split into one fourfold and one eightfold degenerate line. If the magnetic field is switched on, the electric field becomes nonvanishing too. The symmetry is reduced from $O_{\mathrm{h}}$ to $C_{\mathrm{S}}$ [@koster1963properties]. All representations of $C_{\mathrm{S}}$ are one-dimensional, so all degeneracies will be lifted, just as in the case with only a magnetic field. But in contrast to the Faraday configuration, the symmetry is lowered even further, leading to a greater mixture of the states. In fact, all lines become dipole-allowed. To see this, we have to consider the reduction of the irreducible representations of $O_{\mathrm{h}}$ in $C_{\mathrm{S}}$ [@koster1963properties; @abragam2012electron]. The relevent expressions are
$$\begin{aligned}
&\Gamma_{1}^{+} \rightarrow \: \Gamma_{1},& &\Gamma_{1}^{-} \rightarrow \: \Gamma_{2},\\
&\Gamma_{2}^{+} \rightarrow \: \Gamma_{1},& &\Gamma_{2}^{-} \rightarrow \: \Gamma_{2},\\
&\Gamma_{3}^{+} \rightarrow \: \Gamma_{1}\oplus\Gamma_{1},& &\Gamma_{3}^{-} \rightarrow \: \Gamma_{2}\oplus\Gamma_{2},\\
&\Gamma_{4}^{+} \rightarrow \: \Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{2},& &\Gamma_{4}^{-} \rightarrow \: \Gamma_{2}\oplus\Gamma_{1}\oplus\Gamma_{1},\\
&\Gamma_{5}^{+} \rightarrow \: \Gamma_{1}\oplus\Gamma_{2}\oplus\Gamma_{2},& &\Gamma_{5}^{-} \rightarrow \: \Gamma_{2}\oplus\Gamma_{1}\oplus\Gamma_{1}.\end{aligned}$$
The dipole operator belongs to $\Gamma^-_4$ in $O_{\mathrm{h}}$ [@koster1963properties] and its reduction therefore includes all appearing representations. Thus, all $4n^2$ lines receive nonvanishing oscillator strength, the only limitation being given by the polarization of the incident light, i.e., a given state can either be excited by radiation polarized in the $z$-direction ($\Gamma_{2}$) or by radiation polarized in the $x$-$y$-plane ($\Gamma_{1}$).
Summary\[sec:Summary\]
======================
We extended the previous work by Schweiner *et al.*[@frankmagnetoexcitonscuprousoxide] on the optical spectra of magnetoexcitons in cuprous oxide to the Voigt configuration and showed that the nonvanishing exciton momentum perpendicular to the magnetic field leads to the appearance of an effective Magneto-Stark field. Including the valance band structure and taking into account central cell corrections as well as the Haken potential allowed us to produce numerical results in good agreement with experimental absorption spectra. We observe a significant increase in the number of visible lines in both our experimental as well as our numerical data as compared to the Faraday configuration. Using group theoretical methods, we show that this is related to the Magneto-Stark field increasing the mixing between states. While their positions remain relatively unaffected, the mixing of states leads to finite oscillator strength of, at least in principle, all lines.
This work was supported by Deutsche Forschungsgemeinschaft (Grants No. MA1639/13-1 and No. AS459/3-1).
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[^1]: Present address: MINT-Kolleg, Universität Stuttgart
|
---
author:
- Christopher John Jerdonek
title: The Girth of a Heegaard Splitting
---
The Girth of a Heegaard Splitting
\
By\
CHRISTOPHER JOHN JERDONEK,\
A.B. (Harvard University) 1998,\
a DISSERTATION\
submitted in partial satisfaction of the requirements for the degree of\
DOCTOR OF PHILOSOPHY\
in\
MATHEMATICS\
in the\
OFFICE OF GRADUATE STUDIES\
of the\
UNIVERSITY OF CALIFORNIA,\
DAVIS.\
Approved:\
William P. Thurston\
------------------------------------------------------------------------
\
Michael Kapovich\
------------------------------------------------------------------------
\
Greg Kuperberg\
------------------------------------------------------------------------
\
Joel Hass\
------------------------------------------------------------------------
\
Abigail Thompson\
------------------------------------------------------------------------
\
Committee in Charge\
2005\
Abstract {#abstract .unnumbered}
========
We construct simple curves from immersed curves in the setting of handlebodies and Heegaard splittings. We define a measure of complexity we call *girth* for closed curves in a handlebody. We extend this complexity to Heegaard splittings and pose a conjecture about all Heegaard splittings. We prove a test case of this conjecture. Let $S$ be a compact surface embedded in the boundary of a handlebody $H$. Then the minimum girth over all curves in $S$ can be achieved by a simple closed curve. We also present algorithms to compute the girth of curves and surfaces.
Acknowledgments {#acknowledgments .unnumbered}
===============
Introduction
============
Handlebodies, Trees, and Girth {#trees}
==============================
Free Groups and Girth {#free groups}
=====================
Girth of Simple Closed Curves {#handlebodygirth}
=============================
Girth of Heegaard Splittings {#heegaardgirth}
============================
Discs and $\dd$-Interfaces {#subordination}
==========================
Computing the Girth of a Conjugacy Class {#curvealgorithm}
========================================
\[computecurve\]
Main Theorem {#maintheorem}
============
Main Lemma {#mainlemma}
==========
Computing the Girth of a Subsurface {#surfacealgorithm}
===================================
\[computesurface\]
|
---
abstract: |
Labeling schemes seek to assign a short label to each node in a network, so that a function on two nodes (such as distance or adjacency) can be computed by examining their labels alone. For the particular case of trees, following a long line of research, optimal bounds (up to low order terms) were recently obtained for adjacency labeling \[FOCS ’15\], nearest common ancestor labeling \[SODA ’14\], and ancestry labeling \[SICOMP ’06\]. In this paper we obtain optimal bounds for distance labeling. We present labels of size $1/4\log^2n+o(\log^2n)$, matching (up to low order terms) the recent $1/4\log^2n-{{O}}(\log n)$ lower bound \[ICALP ’16\].
Prior to our work, all distance labeling schemes for trees could be reinterpreted as [*universal trees*]{}. A tree $T$ is said to be universal if any tree on $n$ nodes can be found as a subtree of $T$. A universal tree with $|T|$ nodes implies a distance labeling scheme with label size $\log |T|$. In 1981, Chung et al. proved that any distance labeling scheme based on universal trees requires labels of size $1/2\log^2 n -\log n \cdot \log\log n+{{O}}(\log n)$. Our scheme is the first to break this lower bound, showing a separation between distance labeling and universal trees.\
The $\Theta (\log^2 n)$ barrier for distance labeling in trees has led researchers to consider distances bounded by $k$. The size of such labels was improved from $\log n+{{O}}(k\sqrt{\log n})$ \[WADS ’01\] to $\log n+{{O}}(k^2(\log(k\log n))$ \[SODA ’03\] and finally to $\log n+{{O}}(k\log(k\log(n/k)))$ \[PODC ’07\]. We show how to construct labels whose size is the minimum between $\log n+{{O}}(k\log((\log n)/k))$ and ${{O}}(\log n \cdot \log(k/\log n))$. We complement this with almost tight lower bounds of $\log n+\Omega(k\log(\log n / (k\log k)))$ and $\Omega(\log n \cdot \log(k/\log n))$. Finally, we consider $(1+{\varepsilon})$-approximate distances. We show that the recent labeling scheme of \[ICALP ’16\] can be easily modified to obtain an ${{O}}(\log(1/{\varepsilon})\cdot \log n)$ upper bound and we prove a matching $\Omega(\log(1/{\varepsilon})\cdot \log n)$ lower bound.
author:
- 'Ofer Freedman[^1]'
- Paweł Gawrychowski
- 'Patrick K. Nicholson'
- Oren Weimann
bibliography:
- 'main.bib'
title: Optimal Distance Labeling Schemes for Trees
---
Introduction
============
Labeling schemes seek to assign a short label to each vertex in a network, so that a function on two nodes (such as distance or adjacency) can be computed by examining their labels alone. This is particularly desirable in distributed settings, where nodes are often processed using only some locally stored data. Recently, with the rise in popularity of distributed computing platforms such as Spark and Hadoop, labeling schemes have found renewed interest. Indeed, the goal of minimizing the size of the maximal label has been the subject of a great deal of recent research [@alstrup2005labeling; @abiteboul2006compact; @fischer2009short; @fraigniaud2010compact; @alstrup2014near; @alstrup2015adjacency; @alstrup2015optimal; @petersen2015near; @alstrup2015distance; @alstrup2016simpler]. For the particular case of trees, the functions that have been studied are distance [@peleg2000proximity; @gavoille2004distance; @alstrup2005labeling; @alstrup2015distance; @gavoille2007distributed], adjacency [@alstrup2015optimal; @alstrup2002small; @bonichon2007short], nearest common ancestor [@fischer2009short; @alstrup2014near], and ancestry [@abiteboul2006compact; @fraigniaud2010compact] (a recent survey of these results can be found here [@rotbart2016new]). Tree labeling schemes have recently found new uses in large scale graph processing. For example, distance oracles for general graphs use distance labelings for spanning trees rooted at judiciously chosen vertices [@akiba2012treewidth; @akiba2013distance; @ajwani2015oracle].
#### Universal trees.
A particularly clean way of looking at labeling schemes is through *universal graphs*. A graph $G$ is said to be [*universal*]{} for a given family of graphs, if every graph in the family is an induced subgraph of $G$. Similarly, a tree $T$ is said to be universal for [*all*]{} trees on $n$ nodes if any tree on $n$ nodes can be found as a subtree of $T$. For [*adjacency labeling in graphs*]{}, Kannan et al. [@Kannan] observed that if a family of graphs has a universal graph with $|G|$ vertices then it has an adjacency labeling scheme with label size $\log |G|$, and vice versa. For [*distance labeling in trees*]{}, until the present work, this statement was only known to be true in one direction. Namely, a universal tree $T$ of all trees on $n$ nodes implies a distance labeling scheme with label size $\log |T|$. We prove that the converse is in fact not true.
The use of universal trees is powerful, but it is limited. Already 50 years ago, Goldberg and Livshits [@gol1968minimal] showed how to construct a universal tree $T$ that is of size $|T| = n^{(\log n - 2\log \log n + {{O}}(1))/2}$ which was shown by Chung et al. [@graham1981trees] to be the smallest possible up to the ${{O}}(1)$ error term. This shows the first limitation of using universal trees for distance labeling: there is a lower bound of $\log |T| = 1/2\log^2 n -\log n \cdot \log\log n+{{O}}(\log n)$ on the label size. The second limitation is the query time. The universal tree construction of Goldberg and Livshits was given before labeling schemes were ever invented. Of course, one could naively use their universal tree $T$ for distance labeling of an arbitrary tree on $n$ nodes by finding its isomorphic subtree in $T$ and assigning labels which are just the IDs of the nodes in $T$. However, such a non-algorithmic labeling would require prohibitive query time and space since $T$ needs to be computed. This latter limitation was overcome by algorithmic labeling schemes achieving logarithmic query time: An upper bound of ${{O}}(\log^2 n)$ bits on the label size was first shown by Peleg [@peleg2000proximity] and a lower bound of $1/8 \log^2n - {{O}}(\log n)$ bits was shown by Gavoille et al. [@gavoille2004distance]. Very recently, Alstrup et al. [@alstrup2015distance] improved the lower bound to $1 / 4 \log^2 n - {{O}}(\log n)$ and observed that the upper bound can be improved to $1/2 \log^2 n + {{O}}(\log n)$ with a somewhat straightforward use of a nearest common ancestor labeling scheme.
All the above labeling schemes can be reinterpreted as building a universal tree, and are therefore subject to the $1/2\log^2 n -\log n \cdot \log\log n+{{O}}(\log n)$ lower bound of Chung et al. In other words, the scheme of Alstrup et al. is optimal (up to low order terms) amongst all schemes that translate to universal trees. To see why the scheme of Alstrup et al. indeed translates to a universal tree, we show in \[sec:lvl-anc\] that their scheme can be casted as a level-ancestry scheme and we show in \[sec:lb\] that every level-ancestry scheme translates to a universal tree.
We give the first distance labeling scheme that does not translate to a universal tree. This enables us to circumvent the Chung et al. [@graham1981trees] lower bound for labels based on universal trees and to match the general lower bound of Alstrup et al. [@alstrup2015distance]. Namely, in \[sec:ub\] we prove the following:
\[thm:ub-tree-dist\] There is a scheme for tree distance labeling with $1/4 \log^2 n + o(\log^2 n)$ bit labels and constant query time.
The above theorem means that universal trees capture more than is required for distance labeling. To illustrate this, we need to describe the related problem of [*level-ancestor labelings*]{}.
#### Labeling schemes for level-ancestors.
In this problem, we are given a rooted tree and seek to assign labels so that we can compute (the label of) any $k$-th ancestor of a node from its label alone. Notice that here a query receives a single label and a value $k$, and that all labels must be distinct (no scheme which uses the same label twice can be correct).
It is not hard to see that labels supporting level-ancestor queries can be used to answer distance queries. Thus, any lower bound for tree distance labeling immediately applies to level-ancestor labeling, but the converse is not true. Nevertheless, it turns out that all previous distance labeling schemes are also level-ancestor schemes. Like the labeling scheme of Alstrup et al. [@alstrup2015distance], our scheme is also based on a *heavy path decomposition* of the tree, which can be seen as a way of transforming an arbitrary tree into an edge-weighted tree of logarithmic depth. However, while the labels in [@alstrup2015distance] store the weights of every edge on the path to the root (thus allowing for level-ancestor queries), we show that it is possible to carefully distribute the bits between the labels so that the distance can be computed given any pair of labels, yet a single label is not enough to extract the level-ancestors.
We determine this separation between tree distance labeling and level-ancestor labeling by proving that labeling for distances is roughly half as expensive as labeling for level-ancestors:
\[thm:lb-lvl-anc\] Any scheme for level-ancestor labeling must use at least $1/2 \log^2 n - \log n \log \log n$ bits for the maximum length label.
We prove the above theorem in \[sec:lb\] by showing that, as opposed to distance labeling, no level-ancestor labeling scheme can do better than the one based on universal trees. Namely, we prove that any level-ancestor labeling scheme with labels of length $L$ implies a universal rooted tree of size ${{O}}(2^{L})$, and then invoke the known lower bound for universal trees [@graham1981trees; @gol1968minimal]. In particular, it means that for level-ancestor queries, the scheme of Alstrup et al. [@alstrup2015distance] is optimal (after some modifications described in \[sec:lvl-anc\]).
#### Labeling schemes for bounded distances.
The $\Theta(\log^2 n)$ barrier on distance labeling in trees has initiated a line of research that improves the label size when the distances are bounded: In [*$k$-distance labeling*]{}, we are given the labels of $u$ and $v$ and need to decide if the length of the $u$-to-$v$ path is at most $k$, and if so return it. For $k=1$, this is exactly adjacency labeling, which was recently shown by Alstrup et al. [@alstrup2015optimal] to require only $\log n+{{O}}(1)$ bits. For $k\ge 2$, this was first considered by Kaplan and Milo [@kaplan2001short] who showed how to construct labels of length $\log n+{{O}}(k\sqrt{\log n})$. The query time was not explicitly specified in their implementation, but appears to be ${{O}}(k)$. A shorter label of $\log n+{{O}}(k^2\log(k\log n))$ bits was then given by Alstrup, Bille, and Rauhe [@alstrup2005labeling] who also proved that any scheme for $k\geq 2$ (i.e., the scheme is able to answer “ancestor or sibling” queries) requires $\log n+\Omega(\log\log n)$ bits. Hence the ${{O}}(\log\log n)$ addend cannot be avoided, but it remained unclear what should be the exact dependency on $k$ nor the query time (Alstrup, Bille, and Rauhe considered constant $k$ in which case their bounds are tight and their $O(k^2)$ query time is constant). The labeling scheme of Alstrup, Bille, and Rauhe was then improved by Gavoille and Labourel [@gavoille2007distributed] who presented a bound of $\log n+{{O}}(k\log(k\log(n/k)))$ bits and ${{O}}(k)$ query time solution.
In \[sec:k\] we show how to construct a labeling scheme with improved label size and constant query time, and prove an almost matching lower bound. Formally, we prove:
\[thm:krestricted\] For $k < \log n$, there is a $k$-distance labeling scheme with labels of length $\log n+{{O}}(k\log((\log n)/k))$ bits, and any such scheme requires $\log n+\Omega(k\log(\log n/(k \log k))$ bits.\
For $k \ge \log n$, there is a $k$-distance labeling scheme with labels of length ${{O}}(\log n\cdot \log(k/\log n))$ bits, and any such scheme requires $\Omega(\log n\cdot \log(k/\log n))$ bits. In both cases, the query time is constant.
For the upper bound, our starting point is the scheme of Alstrup, Bille, and Rauhe [@alstrup2005labeling]. We observe that, instead of storing the same information for each of the nearest $k$ heavy paths above a node, it is possible to store all information for the topmost of these heavy paths and less information for all the rest. To improve the query time, we show that only a subtle change is needed in the definition of the so-called significant preorder numbers. The new definition retains all the nice properties of the previous while being much easier to operate on. Our constant query time assumes the standard word-RAM model with word size $\Omega(\log n)$.
For the lower bounds we take two different approaches. For $k < \log n$, we show how to construct a family of trees such that, in any $k$-distance labeling scheme, different trees can share some labels but every tree has to introduce many additional unique labels. For $k \ge \log n$, we use the clever lower bound technique from (unbounded) distance labelings, that was introduced by Gavoille et al. [@gavoille2004distance] and refined by Alstrup et al. [@alstrup2015distance]. It is based on constructing a *weighted* almost complete binary tree, where all the leaves are at the same distance from the root. After arguing that the labels of nodes in such a tree must be long, the weights are removed by subdividing edges while not increasing the size of the tree by too much. We show that only a small tweak is required to this known lower bound for distance labeling in order to get a lower bound for $k$-distance labeling.
#### Labeling schemes for approximate distances.
Finally, we consider [*$(1+{\varepsilon})$-approximate distance labeling*]{}, where given the labels of $u$ and $v$ we need to output a value that is at least ${\mathsf{d}}(u,v)$ and at most $(1+{\varepsilon})\cdot{\mathsf{d}}(u,v)$. For the case ${\varepsilon}\in [1/\log n, 1)$, Gavoille et al. [@GavoilleKKPP01] proved a tight bound of $\Theta(\log(1/{\varepsilon})\cdot\log n)$. Very recently, Alstrup et al. [@alstrup2015distance] considered the general trade-off and designed, for any constant ${\varepsilon}\leq 1$, an ${{O}}(\log n)$ bit labeling scheme. In \[sec:approx\] we show that their solution can be easily made to produce labels of size ${{O}}(\log(1/{\varepsilon})\cdot \log n)$ and that this is the best possible:
\[thm:approximate\] For any ${\varepsilon}\le 1$, there is a $(1+{\varepsilon})$-approximate distance labeling scheme with labels of length ${{O}}(\log(1/{\varepsilon})\cdot \log n)$, and any such scheme requires $\Omega(\log(1/{\varepsilon})\cdot \log n)$ bits.
The lower bound is obtained by reducing exact distance labeling to $(1+{\varepsilon})$-approximate distance labeling. This is achieved by appropriately stretching the lengths of the edges in the lower bound instances of Gavoille et al. [@gavoille2004distance]. For the upper bound, we slightly modify the scheme of Alstrup et al. [@alstrup2015distance], which originally stored a sequence of integers using simple unary encoding. Such an encoding requires ${{O}}(1/{\varepsilon}\cdot \log n)$ bits. We show that with a more complicated binary encoding we can obtain a scheme with ${{O}}(\log(1/{\varepsilon})\cdot \log n)$ bits and a constant query time.
We conclude this section with the following table summarizing our contribution.
-- -------------------------------------------------------------------------------------------------------------- -- --
**Upper bound & **Lower bound\
& $1/4\log^2 n + o(\log^2 n)$ & $1/4\log^2 n - O(\log n)$ [@alstrup2015distance]\
& ${{O}}(\log(1/{\varepsilon})\cdot \log n)$ & $\Omega(\log(1/{\varepsilon})\cdot \log n)$\
& $k \geq \log n$ & $O(\log n\cdot\log \frac {k} {\log n})$ & $\Omega(\log n\cdot\log \frac{k}{\log n})$\
& $k < \log n$ & $\log n + O(k\log \frac {\log n} {k})$ & $\log n + \Omega(k\log \frac {\log n} {k\log k})$\
****
-- -------------------------------------------------------------------------------------------------------------- -- --
This lower bound only holds for $k = o(\frac{\log n}{\log\log n})$.
Preliminaries {#sec:prelim}
=============
We consider a rooted tree $T$, or we arbitrarily root it. We denote the root by ${\mathsf{root}}(T)$, and the distance between node $v$ to ${\mathsf{root}}(T)$ by ${\mathsf{root}\text{-}\mathsf{distance}}(v)$. We denote the subtree rooted at $u$ as $T_u$, and the number of nodes of $T$ by ${\left| T \right|}$, or simply $n$ if $T$ is known from the context. For two nodes $u,v$, we denote their distance by ${\mathsf{d}}(u,v)$, their nearest common ancestor by ${\mathsf{NCA}}(u,v)$. First, observe that: $${\mathsf{d}}(u,v) = {\mathsf{root}\text{-}\mathsf{distance}}(u) + {\mathsf{root}\text{-}\mathsf{distance}}(v) - 2\cdot{\mathsf{root}\text{-}\mathsf{distance}}({\mathsf{NCA}}(u,v)).$$ This means that a labeling scheme for ${\mathsf{root}\text{-}\mathsf{distance}}({\mathsf{NCA}}(u,v))$ that assumes $u\ne v$ can actually be used for ${\mathsf{d}}(u,v)$ queries with only $O(\log n)$ additional bits to the label size (the additional bits are simply the distance to the root). Next, observe that although our input tree is unweighted (i.e., all edges have weight 1), if our distance labeling scheme can handle edges-weights in $\{0,1\}$ then we can assume the input tree is binary and that the queries are on leaves only. This can be achieved by connecting every internal node $u$ to a leaf node $u_\ell$ with an edge of weight 0, and then standardly binarizing the tree (by inserting ${{O}}(n)$ intermediate nodes with edge-weights 0 connecting them).
#### Heavy path decompositions.
We apply a variant of heavy path decompositions [@thorup2001compact]. We start at the root of the tree $T$ and repeatedly descend from the current node to its (unique) child whose subtree is of size at least $|T|/2$ as long as possible, that is, we terminate when there is no such child. Note that this is different than the more common versions in which we descend from the current node $u$ to its child $v$ with the largest subtree until (depending on the version) we reach a leaf or $|T_v| < |T_u|/2$. This gives us a *heavy path* $P$ starting at ${\mathsf{root}}(T)$ and many subtrees hanging off the heavy path. We call the edges of $P$ *heavy*, and all other edges outgoing from the nodes of $P$ *light*. The construction is then applied recursively to all subtrees hanging off the heavy path. In the end, each node $u\in T$ has at most one heavy child, denoted ${\mathsf{heavy}}(u)$, so we obtain a decomposition of $T$ into disjoint heavy paths (some of which consist of a single node). The light depth of a node $u\in T$, denoted ${\mathsf{lightdepth}}(u)$, is the number of light edges on the path from $u$ to the root and is at most $\log n$ [@sleator1983data]. We order the children of every node $u$ so that ${\mathsf{heavy}}(u)$ is the rightmost child and assign preorder numbers ${\mathsf{pre}}(u)$ to every node $u$. Then, for any node $v\in T_u$, we have that ${\mathsf{pre}}(v) \in [{\mathsf{pre}}(u), {\mathsf{pre}}(u)+{\left| T_u \right|})$.
#### The collapsed tree.
Given the heavy path decomposition of a [*binary*]{} tree $T$, we define its *collapsed tree*, denoted ${\mathcal{C}}(T)$, whose nodes correspond to heavy paths in $T$. The heavy path starting at ${\mathsf{root}}(T)$ corresponds to the root of the collapsed tree. Every light edge hanging off this heavy path corresponds to an edge outgoing from the root of ${\mathcal{C}}(T)$, and so on. The children of every node in ${\mathcal{C}}(T)$ are ordered according to the top-to-bottom order on the hanging subtrees (i.e, if two subtrees connect to the same heavy path $P$ then the one connecting at a lower depth is to the left of the other). Since $T$ is binary, ties can only happen at the last node of the heavy path $P$, in which case we set the right subtree to be the subtree of maximum size, and call the light edge branching to the right subtree the *exceptional* edge associated with heavy path $P$. See \[fig:heavy-path-decomp\] (right). Note that the height of the collapsed tree is at most $\log n$.
Every heavy path $P$ in $T$ is associated with a node $u'$ in ${\mathcal{C}}(T)$ and every node $u \in P$ is said to be *associated* with $u'$. We refer to the node $u \in P$ closest to the root of $T$ as the *head* of $P$ and denote it as ${\mathsf{head}}(P)$ or ${\mathsf{head}}(u')$. We use ${{\mathsf{lightdepth}}}(u,v)$ to denote ${\mathsf{lightdepth}}({\mathsf{NCA}}(u,v))$. Finally, we say that $u$ *dominates* $v$ if the inorder number of $u$’s associated node in ${\mathcal{C}}(T)$ is smaller than that of $v$’s. Observe that (1) If the ${\mathsf{NCA}}(u,v)$-to-$u$ path in $T$ starts with a light edge and the ${\mathsf{NCA}}(u,v)$-to-$v$ path starts with a heavy edge then $u$ dominates $v$, and (2) If both these paths start with a light edge then the dominated vertex is the one whose path starts with the exceptional edge.
#### Labeling schemes for NCA.
A nearest common ancestor scheme assigns a unique label to every node, so that given the labels of nodes $u,v$ we can return the label of ${\mathsf{NCA}}(u,v)$. Alstrup et al. [@alstrup2014near] design such a scheme with labels of length ${{O}}(\log n)$ bits. They use a heavy path decomposition that slightly differs from ours, but it can be easily verified that the following lemma still holds:
\[lem:NCA scheme\] There is an ${\mathsf{NCA}}$ labeling scheme with label size ${{O}}(\log n)$, which given the labels of $u$ and $v$ returns the label of ${\mathsf{NCA}}(u,v)$ as well as ${\mathsf{lightdepth}}(u,v)$ in constant time.
#### Encoding integers.
To store a single integer $x$, we use Elias $\delta$ codes [@elias1975universal] that require $\log x + {{O}}(\log \log x)$ bits. This encoding is [*self-delimiting*]{}, meaning we can concatenate multiple variable-length values into a single label in a way that each individual value can be decoded later. To store a monotone sequence of integers we use the following:
\[lem:encodingsequence\] A monotone sequence of $s$ integers in $[0,M]$ can be encoded with ${{O}}(s\cdot \max{\{ 1,\log\frac{M}{s} \}})$ bits, so that we can:
1. extract the $k^\text{th}$ number in the sequence,
2. find the position of the successor of a given integer in the sequence,
3. given the representation of two sequences, find the longest common suffix of two specified prefixes.
The first operation takes constant time, and the second and third take constant time if both $s$ and $M$ are ${{O}}(\log n)$.
Let the sequence be $0\leq x_1 \leq x_2 \leq \ldots \leq x_s \leq M$. The encoding consists of $x_1$ and the differences $x_2-x_1, x_3-x_2,\ldots,x_s-x_{s-1}$. Each number is encoded using the Elias $\gamma$ code, so the total size of the encoding becomes ${{O}}(s+\sum_{i=1}^s \log (x_i-x_{i-1}))$, where $x_0=0$. By Jensen’s inequality, this is maximized when all numbers are equal, so the total size of the encoding is $L={{O}}(s\cdot \max{\{ 1,\log\frac{M}{s} \}})$.
To provide constant time access to every $x_i$, we need to store some auxiliary data. We partition the universe $[0,M]$ into blocks of length $b=\frac{M}{s}$. For each $x_{i}$, we store $x_{i}\bmod b$. This is done by reserving $\lceil\log b\rceil$ bits for every $i=1,2,\ldots,s$ and arranging them one after another. We also store $b$ encoded using the Elias $\gamma$ code, so that in constant time we can calculate where the $\lceil\log b\rceil$ bits storing $x_{i}\bmod b$ are. This takes ${{O}}(\log b+s+s\log b)={{O}}(L)$ space so far. It remains to show how to encode $y_{i}=x_{i}{ \nonscript\mskip-\medmuskip\mkern5mu \mathbin{\operator@font div}\penalty900\mkern5mu \nonscript\mskip-\medmuskip
}b$. Notice that $0\leq y_{1}\leq y_{2}\leq\ldots\leq y_{s}\leq s$, so this is a monotone sequence of $s$ integers from $[0,s]$. We encode it with a single bit vector of length at most $2s$, which is the concatenation of $0^{y_{i}-y_{i-1}}1$ for $i=1,2,\ldots,s$ (and $y_{0}=0$). Then, to extract $y_{i}$ we need to find the position $p$ of the $i^\text{th}$ bit set to 1 in the bit vector and then return $p-i+1$. By augmenting the bit vector with a select structure of Clark [@Clark Chapter 2.2], which takes $o(s)$ additional bits of space, we can retrieve the $i^\text{th}$ bit set to 1 in constant time. Thus, in $O(L)$ additional space we can encode $x_{i}\bmod b$ and $x_{i}{ \nonscript\mskip-\medmuskip\mkern5mu \mathbin{\operator@font div}\penalty900\mkern5mu \nonscript\mskip-\medmuskip
}b$, and then recover $x_{i}$ in constant time.
To provide constant time successor queries (when both $s$ and $M$ are ${{O}}(\log n)$), we remove all duplicates and store the resulting sequence $y_1 < y_2 < \ldots < y_r$ in an additional predecessor structure from the second branch of Pǎtraşcu and Thorup [@PatrascuPredecessor]. This structure uses ${{O}}(r \cdot \log M)$ bits and answers queries in ${{O}}(\log\frac{\log M}{\log \log n})={{O}}(1)$ time. The space can be actually improved to ${{O}}(r \cdot \log \frac{M}{r}))={{O}}(s\cdot \max{\{ 1,\log\frac{M}{s} \}})$ as explained in detail by Belazzougui and Navarro [@DjamalSequences].
Finally, to compute the longest common suffixes of two specified prefixes given the encodings of $x_1 \leq x_2 \leq \ldots \leq x_s$ and $y_1 \leq y_2 \leq \ldots \leq y_s$, we observe that for $s,M={{O}}(\log n)$ the encodings fit in a constant number of machine words. Hence, we can first shift both encodings (in constant time) to reduce the problem to computing the longest common suffix. First, we check if $x_s=y_s$. If not, we are done. Otherwise, we only need to find the longest common suffix of the sequences of differences. This can be done by first calculating the longest common suffix of their encodings, and then counting how many differences have their encodings fully in the common suffix. The former can be done in constant time using the standard word-RAM operations. The latter can be done by storing an additional bit vector of length $L$, where we mark the starting position of the encoding of each $x_{i}$ with a bit set to 1. The bit vector is augmented with the rank structure of Jacobson [@Jacobson], which takes $o(L)$ additional bits and allows us to count bits set to 1 in any prefix in constant time.
#### $(h,M)$-trees.
To obtain a lower bound for distance labeling, Gavoille et al. [@gavoille2004distance] consider a family of rooted binary trees called $(h,M)$-trees. The trees are weighted and the weight of every edge is in $[0,M]$. For $h=0$ the tree is a single node. For $h\geq 1$, the tree consists of a root connected to its single child with an edge of length $M-x$ for some $x\in [0,M)$, and the child is connected to two (possibly different) $(h-1,M)$-trees with edges of length $x$. See \[fig:htree\]. A lower bound for tree distance labeling is implied by the following lemma:
![A $(3,M)$-tree, where $x_1,\ldots,x_7\in [0,M)$.[]{data-label="fig:htree"}](htree)
\[lem:tree\_lower\_bound\] For $h\geq 1$ and $M\geq 2$, any scheme for distance labeling in $(h,M)$-trees requires labels of at least $h/2 \cdot \log M$ bits, even if we only query leaves.
Distance Labeling {#sec:ub}
=================
In this section we prove \[thm:ub-tree-dist\]. In \[sec:review\] we review the labeling scheme framework of the existing solutions (in a slightly different way), and in \[sec:modified\] we describe our improved solution and its analysis.
Distance Arrays {#sec:review}
---------------
We now review the general framework for distance labeling. For each node $u \in T$, consider the set of light edges $\ell_1(u), \ldots, \ell_k(u)$ along the root-to-$u$ path. For any light edge $e$ in the collapsed tree ${\mathcal{C}}(T)$ branching from $u'$ to its child $v'$ let ${\mathsf{d}}(e) = {\mathsf{d}}({\mathsf{head}}(u'),{\mathsf{head}}(v'))$. That is, the distance along the heavy path represented by $u'$ to the endpoint where the light edge branches and to its other end. Let $D(u)$ denote the list $[ {\mathsf{d}}(\ell_1(u)), \ldots, {\mathsf{d}}(\ell_k(u)) ]$, which we call the *distance array* of $u$. The next lemma shows that designing an efficient distance labeling scheme boils down to efficiently encoding distance arrays.
\[lem:dist-arrays\] If we can access the elements of the distance arrays $D(u)$ and $D(v)$ then with additional ${{O}}(\log n)$ bits we can compute ${\mathsf{d}}(u,v)$.
We first describe the additional ${{O}}(\log n)$-bits. They are composed of:
1. \[list:exact\_root\_dist\] ${\mathsf{root}\text{-}\mathsf{distance}}(u)$,
2. \[list:exact\_nca\] the NCA label of $u$ generated by \[lem:NCA scheme\],
3. \[list:exact\_inorder\] the inorder number of the node $u'$ corresponding to $u$ in ${\mathcal{C}}(T)$ (so that given $u,v\in T$ we can determine which node dominates the other).
Now, suppose that $u$ is associated with $u' \in {\mathcal{C}}(T)$ and $v$ with $v' \in {\mathcal{C}}(T)$. We use (\[list:exact\_nca\]) to determine $j = {{\mathsf{lightdepth}}}(u,v) + 1$. We assume that the inorder number of $u'$ is smaller than that of $v'$ (using (\[list:exact\_inorder\]) we can verify this and swap $u$ with $v$ otherwise). Thus, $u$ dominates $v$, which implies that ${\mathsf{root}\text{-}\mathsf{distance}}({\mathsf{NCA}}(u,v))=\sum_{i=1}^{j} {\mathsf{d}}(\ell_i(u)) - 1$. Recall from \[sec:prelim\] that ${\mathsf{root}\text{-}\mathsf{distance}}({\mathsf{NCA}}(u,v))$ together with ${\mathsf{root}\text{-}\mathsf{distance}}(u)$ and ${\mathsf{root}\text{-}\mathsf{distance}}(v)$ suffice to compute ${\mathsf{d}}(u,v)$.
Modified Distance Arrays {#sec:modified}
------------------------
The main challenge remaining, is how to efficiently encode $D(u)$ for an arbitrary node $u$. This can clearly be done using $\Theta(\log^2 n)$ bits. By using properties of the heavy path decomposition, Alstrup et al. [@alstrup2015distance] gave a more precise bound of: $\sum_{i=1}^{k} \log {\mathsf{d}}(\ell_i(u)) = \sum_{i=1}^{\log n} \log(n/2^i) = 1/2\log^2 n + {{O}}(\log n).$ In their description, sums of the suffixes of $D(u)$ are stored instead of $D(u)$ itself, but this is essentially the same. Furthermore, distance arrays must be made self-delimiting by adding an additional ${{O}}(\log n \log \log n)$-bits, so we get an overall space bound of $1/2\log^2 n + {{O}}(\log n \log \log n)$.
In this section, we present an improved method and analysis for encoding the distance arrays. We show that our encoding uses less space, but in the process we lose the ability to compute the sum $\sum_{i=1}^{j} {\mathsf{d}}(\ell_i(u))$, which is used to answer the query. However, in \[sec:adjustment\] we show that in fact a query can still be answered by adding only a small amount of auxiliary information. Our *modified distance array* $\hat{D}(u)$ will have the following key property, which is weaker than that of the original distance array:
\[prop:mod\] Given the modified distance arrays $\hat{D}(u)$ and $\hat{D}(v)$ for leaves $u,v \in T$ such that $u$ dominates $v$, we can compute the value ${\mathsf{d}}(\ell_j(u))$ where $j = {{\mathsf{lightdepth}}}(u,v) + 1$.
At a high level, the main idea behind the modified distance array is that, to reduce the number of bits stored for each distance ${\mathsf{d}}(\ell_1(u)), \ldots, {\mathsf{d}}(\ell_k(u))$ at node $u$, we potentially push some of the bits to labels of nodes dominated by $u$. This is acceptable if our goal is to satisfy \[prop:mod\] since we need only compute ${\mathsf{d}}(\ell_i(u))$ if the other queried node $v$ is dominated by $u$. An important observation for the analysis later is that if $\ell_i(u)$ is [*exceptional*]{}, we need not store ${\mathsf{d}}(\ell_i(u))$ at all in order to satisfy \[prop:mod\]. The modified distance array consists of two parts:
1. a list of *truncated distances* $\hat{{\mathsf{d}}}(\ell_1(u)), \ldots, \hat{{\mathsf{d}}}(\ell_k(u))$;
2. a list of *accumulators* ${\mathsf{a}}(\ell_1(u)), \ldots, {\mathsf{a}}(\ell_k(u))$.
Accumulator ${\mathsf{a}}(\ell_i(u))$ will potentially (but not necessarily) store some of the bits of the distances ${\mathsf{d}}(\ell_i(v))$ where $v$ is a node that dominates $u$, and $i = {{\mathsf{lightdepth}}}(u,v) + 1$.
The construction of the labels is recursive: Consider the heavy path $P$ extending from the root of $T$. Let $n_1,\ldots,n_{m+1}$ be the sizes of the subtrees $T_1, \ldots, T_{m+1}$ hanging from $P$ via light edges $e_1, \ldots, e_{m+1}$, where $e_{m+1}$ is the *exceptional* edge. The edges $e_1, \ldots, e_{m+1}$ are ordered according to their left-to-right order in the collapsed tree, and we use $w_1, \ldots, w_m$ to denote the nodes in $P$ from which $e_1, \ldots, e_m$ branch ($e_{m+1}$ also branches from $w_m$). See \[fig:analysis\]. We use $n_1', \ldots, n_m'$ to denote the sizes of the subtrees $T_1', \ldots, T_{m}'$ rooted at nodes $w_1, \ldots, w_{m}$. For consistency, $n_{m+1}'$ denotes the size of $T_{m+1}'=T_{m+1}$. Note that, for an arbitrary node $u \in T_i$ we have that $\ell_1(u) = e_i$.
![\[fig:analysis\]A heavy path $P$ and the subtrees $T_1, \ldots, T_{m+1}$ and $T_1', \ldots, T_{m}'$. The edge $e_5$ is exceptional.](analysis.pdf)
For an arbitrary node $u \in T_i$ where $i \in [1,m]$, we assume that we have some encoding of its modified distance array excluding the encoding of ${\mathsf{d}}(e_i) = {\mathsf{d}}(\ell_1(u))$ using $(\log^2 n_i+\log n_i \log q)/4$ bits, where $q$ is a parameter to be fixed later. We call this encoding the *recursive problem*, and the problem of encoding ${\mathsf{d}}(\ell_1(u))$ the *top-level problem*. Recall that if $i=m+1$ (i.e., $u \in T_{m+1}$) we need not encode the distance ${\mathsf{d}}(e_{m+1})$, since that edge is *exceptional*.
We analyze the space of the top-level problem for $T'_i$ for $i =m,m-1,\ldots,1$ (i.e., from bottom to top), bounding the overall label size in terms of $n_i'$. The goal of each iteration is to produce labels for $T_i'$ of size $(\log^2 n_i' + \log n_i' \log q)/4$. Consider the labels generated in the recursive problem for nodes in $T_i$ and in the previous iteration for $T_{i+1}'$ (or, if $i=m$, in the recursive problem for $T_{m+1}$). The following two lemmas show how many bits we can spend to generate the labels for nodes in $T_i'$. Note that these lemmas ignore the cost of making the encoding self-delimiting, as well the fact that we must take the ceiling of the bound because we cannot store a fraction of a bit. We handle these issues later.
\[lem:slack\] Assume that the recursive problem for nodes in $T_i$ can be solved by storing an encoding of size $(\log^2 n_i + \log n_i \log q)/4$ bits for some parameter $q$. If $n_i = p\cdot n_i'$ and $p \ge 1/q$ then we can spend additional $1/2\log(1/p)\log n_i'$ bits on the top-level problem for nodes in $T_i$ to obtain an encoding of size $(\log^2 n_i' + \log n_i' \log q)/4$ bits.
To prove the lemma it is enough to calculate the difference between the size of the final encoding and the encoding for the recursive problem: $$\begin{aligned}
& = (\log^2 n_i' + \log q \log n_i')/4 - (\log^2(p\cdot n_i') + \log q \log (p\cdot n_i'))/4 \\
& = (\log^2 n_i' + \log q \log n_i' - (\log p + \log n_i')^2 - \log q \log p - \log q \log n_i')/4 \\
& = (\log^2 n_i' + \log q \log n_i' - \log^2 p -2\log p\log n_i' - \log^2 n_i' - \log q \log p - \log q \log n_i')/4 \\
& = (-\log^2 p -2\log p\log n_i'- \log q \log p)/4 \\
& = (2\log(1/p)\log n_i' + \log q \log(1/p) - \log^2(1/p) )/4 \\
& = (2\log(1/p)\log n_i' + \log(1/p)(\log q - \log(1/p))/4 \\
& \geq 1/2\log(1/p)\log n_i'. &
\qedhere\end{aligned}$$
Additionally, we have the following:
\[lem:thin\] Assume that the recursive problem for nodes in $T_i$ can be solved by storing an encoding of size $(\log^2 n_i + \log n_i \log q) / 4$ bits for some parameter $q \ge 2$. If $n_i = p\cdot n_i'$ and $p\le 1/2^8$ then we can spend additional $2\log n_i'$ bits on the top-level problem for nodes in $T_i$ to obtain an encoding of size $(\log^2 n_i' + \log n_i' \log q) / 4$ bits.
Similarly as in the proof of \[lem:slack\], we calculate the difference: $$\begin{aligned}
& = (\log^2 n_i' + \log q \log n_i')/4 - (\log^2(p\cdot n_i') - \log q \log(p\cdot n_i'))/4 \\
& = (2\log(1/p)\log n_i' + \log(1/p)(\log q - \log(1/p))/4.\end{aligned}$$ Now, assuming that $2\log n_i'$ is larger than the difference and using that $p\geq 1/n_i'$ we obtain: $$\begin{aligned}
2\log n_i' &> 1/2\log(1/p)\log n_i' + 1/4\log(1/p)\log q - 1/4\log^2(1/p) \\
& \geq 1/2\log(1/p)\log n_i' - 1/4\log^2(1/p) \\
& \geq 1/2\log(1/p)\log n_i' - 1/4\log(1/p)\log n_i' \\
& = 1/4\log(1/p)\log n_i'\end{aligned}$$ so, after dividing by $\log n_i'$, $8 > \log(1/p)$ and $p > 1/2^8$. Hence for $p \leq 1/2^8$ we can indeed use $2\log n_i'$ additional bits.
We call $T_i$ *thin* if $n_i\le n_i'/2^8$, and *fat* otherwise. We observe that, by the definition of the heavy path decomposition, $\log n \le \log(2n_i') \le 2\log n_i'$. Thus, an immediate consequence of \[lem:thin\] is that if $T_i$ is thin, then we can afford to store ${\mathsf{d}}(e_i)$ explicitly as $\hat{{\mathsf{d}}}(e_i)$, without having to push any bits to the accumulators of nodes in $T_{i+1}, \ldots, T_{m+1}$. However, if $T_i$ is fat, \[lem:slack\] indicates that we do not have enough *slack* to store all the bits of ${\mathsf{d}}(e_i)$. Instead, we store as many bits as the slack allows (rounding up to the nearest bit) in the labels of nodes $u$ in $T_i$. We then append all the remaining bits to the accumulators ${\mathsf{a}}(\ell_i(v))$ of all nodes $v \in \bigcup_{j = i+1}^{m+1} T_{j}$ (i.e., nodes dominated by $u$).
Because $T_i$ is fat, by the slack lemma for nodes in $T_i$ we have slack $1/2\log (n_i'/n_i) \log n_i' $ (the assumption that $T_i$ is fat allows us to adjust the constant $q$). On the other hand, using the same calculations as in the slack lemma, the nodes in $T_{i+1}'$ have slack $1/2\log (n_{i}'/n_{i+1}')\log n_i' $: note that the size of $T_{i+1}'$ is larger than $n_i'/2^8$ by the properties of the heavy path decomposition, as either $i<m$ and $n'_{i+1} \geq n/2$, or $i=m$ and then $n_{i+1} \geq n_i$ so $n'_{i+1} \geq n/4-1/2$. Since $n_i' > n_i + n_{i+1}'$, we have that the sum $1/2(\log(n_i'/n_i) + \log(n_i'/n_{i+1}'))\log n_i'$ can be lower bounded by the minimum of $1/2(\log(1+x) + \log(1+x^{-1}))\log n_i$ for $x \in (0,\infty)$. Thus, the slack is at least $\log n_i'$ bits in total. However, the distance $d(\ell(u)_1)$ occupies $\log n$ bits, rather than $\log n_i'$. As before, we can use the properties of the heavy path decomposition to bound $\log n \le \log(2n_i') = 1+\log n_i'$. Thus, ${\mathsf{d}}(\ell_1(u))$ occupies one extra bit more than we have accounted for with the slack. We store this extra bit in the truncated distance $\hat{{\mathsf{d}}}(\ell_1(u))$. Therefore, the truncated distance $\hat{{\mathsf{d}}}(\ell_1(u))$ consists of the most significant $\lceil 1/2\log (n_i'/n_i) \log n_i' \rceil + 1$ bits of ${\mathsf{d}}(\ell_i(u))$. The remaining least significant $\lfloor 1/2\log (n_i'/n_{i+1}') \log n_i' \rfloor$ bits are concatenated to the accumulators of the nodes dominated by $u$ in $T_{i+1}, \ldots, T_{m+1}$.
For each entry in the modified distance array for a node $u$, we are pushing at most two extra bits beyond those accounted for in the slack lemma. Thus, this works out to an additional ${{O}}(\log n)$-bits in total, per label. We make both parts of the modified distance array (the accumulators and truncated distances) self-delimiting, and also record, for each truncated distance, the number of bits pushed to the accumulators of dominated nodes. Overall, we end up with the following:
\[lem:mod-dist-array\] The modified distance array $\hat{D}(u)$ occupies at most $1/4\log^2 n + {{O}}(\log n \log \log n)$ bits.
It remains to show that these modified distance arrays satisfy \[prop:mod\]. To see this, consider the modified distance array for $u$ and $v$, where $j = {{\mathsf{lightdepth}}}(u,v) + 1$, and $u$ dominates $v$. We have stored the number of bits that were pushed to the accumulator ${\mathsf{a}}(\ell_j(v))$ explicitly. The starting position of this contiguous range of bits can be found by noticing that the accumulator ${\mathsf{a}}(\ell_j(u))$ is a suffix of ${\mathsf{a}}(\ell_j(v))$, since all nodes that dominate $u$ also dominate $v$. Hence, knowing the length of the accumulator ${\mathsf{a}}(\ell_j(u))$ allows us to determine the starting position and, together with the explicitly stored number of pushed bits, recover the bits themselves. By combining them with $\hat{{\mathsf{d}}}(\ell_j(u))$ we can reconstruct ${\mathsf{d}}(\ell_j(u))$.
We have therefore satisfied \[prop:mod\]. It remains to show why this is enough for a distance query. In \[sec:adjustment\] we show that it is, with only additional lower order terms to the label size.
Wrapping Up the Proof of Theorem \[thm:ub-tree-dist\] {#sec:adjustment}
-----------------------------------------------------
In order to prove \[thm:ub-tree-dist\] we need to show how to answer a distance query without inflating the space of \[lem:mod-dist-array\] by more than lower order terms.
Let $u$ be some node contained in the heavy path mapped to $u' \in {\mathcal{C}}(T)$, and consider the path from $u'$ to the root of ${\mathcal{C}}(T)$. We partition this path into $B = \sqrt{\log n}$ *fragments*. The first fragment is the prefix starting at the root, denoted $f_0(u)$, and terminating at the first node $f_1(u)$ such that the subtree rooted at ${\mathsf{head}}(f_1(u))$ has size at most $n/2^B$. The $i$-th fragment is defined recursively from $f_{i-1}(u)$, ending at a node $f_i(u)$ such that the subtree rooted at ${\mathsf{head}}(f_i(u))$ has size at most $n/2^{iB}$, for $i \in [1, h]$, where $h={{O}}(\sqrt{\log n})$. We explicitly store the distances ${\mathsf{d}}(f_i(u), {\mathsf{root}}(T))$ for each $i \in [1,h]$ as the *fragment distance array* $F(u)$.
Next, consider a light edge $e$ in ${\mathcal{C}}(T)$ that branches from the heavy path corresponding to $u'$ to the heavy path corresponding to $v'$. Recall that, in bounding the number of bits for the modified distance arrays, we used the fact that if the subtree rooted at ${\mathsf{head}}(u')$ has size $n$, then the distance, $r = {\mathsf{d}}({\mathsf{head}}(u'),{\mathsf{head}}(v'))$, associated with the light edge $e$ is bounded by $n$. Instead of recording this distance $r$, for each node $u$ that stores $r$ we instead record the distance $r' = {\mathsf{d}}( {\mathsf{head}}(f_j(u)), {\mathsf{head}}(v'))$, where $j$ is the largest index such that the subtree rooted at ${\mathsf{head}}(f_j(u))$ contains node ${\mathsf{head}}(v')$.
Obviously, $r'$ requires more bits to store than $r$, ${{O}}(\sqrt{\log n})$ additional bits to be precise. However, since there are at most $\log n$ truncated distances in $\hat{D}(u)$, we can afford to inflate each of these by ${{O}}(\sqrt{\log n})$ bits. This only increases the lower order space term to ${{O}}(\log^{1.5} n)$ bits. Furthermore, for each truncated distance, we can also afford to store the corresponding index $j$ from the fragment array using ${{O}}(\log n \log \log n)$ extra bits. Thus, since \[prop:mod\] still holds after expanding the truncated distances, we can now recover $r'$ and read ${\mathsf{d}}(f_j(u), {\mathsf{root}}(T))$ from $F(u)$. These two values sum to ${\mathsf{d}}( {\mathsf{head}}(u'), {\mathsf{root}}(T))$, which is exactly what we wanted to compute with distance arrays.
The proof of \[thm:ub-tree-dist\] follows from the above $1/4 \log^2 n + o(\log^2 n)$-bit labeling scheme and the fact that we only need to label leaves and can assume $T$ is binary (see \[sec:prelim\]).
Query Time Analysis
-------------------
Up until now we have not discussed how long it takes to compute the distance given two labels for nodes $u$ and $v$. Let us summarize the steps that are required to answer a query:
1. Ensure $u$ dominates $v$, and swap them if that is not the case. This can be done by examining the inorder number for $u$ and $v$, which are explicitly stored; Lemma \[lem:dist-arrays\] item (\[list:exact\_inorder\]).
2. Extract the explicitly stored distances of $u$ and $v$ from the root; Lemma \[lem:dist-arrays\] item (\[list:exact\_root\_dist\]).
3. Compute the index $j = {{\mathsf{lightdepth}}}(u,v) + 1$, this is done using the explicitly stored NCA encoding; Lemma \[lem:dist-arrays\] item (\[list:exact\_nca\]).
4. Extract the truncated distance $\hat{{\mathsf{d}}}(\ell_j(u))$ from array $\hat{D}(u)$. Note that $\hat{D}(u)$ contains ${{O}}(\log n)$ values, and has length ${{O}}(\log^2 n)$ bits.
5. Extract the accumulator values ${\mathsf{a}}(\ell_j(u))$ from array $\hat{D}(u)$ and ${\mathsf{a}}(\ell_j(v))$ from array $\hat{D}(v)$.
6. Extract explicitly stored lengths of accumulator values ${\mathsf{a}}(\ell_j(u))$ and ${\mathsf{a}}(\ell_j(v))$. Note that there are ${{O}}(\log n)$ explicitly stored lengths, and these lengths occupy ${{O}}(\log n \log \log n)$ bits.
7. Use bitwise arithmetic to extract the relevant bits of ${\mathsf{a}}(\ell_j(v))$ which are then concatenated with $\hat{{\mathsf{d}}}(\ell_j(u))$. This can be done with a constant number of shifts, bitwise and/or operations, and subtractions.
8. Extract the fragment number for $j$, as well as the fragment distance from array $F(u)$. There are ${{O}}(\log n)$ fragment numbers, occupying a total of ${{O}}(\log n \log \log n)$ bits, and a total of ${{O}}(\sqrt{\log n})$ fragment distances, occupying a total of ${{O}}(\log^{1.5} n)$ bits.
9. Compute the overall distance using addition and subtraction.
With the exception of accessing the values stored in the various arrays just mentioned, all steps clearly take constant time. It remains to show how to access each array element in constant time (without increasing the space bound by more than a lower order term). First, we explicitly store the offsets of each of the (constant number of) data structures mentioned above (arrays, individual values, and the NCA labeling) for each label in a header, which is encoded using Elias $\delta$ codes in order to be self-delimiting. This header occupies at most ${{O}}(\log n)$ bits, and provides constant time access to each data structure. Next we discuss how to access the array elements in constant time. Earlier, we mentioned that we used Elias $\delta$ codes to delimit each array element and then concatenate their encodings. Now, for each array that occupies $x$ bits in total and stores $y$ elements, let $p_{1} < p_{2} < \ldots < p_{y}$ be the positions of the first bit of the encoding of each element in the concatenation. We apply Lemma \[lem:encodingsequence\] to this sequence. This takes ${{O}}(x\cdot \max{\{ 1,\log\frac{y}{x} \}})$ and allows us to calculate the first and the last bit of the encoding of any element in constant time. For each of our arrays, $x = {{O}}(\log n)$ and $y = {{O}}(\log^{2}n)$, so storing the sequences increases the total space by only ${{O}}(\log n \log\log n)$ bits. Since there is a constant number of arrays, we can afford to mark the location of their corresponding sequences in the header using ${{O}}(\log n)$ bits. Then, each array access can be performed in constant time.
Lower bound for the Level-Ancestor Problem {#sec:lb}
------------------------------------------
In this section we prove \[thm:lb-lvl-anc\]. The main idea of the proof is to show a lower bound for the parent problem, where the goal is to assign a distinct label to every $u\in T$ so that given the label of $u$ we can return the label of its parent (or a special value $\bot$ denoting that $u={\mathsf{root}}(T)$. This is clearly a special case of the level-ancestor problem. The lower bound is obtained by showing a correspondence between the parent problem and the following *universal tree* problem: what is the size of the smallest rooted tree containing any rooted tree on $n$ nodes as a subtree? The connection between these two problems is captured by the following lemma.
\[lem:uni-tree\] If there exists a labeling scheme for the parent problem on trees of size $n$ that produces labels of size at most $S(n)$, then there exists a universal rooted tree containing all rooted trees on up to $n$ nodes as subtrees of size ${{O}}(2^{S(n)})$.
The proof is by construction. Let $V$ be the set of all possible labels generated by the labeling scheme, and $E$ be the directed edges between these labels defined as follows: if, a label $u$ is assigned to a node of some tree, and $v\not=\bot$ is the label returned by the scheme for $u$, then $(u,v)$ belongs to $E$. Note that $v$ is determined solely from the bits of $u$, hence the graph $G = (V,E)$ consists of one or more directed cycles. See \[fig:function-to-tree\] (left) for an example of such a graph. It is clear that $G$ must contain any tree $T$ on up to $n$ nodes as a subgraph, since the labeling scheme works for all trees on $n$ nodes or less. $G$ is not necessarily a tree itself, but we now describe a general procedure that converts $G$ into a new graph $G' = (V',E')$ that itself is a rooted tree, and is such that that $|V'| \le 2|V| + 1$.
Each weakly connected component of $G$ is either already a tree, or contains a cycle. In the latter case, we arbitrarily remove an edge $(u,v)$ from the cycle (in the figure the chosen edge is intersected by the dashed line). After deleting $(u,v)$ we duplicate the entire weakly connected component, and add a new edge $(u,v')$ where $v'$ is the duplicate of $v$. After doing this for each weakly connected component, we have increased the number of vertices to at most $2|V|$, and the resultant graph $G'$ is a forest of rooted trees. We add a single global root to make $G'$ a rooted tree. The total number of nodes in $G'$ is hence at most $2|V|+1$.
Since $G$ was a universal graph for rooted trees on $n$ nodes, any rooted tree not containing the deleted edge clearly appears as a subgraph in $G'$. Moreover, for any rooted tree $T$ containing $(u,v)$, there exists some subpath of the cycle in $G$ which was in $T$. Since we duplicated each node in the cycle, it is clear that any such subpath also exists in $G'$ (together with any trees rooted at nodes in the subpath), thus, $T$ appears as a subtree in $G'$.
The final detail is to consider the maximum length label output by the labeling scheme, which consists of $S(n)$ bits. Hence, there are at most $2^{S(n)}$ nodes in $G$ and therefore at most ${{O}}(2^{S(n)})$ nodes in $G'$.
![\[fig:function-to-tree\]Converting a weakly connected component to a rooted tree $G'$ by duplicating the path at the dotted line.](function_to_tree.pdf){width="95.00000%"}
Equipped with the previous lemma, we immediately get a lower bound on $S(n)$, provided we have a lower bound on the number of nodes in such a rooted universal tree. Goldberg and Lifschitz [@gol1968minimal] have proved very accurate bounds on the number of nodes in such rooted universal trees (see [@graham1981trees] for the bound as we state it):
\[lem:goldberg\] The smallest rooted tree containing all rooted trees on up to $n$ nodes as subtrees has size $n^{(\log n - 2\log \log n + {{O}}(1))/2}$.
By combining Lemmas \[lem:uni-tree\] and \[lem:goldberg\], \[thm:lb-lvl-anc\] follows immediately.
Effective Level-Ancestor Scheme {#sec:lvl-anc}
-------------------------------
While Alstrup et al. [@alstrup2015distance] describe their scheme in terms of labeling for distance queries, in fact it is not difficult to tweak it to obtain a scheme for level-ancestor queries. We describe the necessary modifications to obtain a scheme for parent queries, i.e., assign distinct labels to every node $u\in T$ so that given the label of $u$ we can return the label of its parent. This immediately implies a scheme for level-ancestor queries by repeatedly moving to the parent as long as necessary.
The labeling consists of three parts. For a node $u$ on a heavy path $P$ we store:
1. ${\mathsf{d}}(u, {\mathsf{root}}(T))$,
2. the ${{O}}(\log n)$ label generated by \[lem:NCA scheme\] applied on ${\mathsf{head}}(P)$,
3. the array $D(u)$ and, additionally, ${\mathsf{d}}(u,{\mathsf{head}}(P))$. (This is differently phrased but essentially equivalent to what the original labeling stores.)
The labels in the NCA scheme are required to be distinct, so the labels of nodes belonging to different heavy paths are distinct. For two nodes on the same heavy path, storing ${\mathsf{d}}(u,{\mathsf{head}}(P))$ explicitly ensures that their labels are not the same. Each label consists of $1/2\log^2 n+{{O}}(\log n)$ bits, because of the bound on the encoding of $D(u)$. We need to argue that given the label of $u \not=\bot$ we can construct the label of its parent.
The NCA labeling scheme from \[lem:NCA scheme\] has the property that the label of every node $u$ is a concatenation of heavy and light labels $h_0.\ell_1.h_1.\ell_2 \ldots \ell_k.h_k$. These labels uniquely determine the path from the root to $u$: $h_0$ encodes how far along the heavy path starting at the root we should continue. Then, either $k=0$ and $u$ in fact lies on the heavy path starting at the root, or $\ell_1$ encodes which light edge outgoing from the current node should be followed. Finally, $h_1.\ell_2\ldots \ell_k.h_k$ recursively encodes the remaining part of the path to $u$ in the subtree hanging off the heavy path starting at the root. It is not necessarily true that given the NCA label of a node $u$ we can determine the NCA label of its parent. However, by truncating the NCA label of $u$ we can construct the NCA label of the parent of ${\mathsf{head}}(P)$.
Given the label of $u$, we construct the label of its parent $u'$ as follows. ${\mathsf{d}}(u, {\mathsf{root}}(T))$ needs to be decreased by 1. Then we inspect ${\mathsf{d}}(u,{\mathsf{head}}(P))$. If $u\neq {\mathsf{head}}(P)$, we decrease ${\mathsf{d}}(u,{\mathsf{head}}(P))$ by 1 and are done. Otherwise, we can use the NCA label of $u$ to determine the label of its parent as explained above. Let $P'$ be the heavy path of $u'$. The last element of the array $D$ is ${\mathsf{d}}(u,{\mathsf{head}}(P'))$, so by subtracting 1 we obtain ${\mathsf{d}}(u',{\mathsf{head}}(P'))$. Finally, we remove the last element of $D$.
$k$-Distance Labeling {#sec:k}
=====================
In this section we prove \[thm:krestricted\]. Recall that in $k$-distance labeling, given the labels of $u$ and $v$ we need to decide if the length of the $u$-to-$v$ path is at most $k$, and if so return it.
Lower Bound for Small $k$ {#sec:smallk}
-------------------------
We define a family of trees and show that labeling the leaves of all trees in that family for $k$-distance queries requires $\log n + \Omega (k \cdot \log \frac{\log n}{k\log k})$-bits.
An $\vec{x}$-regular tree, where $\vec{x} = (x_1,\cdots,x_k) \in \mathbb{N}^k$, is a rooted tree of height $k$ where all depth-$i$ nodes have the same degree $x_{i+1}$. An $(\vec{x}, h, d)$-regular tree, where $(x_1,\cdots,x_k) = \vec{x} \in [h]^k$, is a $\vec{y}$-regular tree with $\vec{y} = (d^{x_1}, d^{h - x_1},d^{x_2}, d^{h - x_2},\cdots, d^{x_k}, d^{h-x_k})$. The total number of leaves in such a tree is $d^{k\cdot h}$. See \[fig:reg2\] for an example.
We consider $(\vec{x},h,d)$-regular trees for some parameters $h$ and $d$ to be chosen later. Consider a labeling scheme that assigns a label to every leaf of such a tree for $2k$-distance queries. The following lemma shows that a $(\vec{x},h,d)$-regular tree and a $(\vec{y},h,d)$-regular tree cannot share many identical labels. More formally, let ${\mathsf{common}}(\vec{x},\vec{y})$ denote the maximum number of labels that can be used in both trees. The following is an upper bound on the sum of ${\mathsf{common}}(\vec{x},\vec{y})$.
\[lem:common\] $\sum_{\vec{x}, \vec{y} \in [h]^k}{{\mathsf{common}}(\vec{x},\vec{y})} \le \left(h\cdot d^h \left(1 + \frac{2}{d-1}\right)\right)^k$.
=-12pt We first prove that ${\mathsf{common}}(\vec{x},\vec{y}) \leq \prod_{i=1}^{k}{d^{\min{\{ x_i, y_i \}}}d^{h - \max{\{ x_i,y_i \}}}}$.
By asking all $2k$-distance queries between a specified subset $S$ of leaves of the $(\vec{x},h,d)$-regular tree we can recover the shape of the subtree induced by $S$. Hence, if two trees share ${\mathsf{common}}(\vec{x},\vec{y})$ labels, then they must have a common isomorphic subtree on ${\mathsf{common}}(\vec{x},\vec{y})$ leaves. To bound the maximum number of leaves in such a subtree, observe that the degree of a node at depth $2i-2$ is at most $\min{\{ d^{x_i},d^{y_i} \}}$, and the degree of a node at depth $2i-1$ is at most $\min{\{ d^{h-x_i},d^{h-y_i} \}}$. The maximum number of shared labels is hence the product of all these quantities over $i=1,\ldots,k$. We conclude that that ${\mathsf{common}}(\vec{x},\vec{y}) \leq \prod_{i=1}^{k}{d^{\min{\{ x_i, y_i \}}}d^{h - \max{\{ x_i,y_i \}}}}$. It then follows that $$\begin{aligned}
\sum_{\vec{x}, \vec{y} \in [h]^k}{{\mathsf{common}}(\vec{x},\vec{y})} & =
\sum_{\vec{x}, \vec{y} \in [h]^k}\prod_{i=1}^{k}{d^{\min{\{ x_i, y_i \}}}d^{h - \max{\{ x_i,y_i \}}}} \\
& = \prod_{i=1}^{k}{\sum_{1 \leq x,y \leq h}{d^{\min{\{ x, y \}}}d^{h - \max{\{ x,y \}}}}}
\\
& = \prod_{i=1}^{k}{(h\cdot d^h + 2\sum_{x < y}{d^x d^{h - y}})} \\
& = \prod_{i=1}^{k}{(h\cdot d^h + 2\sum_{x = 1}^{h-1}{d^x\sum_{y=0}^{h-x-1}{d^{y}}})}
\\
& = \prod_{i=1}^{k}{(h\cdot d^h + 2\sum_{x = 1}^{h-1}{d^x\frac{d^{h-x}-1}{d-1}})}
\\
& \leq \prod_{i=1}^{k}({h\cdot d^h + 2\cdot h \frac{d^h}{d-1})}
\\
& = \left(h\cdot d^h \left(1 + \frac{2}{d-1}\right)\right)^k.\end{aligned}$$
Since the total number of leaves in the $(\vec{x},h,d)$-regular trees family is $d^{k\cdot h} \cdot h^k$, the number of distinct labels required to label them is thus at least:
$$d^{k\cdot h} \cdot h^k -\sum_{\vec{x} < \vec{y}}{{\mathsf{common}}(\vec{x},\vec{y})} =
d^{k\cdot h} \cdot h^k - \frac{1}{2}\sum_{\vec{x} \neq \vec{y}}{{\mathsf{common}}(\vec{x},\vec{y})}
= \frac{3}{2} \cdot d^{k\cdot h} \cdot h^k - \frac{1}{2}\sum_{\vec{x}, \vec{y} \in [h]^k}{{\mathsf{common}}(\vec{x},\vec{y})}.$$
Now we set $d=2k+1$, and since $(1 + \frac{1}{k}) \leq e^{\frac{1}{k}}$ we have from \[lem:common\] that $\sum_{\vec{x}, \vec{y}}{{\mathsf{common}}(\vec{x},\vec{y})} \leq e \cdot d^{k\cdot h} \cdot h^k$, so the number of unique labels is at least: $(3/2-e/2) \cdot d^{k\cdot h} \cdot h^k > 0.1 \cdot d^{k\cdot h} \cdot h^k$. Setting $n=d^{k\cdot h}$ this becomes $0.1 \cdot h^k \cdot n$, making the number of required bits at least: $$\log n + k \log h - {{O}}(1) = \log n + k \log \frac{\log n}{k\log d}-{{O}}(1) = \log n + \Omega(k \cdot \log \frac{\log n}{k \log k}).$$ Note that for the above calculation to make sense, we need that $d^{k} \leq n$.
Lower Bound for Large $k$ {#sec:largek}
-------------------------
The lower bound from \[sec:smallk\] is not meaningful for large values of $k\in [\log n,n]$. In this section we show that the lower bound of Gavoille et al. [@gavoille2004distance] for general distance queries, can be translated into a lower bound of $\Omega(\log n\cdot \log(k/\log n) )$ for $k$-distance queries.
The lower bound uses the family of ${(h,M)\text{-tree}}$s (see \[sec:prelim\]). Recall that every edge of an ${(h,M)\text{-tree}}$ has a weight from $[0,M]$. It is easy to verify that the number of nodes in such a tree is $3\cdot 2^h - 2$, hence the distance between any two leaves is no more than $2hM$.
If $M\leq k/(2h)$ then, because the distance between any two leaves in the tree is at most $2hM\leq k$, any labeling of the leaves for $k$-distance can be used for general distance labeling. By \[lem:tree\_lower\_bound\], such a labeling scheme would require labels of at least $h/2 \cdot \log M$-bits. We set $h=\log \sqrt{n/3}$ and $M = \min{\{ k/2h,2^h \}}$. Then, by subdividing the edges of an ${(h,M)\text{-tree}}$ we obtain an unweighted tree on at most $n$ nodes. Labeling the leaves of such a tree for $k$-distance can be used for general distance labeling of the ${(h,M)\text{-tree}}$, so we obtain the following lower bounds:
1. if $\frac{k}{2h} \leq 2^h$, the number of required bits is $\frac{h}{2}\cdot \log\frac{k}{2h} = \Omega(\log n\cdot \log\frac{k}{\log n})$;
2. if $\frac{k}{2h} > 2^h$, the number of required bits is $\frac{h}{2}\cdot h = \Omega(\log^2 n)$, so $\Omega(\log n\cdot \log\frac{k}{\log n})$ for $k\leq n$.
Upper Bound
-----------
In this section we present our improved upper bound for $k$-distance labeling. We build upon the ideas of Alstrup, Bille, and Rauhe [@alstrup2005labeling], who presented an $\log n+{{O}}(k^2\log(k\log n))$ bits labeling scheme. As a preliminary step, we will show an ${{O}}(\log k\cdot \log n)$ bits scheme for $k \geq \log n$, and then move to the more complicated $\log n + {{O}}(k\log\frac{\log n}{k})$ bits scheme for $k < \log n$.
Consider the heavy path decomposition of $T$. We define the *light range* of $u$, denoted ${\mathsf{L}_{u}}$, to contain the preorder number of all nodes in $T_u$ if $u$ has no heavy child, and all nodes in $T_u\setminus T_{{\mathsf{heavy}}(u)}$ otherwise. We say that $v$ is a [*significant ancestor*]{} of $u$ if ${\mathsf{pre}}(u) \in {\mathsf{L}_{v}}$. For example, in \[fig:heavy-path-decomp\] $v$ is a significant ancestor of $u$ since the light range of $v$ is ${\mathsf{L}_{v}} = [5,23)$. The number of significant ancestors of $u$ is equal to ${\mathsf{lightdepth}}(u)={{O}}(\log n)$. The nearest common significant ancestor of $u$ and $v$, denoted ${\mathsf{NCSA}}(u,v)$, is $w$ such that ${\mathsf{pre}}(w)$ is as large as possible and $w$ is a significant ancestor of both $u$ and $v$. In other words, $w$ is the first significant ancestor on the path from $u$ to the root, which is also a significant ancestor of $v$. The heavy path $P$ such that ${\mathsf{head}}(P)$ is a child of ${\mathsf{NCSA}}(u,v)$ is called the nearest common heavy path of $u$ and $v$ and denoted ${\mathsf{NCH}}(u,v)$. When there is no common significant ancestor for $u$ and $v$ we set ${\mathsf{NCSA}}(u,v)$ to ${\mathsf{nil}}$ and ${\mathsf{NCH}}(u,v)$ to be the heavy path starting at the root.
Let the significant ancestors of $u$ and $v$ on ${\mathsf{NCH}}(u,v)$ be $u'$ and $v'$, respectively. Then ${\mathsf{d}}(u,v)={\mathsf{d}}(u,u')+{\mathsf{d}}(u',v')+{\mathsf{d}}(v,v')$. Computing ${\mathsf{d}}(u,v)$ consists of two steps:
1. identifying ${\mathsf{NCH}}(u,v)$, $u'$ and $v'$, and computing ${\mathsf{d}}(u,u')$ and ${\mathsf{d}}(v,v')$,
2. computing ${\mathsf{d}}(u',v')$.
We describe these steps separately, and then describe how to implement them in constant time.
#### Identifying ${\mathsf{NCH}}(u,v)$.
For an integer range $A = [a,b]\subset [1,n]$ we define its identifier ${\mathsf{id}}(A)$ by considering a binary trie representing all words of length ${\left\lceil \log n \right\rceil}$. The label of a node $u$ in the trie is the concatenation of the labels of the edges on the path from the root to $u$. Every integer $x\in [1,n]$ corresponds to a leaf $u$ in the trie, such that the label of $u$ is the binary expansion of $x$. Then, ${\mathsf{NCA}}(a,b)$ is the nearest common ancestor of the leaves corresponding to $a$ and $b$ in the trie, ${\mathsf{height}}(A)$ is the height of the subtree rooted at ${\mathsf{NCA}}(a,b)$, and finally ${\mathsf{id}}(A)$ is the label of ${\mathsf{NCA}}(a,b)$.
\[ob:range\_id\] For any range $A$:
1. \[ob:range\_id:compute\] ${\mathsf{id}}(A)$ can be computed given ${\mathsf{height}}(A)$ and any $x\in A$,
2. \[ob:range\_id:disjoint\] $A \cap B = \emptyset \implies {\mathsf{id}}(B) \neq {\mathsf{id}}(A)$.
Alstrup, Bille, and Rauhe [@alstrup2005labeling] use the notion of *significant preorder numbers*. We replace it with our notion of range identifier, that has very similar properties, yet is somewhat easier to operate on (and hence we are able to achieve much better query time). For any node $u\in T$, let ${\mathsf{id}}(u)=({\mathsf{id}}({\mathsf{L}_{u}}), {\mathsf{lightdepth}}(u))$.
\[lem:uniqueid\] For any nodes $u,v\in T$, if $u\neq v$ then ${\mathsf{id}}(u)\neq {\mathsf{id}}(v)$.
If ${\mathsf{lightdepth}}(u) \neq {\mathsf{lightdepth}}(v)$ then we are done. Otherwise, ${\mathsf{L}_{v}}$ and ${\mathsf{L}_{u}}$ are disjoint, so by \[ob:range\_id\].\[ob:range\_id:disjoint\] ${\mathsf{id}}({\mathsf{L}_{v}}) \neq {\mathsf{id}}({\mathsf{L}_{u}})$ and we are also done.
Consider a node $u\in T$ and let $u=u_0,u_1,u_2,\ldots$ be all of its significant ancestors in the order in which they appear on the path from $u$ to ${\mathsf{root}}(T)$. Let $u_{r}$ be the last of these ancestors such that ${\mathsf{d}}(u,u_{r}) \leq k$. We call $u_{r}$ the top significant ancestor of $u$. The label of $u$ consists of ${\mathsf{pre}}(u)$, ${\mathsf{lightdepth}}(u)$, and an encoding of ${\mathsf{height}}({\mathsf{L}_{u_i}})$ for every $i=0,1,\ldots,r$. By \[ob:range\_id\].\[ob:range\_id:compute\] this is enough to compute ${\mathsf{id}}(u_i)$ for every $i=0,1,\ldots,r$. Consequently, given the labels of $u$ and $v$, we can either detect that the distance from $u$ or $v$ to ${\mathsf{NCA}}(u,v)$ exceeds $k$, or calculate ${\mathsf{lightdepth}}({\mathsf{NCSA}}(u,v))$.
To encode ${\mathsf{height}}({\mathsf{L}_{v_i}})$ for every $i=0,1,\ldots,r$, we observe that ${\mathsf{L}_{v_i}}\subseteq {\mathsf{L}_{v_{i+1}}}$ and that $r\leq \min{\{ \log n,k \}}$. Hence, we need to encode a non-decreasing sequence of $\min{\{ \log n,k \}}$ numbers from $[0,\log n]$. By \[lem:encodingsequence\], for $k < \log n$ this can be done using ${{O}}(k\log\frac{\log n}{k})$ bits and for $k \geq \log n$ using ${{O}}(\log n)$ bits, and allows us to calculate ${\mathsf{lightdepth}}({\mathsf{NCSA}}(u,v))$ or detect that ${\mathsf{d}}(u,v)>k$.
We encode in the label of $u$ the distance from $u$ to $u_i$ for every $i=0,1,\ldots,r-1$. Because $0=d(u,u_0)<d(u,u_1)<\cdots < d(u,u_{r-1})\leq k$ we need to encode an increasing sequence of $\min{\{ \log n,k \}}$ numbers from the range $[0,k]$. By \[lem:encodingsequence\], if $k<\log n$ this can be done using ${{O}}(k)$ bits and if $k\geq \log n$ using ${{O}}(\log n\cdot\log \frac{k}{\log n})$ bits. Then, after having found ${\mathsf{lightdepth}}({\mathsf{NCSA}}(u,v))$ we can compute ${\mathsf{d}}(u,u')$ and ${\mathsf{d}}(v,v')$.
![$w={\mathsf{NCSA}}(u,v)$, $u'$ and $v'$ are the significant ancestors of $u$ and $v$ on ${\mathsf{NCH}}(u,v)$, respectively. Significant ancestors are white, heavy edges are solid, and light edges are dashed.[]{data-label="fig:ncsa"}](NCSA.pdf)
#### Computing ${\mathsf{d}}(u',v')$.
Recall that $u'$ and $v'$ are the significants ancestors on the ${\mathsf{NCH}}(u,v)$ of $u$ and $v$, respectively. We want to compute ${\mathsf{d}}(u',v')$. If $u'$ is not the top significant ancestor of $u$ and $v'$ is not the top significant ancestor of $v'$ then from the distances encoded in the labels of $u$ and $v$ we can retrieve ${\mathsf{d}}(u',{\mathsf{NCSA}}(u,v))$ and ${\mathsf{d}}(v',{\mathsf{NCSA}}(u,v))$, and return their absolute difference as ${\mathsf{d}}(u',v')$. Now consider the case that $u'$ is the top significant ancestor of $u$, but $v'$ is not the top significant ancestor of $v$. To deal with this case, the label of $u$ should also encode the distance $\alpha$ from $u'$ to the head of it’s heavy path. This distance might be very large (even up to $n$), so we cap it at $2k+1$ to use only ${{O}}(\log k)$ bits. Since $v'$ is not the top significant ancestor of $v$, we can retrieve $\beta={\mathsf{d}}(v',{\mathsf{NCSA}}(u,v))$ as in the previous case. We know that $\beta \leq k$ because otherwise $v'$ would be the top significant ancestor of $v$. Recall that ${\mathsf{d}}(u',v')$ is equal to the absolute difference between ${\mathsf{d}}(u',{\mathsf{NCSA}}(u,v))$ and ${\mathsf{d}}(v',{\mathsf{NCSA}}(u,v))$. If $\alpha=2k+1$ then this value must exceed $k$, so we terminate. Otherwise, we return $|\alpha-\beta|$.
The remaining and most complicated case is when $u'$ is the top significant ancestor of $u$ and $v'$ is the top significant ancestor of $v'$. If $k > \log n$, the solution is simple, as we can afford to store the distance from the top significant ancestor to the head of its heavy path for every node (i.e., ${\mathsf{d}}(u,u_{r+1})$) using ${{O}}(\log n)$ bits. The rest of this section is dedicated for solving $k\leq \log n$.
To make the further exposition more concise, we define the 2-approximation of an integer $x$, denoted $\appx{x}$, as the largest power of 2 not exceeding $x$. That is, $\appx{x} = 2^{{\left\lfloor \log x \right\rfloor}}$. Clearly, 2-approximation is monotone, meaning that $x\leq y$ implies $\appx{x}\leq \appx{y}$, and furthermore $\appx{x} < \appx{2x}$.
\[lem:ranges\] Let $A,B,C$ be three open intervals such that $A\cap B=\emptyset$ and $A,B\subseteq C$. Then $\appx{|C|}\neq \appx{|A|}$ or $\appx{|C|}\neq \appx{|B|}$.
Assume that $|B| \leq |A|$. Then $2|B| \leq |A|+|B| \leq |C|$ and by the properties of 2-approximation $\appx{|B|} < \appx{|C|}$, so indeed $\appx{|B|}\neq \appx{|C|}$. Symmetrically, if $|A| \leq |B|$ then $\appx{|A|}\neq \appx{|C|}$.
The following lemma captures the essence of the $k$-distance scheme of Alstrup, Bille, and Rauhe [@alstrup2005labeling], while being optimized so that we can obtain our improvement.
\[lem:monotone\_sequence\_scheme\] Consider an increasing sequence of integers $a_1 < a_2 < \ldots < a_s$. Given $a_i < a_j$, $i' = i\bmod{k}$, $j' = j\mod{k}$, and $\appx{a_{i+t} - a_i}$ and $\appx{a_j - a_{j-t}}$ for every $t=1,2,\ldots,k$ we can calculate $j-i$ or determine that $j-i > k$ in constant time.
We start by setting $t = j'-i'$. Now either $t = j - i$ or $j - i \geq k+t$. Hence we only need to distinguish between these two cases.
Consider three intervals $(a_i, a_{i+t})$, $(a_{j-t}, a_j)$ and $(a_i, a_j)$. If $t=j-i$ then these three intervals are equal and so are $\appx{a_{i+t}-a_i}, \appx{a_j-a_{j-t}}$ and $\appx{a_j-a_i}$. Otherwise $j-i \geq k+t > 2t$, so $(a_i, a_{i+t})$ and $(a_{j-t},a_j)$ are two disjoint intervals contained in $(a_i, a_j)$. Hence by \[lem:ranges\] either $\appx{a_{i+t}-a_i}\neq \appx{a_j-a_i}$ or $\appx{a_j-a_{j-t}}\neq \appx{a_j-a_i}$. Therefore after retrieving $\appx{a_{i+t}-a_i}$ and $\appx{a_j-a_{j-t}}$ and calculating $\appx{a_j-a_i}$ we can distinguish between the two cases and either return $j-i$ or report that $j-i > k$. Notice that $\appx{a_j-a_i}$ can be calculated in constant time using standard word-RAM operations.
We need to show that, for every heavy path, we store enough information for applying \[lem:monotone\_sequence\_scheme\]. Consider a heavy path $u_1 - u_2 - \ldots - u_s$, where $u_1$ is the head. By the properties of the heavy path decomposition, ${\mathsf{id}}({\mathsf{L}_{u_1}})<{\mathsf{id}}({\mathsf{L}_{u_2}})<\ldots<{\mathsf{id}}({\mathsf{L}_{u_s}})$. The label of every $u\in T$ such that $u_i$ is the top significant ancestor of $u$ encodes the following:
1. ${\mathsf{id}}({\mathsf{L}_{u_i}})$;
2. $\appx{{\mathsf{id}}({\mathsf{L}_{u_{i+t}}}) - {\mathsf{id}}({\mathsf{L}_{u_i}})}$ and $\appx{{\mathsf{id}}({\mathsf{L}_{u_i}}) - {\mathsf{id}}({\mathsf{L}_{u_{i-t}}})}$ for every $t=1,2,\ldots,k$;
3. $i\bmod{k}$.
To encode ${\mathsf{id}}({\mathsf{L}_{u_i}})$, we store ${\mathsf{height}}({\mathsf{L}_{u_i}})$ using ${{O}}(\log\log n)$ bits. Encoding $\appx{{\mathsf{id}}({\mathsf{L}_{u_{i+t}}}) - {\mathsf{id}}({\mathsf{L}_{u_i}})}$ and $\appx{{\mathsf{id}}({\mathsf{L}_{u_i}}) - {\mathsf{id}}({\mathsf{L}_{u_{i-t}}})}$ for every $t=1,2,\ldots,k$ reduces to encoding two non-decreasing sequences of $k$ integers from $[0,\log n]$. By \[lem:encodingsequence\], such a sequence can be stored using ${{O}}(k\log\frac{\log n}{k})$-bits. Finally, $i\bmod{k}$ is encoded using ${{O}}(\log k)$ bits. Notice that both ${{O}}(\log\log n)$ and ${{O}}(\log k)$ are absorbed by ${{O}}(k\log\frac{\log n}{k})$.
To conclude, given the labels of $u$ and $v$, whose significant ancestors $u'$ and $v'$ are on ${\mathsf{NCH}}(u,v)=u_1-u_2-\ldots -u_s$ and are both the top significant ancestors, we can now calculate ${\mathsf{d}}(u',v')$ or detect that it exceeds $k$ by retrieving the necessary information from the labels of $u$ and $v$ and then applying \[lem:monotone\_sequence\_scheme\]. Finally, in the following section (\[sec:query-time\]) we show that queries can be supported in constant time. The gist of the improvement in the query time is that ${\mathsf{id}}({\mathsf{L}_{u_i}})$ can be obtained from ${\mathsf{pre}}(u)$ by truncating the last ${\mathsf{height}}({\mathsf{L}_{u_i}})$ trailing bits and setting the ${\mathsf{height}}({\mathsf{L}_{u_i}})^\text{th}$ bit to $1$.
Query Time Analysis {#sec:query-time}
-------------------
We now show how to implement the query in constant time. The main difficulty is in determining ${\mathsf{lightdepth}}({\mathsf{NCSA}}(u,v))$ efficiently. Once it is known, from ${\mathsf{lightdepth}}(u)$ and the encoding of the distances from $u$ to its significant ancestors implemented with \[lem:encodingsequence\] we obtain ${\mathsf{d}}(u,u')$ in constant time, and similarly for ${\mathsf{d}}(v,v')$ (or conclude that ${\mathsf{d}}(u,v)$ exceeds $k$). Calculating ${\mathsf{d}}(u',v')$ requires invoking \[lem:monotone\_sequence\_scheme\] while providing access to the stored non-decreasing sequences of 2-approximations with \[lem:encodingsequence\], so also takes only constant time.
Recall that the label of $u$ contains ${\mathsf{pre}}(u)$, ${\mathsf{lightdepth}}(u)$, and an encoding of the sequence ${\mathsf{height}}({\mathsf{L}_{u_0}}) \leq {\mathsf{height}}({\mathsf{L}_{u_1}}) \leq \dots \leq {\mathsf{height}}({\mathsf{L}_{u_r}})$ implemented with \[lem:encodingsequence\]. Similarly, the label of $v$ contains ${\mathsf{pre}}(v)$, ${\mathsf{lightdepth}}(v)$, and ${\mathsf{height}}({\mathsf{L}_{v_0}}) \leq {\mathsf{height}}({\mathsf{L}_{v_1}}) \leq \dots \leq {\mathsf{height}}({\mathsf{L}_{v_s}})$. We want to calculate ${\mathsf{lightdepth}}({\mathsf{NCSA}}(u,v))$. For now, we assume that $r=s$ and ${\mathsf{lightdepth}}(u_i)={\mathsf{lightdepth}}(v_i)$ for every $i=0,1,\ldots,r$. Then, calculating ${\mathsf{lightdepth}}({\mathsf{NCSA}}(u,v))$ reduces to finding the smallest $i$ such that $u_i=v_i$. Notice that then $u_j=v_j$ for every $j=i,i+1,\ldots,r$. If $u_j=v_j$ then clearly ${\mathsf{height}}({\mathsf{L}_{u_j}})={\mathsf{height}}({\mathsf{L}_{v_j}})$, so we start with locating the smallest $i'$ such that ${\mathsf{height}}({\mathsf{L}_{u_j}})={\mathsf{height}}({\mathsf{L}_{v_j}})$ for every $j=i',i'+1,\ldots,r$. This can be done in constant time by computing the longest common suffix of both sequences.
Because ${\mathsf{lightdepth}}(u_j)={\mathsf{lightdepth}}(v_j)$ for every $j=0,1,\ldots,r$, it remains to find the smallest $i \geq i'$ such that ${\mathsf{id}}({\mathsf{L}_{u_j}})={\mathsf{id}}({\mathsf{L}_{v_j}})$ for every $j=i,i+1,\ldots,r$. Observe that ${\mathsf{id}}({\mathsf{L}_{u_j}})$ is obtained by clearing all ${\mathsf{height}}({\mathsf{L}_{u_j}})$ least significant bits of ${\mathsf{pre}}(u)$ and, if ${\mathsf{height}}({\mathsf{L}_{u_j}})>0$, setting the ${\mathsf{height}}({\mathsf{L}_{u_j}})^\text{th}$ bit to 1, and similarly for ${\mathsf{id}}({\mathsf{L}_{v_j}})$. Without loss of generality, assume that ${\mathsf{height}}({\mathsf{L}_{u_{i'}}})={\mathsf{height}}({\mathsf{L}_{v_{i'}}})>0$ (if not, $i=i'$ is checked separately in constant time). We find the longest common prefix of the binary expansions of ${\mathsf{pre}}(u)$ and ${\mathsf{pre}}(v)$, i.e., the smallest $\ell \geq 0$ such that their binary expansions are the same after truncating the $\ell$ least significant bits. $\ell$ can be found in constant time using standard word-RAM operations $\text{MSB}({\mathsf{pre}}(u) \text{ XOR } {\mathsf{pre}}(v))$. Then, for ${\mathsf{id}}({\mathsf{L}_{u_i}})={\mathsf{id}}({\mathsf{L}_{v_i}})$ to hold, we need to clear at least $\ell$ least significant bits of ${\mathsf{pre}}(u)$ and ${\mathsf{pre}}(v)$. Hence it remains to find the smallest $i \geq i'$ such that ${\mathsf{height}}({\mathsf{L}_{u_i}})={\mathsf{height}}({\mathsf{L}_{v_i}}) \geq i$. Such an $i$ can be found in constant time with a successor query on the encoded sequence.
If $r\neq s$ or ${\mathsf{lightdepth}}(u_0)\neq {\mathsf{lightdepth}}(v_0)$, then essentially the same argument works, except that we need to compute the longest common prefix of suffixes instead of whole sequences.
Approximate Distance Labeling {#sec:approx}
=============================
In this section we prove \[thm:approximate\]. Recall that in $(1+{\varepsilon})$-approximate distance labeling, given the labels of $u$ and $v$ we need to output some value in the interval $[{\mathsf{d}}(u,v),(1+{\varepsilon})\cdot{\mathsf{d}}(u,v)]$.
Lower bound
-----------
To show the lower bound we modify the family of $(h,M)$-trees such that exact distances between leaves can be inferred from their approximate distances. Thereafter, we can invoke \[lem:tree\_lower\_bound\] to establish the lower bound.
An $(h,M)$-tree is modified by first subdividing its edges to obtain an unweighted tree of height $h\cdot M$. The edges of this unweighted tree are then further subdivided: every edge of depth $d\ge 0$ is subdivided into ${\left\lfloor (1 + {\varepsilon})^{hM-d} \right\rfloor}$ edges. Note that in the original $(h,M$)-tree all leaves are at the same distance from the root. Therefore, if the distance between two leaves is $2k$ in the original tree, it is $f(k)=2\sum_{i=1}^{k}{{\left\lfloor (1+{\varepsilon})^i \right\rfloor}}$ in the final tree. A $(1+{\varepsilon})$-approximation of this distance belongs to the interval $[f(k), (1+{\varepsilon})f(k)]$. We next show that these intervals are disjoint, so in fact a $(1+{\varepsilon})$-approximation of $f(k)$ is enough to infer the original distance, $2k$.
Observe that $f(k)$ is monotone, so to prove that the intervals $[f(k), (1+{\varepsilon})f(k)]$ are disjoint, it is enough to show that $(1+{\varepsilon})f(k) < f(k+1)$, or: $$\begin{aligned}
(1+{\varepsilon})\sum_{i=1}^{k}{{\left\lfloor (1+{\varepsilon})^i \right\rfloor}} &< \sum_{i=1}^{k+1}{{\left\lfloor (1+{\varepsilon})^i \right\rfloor}}, \text{ or equivalently} \\
{\varepsilon}\sum_{i=1}^{k}{{\left\lfloor (1+{\varepsilon})^i \right\rfloor}} &< {\left\lfloor (1+{\varepsilon})^{k+1} \right\rfloor}. \\\end{aligned}$$ Since ${\left\lfloor (1+{\varepsilon})^i \right\rfloor} < (1+{\varepsilon})^i$, it is enough to show that: $$\begin{aligned}
{\varepsilon}\sum_{i=1}^{k}{(1+{\varepsilon})^i} &< {\left\lfloor (1+{\varepsilon})^{k+1} \right\rfloor}, \text{ or equivalently}\\
(1+{\varepsilon})^{k+1} - (1+{\varepsilon}) &< {\left\lfloor (1+{\varepsilon})^{k+1} \right\rfloor}.\\ \end{aligned}$$ Since $x - 1 < {\left\lfloor x \right\rfloor}$ is always true, we conclude that the intervals are indeed disjoint. Hence, by \[lem:tree\_lower\_bound\] we obtain that labeling the leaves of the final tree for $(1+{\varepsilon})$-approximate distances requires $h/2 \cdot \log M$ bits. It remains to choose $h$ and $M$ and rephrase this bound in terms of the size of the final tree. The size of the final tree is at most $$\begin{aligned}
2\sum_{i=0}^{h-1}{2^{h-1-i}} \sum_{j=M\cdot i+1}^{M\cdot(i+1)}{\left\lfloor (1 + {\varepsilon})^j \right\rfloor} & =
\sum_{i=0}^{h-1}2^{h-i} \sum_{j=M\cdot i+1}^{M\cdot(i+1)} (1 + {\varepsilon})^j \\
&\leq 2^{h}\sum_{i=0}^{h-1}{2^{-i}}(1+{\varepsilon})^{M\cdot i+1}\frac{(1+{\varepsilon})^{M}-1}{(1+{\varepsilon})-1}\\
&\leq 2^h \frac{1}{{\varepsilon}}\sum_{i=0}^{h-1}{2^{-i}}(1+{\varepsilon})^{M(i+1)+1} \\
&\leq 2^h \frac{1}{{\varepsilon}}(1+{\varepsilon})^{M+1}\sum_{i=0}^{h-1}\left(\frac{(1+{\varepsilon})^{M}}{2}\right)^{i} \\
&= 2^h \frac{1}{{\varepsilon}}(1+{\varepsilon})^{M+1} \frac{(\frac{(1+{\varepsilon})^M}{2})^h-1}{\frac{(1+{\varepsilon})^M}{2}-1}\\
&\leq \frac{2}{{\varepsilon}}\frac{(1+{\varepsilon})^{M+1}}{(1+{\varepsilon})^M-2} (1+{\varepsilon})^{M\cdot h} \\\end{aligned}$$ We set $M=2/{\varepsilon}$. Then, because ${\varepsilon}\leq 1$ and $(1+{\varepsilon})^M \geq 4$, the size is at most: $$\begin{aligned}
&\leq 2\frac{1+{\varepsilon}}{{\varepsilon}} \frac{(1+{\varepsilon})^M}{(1+{\varepsilon})^M-2} e^{2h} \\
&\leq \frac{8}{{\varepsilon}} e^{2h}.\end{aligned}$$ We set $h = \log ({\varepsilon}\cdot n/8) / (2 \log e) = \Theta(\log({\varepsilon}\cdot n))$, and obtain that labeling trees of size $n$ for $(1+{\varepsilon})$-approximate distances requires $\Omega(\log(1/{\varepsilon})\cdot \log ({\varepsilon}\cdot n))$ bits. Now, if ${\varepsilon}> 1 / \sqrt{n}$ this is in fact $\Omega(\log(1/{\varepsilon})\cdot \log n)$ and we are done. Otherwise (${\varepsilon}\leq 1/\sqrt{n}$), we observe that a scheme with such small ${\varepsilon}$ can be used for labeling a tree of size $\sqrt{n}$ for exact distances (by subdividing every edge into $\sqrt{n}$ edges). Such labeling requires $\Omega(\log^2(\sqrt{n}))=\Omega(\log^2n)$ bits, which for ${\varepsilon}\geq 1/n$ is also $\Omega(\log(1/{\varepsilon})\cdot \log n)$ as required.
Upper bound
-----------
We now describe a matching upper bound: a $(1+{\varepsilon})$-approximate distance labeling scheme with label size ${{O}}(\log(1/{\varepsilon})\cdot\log n)$. Our scheme is based on the scheme of Alstrup et al. [@alstrup2015distance] whose label size is ${{O}}(1/{\varepsilon}\cdot\log n)$. For any node $v$, let $v_1, \dots, v_k$ be the significant ancestors of $v$ in the order they appear on the $v$-to-root path. Let ${\left\lceil x \right\rceil}_{1+{\varepsilon}}$ denote the smallest power of $1+{\varepsilon}$ larger than $x$. Observe that ${\left\lceil x \right\rceil}_{1+{\varepsilon}}$ is a $(1+{\varepsilon})$-approximation of $x$.
The label of a node $v$ in [@alstrup2015distance] is composed of the following fields:
1. \[list:root distance\] $d(v,{\mathsf{root}}(T))$,
2. \[list:nca label\] the ${{O}}(\log n)$ label generated by \[lem:NCA scheme\] applied on $v$,
3. \[list:significant ancestor sequence\] the sequence ${\left\lceil {\mathsf{d}}(v,v_1) \right\rceil}_{1+{\varepsilon}}, {\left\lceil {\mathsf{d}}(v,v_2) \right\rceil}_{1+{\varepsilon}}, \dots, {\left\lceil {\mathsf{d}}(v,v_k) \right\rceil}_{1+{\varepsilon}}.$
Let $w = {\mathsf{NCA}}(u,v)$. If $w = v$ or $w=u$, we can extract the exact distance from (\[list:root distance\]). Otherwise, w.l.o.g. we can find the significant ancestor $v_j$ of $v$ such that $v_j=w$ using (\[list:nca label\]), and then find ${\left\lceil {\mathsf{d}}(v,w) \right\rceil}_{1+{\varepsilon}}$ using (\[list:significant ancestor sequence\]). Alstrup et al. show that: $$\begin{aligned}
{\mathsf{d}}(u,v) \leq {\mathsf{d}}(u,{\mathsf{root}}(T)) - {\mathsf{d}}(v,{\mathsf{root}}(T)) + 2\cdot{\left\lceil {\mathsf{d}}(v,w) \right\rceil}_{1+{\varepsilon}} \leq
(1 + 2{\varepsilon})\cdot {\mathsf{d}}(u,v).\end{aligned}$$
This means we can compute a $(1 + {\varepsilon})$-approximation of ${\mathsf{d}}(u,v)$ by replacing ${\varepsilon}$ with ${\varepsilon}/2$. The bottleneck for the size of the label is storing the sequence in (\[list:significant ancestor sequence\]). In [@alstrup2015distance], this sequence is stored using a unary encoding of the sequence ${\left\lceil {\mathsf{d}}(v,v_1) \right\rceil}_{1+{\varepsilon}}, {\left\lceil {\mathsf{d}}(v,v_2) \right\rceil}_{1+{\varepsilon}}-{\left\lceil {\mathsf{d}}(v,v_1) \right\rceil}_{1+{\varepsilon}}, \dots, {\left\lceil {\mathsf{d}}(v,v_k) \right\rceil}_{1+{\varepsilon}}-{\left\lceil {\mathsf{d}}(v,v_{k-1}) \right\rceil}_{1+{\varepsilon}}$ delimited by a single bit between two consecutive values. The maximal length of the path is at most $n$, so such an encoding will require $\log_{1+{\varepsilon}}{n}$ bits and additional $k\leq\log n$ bits for the delimiters. This means that the final label size is $\Theta(\log_{1+{\varepsilon}}{n})$, or $\Theta(1/{\varepsilon}\cdot\log n)$ for small ${\varepsilon}$. Instead, we store the sequence using \[lem:encodingsequence\], which yields a label of size ${{O}}(\log(1/{\varepsilon})\cdot\log{n})$ bits and a constant query time.
[^1]: The research was supported in part by Israel Science Foundation grant 794/13.
|
---
abstract: |
Although the *residual method*, or *constrained regularization*, is frequently used in applications, a detailed study of its properties is still missing. This sharply contrasts the progress of the theory of Tikhonov regularization, where a series of new results for regularization in Banach spaces has been published in the recent years. The present paper intends to bridge the gap between the existing theories as far as possible. We develop a stability and convergence theory for the residual method in general topological spaces. In addition, we prove convergence rates in terms of (generalized) Bregman distances, which can also be applied to non-convex regularization functionals.
We provide three examples that show the applicability of our theory. The first example is the regularized solution of linear operator equations on $L^p$-spaces, where we show that the results of Tikhonov regularization generalize unchanged to the residual method. As a second example, we consider the problem of density estimation from a finite number of sampling points, using the Wasserstein distance as a fidelity term and an entropy measure as regularization term. It is shown that the densities obtained in this way depend continuously on the location of the sampled points and that the underlying density can be recovered as the number of sampling points tends to infinity. Finally, we apply our theory to compressed sensing. Here, we show the well-posedness of the method and derive convergence rates both for convex and non-convex regularization under rather weak conditions.
**Keywords.** Ill-posed problems, Regularization, Residual Method, Sparsity, Stability, Convergence Rates.
**AMS.** 65J20; 47J06; 49J27.
author:
- |
Markus Grasmair, Markus Haltmeier, and Otmar Scherzer\
Computational Science Center, University of Vienna,\
Nordbergstra[ß]{}e 15, Vienna, Austria
title: 'The Residual Method for Regularizing Ill-Posed Problems'
---
Introduction
============
We study the solution of ill-posed operator equations $$\label{eq:F}
\operatorname{\mathbf F}(x) = y\,,$$ where $\operatorname{\mathbf F}\colon {X}\to {Y}$ is an operator between the topological spaces ${X}$ and ${Y}$, and $y \in {Y}$ are given, noisy data, which are assumed to be close to some unknown, noise-free data $y^\dagger \in \operatorname{ran}(\operatorname{\mathbf F})$. If the operator $\operatorname{\mathbf F}$ is not continuously invertible, then may not have a solution and, if a solution exists, arbitrarily small perturbations of the data may lead to unacceptable results.
If ${Y}$ is a Banach space and the given data are known to satisfy an estimate ${{\left\lVerty^\dagger-y\right\rVert}} \le {\beta}$, one strategy for defining an approximate solution of is to solve the *constrained minimization problem* $$\label{eq:constrained}
{\mathcal R}(x) \to \min
\qquad \text{ subject to }
\quad {{\left\lVert\operatorname{\mathbf F}(x)-y\right\rVert}} \le {\beta}\;.$$ Here, the *regularization term* ${\mathcal R}\colon {X}\to [0,+\infty]$ is intended to enforce certain regularity properties of the approximate solution and to stabilize the process of solving . In [@IvaVasTan02; @Tan97], this strategy is called the *residual method*. It is closely related to *Tikhonov regularization* $$\label{eq:tik}
{\cal T}(x) := {{\left\lVert\operatorname{\mathbf F}(x)-y\right\rVert}}^2 + \alpha{\mathcal R}(x) \to \min \,,$$ where $\alpha > 0$ is a regularization parameter. In the case that the operator $\operatorname{\mathbf F}$ is linear and ${\mathcal R}$ is convex, and are basically equivalent, if $\alpha$ is chosen according to Morozov’s discrepancy principle (see [@IvaVasTan02 Chap. 3]).
While the theory of Tikhonov regularization has received much attention in the literature (see for instance [@AcaVog94; @DauDefDem04; @EngHanNeu96; @EngKunNeu89; @Gro84; @HofYam05; @Mor93; @SchEngKun93; @SeiVog89; @TikArs77; @Vas06]), the same cannot be said about the residual method. The existing results are mainly concerned with the existence theory of and with the question of convergence, which asks whether solutions of converge to a solution of as ${{\left\lVerty - y^\dagger\right\rVert}} \le {\beta}\to 0$. These problems have been treated in very general settings in [@Iva69; @Sei81] (see also [@Hei08; @Tan95; @Tan97]). Convergence rates have been derived in [@BurOsh04] for linear equations in Hilbert spaces and later generalized in [@Hei08] to non-linear equations in Banach spaces. Convergence rates have also been derived in [@Can08; @CanRomTao06b] for the reconstruction of sparse sequences.
The problem of stability, however, that is, continuous dependence of the solution of on the input data $y$ and the presumed noise level ${\beta}$, has been hardly considered at all. One reason for the lack of results is that, in contrast to Tikhonov regularization, stability simply does not hold for general non-linear operator equations. But even for the linear case, where we indeed prove stability, so far stability theorems are non-existent in the literature. Though some results have been derived in [@Hei08], they only cover a very weak form of stability, which states that the solutions of with perturbed data stay close to the solution with unperturbed data, if one additionally increases the regularization parameter ${\beta}$ in the perturbed problem by a sufficient amount.
The present paper tries to generalize the existent theory on the residual method as far as possible. We assume that ${X}$ and ${Y}$ are mere topological spaces and consider the minimization of ${\mathcal R}(x)$ subject to the constraint ${\mathcal S}\left(\operatorname{\mathbf F}(x),y\right) \le {\beta}$. Here ${\mathcal S}$ is some distance like functional taking over the role of the norm in . In addition, we discuss the case where the operator $\operatorname{\mathbf F}$ is not known exactly. This subsumes errors due to the modeling process as well as discretizations of the problem necessary for its numerical solution. We provide different criteria that ensure stability (Lemma \[le:stability\], Theorem \[thm:stability:2\] and Proposition \[pr:linear\_stab\]) and convergence (Theorem \[thm:convergence\] and Proposition \[pr:linear\_stab\]) of the residual method. In particular, our conditions also include certain non-linear operators (see Example \[ex:stab\_nonlinear\]).
Section \[se:rates\] is concerned with the derivation of convergence rates, i.e., quantitative estimates between solutions of and the exact data $y^\dag$. Using notions of abstract convexity, we define a generalized Bregman distance that allows us to state and prove rates on arbitrary topological spaces (see Theorem \[thm:rates\]). In Section \[sec:sparsity\] we apply our general results to the case of sparse $\ell^p$-regularization with $p \in (0,2)$. We prove the well-posedness of the method and derive convergence rates with respect to the norm in a fairly general setting. In the case of convex regularization, that is, $p \ge 1$, we derive a convergence rate of order $\mathcal{O}\bigl({\beta}^{1/p}\bigr)$. In the non-convex case $0 < p < 1$, we show that the rate $\mathcal{O}({\beta})$ holds.
Definitions and Mathematical Preliminaries {#sec:defs}
==========================================
Throughout the paper, ${X}$ and ${Y}$ denote sets. Moreover, ${\mathcal R}\colon {X}\to [0,+\infty]$ is a functional on ${X}$, and ${\mathcal S}\colon {Y}\times {Y}\to [0,+\infty]$ is a functional on ${Y}\times {Y}$ such that ${\mathcal S}(y,z) = 0$ if and only if $y = z$.
The Residual Method
-------------------
For given mapping $\operatorname{\mathbf F}\colon {X}\to {Y}$, given data $y \in {Y}$, and fixed parameter ${\beta}\ge 0$, we consider the constrained minimization problem $$\label{eq:min_prob}
{\mathcal R}(x) \to \min
\qquad\text{ subject to } \quad
{\mathcal S}(\operatorname{\mathbf F}(x),y) \le {\beta}\;.$$ For the analysis of the residual method it is convenient to introduce the following notation.
The *feasible set* $\Phi(\operatorname{\mathbf F},y,{\beta})$, the *value* $v(\operatorname{\mathbf F},y,{\beta})$, and the *set of solutions* ${\Sigma}(\operatorname{\mathbf F},y,{\beta}) $ of are defined by $$\begin{aligned}
\Phi(\operatorname{\mathbf F},y,{\beta}) &:= {\left\{x \in {X}: {\mathcal S}(\operatorname{\mathbf F}(x),y)\le {\beta}\right\}}\,,\\
v(\operatorname{\mathbf F},y,{\beta}) &:= \inf{\left\{{\mathcal R}(x):x \in \Phi(\operatorname{\mathbf F},y,{\beta})\right\}} \,, \\
{\Sigma}(\operatorname{\mathbf F},y,{\beta}) &:= {\left\{x \in \Phi(\operatorname{\mathbf F},y,{\beta}): {\mathcal R}(x) = v(\operatorname{\mathbf F},y,{\beta})\right\}} \,.\end{aligned}$$ In particular, $\Phi(\operatorname{\mathbf F},y,0)$ consist of all solutions of the equation $\operatorname{\mathbf F}(x) = y$. The elements of ${\Sigma}(\operatorname{\mathbf F},y,0)$ are therefore referred to as ${\mathcal R}$-*minimizing solutions* of $\operatorname{\mathbf F}(x) = y$.
In addition, for $t \geq 0$, we set $$\label{eq:fr}
\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t) := \Phi(\operatorname{\mathbf F},y,{\beta}) \cap {\left\{x \in {X}: {\mathcal R}(x) \le t\right\}} \,.$$ An immediate consequence of the above definitions is the identity $$\label{eq:Sigma_min}
{\Sigma}(\operatorname{\mathbf F},y,{\beta}) = \Phi_{\mathcal R}\left(\operatorname{\mathbf F},y,{\beta}, v(\operatorname{\mathbf F},y,{\beta})\right) \,.$$
We do not assume a–priori that a solution of the minimization problem exists. Only in the next section shall we deduce the existence of solutions under a compactness assumption on the sets $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t)$, see Theorem \[th:existence\].
\[le:Phi\_prop\] The sets $\Phi_{{\mathcal R}}(\operatorname{\mathbf F},y,{\beta},t)$ defined in satisfy $$\label{eq:Sigma_order}
\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t) \subset \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+\gamma,t+{\varepsilon})$$ for every $\gamma, {\varepsilon}\ge 0$, and $$\label{eq:Sigma_intersect}
\Phi_{{\mathcal R}}(\operatorname{\mathbf F},y,{\beta},t) =
\bigcap_{\gamma,{\varepsilon}> 0} \Phi_{\cal R}(\operatorname{\mathbf F},y,{\beta}+\gamma,t+{\varepsilon})\;.$$
The inclusion follows immediately from the definition of $\Phi_{\cal R}$. For the proof of note that $x \in \bigcap_{\gamma,{\varepsilon}> 0} \Phi_{\cal R}(\operatorname{\mathbf F},y,{\beta}+\gamma,t+{\varepsilon})$ if and only if ${\mathcal S}(\operatorname{\mathbf F}(x),y) \le {\beta}+\gamma$ for all $\gamma > 0$ and ${\mathcal R}(x) \le t+{\varepsilon}$ for all ${\varepsilon}> 0$. This, however, is the case if and only if ${\mathcal S}(\operatorname{\mathbf F}(x),y) \le {\beta}$ and ${\mathcal R}(x) \le t$, which means that $x \in \Phi_{{\mathcal R}}(\operatorname{\mathbf F},y,{\beta},t)$.
Further properties of the value $v$ and the sets $\Phi$, $\Phi_{{\mathcal R}}$, and ${\Sigma}$ are summarized in Appendix \[sec:appendix\].
Convergence of Sets of Solutions
--------------------------------
In the next section we study convergence and stability of the residual method, that is, the behavior of the set of solutions ${\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)$ for ${\beta}_k \to {\beta}$, $y_k \to y$, and $\operatorname{\mathbf F}_k \to \operatorname{\mathbf F}$. In [@EngHanNeu96; @SchGraGroHalLen09], where convergence and stability of Tikhonov regularization have been investigated, the stability results are of the following form: For every sequence $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}} \to y$ and every sequence of minimizers $x_k \in \operatorname*{arg\,min}{\left\{ {{\left\lVert\operatorname{\mathbf F}(x)-y_k\right\rVert}}^2 + \alpha{\mathcal R}(x) \right\}}$ there exists a subsequence of $(x_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ that converges to a minimizer of ${{\left\lVert\operatorname{\mathbf F}(x)-y\right\rVert}}^2 + \alpha{\mathcal R}(x)$. In this paper we prove similar results for the residual method but with a different notation using a type of convergence of sets (see, for example, [@Kur66 §29]).
\[de:Limsup\] Let $\tau$ be a topology on ${X}$ and let $({\Sigma}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ be a sequence of subsets of ${X}$.
1. The *upper limit* of $({\Sigma}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ is defined as $$\operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k := \bigcap_{k\in{{\ensuremath{\mathbb{N}}}}} \left({\operatorname{\tau-cl}}\bigcup_{k'\ge k} {\Sigma}_{k'}\right)\;,$$ where ${\operatorname{\tau-cl}}$ denotes the closure with respect to $\tau$.
2. An element $x \in {X}$ is contained in the *lower limit* of the sequence $({\Sigma}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$, in short, $$x \in \operatorname*{\tau-Lim\,inf}_{k\to \infty} {\Sigma}_k \,,$$ if for every neighborhood $N$ of $x$ there exists $k\in{{\ensuremath{\mathbb{N}}}}$ such that $N\cap {\Sigma}_{k'} \neq \emptyset$ for every $k'\ge k$.
3. If the lower limit and the upper limit of $({\Sigma}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ coincide, we define $$\operatorname*{\tau-Lim}_{k\to\infty} {\Sigma}_k := \operatorname*{\tau-Lim\,inf}_{k\to \infty} {\Sigma}_k = \operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$$ as the *limit* of the sequence $({\Sigma}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$.
\[re:limsup\] As a direct consequence of Definition \[de:Limsup\], an element $x$ is contained in the upper limit $\operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$, if and only if for every neighborhood $N$ of $x$ and every $k \in {{\ensuremath{\mathbb{N}}}}$ there exists $k' \ge k$ such that $N\cap {\Sigma}_{k'} \neq \emptyset$.
Moreover, if ${X}$ satisfies the first axiom of countability, then $x \in \operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$, if and only if there exists a subsequence $({\Sigma}_{k_j})_{j\in{{\ensuremath{\mathbb{N}}}}}$ of $({\Sigma}_{k})_{k\in{{\ensuremath{\mathbb{N}}}}}$ and a sequence of elements $x_j \in {\Sigma}_{k_j}$ such that $x_j \to_\tau x$ (see [@Kur66 §29.IV]). Note that in particular every metric space satisfies the first axiom of countability.
The following proposition clarifies the relation between the stability and convergence results in [@EngHanNeu96; @SchGraGroHalLen09] and the results in the present paper.
Let $({\Sigma}_k)_{k \in {{\ensuremath{\mathbb{N}}}}}$ be a sequence of nonempty subsets of ${X}$, and assume that there exists a compact set $K$ such that ${\Sigma}_k \subset K$ for all $k \in {{\ensuremath{\mathbb{N}}}}$. Then $\operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$ is non-empty.
If, in addition, ${X}$ satisfies the first axiom of countability, then every sequence of elements $x_k \in {\Sigma}_k$ has a subsequence converging to some element $x \in \operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$.
By assumption, the sets $S_k := {\operatorname{\tau-cl}}\bigcup_{k'\ge k}{\Sigma}_k$ form a decreasing family of non-empty, compact sets. Thus also their intersection $\bigcap_{k\in{{\ensuremath{\mathbb{N}}}}} S_k = \operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$ is non-empty (see [@Kel55 Thm. 5.1]).
Now assume that ${X}$ satisfies the first axiom of countability. Then in particular every compact set is sequentially compact (see [@Kel55 Thm. 5.5]). Let now $x_k \in {\Sigma}_k$ for every $k\in{{\ensuremath{\mathbb{N}}}}$. Then $(x_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ is a sequence in the compact set $K$ and therefore has a subsequence $(x_{k_j})_{j\in{{\ensuremath{\mathbb{N}}}}}$ converging to some element $x \in K$. From Remark \[re:limsup\] it follows that $x \in \operatorname*{\tau-Lim\,sup}_{k\to\infty} {\Sigma}_k$, which shows the assertion.
Convergence of the Data
-----------------------
In addition to the convergence of subsets ${\Sigma}_k$ of ${X}$, it is necessary to define a notion of convergence on the set ${Y}$ that is compatible with the distance measure ${\mathcal S}$.
\[de:s\_uniform\] The sequence $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}} \subset {Y}$ converges *${\mathcal S}$-uniformly* to $y \in {Y}$, if $$\sup{\left\{{\left\lvert{\mathcal S}(z,y_k)-{\mathcal S}(z,y)\right\rvert}:z \in {Y}\right\}} \to 0\;.$$ The sequence of mappings $\operatorname{\mathbf F}_k \colon {X}\to {Y}$ converges *locally ${\mathcal S}$-uniformly* to $\operatorname{\mathbf F}\colon {X}\to {Y}$, if $$\sup{\left\{{\left\lvert{\mathcal S}(\operatorname{\mathbf F}_k(x),y)-{\mathcal S}(\operatorname{\mathbf F}(x),y)\right\rvert}: y \in {Y},\, x \in {X},\, {\mathcal R}(x) \le t\right\}} \to 0$$ for every $t \ge 0$.
The ${\mathcal S}$-uniform convergence on ${Y}$ is induced by the extended metric $${\mathcal S}_1(y_1,y_2) := \sup{\left\{{\left\lvert{\mathcal S}(z,y_1)-{\mathcal S}(z,y_2)\right\rvert} : z \in {Y}\right\}} \,.$$ If the distance measure ${\mathcal S}$ itself equals a metric, then ${\mathcal S}_1$ coincides with ${\mathcal S}$. Similarly, local ${\mathcal S}$-uniform convergence of a sequence of mappings $\operatorname{\mathbf F}_k$ equals the uniform convergence of $\operatorname{\mathbf F}_k$ on ${\mathcal R}$-bounded sets with respect to the extended metric $${\mathcal S}_2(y_1,y_2) := \sup{\left\{ {\left\lvert{\mathcal S}(y_1,z)-{\mathcal S}(y_2,z)\right\rvert}:z \in {Y}\right\}} \,.$$
Well-posedness of the Residual Method {#sec:wp}
=====================================
In the following we investigate the existence of minimizers, and the stability and the convergence of the residual method. Throughout the whole section we assume that $\tau$ is a topology on ${X}$, $\operatorname{\mathbf F}\colon {X}\to {Y}$ is a mapping, $y \in {Y}$ are given data and ${\beta}\ge 0$ is a fixed parameter.
Existence
---------
We first investigate under which conditions ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$, the set of solutions of , is not empty.
\[th:existence\] Assume that $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t)$ is $\tau$-compact for every $t\geq 0$ and non-empty for some $t_0 \geq 0$. Then Problem has a solution.
Equation and Lemma \[le:Phi\_prop\] imply the identity $${\Sigma}(\operatorname{\mathbf F},y,{\beta}) = \Phi_{\mathcal R}{\left(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})\right)}
=
\bigcap_{{\varepsilon}> 0} \Phi_{\mathcal R}{\left(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})+{\varepsilon}\right)} \,.$$ Because $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t_0) \neq \emptyset $, the value of satisfies $v(\operatorname{\mathbf F},y,{\beta}) \leq t_0 < \infty$ and therefore $\emptyset \neq \Phi_{\mathcal R}\left(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})+{\varepsilon}\right)$ for every ${\varepsilon}>0$. Consequently, ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ is the intersection of a decreasing family of non-empty $\tau$-compact sets and thus non-empty (see [@Kel55 Thm. 5.1]).
Recall that a mapping $\mathcal{F}\colon {X}\to [0,+\infty]$ is *lower semi-continuous*, if its lower level sets ${\left\{x \in {X}:\mathcal{F}(x) \le t\right\}}$ are closed for every $t \ge 0$. Moreover, the mapping $\mathcal{F}$ is *coercive*, if its lower level sets are pre-compact, see [@Bra02]. (In a Banach space one often calls a functional coercive, if it is unbounded on unbounded sets. The notion used here is equivalent if the Banach space is reflexive, ${\mathcal R}$ is the norm on ${X}$, and $\tau$ is the weak topology.)
\[prop:compact\] Assume that ${\mathcal R}$ and $x \mapsto {\mathcal S}(\operatorname{\mathbf F}(x),y)$ are lower semi-continuous and one of them, or their sum, is coercive. Then $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t)$ is $\tau$-compact for every $t\geq 0$. In particular, Problem has a solution.
If ${\mathcal R}$ and $x \mapsto {\mathcal S}(\operatorname{\mathbf F}(x), y)$ are lower semi-continuous and one of them is coercive, then $$\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t) = {\left\{x: {\mathcal S}(\operatorname{\mathbf F}(x), y) \leq {\beta}\right\}} \cap {\left\{x : {\mathcal R}(x) \le t\right\}}$$ is the intersection of a closed and a $\tau$-compact set and therefore itself $\tau$-compact. In case that only the sum $x \mapsto {\mathcal S}(\operatorname{\mathbf F}(x), y) + {\mathcal R}(x)$ is coercive, the set $$\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t) = {\left\{x: {\mathcal S}(\operatorname{\mathbf F}(x), y) \leq {\beta}\right\}} \cap {\left\{x : {\mathcal R}(x) \le t\right\}} \subset
\{x: {\mathcal S}(\operatorname{\mathbf F}(x), y) + {\mathcal R}(x) \leq {\beta}+t\}$$ is a closed set contained in a $\tau$-compact set and therefore again $\tau$-compact.
The lower semi-continuity of $x \mapsto {\mathcal S}\left(\operatorname{\mathbf F}(x),y\right)$ certainly holds if $\operatorname{\mathbf F}$ is continuous and ${\mathcal S}$ is lower semi-continuous with respect to the first component (for some given topology on ${Y}$). It is, however, also possible to obtain lower semi-continuity, if $\operatorname{\mathbf F}$ is not continuous but the functional ${\mathcal S}$ satisfies a stronger condition:
\[le:Slsc\] Let $\tau'$ be a topology on ${Y}$ such that $z \mapsto {\mathcal S}(z,y)$ is lower semi-continuous and coercive, and assume that $\operatorname{\mathbf F}\colon {X}\to {Y}$ has a closed graph. Then the functional $x \mapsto {\mathcal S}(\operatorname{\mathbf F}(x),y)$ is lower semi-continuous.
Because $\operatorname{\mathbf F}$ has a closed graph, the pre-image under $\operatorname{\mathbf F}$ of every compact set is closed (see [@Iva69 Thm. 4]). This shows that $${\left\{x \in {X}:{\mathcal S}(\operatorname{\mathbf F}(x),y) \le {\beta}\right\}} = \operatorname{\mathbf F}^{-1}\left({\left\{z \in {Y}:{\mathcal S}(z,y) \le {\beta}\right\}}\right)$$ is closed for every ${\beta}$, that is, the mapping $x \mapsto {\mathcal S}(\operatorname{\mathbf F}(x),y)$ is lower semi-continuous.
Stability
---------
Stability is concerned with the continuous dependence of the solutions of of the input data, that is, the element $y$, the parameter ${\beta}$, and, possibly, the operator $\operatorname{\mathbf F}$. Given sequences ${\beta}_k \to {\beta}$, $y_k \to y$, and $\operatorname{\mathbf F}_k \to \operatorname{\mathbf F}$, we ask whether the sequence of sets ${\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)$ converges to ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$. As already indicated in Section \[sec:defs\], we will make use of the upper convergence of sets introduced in Definition \[de:Limsup\]. The topology, however, with respect to which the results are formulated, is stronger than $\tau$.
\[de:tauR\] The topology $\tau_{\mathcal R}$ on ${X}$ is generated by all sets of the form $U \cap \{x \in {X}:s < {\mathcal R}(x) < t\}$ with $U \in \tau$ and $s < t \in {{\ensuremath{\mathbb{R}}}}$.
Note that a sequence $(x_k)_{k\in{{\ensuremath{\mathbb{N}}}}} \subset {X}$ converges to $x$ with respect to $\tau_{\mathcal R}$, if and only if $(x_k)_{k\in{{\ensuremath{\mathbb{N}}}}} $ converges to $x$ with respect to $\tau$ and satisfies ${\mathcal R}(x_k) \to {\mathcal R}(x)$.
For the stability results we make the following assumption:
\[as:stability\]
1. Let ${\beta}\geq 0$, let $y \in {Y}$, and let $\operatorname{\mathbf F}\colon {X}\to {Y}$ be a mapping.
2. Let $({\beta}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ be a sequence of nonnegative numbers, let $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ be a sequence in ${Y}$, and let $(\operatorname{\mathbf F}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ be a sequence of mappings $\operatorname{\mathbf F}_k \colon {X}\to {Y}$.
3. The sequence $({\beta}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges to ${\beta}$, the sequence $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges ${\mathcal S}$-uniformly to $y$, and $(\operatorname{\mathbf F}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges locally ${\mathcal S}$-uniformly to $\operatorname{\mathbf F}$.
4. The sets $\Phi_{\mathcal R}(\operatorname{\mathbf F}_k,w,\gamma,t)$ and $\Phi_{\mathcal R}(\operatorname{\mathbf F},w,\gamma,t)$ are compact for all $w$, $\gamma$, $t$, and $k$, and non-empty for some $t$.
The following lemma is the key result to prove stability of the residual method.
\[le:stability\] Let Assumption \[as:stability\] hold and assume that $$\label{eq:stability:V}
\limsup_{k\to\infty} v(\operatorname{\mathbf F}_k,y_k,{\beta}_k) \le v(\operatorname{\mathbf F},y,{\beta}) < \infty\,.$$ Then, $$\label{eq:stability:lim}
\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,{\beta})\;.$$ If, additionally, the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x_{\beta}$, then $$\label{eq:stability:lim_unique}
\{x_{\beta}\} = \operatorname*{\tau_{{\mathcal R}}-Lim}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)\;.$$
In order to simplify the notation, we define $$\begin{aligned}
\Phi_k(t) &:= \Phi_{\mathcal R}(\operatorname{\mathbf F}_k,y_k,{\beta}_k,t)\,,
& \Phi(t) &:= \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t)\,,\\
v_k &:= v(\operatorname{\mathbf F}_k,y_k,{\beta}_k)\,,
& v & := v(\operatorname{\mathbf F},y,{\beta})\,,\\
{\Sigma}_k &:= {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)\,,
& {\Sigma}&:= {\Sigma}(\operatorname{\mathbf F},y,{\beta})\;.
\end{aligned}$$ Moreover we define the set $T := \operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k$. Because the topology $\tau_{\mathcal R}$ is finer than $\tau$, it follows that $\operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}_k \subset T$. We proceed by showing that $\emptyset \neq T \subset {\Sigma}$ and $T \subset \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}_k$, which then gives the assertion .
The inequality implies that for every ${\varepsilon}> 0$ there exists some $k_0 \in {{\ensuremath{\mathbb{N}}}}$ such that $v_k \le v + {\varepsilon}$ for all $k \ge k_0$. Since ${\beta}_k \to {\beta}$, we may additionally assume that ${\beta}_k \le {\beta}+{\varepsilon}$. Lemma \[le:Phiincl2\] implies, after possibly enlarging $k_0$, $$\label{eq:stability:h1}
\Phi_k(v_k)
\subset \Phi_{\mathcal R}(\operatorname{\mathbf F}_k,y_k,{\beta}+{\varepsilon},v_k)
\subset \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon},v_k)
\subset \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon},v+{\varepsilon})$$ for all $k \ge k_0$. Thus, $$\label{eq:stability:h1a}
T = \operatorname*{\tau-Lim\,sup}_{k\to \infty} {\Sigma}_k
= \bigcap_{k \in {{\ensuremath{\mathbb{N}}}}} \left({\operatorname{\tau-cl}} \bigcup_{k'\ge k} {\Sigma}_{k'}\right)
= \bigcap_{k \ge k_0} \left({\operatorname{\tau-cl}} \bigcup_{k'\ge k} \Phi_{k'}(v_{k'})\right)
\subset \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon},v+{\varepsilon})\;.$$ The sets ${\operatorname{\tau-cl}}\bigcup_{k'\ge k} {\Sigma}_{k'}$ are closed and non-empty and, by assumption, the set $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon},v+{\varepsilon})$ is compact. Thus $T$ is the intersection of a decreasing family of non-empty compact sets and therefore non-empty. Moreover, because holds for every ${\varepsilon}> 0$, we have $$\label{eq:stability:h1b}
\emptyset\neq T
\subset \bigcap_{{\varepsilon}> 0} \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon},v+{\varepsilon})
= \Phi(v)
= {\Sigma}\;.$$
Next we show the inclusion $T \subset \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}_k$. To that end, we first prove that $$\label{eq:stability:limV}
v = \lim_k v_k\;.$$ Recall that Theorem \[th:existence\] implies that $\Phi_k(v_k) = {\Sigma}_k$ is non-empty. Therefore, implies that also $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon},v_k)$ is non-empty, which in turn shows that $v_k \ge v(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon})$ for all $k$ large enough. Consequently, $$\label{eq:stability:h2}
\liminf_{k\to \infty} v_k \ge v(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon})$$ for all ${\varepsilon}> 0$. From Lemma \[le:Hlimit\] we obtain that $v = \sup_{{\varepsilon}> 0} v(\operatorname{\mathbf F},y,{\beta}+2{\varepsilon})$. Together with and this shows .
Let now $x \in T$, let $N$ be a neighborhood of $x$ with respect to $\tau$, let ${\beta}> 0$ and $k_0 \in {{\ensuremath{\mathbb{N}}}}$. Since $T \subset {\Sigma}$ (see ), it follows that ${\mathcal R}(x) = v$. Thus it follows from that there exists $k_1 \ge k_0$ such that $${\left\lvertv_k - {\mathcal R}(x)\right\rvert} < {\beta}$$ for all $k \ge k_1$. In particular, $$\label{eq:stability:N}
{\Sigma}_k \subset {\left\{\tilde{x}\in {X}:{\mathcal R}(x)-{\beta}< {\mathcal R}(\tilde{x}) < {\mathcal R}(x)+{\beta}\right\}}$$ for all $k \ge k_1$. Remark \[re:limsup\] implies that there exists $k_2 \ge k_1$ such that $$\label{eq:stability:V_old}
N \cap {\Sigma}_{k_2} \neq \emptyset\;.$$ Now recall that the sets $N \cap {\left\{\tilde{x}\in {X}:{\mathcal R}(x)-{\beta}< {\mathcal R}(\tilde{x}) < {\mathcal R}(x)+{\beta}\right\}}$ form a basis of neighborhoods of $x$ for the topology $\tau_{\mathcal R}$. Therefore , , and the characterization of the upper limit of sets given in Remark \[re:limsup\] imply that $x \in \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}_k$. Thus the inclusion follows.
If the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x_{\beta}$, then the first part of the assertion implies that for every subsequence $(k_j)_{j\in{{\ensuremath{\mathbb{N}}}}}$ we have $$\operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{j\to\infty} {\Sigma}(\operatorname{\mathbf F}_{k_j},y_{k_j},{\beta}_{k_j}) = \{x_{\beta}\}\;.$$ Thus the assertion follows from Lemma \[le:subseq\].
The crucial condition in Lemma \[le:stability\] is the inequality . Indeed, one can easily construct examples, where this condition fails and the solution of Problem is unstable, see Example \[ex:stability\] below. What happens in this example is that the upper limit $\operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta})$ consists of local minima of ${\mathcal R}$ on $\Phi(\operatorname{\mathbf F},y,{\beta})$ that fail to be global minima of ${\mathcal R}$ restricted to $\Phi(\operatorname{\mathbf F},y,{\beta})$.
![The nonlinear function $\operatorname{\mathbf F}$ from Example \[ex:stability\]. The set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of an interval and the isolated point $\{0\}$.](example-stability.eps){width="60.00000%"}
\[ex:stability\] Consider the function $\operatorname{\mathbf F}\colon {{\ensuremath{\mathbb{R}}}}\to {{\ensuremath{\mathbb{R}}}}$, $\operatorname{\mathbf F}(x) = x^3 - x^2$, and the regularization functional ${\mathcal R}(x) = x^2$. Let $y > 0$ and choose ${\beta}= y$. Then $$\label{eq:stab_ex}
\operatorname*{arg\,min}{\left\{{\mathcal R}(x):{\left\lvert\operatorname{\mathbf F}(x)-y\right\rvert} \le {\beta}\right\}}
= \operatorname*{arg\,min}{\left\{x^2:{\left\lvertx^3-x^2-y\right\rvert} \le y\right\}}
= 0\;.$$ Now let $y_k > y$. Then $$\operatorname*{arg\,min}{\left\{{\mathcal R}(x):{\left\lvert\operatorname{\mathbf F}(x)-y_k\right\rvert} \le {\beta}\right\}}
= \operatorname*{arg\,min}{\left\{x^2:{\left\lvertx^3-x^2-y_k\right\rvert} \le y\right\}}
= x_k\,,$$ where $x_k$ is the unique solution of the equation $\operatorname{\mathbf F}(x) = y_k - y$. Thus, if the sequence $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges to $y$ from above, we have $x_k > 1$ for all $k$ and $\lim_{k\to \infty} x_k = 1$. According to , however, the solution of the limit problem equals zero.
The following two theorems are central results of this paper. They answer the question to which extent we obtain stability results for the residual method similar to the ones known for Tikhonov regularization.
\[thm:ap\_stability\] Let Assumption \[as:stability\] hold. Then there exists a sequence ${\varepsilon}_k \to 0$ such that $$\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k+{\varepsilon}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,{\beta})\;.$$
Define $${\varepsilon}_k := \inf{\left\{{\varepsilon}> 0: \Phi_{\mathcal R}\left(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})\right)
\subset \Phi_{\mathcal R}\left(\operatorname{\mathbf F}_k,y_k,{\beta}_k+{\varepsilon},v(\operatorname{\mathbf F},y,{\beta})\right)\right\}}\;.$$ Lemma \[le:Phiincl2\] and the assumption that ${\beta}_k \to {\beta}$ imply that ${\varepsilon}_k \to 0$. Since by assumption $$\emptyset \neq {\Sigma}(\operatorname{\mathbf F},y,{\beta})
= \Phi_{\mathcal R}\left(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})\right)
\subset \Phi_{\mathcal R}\left(\operatorname{\mathbf F}_k,y_k,{\beta}_k+{\varepsilon}_k,v(\operatorname{\mathbf F},y,{\beta})\right)\,,$$ we obtain that $v(\operatorname{\mathbf F}_k,y_k,{\beta}_k+{\varepsilon}_k) \le v(\operatorname{\mathbf F},y,{\beta})$. Thus the assertion follows from Lemma \[le:stability\].
Theorem \[thm:ap\_stability\], is a stability result in the same spirit as the one derived in [@Hei08]. While it does not assert that, in the general setting described by Assumption \[as:stability\], the residual method is stable in the sense that the solutions depend continuously on the input data, it does state that the solutions of the perturbed problems stay close to the solution of the original problem, if one allows the regularization parameter ${\beta}$ to increase slightly. Apart from the more general, topological setting, the main difference to [@Hei08 Lemma 2.2] is the additional inclusion of operator errors into the result.
The next theorem provides a true stability theorem, including both data as well as operator perturbations.
\[thm:stability:2\] Let Assumption \[as:stability\] hold with $\beta > 0$ and assume that the inclusion $$\label{eq:stability2:lim}
\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t) \subset
\bigcap_{{\beta}> 0} \left({\operatorname{\tau-cl}}\bigcup_{{\varepsilon}> 0} \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t+{\beta})\right)$$ holds for every $t \ge 0$. Then, $$\label{eq:stability2:Limsup}
\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,{\beta})\;.$$ If, additionally, the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x_{\beta}$, then $$\{x_{\beta}\} = \operatorname*{\tau_{{\mathcal R}}-Lim}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)\;.$$
The convergence of $({\beta}_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ to ${\beta}$ and Lemma \[le:Phiincl2\] imply that for every ${\varepsilon}> 0$ and $t \in {{\ensuremath{\mathbb{R}}}}$ there exists $k_0 \in {{\ensuremath{\mathbb{N}}}}$ such that $$\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t)
\subset \Phi_{\mathcal R}(\operatorname{\mathbf F}_k,y_k,{\beta}_k,t)$$ for all $k \ge k_0$. Consequently, $$\begin{gathered}
\limsup_{k\to\infty} v(\operatorname{\mathbf F}_k,y_k,{\beta}_k)
= \limsup_{k\to\infty} \, \Bigl( \inf{\left\{t:\Phi_{\mathcal R}(\operatorname{\mathbf F}_k,y_k,{\beta}_k,t) \neq \emptyset\right\}} \Bigr) \\
\le \inf_{{\varepsilon}> 0} \, \Bigl( \inf {\left\{t:\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t) \neq\emptyset\right\}} \Bigr) \;.
\end{gathered}$$ From we obtain that $$\inf_{{\varepsilon}> 0} \, \Bigl( \inf {\left\{t:\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t) \neq\emptyset\right\}} \Bigr)
\le \inf{\left\{t:\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t)\neq\emptyset\right\}} = v(\operatorname{\mathbf F},y,{\beta})\;.$$ This shows . Now follows from Lemma \[le:stability\].
For Theorem \[thm:stability:2\] to hold, the mapping $x \mapsto {\mathcal S}\bigl(\operatorname{\mathbf F}(x),y\bigr)$ has to satisfy the additional regularity property . This property requires that every $x \in {X}$ for which $\operatorname{\mathbf F}(x) \neq y$ can be approximated by elements $\tilde{x}$ with ${\mathcal S}\bigl(\operatorname{\mathbf F}(\tilde{x}),y\bigr) < {\mathcal S}\bigl(\operatorname{\mathbf F}(x),y\bigr)$ and ${\mathcal R}(\tilde{x}) \le {\mathcal R}(x) + {\beta}$. That is, the function $x \mapsto {\mathcal S}\bigl(\operatorname{\mathbf F}(x),y\bigr)$ does not have local minima in the sets ${\left\{x \in {X}:{\mathcal R}(x)< t\right\}}$. As will be shown in the following Section \[se:linear\], this property is naturally satisfied for linear operators on Banach spaces.
Convergence
-----------
The following theorem states the solutions obtained with the residual method indeed converge to the ${\mathcal R}$-minimizing solution of the equation $F(x) = y$, if the noise level decreases to zero. Recall the set of all ${\mathcal R}$-minimizing solution of the equation $F(x) = y$ is given by ${\Sigma}(\operatorname{\mathbf F},y,0)$.
\[thm:convergence\] Let $y \in {Y}$ be such that there exists $x \in {X}$ with $\operatorname{\mathbf F}(x) = y$ and ${\mathcal R}(x) < \infty$ and assume that $\Phi_{\mathcal R}(\operatorname{\mathbf F},w,\gamma, t)$ is $\tau$-compact for all $w \in {Y}$ and $\gamma, t \geq 0$. If $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges ${\mathcal S}$-uniformly to $y$ and satisfies ${\mathcal S}( y, y_k) \le {\beta}_k \to 0$, then $$\label{eq:convergence_V}
\limsup_{k\to\infty} v(\operatorname{\mathbf F},y_k,{\beta}_k) \le v(\operatorname{\mathbf F},y,0) < \infty\;.$$ In particular, $$\label{eq:convergence}
\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,0)\;.$$ If, additionally, the ${\mathcal R}$-minimizing solution $x^\dagger$ is unique, then $$\label{eq:convergence_2}
\{x^\dagger\} = \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k)\;.$$
By assumption $S(y,y_k) \le {\beta}_k$, which implies that $v(\operatorname{\mathbf F},y_k,{\beta}_k) \le {\mathcal R}(x')$ for all $x'\in \Phi(\operatorname{\mathbf F},y,0)$. This proves . Now and follow from Lemma \[le:stability\].
Linear Spaces {#se:linear}
=============
Now we assume that ${X}$ and ${Y}$ are subsets of topological vector spaces. Then the linear structures allows us to introduce more tangible conditions implying stability of the residual method.
For the following we assume that $\operatorname{\mathbf F}\colon {X}\to {Y}$ and $y \in {Y}$ are fixed.
\[as:linear\] Assume that the following hold:
1. \[it:linear:X\] The set ${X}$ is a convex subset of a topological vector space, and ${Y}$ is a topological vector space.
2. \[it:linear:S\] For all $x_0$, $x_1 \in {X}$ with ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr)$, ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_1),y\bigr) < \infty$, and all $0 < \lambda < 1$ we have $${\mathcal S}\bigl(\operatorname{\mathbf F}(\lambda x_0 + (1-\lambda) x_1),y\bigr)
\le \max\bigl\{ {\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr), {\mathcal S}\bigl(\operatorname{\mathbf F}(x_1),y\bigr)\bigr\}\;.$$ Moreover, the inequality is strict whenever ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr) \neq {\mathcal S}\bigl(\operatorname{\mathbf F}(x_1),y\bigr)$.
3. \[it:linear:domain\] For every ${\beta}\ge 0$ there exists $x \in {X}$ with ${\mathcal S}\bigl(\operatorname{\mathbf F}(x),y\bigr) \le {\beta}$ and ${\mathcal R}(x) < \infty$.
4. \[it:linear:R\] The domain $\operatorname{dom}{\mathcal R}= {\left\{x \in {X}:{\mathcal R}(x) < +\infty\right\}}$ of ${\mathcal R}$ is convex and for every $x_0,\,x_1 \in \operatorname{dom}{\mathcal R}$ the restriction of ${\mathcal R}$ to $$L = {\left\{ \lambda x_0 + (1-\lambda) x_1:0 \le \lambda \le 1\right\}}$$ is continuous.
We now show that Assumption \[as:linear\] implies the main condition of the stability result Theorem \[thm:stability:2\], the inclusion :
\[le:linear:1\] Assume that Assumption \[as:linear\] holds. Then is satisfied.
Let $x_0 \in \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t)$ for some ${\beta}> 0$. We have to show that for every neighborhood $N \subset {X}$ of $x_0$ and every ${\beta}> 0$ there exist ${\varepsilon}> 0$ and $x' \in N$ such that $x' \in \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t+{\beta})$.
Item \[it:linear:domain\] in Assumption \[as:linear\] implies the existence of some $x_1 \in {X}$ satisfying the inequalities ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_1),y\bigr) \le {\beta}/2$ and ${\mathcal R}(x_1) < \infty$. Since we have ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_1),y\bigr) < {\beta}$ and ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr) \le {\beta}$, we obtain from Item \[it:linear:S\] that ${\mathcal S}(\operatorname{\mathbf F}(x),y) < {\beta}$ for every $x \in L := {\left\{\lambda x_0 + (1-\lambda)x_1:0 \le \lambda < 1\right\}}$. Since $x_0$, $x_1 \in \operatorname{dom}{\mathcal R}$, it follows from Item \[it:linear:R\] that ${\mathcal R}$ is continuous on $L$. Consequently $\lim_{\lambda \to 1} {\mathcal R}(\lambda x_0 + (1-\lambda) x_1) = {\mathcal R}(x_0) \le t$. In particular, there exists $\lambda_0 < 1$ such that ${\mathcal R}(\lambda x_0 + (1-\lambda)x_1) \le t+{\beta}$ for all $1 > \lambda > \lambda_0$. Since ${X}$ is a topological vector space (Item \[it:linear:X\]), it follows that $x':=\lambda x_0 + (1-\lambda) x_1 \in N$ for some $1 > \lambda > \lambda_0$. This shows the assertion with ${\varepsilon}:= {\beta}- {\mathcal S}\bigl(\operatorname{\mathbf F}(x'),y\bigr) > 0$.
Lemma \[le:linear:1\] allows us to apply the stability result Theorem \[thm:stability:2\], which shows that Assumption \[as:linear\] implies the continuous dependence of the solutions of on the data $y$ and the regularization parameter ${\beta}$.
\[pr:linear\_stab\] Let Assumption \[as:linear\] hold and assume that the sets $\Phi_{\mathcal R}(\operatorname{\mathbf F},\gamma,w,t)$ are compact for every ${\beta}\ge 0$, $t \in {{\ensuremath{\mathbb{R}}}}$, and $w \in {Y}$. Assume moreover that $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges ${\mathcal S}$-uniformly to $y \in {Y}$, and that ${\beta}_k \to {\beta}$. If ${\beta}= 0$, assume in addition that ${\mathcal S}(y,y_k) \le {\beta}_k$. Then $$\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,{\beta})\;.$$ If, additionally, the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x_{\beta}$, then $$\{x_{\beta}\} = \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k)\;.$$
If ${\beta}= 0$, the assertion follows from Theorem \[thm:convergence\]. In the case ${\beta}> 0$, Lemma \[le:linear:1\] implies that holds. Thus, the assertion follows from Theorem \[thm:stability:2\].
\[pr:linear\_stab2\] Let Assumption \[as:linear\] hold. Assume that $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges ${\mathcal S}$-uniformly to $y \in {Y}$, the mappings $\operatorname{\mathbf F}_k\colon {X}\to {Y}$ converge locally ${\mathcal S}$-uniformly to $\operatorname{\mathbf F}\colon {X}\to {Y}$ (see Definition \[de:s\_uniform\]), and ${\beta}_k \to {\beta}> 0$. Assume that the sets ${\left\{x \in {X}:{\mathcal R}(x) \le t\right\}}$, $\Phi_{\mathcal R}(\operatorname{\mathbf F}_k,\gamma,w,t)$ and $\Phi_{\mathcal R}(\operatorname{\mathbf F},\gamma,w,t)$ are compact for every $\gamma \ge 0$, $t \in {{\ensuremath{\mathbb{R}}}}$, and $w \in {Y}$. Then $$\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,{\beta})\;.$$ If, additionally, the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x_{\beta}$, then $$\{x_{\beta}\} = \operatorname*{\tau_{{\mathcal R}}-Lim}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)\;.$$
Again, Lemma \[le:linear:1\] shows that holds. Thus the assertion follows from Theorem \[thm:stability:2\].
Item \[it:linear:S\] in Assumption \[as:linear\] is concerned with the interplay of the functional $\operatorname{\mathbf F}$ and the distance measure ${\mathcal S}$. The next two examples consider two situations, where this part of the assumption holds. Example \[ex:linear1\] considers linear operators $\operatorname{\mathbf F}$ and convex distance measures ${\mathcal S}$. Example \[ex:stab\_nonlinear\] introduces a class of *non-linear* operators on Hilbert spaces, where Item \[it:linear:S\] is satisfied if the distance measure equals the squared Hilbert space norm.
\[ex:linear1\] Assume that $\operatorname{\mathbf F}\colon {X}\to {Y}$ is linear and ${\mathcal S}$ is convex in its first component. Then Item \[it:linear:S\] in Assumption \[as:linear\] is satisfied. Indeed, in such a situation, $$\begin{gathered}
{\mathcal S}\bigl(\operatorname{\mathbf F}(\lambda x_0 + (1-\lambda)x_1),y\bigr)
= {\mathcal S}\bigl(\lambda \operatorname{\mathbf F}(x_0) + (1-\lambda)\operatorname{\mathbf F}(x_1),y\bigr)\\
\le \lambda {\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr) + (1-\lambda){\mathcal S}\bigl(\operatorname{\mathbf F}(x_1),y\bigr)
\le \max\bigl\{{\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr), {\mathcal S}\bigr(\operatorname{\mathbf F}(x_1),y\bigr)\bigr\}\;.
\end{gathered}$$ If moreover, ${\mathcal S}\bigl(\operatorname{\mathbf F}(x_0),y\bigr) \neq {\mathcal S}\bigr(\operatorname{\mathbf F}(x_1),y\bigr)$ and $0 < \lambda < 1$, then the last inequality is strict.
\[ex:stab\_nonlinear\] Assume that ${Y}$ is a Hilbert space, ${\mathcal S}(y,z) = {{\left\lVerty-z\right\rVert}}^2$, and $\operatorname{\mathbf F}\colon {X}\to {Y}$ is two times Gâteaux differentiable. Then Item \[it:linear:S\] in Assumption \[as:linear\] is equivalent to the assumption that for all $x_0$, $x_1 \in {X}$ the mapping $$t \mapsto \varphi(t;x_0,x_1) := {{\left\lVert\operatorname{\mathbf F}(x_0 + tx_1) - y\right\rVert}}^2$$ has no local maxima. This condition holds, if the inequality $\partial_t^2 \varphi(0;x_0,x_1) > 0$ is satisfied whenever $\partial_t \varphi(0;x_0,x_1) = 0$. The computation of the derivative of $\varphi(\,\cdot\,;x_0,x_1)$ at zero yields that $$\partial_t \varphi(0;x_0,x_1) = 2{\left\langle\operatorname{\mathbf F}'(x_0)(x_1),\operatorname{\mathbf F}(x_0)\right\rangle}$$ and $$\partial_t^2 \varphi(0;x_0,x_1) = 2{\left\langle\operatorname{\mathbf F}''(x_0) (x_1;x_1),\operatorname{\mathbf F}(x_0)\right\rangle} + 2{{\left\lVert\operatorname{\mathbf F}'(x_0)x_1\right\rVert}}^2\;.$$
Consequently, Item \[it:linear:S\] in Assumption \[as:linear\] is satisfied if, for every $x_0$, $x_1 \in {X}$ with $x_1 \neq 0$, the equality ${\left\langle\operatorname{\mathbf F}'(x_0)(x_1),\operatorname{\mathbf F}(x_0)\right\rangle} = 0$ implies that $${\left\langle\operatorname{\mathbf F}''(x_0) (x_1;x_1),\operatorname{\mathbf F}(x_0)\right\rangle} + {{\left\lVert\operatorname{\mathbf F}'(x_0)(x_1)\right\rVert}}^2 > 0\;.$$
Regularization on $L^p$-spaces
------------------------------
Let $p \in (1,\infty)$ and set ${X}= L^p(\Omega,\mu)$ for some $\sigma$-finite measure space $(\Omega,\mu)$. Assume that ${Y}$ is a Banach space and $\operatorname{\mathbf F}\colon {X}\to {Y}$ is a bounded linear operator with dense range. Let ${\mathcal R}(x) = {{\left\lVertx\right\rVert}}_p^p$ and ${\mathcal S}(w,y) = {{\left\lVertw-y\right\rVert}}$. We thus consider the minimization problem $${{\left\lVertx\right\rVert}}_p^p \to \min
\qquad\text{ subject to }\quad
{{\left\lVert\operatorname{\mathbf F}x - y\right\rVert}} \le {\beta}\;.$$
We now show that in this situation the assumptions of Proposition \[pr:linear\_stab\] are satisfied. To that end, let $\tau$ be the weak topology on $L^p(\Omega,\mu)$. As $L^p(\Omega,\mu)$ is reflexive, the level sets ${\left\{x \in {X}:{\mathcal R}(x) \le t\right\}}$ are weakly compact. Moreover, the mapping $x \mapsto {{\left\lVert\operatorname{\mathbf F}x-y\right\rVert}}$ is weakly lower semi-continuous. Thus all the sets $\Phi_{\mathcal R}(\operatorname{\mathbf F},\gamma,w,t)$ are weakly compact. Example \[ex:linear1\] shows that Item \[it:linear:S\] in Assumption \[as:linear\] holds. Item \[it:linear:domain\] follows from the density of the range of $\operatorname{\mathbf F}$. Finally, Item \[it:linear:R\] holds, because ${\mathcal R}$ is norm continuous and convex.
Now assume that $y_k \to y$ and ${\beta}_k \to {\beta}$. If ${\beta}= 0$ assume in addition that ${{\left\lVerty_k-y\right\rVert}} \le {\beta}_k$. The strict convexity of ${\mathcal R}$ and convexity of the mappings $x \mapsto {{\left\lVert\operatorname{\mathbf F}x-y_k\right\rVert}}$ imply that each set ${\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k)$ consists of a single element $x_k$. Similarly, ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x^\dagger$. From Proposition \[pr:linear\_stab\] we now obtain that $(x_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ weakly converges to $x^\dagger$ and ${{\left\lVertx_k\right\rVert}}_p^p \to {{\left\lVertx^\dagger\right\rVert}}_p^p$. Thus, in fact, the sequence $(x_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ strongly converges to $x^\dagger$ (see [@Meg98 Cor. 5.2.19]).
Let ${\beta}> 0$ and assume that $\operatorname{\mathbf F}_k\colon {X}\to {Y}$ is a sequence of bounded linear operators converging to $\operatorname{\mathbf F}$ with respect to the strong topology on $L({X},{Y})$, that is, $\sup\{{{\left\lVert\operatorname{\mathbf F}_k x - \operatorname{\mathbf F}x\right\rVert}}:{{\left\lVertx\right\rVert}} \le 1\} \to 0$. Let again ${\beta}_k \to {\beta}$ and $y_k \to y$, and denote by $x_k$ the single element in ${\Sigma}(\operatorname{\mathbf F}_k,y_k,{\beta}_k)$. Applying Proposition \[pr:linear\_stab2\], we again obtain that $x_k \to x^\dagger$.
The above results rely heavily on the assumption that $p > 1$, which implies that the space $L^p(\Omega,\mu)$ is reflexive. In the case ${X}= L^1(\Omega,\mu)$, the level sets ${\left\{x \in {X}:{{\left\lVertx\right\rVert}}_1 \le t\right\}}$ fail to be weakly compact, and thus even the existence of a solution of Problem need not hold.
The assertions concerning stability and convergence with respect to the norm topology remain valid, if ${X}$ is any uniformly convex Banach space and ${\mathcal R}$ the norm on ${X}$ to some power $p > 1$. Also in this case, weak convergence and convergence of norms imply the strong convergence of a sequence [@Meg98 Thm. 5.2.18]. More generally, this property is called the *Radon–Riesz property* [@Meg98 p. 453]. Spaces satisfying this property are also called *Efimov–Stechkin* spaces in [@Tan97].
Regularization of Probability Measures {#ss:prob}
--------------------------------------
Let $(\Omega, d)$ be a separable, complete metric space with distance $d$ and denote by $\mathcal{P}(\Omega)$ the space of probability measures on the Borel sets of $\Omega$. That is, $\mathcal{P}(\Omega)$ consists of all positive Borel measures $\mu$ on $\Omega$ that satisfy $\mu(\Omega) = 1$. For $p \ge 1$ the *$p$-Wasserstein distance* on $\mathcal{P}(\Omega)$ is defined as $$W_p(\mu,\nu) := \left(\inf{\left\{\int d(x,y)^p\,d\xi:\xi \in \mathcal{P}(\Omega\times\Omega),\
\pi^1_\# \xi = \mu,\ \pi^2_\# \xi = \nu\right\}}\right)^{1/p}\;.$$ Here $\pi^i_\# \xi$ denotes the push forward of the measure $\xi$ by means of the $i$-th projection. In other words, $\pi^1_\# \xi (U) = \xi(U\times \Omega)$ and $\pi^2_\#\xi(U) = \xi(\Omega\times U)$ for every Borel set $U \subset \Omega$.
Recall that the *narrow topology* on $\mathcal{P}(\Omega)$ is induced by the action of elements of $\mathcal{P}(\Omega)$ on continuous functions $u \in C(\Omega)$. That is, a sequence $(\mu_k)_{k\in{{\ensuremath{\mathbb{N}}}}}\subset \mathcal{P}(\Omega)$ converges narrowly to $\mu\in\mathcal{P}(\Omega)$, if $$\int_\Omega u\,d\mu_k = \int_\Omega u\,d\mu \quad \text{ for all } u \in C(\Omega) \,.$$
\[le:wp\] Let $p \ge 1$. Then the Wasserstein distance satisfies, for every $\mu_1$, $\mu_2$, $\nu \in \mathcal{P}(\Omega)$ and $0 \le \lambda \le 1$, the inequality $$\label{eq:wp}
W_p\bigl(\lambda\mu_1 + (1-\lambda)\mu_2,\nu\bigr)^p
\le \lambda W_p(\mu_1,\nu)^p + (1-\lambda)W_p(\mu_2,\nu)^p\;.$$ Moreover it is lower semi-continuous with respect to the narrow topology.
The lower semi-continuity of $W_p$ has, for instance, been shown in [@GivSho84]. In order to show the inequality , let $\xi_1$, $\xi_2 \in \mathcal{P}(\Omega\times\Omega)$ be two measures that realize the infimum in the definition of $W_p(\mu_1,\nu)$ and $W_p(\mu_2,\nu)$, respectively. Then $\pi^1_\#\bigl(\lambda\xi_1+(1-\lambda)\xi_2\bigr) = \lambda\mu_1 + (1-\lambda)\mu_2$ and $\pi^2_\#\bigl(\lambda\xi_1+(1-\lambda)\xi_2\bigr) = \nu$, which implies that the measure $\lambda\xi_1+(1-\lambda)\xi_2$ is admissible for measuring the distance between $\lambda\mu_1+(1-\lambda)\mu_2$ and $\nu$. Therefore $$\begin{aligned}
W_p\bigl(\lambda\mu_1 + (1-\lambda)\mu_2,\nu\bigr)^p
& = \inf{\left\{\int d(x,y)^p\,d\xi:\pi^1_\# \xi = \lambda\mu_1+(1-\lambda)\mu_2,\ \pi^2_\# \xi = \nu\right\}}\\
& \le \int d(x,y)^p\,d(\lambda\xi_1 + (1-\lambda)\xi_2)\\
& = \lambda W_p(\mu_1,\nu)^p + (1-\lambda) W_p(\mu_2,\nu)^p\,,
\end{aligned}$$ which proves the assertion.
Since $\mathcal{P}(\Omega)$ is a convex subset of the space $\mathcal{M}(\Omega)$ of all finite Radon measures on $\Omega$, and the narrow topology on $\mathcal{P}(\Omega)$ is the restriction of the weak$^*$ topology on $\mathcal{M}(\Omega)$ considered as the dual of $C_b(\Omega)$, the space of bounded continuous functions on $\Omega$, it is possible to apply the results of this section also to the situation where ${Y}= \mathcal{P}(\Omega)$ and ${\mathcal S}= W_p$. As an easy example, we consider the problem of density estimation from a finite number of measurements.
Let $\Omega \subset {{\ensuremath{\mathbb{R}}}}^n$ be an open domain. Given a finite number of measurements $\{y_1,\ldots,y_k\}\subset \Omega$, the task of density estimation is the problem of finding a simple density function $u$ on $\Omega$ in such a way that the measurements look like a typical sample of the distribution defined by $u$. Interpreting the measurements as a normalized sum of delta peaks, that is, equating $\{y_1,\ldots,y_k\}$ with the measure $\mathbf{y} := \frac{1}{k}\sum_i {\beta}(y_i) \in \mathcal{P}(\Omega)$, we can easily translate the problem into the setting of this paper.
We set ${X}:= {\left\{u\in L^1(\Omega):u \ge 0 \text{ and } {{\left\lVertu\right\rVert}}_1 = 1\right\}}$, which is a convex and closed subset of $L^1(\Omega)$, ${Y}:= \mathcal{P}(\Omega)$, and consider the embedding $\operatorname{\mathbf F}\colon {X}\to \mathcal{P}(\Omega)$, $u \mapsto u\,\mathcal{L}^n$. Then $\operatorname{\mathbf F}$ is continuous with respect to the weak topology on ${X}$ and the narrow topology on $\mathcal{P}(\Omega)$. We now consider the distance measure ${\mathcal S}= W_p$ for some $p \ge 1$ and the Euclidean distance $d$ on $\Omega$. Then Lemma \[le:wp\] implies that, for every $\mu \in \mathcal{P}(\Omega)$, the mapping $u \mapsto W_p(Fu,\mu)$ is weakly lower semi-continuous.
There are several possibilities for choosing a regularization functional on ${X}$. If $\Omega$ is bounded (or at least $\mathcal{L}^n(\Omega) < \infty$), one can, for instance, use the Boltzmann–Shannon entropy defined by $${\mathcal R}(u) := \int_\Omega u\,\log(u)\,dx
\qquad
\text{ for } u \in {X}\;.$$ Then the Theorems of De la Vallée Poussin and Dunford–Pettis (see [@FonLeo07 Thms. 2.29, 2.54]) show that the lower level sets of ${\mathcal R}$ are weakly pre-compact in $L^1(\Omega)$. Moreover, the functional ${\mathcal R}$ is convex and therefore weakly lower semi-continuous (see [@FonLeo07 Thm. 5.14]). Using Proposition \[prop:compact\], we therefore obtain that the compactness required in Assumption \[as:stability\] holds. Also, Lemma \[le:wp\] shows that Item \[it:linear:S\] in Assumption \[as:linear\] holds. Items \[it:linear:X\] and \[it:linear:R\] are trivially satisfied. Finally, Item \[it:linear:domain\] follows from the density of $\operatorname{dom}{R}$ in ${X}$ and the density (with respect to the narrow topology) of $\operatorname{ran}\operatorname{\mathbf F}$ in $\mathcal{P}(\Omega)$. In addition, it has been shown in [@Vis84] that the weak convergence of a sequence $(u_k)_{k\in{{\ensuremath{\mathbb{N}}}}} \subset L^1(\Omega)$ to $u \in L^1(\Omega)$ together with the convergence ${\mathcal R}(u_k) \to {\mathcal R}(u)$ imply that ${{\left\lVertu_k-u\right\rVert}}_1 \to 0$. Thus the topology $\tau_{{\mathcal R}}$ coincides with the strong topology on ${X}$.
Proposition \[pr:linear\_stab\] therefore implies that the residual method is a stable and convergent regularization method with respect to the strong topology on ${X}$. More precisely, given a sample $\mathbf{y} = \frac{1}{k} \sum_i {\beta}(y_i)$, the density estimate $u$ depends continuously on the positions $y_i$ of the measurements and on the regularization parameter ${\beta}$. In addition, if the number of measurements increases, then the Wasserstein distance between the sample and the true probability converges almost surely to zero. Thus also the reconstructed density converges to the true underlying density, provided the regularization parameters decrease to zero slowly enough.
Convergence Rates {#se:rates}
=================
In this section we derive quantitative estimates (convergence rates) for the difference between regularized solutions $x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta})$ and the exact solution of the equation $\operatorname{\mathbf F}(x^\dagger) = y^\dagger$.
For Tikhonov regularization, convergence rates have been derived in [@BotHof10; @BurOsh04; @FleHof10; @HofKalPoeSch07; @Res05; @ResSch06] in terms of the *Bregman distance*. However, its classical definition, $$\label{eq:bregman_dist}
D_\xi(x,x^\dagger) = {\mathcal R}(x) - {\mathcal R}(x^\dagger) + {\left\langle\xi,x^\dagger - x\right\rangle}_{{X}^*,{X}} \,,$$ where $\xi \in \partial{\mathcal R}(x^\dagger) \subset {X}^*$, requires the space ${X}$ to be linear and the functional ${\mathcal R}$ to be convex, as the (standard) subdifferential $\partial{\mathcal R}(x^\dagger)$ is only defined for convex functionals. In the sequel we will extend the notion of subdifferentials and Bregman distances to work for arbitrary functionals ${\mathcal R}$ on arbitrary sets ${X}$. To that end, we make use of a generalized notion of convexity, which is not based on the duality between a Banach space ${X}$ and its dual ${X}^*$ but on more general pairings (see [@Sin97]). The same notion has recently been used in [@Gra10b] for the derivation of convergence rates for non-convex regularization functionals.
\[def:bregman\] Let $W$ be a set of functions $w\colon {X}\to {{\ensuremath{\mathbb{R}}}}$, let ${\mathcal R}\colon {X}\to {{\ensuremath{\mathbb{R}}}}\cup\{+\infty\}$ be a functional and let $x^\dag \in X$.
1. The functional ${\mathcal R}$ is *convex at $x^\dag$ with respect to $W$*, if $$\label{eq:conj}
{\mathcal R}(x^\dag ) = {\mathcal R}^{**}(x^\dag)
:= \sup_{w \in W} \,
\left( \inf_{x\in {X}} \left( {\mathcal R}(x) - w(x) + w(x^\dag) \right) \right) \,.$$
2. Let ${\mathcal R}$ be convex at $x^\dag$ with respect to $W$. The *subdifferential at $x^\dag$ with respect $W$* is defined as $$\partial_W {\mathcal R}(x^\dag)
:= {\left\{w \in W: {\mathcal R}(x) \ge {\mathcal R}(x^\dag) + w \bigl(x\bigr) - w(x^\dag)
\text{ for all } x \in {X}\right\}}\,.$$
3. Let ${\mathcal R}$ be convex at $x^\dag$ with respect to $W$. For $w \in \partial_W {\mathcal R}(x^\dag)$ and $x \in X$, the *Bregman distance between $x^\dag$ and $x$ with respect to $w$* is defined as $$\label{eq:bregman_dist2}
D_w (x,x^\dag) := {\mathcal R}(x) - {\mathcal R}(x^\dag) - w(x) + w(x^\dag) \,.$$
\[rem:bregman-lc\] Let ${X}$ be a Banach space and set $W = {X}^*$. Then a functional ${\mathcal R}\colon {X}\to {{\ensuremath{\mathbb{R}}}}\cup\{+\infty\}$ is convex with respect to $W$, if and only if it is lower semi-continuous and convex in the classical sense. Moreover,at every $x^\dag \in {X}$, the subdifferential with respect $W$ coincides with the classical subdifferential $\partial{\mathcal R}(x^\dagger) \subset {X}^*$. Finally, the standard Bregman distance, defined by , coincides with the Bregman distance obtained by means of Definition \[def:bregman\].
In the following, let $W$ be a given family of real valued functions on ${X}$. Convergence rates in Bregman distance with respect to $W$ will be derived under the following assumption:
\[as:rates\]
1. There exists a monotonically increasing function $\psi \colon [0, \infty) \to [0, \infty) $ such that $$\label{eq:triangle}
{\mathcal S}(y_1,y_2) \le \psi \left( {\mathcal S}(y_1,y_3) + {\mathcal S}(y_2,y_3) \right)
\qquad \text{ for all } y_1, y_2, y_3 \in {Y}\,.$$
2. The functional ${\mathcal R}\colon {X}\to {{\ensuremath{\mathbb{R}}}}\cup\{+\infty\}$ is convex at $x^\dag \in {X}$ with respect to $W$.
3. There exist $w \in \partial_W {\mathcal R}(x^\dagger)$ and constants $\gamma_1 \in [0,1)$, $\gamma_2 \ge 0$ such that $$\label{eq:bregman_est}
w(x^\dagger)-w(x)
\le \gamma_1 D_w(x,x^\dagger) + \gamma_2 {\mathcal S}\left( \operatorname{\mathbf F}(x), \operatorname{\mathbf F}(x^\dagger) \right)$$ for every $x \in \Phi_{\mathcal R}\left(\operatorname{\mathbf F},\operatorname{\mathbf F}(x^\dagger),\psi(2{\beta}),{\mathcal R}(x^\dagger)\right)$.
In a Banach space setting, the *source inequality* has already been used in [@HofKalPoeSch07; @SchGraGroHalLen09] to derive convergence rates for Tikhonov regularization with convex functionals and in [@Hei08] for multiparameter regularization. Equation is an alternate for the missing triangle inequality in the non-metric case.
\[thm:rates\] Let Assumption \[as:rates\] hold and let $y \in {Y}$ satisfy ${\mathcal S}\left(\operatorname{\mathbf F}(x^\dagger),y\right) \le {\beta}$. Then, the estimate $$\label{eq:rate_est}
D_w(x_{\beta}, x^\dagger)
\le
\frac{\gamma_2 }{1-\gamma_1} \, \psi\left({\beta}+ {\mathcal S}\left(\operatorname{\mathbf F}(x^\dagger),y\right)\right)$$ holds for all $x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta})$.
Let $x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta})$. This, together with and the assumption that ${\mathcal S}\left(\operatorname{\mathbf F}(x^\dagger),y\right)\le{\beta}$, implies that $${\mathcal S}\left(\operatorname{\mathbf F}(x_{\beta}),\operatorname{\mathbf F}(x^\dagger)\right)
\le \psi \left( {\mathcal S}\left(\operatorname{\mathbf F}(x_{\beta}),y\right) + {\mathcal S}\left(\operatorname{\mathbf F}(x^\dag),y\right) \right)
\le \psi(2{\beta})\,.$$ Together with it follows that $$\begin{gathered}
D_w(x_{\beta},x^\dagger)
= {\mathcal R}(x_{\beta}) - {\mathcal R}(x^\dagger) - w(x_{\beta}) + w(x^\dagger)\\
\le {\mathcal R}(x_{\beta}) - {\mathcal R}(x^\dagger) + \gamma_1 D_w(x_{\beta},x^\dagger)
+ \gamma_2 {\mathcal S}\left(\operatorname{\mathbf F}(x_{\beta}),\operatorname{\mathbf F}(x^\dagger)\right)\;.
\end{gathered}$$ The assumption $\gamma_1 \in [0,1)$ implies the inequality $$\label{eq:rates:hlp1}
D_w(x_{\beta},x^\dagger)
\leq \frac{\gamma_1}{1-\gamma_1}\left({\mathcal R}(x_{\beta}) - {\mathcal R}(x^\dagger)\right)
+ \frac{\gamma_2}{1-\gamma_1}\,{\mathcal S}\left(\operatorname{\mathbf F}(x_{\beta}),\operatorname{\mathbf F}(x^\dagger)\right)\;.$$ Since ${\mathcal S}\left(\operatorname{\mathbf F}(x^\dagger),y \right) \le {\beta}$, we have ${\mathcal R}(x^\dagger) \le {\mathcal R}(x_{\beta})$. Therefore and imply $$D_w(x_{\beta}, x^\dagger)
\le \frac{\gamma_2}{1-\gamma_1}\,{\mathcal S}\left(\operatorname{\mathbf F}(x_{\beta}),\operatorname{\mathbf F}(x^\dagger)\right)
\leq \frac{\gamma_2}{1-\gamma_1} \, \psi\left({\beta}+ {\mathcal S}\left(\operatorname{\mathbf F}(x^\dagger),y\right)\right) \;,$$ which concludes the proof.
Typically, convergence rates are formulated in a setting which slightly differs from the one of Theorem \[thm:rates\], see [@BurOsh04; @EngHanNeu96; @HofKalPoeSch07; @SchGraGroHalLen09]). There one assumes the existence of an ${\mathcal R}$-minimizing solution $x^\dagger \in {X}$ of the equation $\operatorname{\mathbf F}(x^\dagger) = y^\dagger$, for some exact data $y^\dagger \in \operatorname{ran}(\operatorname{\mathbf F})$. Instead of $y^\dagger$, only noisy data $y \in {Y}$ and the error bound ${\mathcal S}(y^\dagger,y) \le \beta$ are given.
For this setting, implies the rate $$D_w( x_{\beta}, x^\dagger)
\le \frac{\gamma_2}{1-\gamma_1} \, \psi(2 {\beta})
= \mathcal{O}\bigl( \psi(2{\beta}) \bigr)
\qquad
\text{ as } {\beta}\to 0 \,,$$ where $x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta})$ denotes any regularized solution.
\[re:bregman\_est\_var\] The inequality is equivalent to the existence of $\eta_1$, $\eta_2 > 0$ such that $$\label{eq:bregman_est_var}
w(x^\dagger) - w(x)
\le \eta_1 \left({\mathcal R}(x)-{\mathcal R}(x^\dagger)\right) + \eta_2 \, {\mathcal S}\left(\operatorname{\mathbf F}(x),\operatorname{\mathbf F}(x^\dagger)\right)\;.$$ Indeed, we obtain from by setting $\eta_1 := \gamma_1/(1-\gamma_1)$ and $\eta_2 := \gamma_2/(1-\gamma_1)$. Conversely, implies by taking $\gamma_1 := \eta_1/(1+\eta_1)$ and $\gamma_2 := \eta_2/(1+\eta_1)$.
Convergence Rates in Banach spaces
----------------------------------
In the following, assume that ${X}$ and ${Y}$ are Banach spaces with norms ${{\left\lVert\cdot\right\rVert}}$ and ${{\left\lVert\cdot\right\rVert}}$, and assume that ${\mathcal R}$ is a convex and lower semi-continuous functional on ${X}$. We set ${\mathcal S}(y,z):= {{\left\lVerty-z\right\rVert}}$ and let $D_{\xi}$ with $\xi \in \partial{\mathcal R}(x^\dagger)$ denote the classical Bregman distance (see Remark \[rem:bregman-lc\]).
If $x^\dagger$ satisfies the inequality $$\label{eq:bregman_est_banach}
{\left\langle\xi,x^\dag-x\right\rangle}
\le \gamma_1 D_{\xi}(x,x^\dagger) +
\gamma_2 {\mathcal S}\bigl( \operatorname{\mathbf F}(x), \operatorname{\mathbf F}(x^\dagger) \bigr) \,.$$ and $y$ are given data with ${{\left\lVert\operatorname{\mathbf F}(x^\dagger) - y\right\rVert}} \leq {\beta}$, then Theorem \[thm:rates\] implies the convergence rate $D_\xi(x_{\beta}, x^\dagger) = \mathcal O({\beta})$. In the special case where ${X}$ is a Hilbert space and ${\mathcal R}(x) = {{\left\lVertx\right\rVert}}^2$ we have $D_\xi(x, x^\dagger) = {{\left\lVertx-x^\dagger\right\rVert}}^2$, which implies the convergence rate ${{\left\lVertx-x^\dagger\right\rVert}} = \mathcal O\bigl({\beta}^{1/2}\bigr)$ in the norm. In Proposition \[pr:rates\_norm\] below we show that the same convergence rate holds on any $2$-convex space. For $r$-convex Banach spaces with $r > 2$, we derive the rate $\mathcal{O}\bigl({\beta}^{1/r}\bigr)$.
The Banach space ${X}$ is called $r$-convex (or is said to have modulus of convexity of power type $r$), if there exists a constant $C>0$ such that $$\inf {\left\{1- {{\left\lVert(x+y)/2\right\rVert}} :{{\left\lVertx\right\rVert}} = {{\left\lVerty\right\rVert}} =1,\, {{\left\lVertx-y\right\rVert}}\ge \epsilon\right\}} \ge C {\varepsilon}^r$$ for all ${\varepsilon}\in [0,2]$.
Note that every Hilbert space is $2$-convex and that there is *no* Banach space (with $\dim({X})\ge 2$) that is $r$-convex for some $r<2$ (see [@LinTza79 pp. 63ff]).
\[pr:rates\_norm\] Let ${X}$ be an $r$-convex Banach space with $r \ge 2$ and let ${\mathcal R}(x) := {{\left\lVertx\right\rVert}}^r/r$. Assume that there exists $x^\dagger \in {X}$, a subgradient $\xi \in \partial {\mathcal R}(x^\dagger)$, and constants $\gamma_1 \in [0,1)$, $\gamma_2 \ge 0$, ${\beta}_0 >0$ such that holds for every $x \in \Phi_{\mathcal R}\bigl(\operatorname{\mathbf F},\operatorname{\mathbf F}(x^\dagger),2{\beta}_0,{\mathcal R}(x^\dagger)\bigr)$.
Then there exists a constant $c>0$ such that $$\label{eq:rate_est_banach}
{{\left\lVertx_{\beta}- x^\dagger\right\rVert}}
\le
c \left({\beta}+ {{\left\lVert\operatorname{\mathbf F}(x^\dagger)- y\right\rVert}} \right)^{1/r}$$ for all ${\beta}\in [0, \beta_0]$, all $y \in {Y}$ with ${{\left\lVert\operatorname{\mathbf F}(x^\dagger)- y \right\rVert}} \le {\beta}$, and all $x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta})$.
Let $J_r: {X}\to 2^{{X}^*}$ denote the duality mapping with respect to the weight function $s \mapsto s^{r-1}$. In [@XuRoa91 Equation $(2.17)'$] it is shown that there exists a constant $K > 0$ such that $$\label{eq:xuroach}
{{\left\lVertx^\dagger + z\right\rVert}}^r \ge
{{\left\lVertx^\dagger\right\rVert}}^r + r {\left\langlej_r(x^\dagger),z\right\rangle}_{{X}^*,{X}}
+
K {{\left\lVertz\right\rVert}}^r$$ for all $x^\dagger$, $z \in {X}$ and $j_r(x^\dagger) \in J_r(x^\dagger)$. By Asplund’s theorem [@Cio90 Chap. 1, Thm. 4.4], the duality mapping $J_r$ equals the subgradient of ${\mathcal R}= {{\left\lVert\cdot\right\rVert}}^r/r$. Therefore, by taking $z = x-x^\dagger$ and $j_r(x^\dagger)=\xi$, inequality implies $$\label{eq:bregam-norm}
D_\xi(x, x^\dagger) \ge \frac{K}{r} \ {{\left\lVertx-x^\dagger\right\rVert}}^r
\qquad \text{ for all } x^\dagger, x \in {X}\text{ and } \xi \in \partial {\mathcal R}(x^\dagger)\,.$$ Consequently, follows from Theorem \[thm:rates\].
Exact values for the constant $K$ in (and thus for the constant $c$ in ) can be derived from [@XuRoa91]. Bregman distances satisfying are called $r$-coercive in [@HeiHof09]. This $r$-coercivity has already been applied in [@BonKazMaaSchoSchu08] for the minimization of Tikhonov functionals in Banach spaces.
The spaces ${X}= L^p(\Omega, \mu)$ for $p \in (1,2]$ and some $\sigma$-finite measure space $(\Omega,\mu)$ are examples of 2-convex Banach spaces (see [@LinTza79 p. 81, Remarks following Theorem 1.f.1.]). Consequently we obtain for these spaces the convergence rate $\mathcal O\bigl({\beta}^{1/2}\bigr)$. The spaces ${X}= L^p(\Omega, \mu)$ for $p > 2$ are only $p$-convex, leading to the rate $\mathcal O\bigl({\beta}^{1/p}\bigr)$ in those spaces.
\[rem:rates:banach\] The book [@SchGraGroHalLen09 pp. 70ff] clarifies the relation between and the source conditions used to derive convergence rates for convex functionals on Banach spaces. In particular, it is shown that, if $\operatorname{\mathbf F}$ and ${\mathcal R}$ are Gâteaux differentiable at $x^\dagger$ and there exist $\gamma > 0$ and $\omega \in {Y}^*$ such that $\gamma {{\left\lVert\omega\right\rVert}} < 1$ and $$\begin{gathered}
\xi = \operatorname{\mathbf F}'(x^\dagger)^*\,\omega \in \partial \mathcal R(x^\dagger) \,, \label{eq:cond-a}
\\ \label{eq:cond-b}
{{\left\lVert\operatorname{\mathbf F}(x) - \operatorname{\mathbf F}(x^\dagger)- \operatorname{\mathbf F}'(x^\dagger)(x-x^\dagger)\right\rVert}} \leq \gamma D_\xi(x,x^\dagger)
\end{gathered}$$ for every $x \in {X}$, then holds on ${X}$. (Here $\operatorname{\mathbf F}'(x^\dagger)^* : {Y}^* \to {X}^*$ is the adjoint of $\operatorname{\mathbf F}'(x^\dagger)$.) Conversely, if $\xi \in \partial {\mathcal R}(x^\dagger)$ satisfies , then holds for every $x \in {X}$.
In the particular case that $\operatorname{\mathbf F}\colon {X}\to {Y}$ is linear and bounded, the inequality is trivially satisfied with $\gamma =0$. Thus, is equivalent to the sourcewise representability of the subgradient, $ \xi \in \partial {\mathcal R}(x^\dagger) \cap \operatorname{ran}(\operatorname{\mathbf F}^*)$.
Sparse Regularization {#sec:sparsity}
=====================
Let $\Lambda$ be an at most countable index set, define $$\ell^2(\Lambda) := {\left\{x = (x_\lambda)_{\lambda \in \Lambda} \subset {{\ensuremath{\mathbb{R}}}}: \sum_{\lambda \in \Lambda} {\left\lvertx_\lambda\right\rvert}^2 < \infty \right\}}\,,$$ and assume that $\operatorname{\mathbf F}\colon {X}:=\ell^2(\Lambda) \to {Y}$ is a bounded linear operator with dense range in the Hilbert space ${Y}$. We consider for $p \in (0,2)$ the minimization problem $$\label{eq:min_prob_sparse}
{\mathcal R}_{p}(x):= {{\left\lVertx\right\rVert}}_{\ell^p(\Lambda)}^p := \sum_{\lambda \in \Lambda} {\left\lvertx_\lambda\right\rvert}^p \to \min
\qquad\text{ subject to }\quad {{\left\lVert\operatorname{\mathbf F}x - y\right\rVert}}^2 \le {\beta}\;.$$ For $p>1$, the subdifferential $\partial{\mathcal R}_p(x^\dagger)$ is at most single valued and is identified with its single element.
In a finite dimensional setting with $p = 1$, the minimization problem has received a lot of attention during the last years under the name of *compressed sensing* (see [@Can08; @CanRomTao06; @CanTao06; @Don06c; @Don06b; @DonElaTem06; @Ela10; @Fuc05b; @Tro06]). Under some assumptions, the solution of with $y = \operatorname{\mathbf F}x^\dagger$ and ${\beta}= 0$ has been shown to recover $x^\dagger$ *exactly* provided the set ${\left\{\lambda\in \Lambda :x^\dagger_\lambda \neq 0\right\}}$ has sufficiently small cardinality (that is, it is sufficiently sparse). Results for $p<1$ can be found in [@Cha07; @DavGri09; @FouLai09; @SaaChaYil08].
In this section we prove well-posedness of and derive convergence rates in a possibly infinite dimensional setting. This inverse problems point of view has so far only been treated for the case $p=1$ (see [@GraHalSch11]). The more general setting has only been considered for Tikhonov regularization $${{\left\lVert\operatorname{\mathbf F}x - y\right\rVert}}^2 + \alpha {\mathcal R}_{p}(x) \to \min$$ (see [@ChaComPesWaj07; @DauDefDem04; @Gra09b; @GraHalSch08; @Lor08; @Zar09]).
Well-Posedness
--------------
In the following, $\tau$ denotes the weak topology on $\ell^2(\Lambda)$, and $\tau_p := \tau_{{\mathcal R}_p}$ denotes the topology as in Definition \[de:tauR\]. Then a sequence $(x_k)_{k \in {{\ensuremath{\mathbb{N}}}}} \subset \ell^2(\lambda)$ converges to $x \in \ell^2(\lambda)$ with respect to $\tau_p$ if and only if $x_k \to x$ and ${\mathcal R}_p(x_k) \to {\mathcal R}_p(x)$.
\[pr:lp:well\_posed\] Let $\operatorname{\mathbf F}\colon \ell^2(\Lambda) \to {Y}$ be a bounded linear operator with dense range. Then constrained $\ell^p$ regularization with $0 < p < 2$ is well-posed:
1. [*Existence:*]{} For every ${\beta}> 0$ and $y \in {Y}$, the set of regularized solutions ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ is non-empty.
2. [*Stability:*]{}\[wp-sparse:2\] Let $({\beta}_k)$ and $(y_k)$ be sequences with ${\beta}_k \to {\beta}> 0$ and $y_k \to y \in {Y}$. Then $\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k) \subset
{\Sigma}(\operatorname{\mathbf F},y,{\beta})$.
3. [*Convergence:*]{}\[wp-sparse:3\] Let ${{\left\lVerty_k - y\right\rVert}} \le {\beta}_k \to 0$ and assume that the equation $\operatorname{\mathbf F}x = y$ has a solution in $\ell^p(\Lambda)$. Then we have $\emptyset \neq \operatorname*{\tau_{{\mathcal R}}-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k) \subset {\Sigma}(\operatorname{\mathbf F},y,0)$. Moreover, if the equation $\operatorname{\mathbf F}x = y$ has a unique ${\mathcal R}_p$-minimizing solution $x^\dagger$, then we have $\operatorname*{\tau_p-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k) = \{x^\dagger\}$.
In order to prove the existence of minimizers, we apply Theorem \[th:existence\] by showing that $\Phi_{\mathcal R}( \operatorname{\mathbf F},y,{\beta}, t)$ is compact with respect to the weak topology on $\ell^2(\Lambda)$ for every $t>0$ and is nonempty for some $t$. Because $\operatorname{\mathbf F}$ has dense range, the set $$\Phi_{\mathcal R}( \operatorname{\mathbf F},y,{\beta}, t) = {\left\{x\in \ell^2(\Lambda):{\mathcal R}_p(x) \le t, {{\left\lVert\operatorname{\mathbf F}(x) - y\right\rVert}}^2 \le {\beta}\right\}}$$ is non-empty for $t$ large enough. It remains to show that the sets $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}, t)$ are weakly compact on $\ell^2(\lambda)$ for every positive $t$.
The functional ${\mathcal R}_{p}(x) = \sum_{\lambda\in \Lambda} {\left\lvertx_\lambda\right\rvert}^p$ is weakly lower semi-continuous (on $\ell^2(\lambda)$) as the sum of non-negative and weakly continuous functionals (see [@EkeTem76]). Moreover, the mapping $\operatorname{\mathbf F}$ is weakly continuous, and therefore $x \mapsto {{\left\lVert\operatorname{\mathbf F}x - y\right\rVert}}^2$ is weakly lower semi-continuous, too. The estimate ${\mathcal R}_p(x) \ge {{\left\lVertx\right\rVert}}_{\ell^2(\Lambda)}^p$ (see [@GraHalSch08 Equation (5)]) shows that ${\mathcal R}_p$ is weakly coercive. Therefore the sets $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}, t)$ are weakly compact for all $t>0$, see Proposition \[prop:compact\].
Taking into account Example \[ex:linear1\], it follows that ${\mathcal R}_p$, ${\mathcal S}$, and $\operatorname{\mathbf F}$ satisfy Assumption \[as:linear\]. Consequently, Items \[wp-sparse:2\] and \[wp-sparse:3\] follow from Proposition \[pr:linear\_stab\].
In the case $p>1$, the functional ${\mathcal R}_p$ is strictly convex, and therefore the ${\mathcal R}_p$-minimizing solution $x^\dag$ of $\operatorname{\mathbf F}x = y$ is unique. Consequently the equality $$\operatorname*{\tau_p-Lim\,sup}_{k\to\infty} {\Sigma}(\operatorname{\mathbf F},y_k,{\beta}_k) = {\left\{x^\dagger\right\}}$$ holds for every $y$ in the range of the operator $\operatorname{\mathbf F}$.
For the convex case $p\geq 1$, it is shown in [@GraHalSch08 Lemma 2] that the $\tau_p$ convergence of a sequence $x_k$ already implies ${\mathcal R}_p(x_k-x) \to 0$. In particular, the topology $\tau_p$ is stronger than the topology induced by ${{\left\lVert\cdot\right\rVert}}_{\ell^2(\Lambda)}$. A similar result for $0 < p < 1$ has been derived in [@Gra10].
Convergence Rates {#convergence-rates}
-----------------
In the following, we derive two types of convergence rates results with respect the $\ell^2$-norm: The rate $\mathcal O\bigl({\beta}^{1/2}\bigr)$ (for $p \in (1,2)$), and the rate $\mathcal O\bigl({\beta}^{\min\{1,1/p\}}\bigr)$ (for every $p \in (0,2)$) for sparse sequences—here and in the following, $x^\dagger \in \ell^2(\Lambda)$ is called sparse, if $$\operatorname{supp}(x^\dagger) := {\left\{\lambda \in \Lambda: x^\dagger_\lambda \neq 0\right\}}$$ is finite. The convergence rates results for constrained $\ell^p$ regularization, derived in this section, are summarized in Table \[tb:rates\].
[l |@ l |@ l |@ l ]{} & [Norm]{} & [Premises (besides $\operatorname{ran}(\operatorname{\mathbf F}^*) \cap \partial {\mathcal R}_p \neq \emptyset)$]{} & Result\
${\beta}^{1/2}$& ${{\left\lVert\cdot\right\rVert}}_{\ell^2} $ & $p\in (1,2)$ & Prop. \[pr:rate\_lp\]\
${\beta}^{1/2}$& ${{\left\lVert\cdot\right\rVert}}_{\ell^p} $ & $p\in (1,2)$ & Rem. \[re:rate\_lp\]\
${\beta}^{1/p}$& ${{\left\lVert\cdot\right\rVert}}_{\ell^2} $ & $p\in [1,2)$, sparsity, injectivity on $V$ & Prop. \[pr:rate\_lp\_sparse\]\
${\beta}$ & ${{\left\lVert\cdot\right\rVert}}_{\ell^2} $ &
------------------------------------------
$p\in (0,1)$, uniqueness of $x^\dagger$,
sparsity, injectivity on $V$
------------------------------------------
: Convergence rates for constrained $\ell^p$ regularization.\[tb:rates\]
& Prop. \[pr:rate\_lp\_sparse\]\
For $p \ge 1$, the same type of results (Propositions \[pr:rate\_lp\], \[pr:rate\_lp\_sparse\]) has also been obtained for $\ell^p$-Tikhonov regularization in [@GraHalSch08; @SchGraGroHalLen09]. The results for the non-convex case, $p\in (0,1)$, are based on [@Gra10], where the same rate for non-convex Tikhonov regularization with a–priori parameter choice has been derived (see also [@Gra10b]). Similar, but weaker, results have been already been derived in [@BreLor09; @Gra09b; @Zar09] in the context of Tikhonov regularization. In [@Zar09], the conditions for the convergence rates result for non-convex regularization are basically the same as in Proposition \[pr:rate\_lp\_sparse\], but only a rate of order $\mathcal O\bigl({\beta}^{1/2}\bigr)$ has been obtained. In [@BreLor09; @Gra09b], a linear convergence rate $\mathcal{O}({\beta})$ is proven, but with a considerably stronger range condition: each standard basis vector $e_\lambda$, $\lambda \in \Lambda$, has to satisfy $e_\lambda \in \operatorname{ran}\operatorname{\mathbf F}^*$.
\[pr:rate\_lp\] Let $1< p < 2$, $x^\dagger = (x^\dagger_\lambda)_{\lambda \in \Lambda} \in \ell^{2}(\Lambda)$, and let $\operatorname{\mathbf F}\colon \ell^2(\Lambda) \to {Y}$ be a bounded linear operator. Moreover, assume that there exists $\omega \in {Y}$ with $\partial {\mathcal R}_p( x^\dagger ) = \operatorname{\mathbf F}^* \omega$. Then the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta}) =: {\left\{x_{\beta}\right\}}$ consists of a single element and there exists a constant $d_p>0$ only depending on $p$, such that $$\label{eq:rate_est_lp}
{{\left\lVertx_{\beta}- x^\dagger\right\rVert}}_{\ell^{2}(\Lambda)}^2
\le
\frac{d_p {{\left\lVert\omega\right\rVert}}}{3 + 2{\mathcal R}_p(x^\dag)}\,
\left( {\beta}+ {{\left\lVert\operatorname{\mathbf F}x^\dagger - y \right\rVert}} \right)$$ for all ${\beta}>0$ and $y \in {Y}$ with ${{\left\lVert \operatorname{\mathbf F}(x^\dagger)-y \right\rVert}} \le {\beta}$.
The assumption $\partial {\mathcal R}_p(x^\dagger) = \operatorname{\mathbf F}^*\omega$ then implies that is satisfied with $W = {X}^*$, $\gamma_1=0$ and $\gamma_2 = {{\left\lVert\omega\right\rVert}}$. Theorem \[thm:rates\] therefore implies the inequality $$\label{eq:rate_lp:2}
\sup {\left\{D_{\partial {\mathcal R}_p (x^\dagger)}(x_{\beta},x^\dagger):x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta}) \right\}}
\leq
{{\left\lVert\omega\right\rVert}} \left({\beta}+{\bigl\lVert\operatorname{\mathbf F}x^\dagger - y\bigr\rVert}\right)\;.$$ From [@GraHalSch08 Lemma 10] we obtain the inequality $$\label{eq:rate_lp:3}
{\bigl\lVertx-x^\dagger\bigr\rVert}_{\ell^{2}(\Lambda)}^2
\le
\frac{d_p}{3 + 2{\mathcal R}_p(x^\dagger) + {\mathcal R}_p(x)}
\, D_{\partial {\mathcal R}_p (x^\dagger)} \left(x,x^\dagger\right)$$ for all $x \in \operatorname{dom}({\mathcal R}_p)$. Now, follows from and .
\[re:rate\_lp\] Since $\ell^p(\Lambda)$ is 2-convex (see [@LinTza79]) and continuously embedded in $\ell^2(\Lambda)$, Proposition \[pr:rates\_norm\] provides an alternative estimate for $x_{\beta}- x^\dagger$ in terms of the stronger distance ${{\left\lVert\cdot\right\rVert}}_{\ell^{p}(\Lambda)}$. The prefactor in , however, is constant, whereas the prefactor in tends to $0$ as ${\mathcal R}_p(x^\dagger)$ increases. Thus the two estimates are somehow independent from each other.
\[pr:rate\_lp\_sparse\] Let $p \in (0,2)$, let $x^\dagger = (x^\dagger_\lambda)_{\lambda \in \Lambda} \in \ell^{2}(\Lambda)$ be sparse, and let $\operatorname{\mathbf F}\colon \ell^2(\Lambda) \to {Y}$ be bounded linear. Assume that one of the following conditions holds:
- We have $p \in (1,2)$, there exists $\omega \in {Y}$ with $\partial{\mathcal R}_p(x^\dagger) = \operatorname{\mathbf F}^*\omega$, and $\operatorname{\mathbf F}$ is injective on $$V = {\left\{x\in \ell^2(\Lambda):\operatorname{supp}(x) \subset \operatorname{supp}(x^\dagger)\right\}}\;.$$
- We have $p = 1$, there exist $\xi = (\xi_\lambda)_{\lambda \in \Lambda}\in \partial{\mathcal R}_{1}(x^\dagger)$ and $\omega \in {Y}$ with $\xi = \operatorname{\mathbf F}^* \omega$, and $\operatorname{\mathbf F}$ is injective on $$V = {\left\{x\in \ell^2(\Lambda): \operatorname{supp}(x)\subset {\left\{\lambda \in \Lambda : {\left\lvert\xi_\lambda\right\rvert} = 1 \right\}} \right\}} \,.$$
- We have $p \in (0,1)$, $x^\dagger$ is the unique ${\mathcal R}_p$-minimizing solution of $\operatorname{\mathbf F}x = \operatorname{\mathbf F}x^\dagger$, and $\operatorname{\mathbf F}$ is injective on $$V = {\left\{x \in \ell^2(\Lambda) : \operatorname{supp}(x) \subset \operatorname{supp}(x^\dagger)\right\}}\;.$$
Then $$ \sup{\left\{ {\bigl\lVertx_{\beta}- x^\dagger\bigr\rVert}_{\ell^{2}(\Lambda)} :x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta}),\,
{\bigl\lVert\operatorname{\mathbf F}x^\dagger-y\bigr\rVert}\le{\beta}\right\}}
= \mathcal O \left({\beta}^{\min{\left\{1,1/p\right\}}} \right) \ \text{ as } {\beta}\to 0 \;.$$
Assume first that $p \in (1,2)$. Define $W := {\left\{w(x) := -c{{\left\lVertx-\tilde{x}\right\rVert}}^p:\tilde{x} \in {X},\ c > 0\right\}}$. Then the functional ${\mathcal R}_p$ is convex at $x^\dagger$ with respect to $W$. Moreover it has been shown in [@GraHalSch08 Proof of Thm. 14] that there exists $w(x) = -c{{\left\lVertx-x^\dagger\right\rVert}}^p \in \partial_W(x^\dagger) \subset W$ such that for some $\eta_1$, $\eta_2 > 0$ the inequality $$\label{eq:rate_sparse_h1}
-w(x) = c{\bigl\lVertx-x^\dagger\bigr\rVert}^p \le \eta_1 \bigl({\mathcal R}_p(x)-{\mathcal R}_p(x^\dagger)\bigr) + \eta_2{\bigl\lVert\operatorname{\mathbf F}(x-x^\dagger)\bigr\rVert}$$ holds on ${\Sigma}(2{\beta},y^\dagger,\operatorname{\mathbf F})$ for ${\beta}$ small enough. Using Remark \[re:bregman\_est\_var\], Theorem \[thm:rates\] therefore implies the rate $$\sup{\left\{D_w(x_{\beta},x^\dagger):x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta}),\, {\bigl\lVert\operatorname{\mathbf F}x^\dagger-y\bigr\rVert}\le{\beta}\right\}}
= \mathcal{O}({\beta})
\ \text{ as } {\beta}\to 0\;.$$ The assertion then follows from the fact that the norm on $\ell^2(\Lambda)$ can be bounded by the Bregman distance $D_w$.
The proofs for $p = 1$ and $p \in (0,1)$ are similar; the required estimate has been shown for $p=1$ in [@GraHalSch08 Proof of Thm. 15] and for $p \in (0,1)$ in [@Gra10 Eq. (7)].
Conclusion
==========
Due to modeling, computing, and measurement errors, the solution of an ill-posed equation $\operatorname{\mathbf F}( x ) = y$, even if it exists, typically yields unacceptable results. The residual method replaces the exact solution by the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta}) = \operatorname*{arg\,min}{\left\{\mathcal R(x) :\mathcal S( \operatorname{\mathbf F}(x), y ) \leq {\beta}\right\}}$, where $\mathcal R$ is a stabilizing functional and $\mathcal S$ denotes a distance measure between $\operatorname{\mathbf F}(x)$ and $y$. This paper shows that in a very general setting ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ is stable with respect to perturbations of the data $y$ and the operator $\operatorname{\mathbf F}$ (Lemma \[le:stability\] and Theorem \[thm:stability:2\]), and the regularized solutions converge to ${\mathcal R}$-minimizing solutions of $\operatorname{\mathbf F}(x)=y$ as ${\beta}\to 0$ (Theorem \[thm:convergence\]). In particular the stability issue has hardly been considered so far in the literature.
In the case where $\operatorname{\mathbf F}$ acts between linear spaces ${X}$ and ${Y}$, stability and convergence have been shown under a list of reasonable properties (see Assumption \[as:linear\]). These assumptions are satisfied for bounded linear operators, but also for a certain class of nonlinear operators (Example \[ex:stab\_nonlinear\]). If ${Y}$ is reflexive, ${X}$ satisfies the Radon–Riesz property, $\operatorname{\mathbf F}$ is a closed linear operator, and $\mathcal R$ and $\mathcal S$ are given by powers of the norms on ${X}$ and ${Y}$, the set ${\Sigma}(\operatorname{\mathbf F},y,{\beta})$ consists of a single element $x_{\beta}$. This element is shown to converge strongly to the minimal norm solution $x^\dagger$ as ${\beta}\to 0$. In this special situation, norm convergence has also been shown in [@IvaVasTan02 Theorem 3.4.1].
In Section \[se:rates\] we have derived quantitative estimates (convergence rates) for the difference between $x^\dagger$ and minimizers $x_{\beta}\in {\Sigma}(\operatorname{\mathbf F},y,{\beta})$ in terms of a (generalized) Bregman distance. All these estimates hold provided ${\mathcal S}(\operatorname{\mathbf F}(x^\dagger), y) \le {\beta}$ and a source inequality introduced in [@HofKalPoeSch07] is satisfied. For linear operators, the required source inequality follows from a source wise representation of a subgradient of ${\mathcal R}$ at $x^\dagger$. This carries on the result of [@BurOsh04] for constrained regularization. In the case that ${X}$ is an $r$-convex Banach space with $r \geq 2$ and ${\mathcal R}$ is the $r$-th power of the norm on ${X}$, we have obtained convergence rates $\mathcal O({\beta}^{1/r})$ with respect to the norm. The spaces ${X}= L^p(\Omega)$ for $p \in (1,2]$ are examples of 2-convex Banach spaces, leading to the rate $\mathcal O\bigl(\sqrt {\beta}\bigr)$ in those spaces.
As an application for our rather general results we have investigated sparse $\ell^p$ regularization with $p \in (0,2)$. We have shown well-posedness in both the convex ($p\geq 1$) and the non-convex case ($p < 1$). In addition, we have studied the reconstruction of sparse sequence. There we have derived the improved convergence rates $\mathcal O\bigl({\beta}^{1/p}\bigr)$ for the convex and $\mathcal O({\beta})$ for the non-convex case.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work has been supported by the Austrian Science Fund (FWF), projects 9203-N12 and project S10505-N20.
Auxiliary results {#sec:appendix}
=================
\[le:Phiincl2\] Assume that $(y_k)_{k\in{{\ensuremath{\mathbb{N}}}}}$ converges ${\mathcal S}$-uniformly to $y \in {Y}$ and the mappings $\operatorname{\mathbf F}_k\colon {X}\to {Y}$ converge locally ${\mathcal S}$-uniformly to $\operatorname{\mathbf F}\colon {X}\to {Y}$.
Then, for every ${\beta}> 0$, $t > 0$ and ${\varepsilon}> 0$, there exists some $k_0 \in {{\ensuremath{\mathbb{N}}}}$ such that $$\label{eq:Phiincl2}
\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t') \subset \Phi_{\mathcal R}({\beta},y_k,\operatorname{\mathbf F}_k,t')
\subset \Phi_{\mathcal R}({\beta}+{\varepsilon},y,\operatorname{\mathbf F},t')$$ for every $t' \le t$ and $k \ge k_0$.
Since $y_k \to y$ ${\mathcal S}$-uniformly and $\operatorname{\mathbf F}_k \to \operatorname{\mathbf F}$ locally ${\mathcal S}$-uniformly, there exists $k_0 \in {{\ensuremath{\mathbb{N}}}}$ such that $$\label{eq:Phiincl2:1}
\begin{aligned}
{\left\lvert{\mathcal S}(\operatorname{\mathbf F}_k(x),y_k) - {\mathcal S}(\operatorname{\mathbf F}_k(x),y)\right\rvert} &\le {\varepsilon}/2\,,\\
{\left\lvert{\mathcal S}(\operatorname{\mathbf F}_k(x),y) - {\mathcal S}(\operatorname{\mathbf F}(x),y)\right\rvert} &\le {\varepsilon}/2\,,
\end{aligned}$$ for all $x \in {X}$ with ${\mathcal R}(x) \le t$ and $k \ge k_0$.
Now let $x \in \Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta}-{\varepsilon},t)$. Then implies that $${\left\lvert{\mathcal S}(\operatorname{\mathbf F}_k(x),y_k) - {\mathcal S}(\operatorname{\mathbf F}(x),y)\right\rvert}
\le {\left\lvert{\mathcal S}(\operatorname{\mathbf F}_k(x),y_k) - {\mathcal S}(\operatorname{\mathbf F}_k(x),y)\right\rvert}
+ {\left\lvert{\mathcal S}(\operatorname{\mathbf F}_k(x),y) - {\mathcal S}(\operatorname{\mathbf F}(x),y)\right\rvert}
\le {\varepsilon}\,,$$ and thus $${\mathcal S}(\operatorname{\mathbf F}_k(x),y_k) \le {\mathcal S}(\operatorname{\mathbf F}(x),y) + {\varepsilon}\le {\beta}\,,$$ that is, $x \in \Phi_{\mathcal R}({\beta},y_k,\operatorname{\mathbf F}_k,t)$, which proves the first inclusion in . The second inclusion can be shown in a similar manner.
The following lemma states that the value of the minimization problem behaves well as the parameter ${\beta}$ decreases.
\[le:Hlimit\] Assume that $\Phi_{\mathcal R}(\gamma,y,\operatorname{\mathbf F},t)$ is $\tau$-compact for every $\gamma$ and every $t$. Then the value $v$ of the constraint optimization problem is right continuous in the first variable, that is, $$\label{eq:Hlimit}
v(\operatorname{\mathbf F},y,{\beta}) = \lim_{{\varepsilon}\to 0+} v({\beta}+{\varepsilon},y,\operatorname{\mathbf F}) = \sup_{{\varepsilon}> 0} v({\beta}+{\varepsilon},y,\operatorname{\mathbf F})\;.$$
Since $\Phi_{\mathcal R}(\operatorname{\mathbf F},y,{\beta},t) \subset \Phi_{\mathcal R}({\beta}+{\varepsilon},y,\operatorname{\mathbf F},t)$, it follows that $v(\operatorname{\mathbf F},y,{\beta}) \ge v({\beta}+{\varepsilon},y,\operatorname{\mathbf F})$ for every ${\varepsilon}> 0$, and therefore $v(\operatorname{\mathbf F},y,{\beta}) \ge \sup_{{\varepsilon}> 0} v({\beta}+{\varepsilon},y,\operatorname{\mathbf F})$.
In order to show the converse inequality, let ${\beta}> 0$. Then the definition of $v(\operatorname{\mathbf F},y,{\beta})$ implies that $\Phi_{\mathcal R}\bigl(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})-{\beta}\bigr) = \emptyset$. Since (cf. Lemma \[le:Phi\_prop\]) $$\label{eq:Hlimit:1}
\emptyset = \Phi_{\mathcal R}\bigl(\operatorname{\mathbf F},y,{\beta},v(\operatorname{\mathbf F},y,{\beta})-{\beta}\bigr)
= \bigcap_{{\varepsilon}> 0} \Phi_{\mathcal R}\bigl({\beta}+{\varepsilon},y,\operatorname{\mathbf F},v(\operatorname{\mathbf F},y,{\beta})-{\beta}\bigr)$$ and the right hand side of is a decreasing family of compact sets. It follows that already $\Phi_{\mathcal R}\bigl(\operatorname{\mathbf F},y,{\beta}+{\varepsilon}_0,v(\operatorname{\mathbf F},y,{\beta})-{\beta}\bigr) = \emptyset$ for some ${\varepsilon}_0 > 0$, and thus $$\sup_{{\varepsilon}> 0} v({\beta}+{\varepsilon},y,\operatorname{\mathbf F}) \geq v(\operatorname{\mathbf F},y,{\beta}+{\varepsilon}_0) \ge v(\operatorname{\mathbf F},y,{\beta})-{\beta}\;.$$ Since ${\beta}$ was arbitrary, this shows the assertion.
\[le:subseq\] Let $({\Sigma}_k)_{k \in {{\ensuremath{\mathbb{N}}}}}$ be a sequence of subsets of ${X}$. Then $U = \operatorname*{\tau-Lim}_{k\to\infty} {\Sigma}_k$, if and only if every subsequence $({\Sigma}_{k_j})_{j\in{{\ensuremath{\mathbb{N}}}}}$ satisfies $U = \operatorname*{\tau-Lim\,sup}_{j\to\infty} {\Sigma}_{k_j}$.
See [@Kur66 §29.V].
\[1\]
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[**Maxim Pospelov$^{\,(a,b)}$ and Josef Pradler$^{\,(a)}$**]{}
$^{\,(a)}$[*Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada*]{}
$^{\,(b)}$[*Department of Physics and Astronomy, University of Victoria,\
Victoria, BC, V8P 1A1 Canada*]{}
0.2in
**Abstract**
The coupling of the baryonic current to new neutrino states [$\nu_{b}$]{} with strength in excess of the weak interactions is a viable extension of the Standard Model. We analyze the signature of [$\nu_{b}$]{} appearance in the solar neutrino flux that gives rise to an elastic scattering signal in dark matter direct detection and in solar neutrino experiments. This paper lays out an in-depth study of [$\nu_{b}$]{} detection prospects for current and future underground rare event searches. We scrutinize the model as a possible explanation for the reported anomalies from DAMA, CoGeNT, and CRESST-II and confront it with constraints from other null experiments.
Introduction
============
The phenomenon of neutrino mass-mixing and thereby induced lepton-flavor oscillations constitute the only conclusively detected deviation from the Standard Model (SM) to date. Though the neutrino sector is certainly the most elusive part of the SM, a tremendous and sometimes painstaking experimental effort has firmly established the patterns of mass splitting and mixing for the three SM neutrinos, $\nu_e,\,\nu_{\mu},\,\nu_{\tau}$. The multitude of data is consistent with neutrino flavor eigenstates being the linear combination of—at least—three massive ones $\nu_1,\,\nu_2,\,\nu_3$, with mass-squared differences $\Delta m_{21}^2 = \mathcal{O}(10^{-5})\,{\ensuremath{\mathrm{eV}}}^2$ and $
\Delta m_{31}^2 \sim \Delta m_{32}^2 = \pm\mathcal{O}(10^{-3})\,{\ensuremath{\mathrm{eV}}}^2$ and mixing angles with magnitudes $\sin^22\theta_{13}\sim 0.1$, $\theta_{23}\sim \pi/4$, $\tan^2\theta_{12} \sim 1/2
$ [@Nakamura:2010zzi; @An:2012eh]. Neutrinos are predominantly detected via their charged current (CC) interaction on matter, and elastic scattering (ES) on electrons. The only pure neutral current (NC) process for solar neutrinos with participation of all active neutrino flavors observed remains the deuteron breakup reaction $d+\nu_x\to p
+n +\nu_x$ in the SNO experiment [@Ahmad:2002jz; @Aharmim:2011vm].
On the theoretical side, the interest to the elastic scattering of neutrinos on nuclei dates back to Drukier and Stodolsky [@PhysRevD.30.2295], who outlined their vision of a true neutrino observatory: neutrinos with MeV-scale energies as they emerge from the interior of the sun, from nuclear reactors, from spallation beam experiments, or from supernovae explosions can scatter elastically and with $N^2$ enhancement in the cross section on keV-scale recoiling nuclei of $N$ neutrons [@Freedman:1977xn]. Even though the energy transfer to the target nucleus is small, the idea was that its effect in a detector at cryogenic temperatures may be macroscopic given that the specific heat of the material can be minute. The underlying principle of NC neutrino scattering on nuclei has found its proliferation in Dark Matter (DM) searches for weakly interacting massive particles (WIMPs) [@PhysRevD.31.3059].
Direct detection experiments seek for evidence of DM via its elastic scattering on various targets such as crystals made from germanium or liquefied nobles gases such as xenon. No univocal evidence for DM has yet been found but upper limits as low as $7\times 10^{-45}\,{\ensuremath{\mathrm{cm}}}^2$ [@Aprile:2011hi] in the WIMP-nucleon cross section have been inferred for the optimal range of recoil energies. These experiments have reached a level in sensitivity such that neutrino coherent scattering on nuclei is being discussed as potential background for future ton-scale experiments; see *e.g.* [@Monroe:2007xp; @Strigari:2009bq]. The latter process has a cross section $\sigma_{\nu}/N^2 \simeq
4\times 10^{-43} (E_{\nu}/10\,{\ensuremath{\mathrm{MeV}}})^2\,{\ensuremath{\mathrm{cm}}}^2$ where $E_{\nu}$ is the neutrino energy. Indeed, the average flux of neutrinos at earth, dominated by the solar $pp$-chain of thermonuclear reactions, $\Phi_{\nu}^{pp} \simeq 6\times 10^{10} \,
{\ensuremath{\mathrm{cm}}}^{-2}{\ensuremath{\mathrm{s}}}^{-1}$ [@RevModPhys.60.297], exceeds the flux expected from WIMPs, $\Phi_{{\ensuremath {\mathrm{DM}}}} \sim 10^5 \left( 100\,{\ensuremath{\mathrm{GeV}}}/m_{{\ensuremath {\mathrm{DM}}}} \right)\,
{\ensuremath{\mathrm{cm}}}^{-2}{\ensuremath{\mathrm{s}}}^{-1} $, by many orders of magnitude. The reason why neutrinos are *not* copiously detected via their NC interaction in DM experiments lies in the soft recoil spectrum they induce, $ E_R
\leq {2 E_{\nu}^2}/{m_N} \simeq 2\,{\ensuremath{\mathrm{keV}}}\times \left( {100}/{A}
\right) \left( {E_{\nu}}/{10\,{\ensuremath{\mathrm{MeV}}}} \right)^2$, where $m_N$ and $A$ are the mass and atomic mass number of the target nucleus. Whereas these experiments fall short in sensitivity to SM neutrinos, they may nonetheless be powerful probes of an extended neutrino sector [@Pospelov:2011ha].
This paper surveys potential signatures of the existence of yet another “sterile-active” neutrino state, [$\nu_{b}$]{}, where active is understood in the sense that [$\nu_{b}$]{} shall interact via a *new* neutral current interaction with baryons (NCB) [@Pospelov:2011ha]. This “baryonic” neutrino is also “sterile” in that it shares no SM NC or CC interactions. In particular, this implies that [$\nu_{b}$]{} does not scatter elastically on electrons. Remarkably, [$\nu_{b}$]{} can then be coupled to baryons with a strength $G_B$ which exceeds that of the weak interactions substantially, $G_B/G_F = (10^2-10^3)$; $G_F$ is the Fermi constant. This, in turn, implies the existence of a force mediator with the mass much smaller than $m_{W}$. The key observation made in [@Pospelov:2011ha] is that for MeV-scale energies of [$\nu_{b}$]{} the ratio of elastic to inelastic cross section with nuclei scales as $$\begin{aligned}
\label{eq:scaling}
\frac{\sigma_{{\ensuremath {\nu_{b}}}N}(\mathrm{elastic})}{\sigma_{{\ensuremath {\nu_{b}}}N}(\mathrm{inelastic}) } \sim 10^8 \times \, \left( \frac{A}{100}
\right)^2 \left( \frac{10\,{\ensuremath{\mathrm{MeV}}}}{E_{\nu}} \right)^4 \left(
\frac{2\,{\ensuremath{\mathrm{fm}}}}{R_N} \right)^4 ,\end{aligned}$$ where $R_N$ denotes a nuclear radius. This tremendous enhancement opens an exciting phenomenological window of opportunity: [$\nu_{b}$]{} states can be searched for in the growing number of direct detection dark matter experiments. The less constrained choice of parameters for this model is when the oscillation length for $\nu_{SM}\to \nu_b$ transition is long enough not to create any significant fluxes of reactor/beam/atmospheric neutrinos, while for solar neutrinos one can have a sizable fraction of $\nu_b$. In particular, solar [${}^8\mathrm{B}$]{}neutrinos have the best combination of large flux and high end-point energy for producing an observable signal in rare event searches. Relation (\[eq:scaling\]) makes even small scale DM searches competitive with dedicated large target-mass neutrino experiments in their sensitivity to [$\nu_{b}$]{} that originate from $^8$B neutrinos.
Moreover, as proposed in [@Pospelov:2011ha], if the oscillation length of [$\nu_{b}$]{} is on the order of the earth-sun distance, the signal may be annually modulated in a non-trivial way, with a possibility of a bigger flux in the summer, and with modulation amplitude larger than naively expected. In this paper we analyze this corner of the parameter space in great detail, as it could offer a $\nu_b$-scattering explanation to the DAMA signal [@Bernabei:2008yi; @Bernabei:2010mq] while still being compatible with other null-results from DM experiments; for a relevant collider study in this context cf. [@Friedland:2011za]. The long-standing DAMA “anomaly” has very recently received some additional impetus by the reports [@Aalseth:2010vx; @Aalseth:2011wp] of the CoGeNT collaboration on 1) an unexpected rise in observed events at low nuclear recoils and 2) an indication of annual modulation, see *e.g.* [@Aalseth:2011wp; @Hooper:2011hd; @Fox:2011px; @HerreroGarcia:2011aa]. Finally, the CRESST-II experiment has published results of its latest run, which had some population of unexpected events on top of existing backgrounds [@Angloher:2011uu].
The attempts to explain positive signals (DAMA) and possible hints on non-zero signals (CoGeNT, CRESST) while staying consistent with null results of other groups, are widespread in the WIMP literature, see *e.g* [@Chang:2010yk; @Feng:2011vu; @Cline:2011zr; @Schwetz:2011xm; @Farina:2011pw; @Frandsen:2011ts; @Kopp:2011yr; @Kelso:2011gd; @Shoemaker:2011vi] and references therein. The models that fare best feature $\sim 10$ GeV WIMP masses, although the overall consistency of the “light WIMP” explanation for the “DM anomalies” remains doubtful. In this paper, we provide an in-depth critical assessment of ${\ensuremath {\nu_{b}}}$ models with regard to their potential to explain positive signals, and make predictions for the upcoming experiments, paying attention to those that could potentially differentiate between [$\nu_{b}$]{}and light dark matter recoils. We leave other phenomenological aspects of this interesting, but to date a relatively poorly explored model to subsequent work.
This paper is organized as follows: in the next section, the baryonic neutrino model is reviewed. In Sec. \[sec:dd\] we cover the current and future sensitivity of DM searches to [$\nu_{b}$]{} and study the potential explanation of the DAMA, CoGeNT, and CRESST-II signals. In Sec. \[sec:ns\] the elastic scattering of [$\nu_{b}$]{} in neutrino experiments is considered, and in Sec. \[sec:conclusions\] we reach our conclusions.
Baryonic neutrinos {#sec:nub}
==================
In the baryonic neutrino model [@Pospelov:2011ha], the SM gauge group is extended by an abelian factor U(1)$_B$. The new neutrino is a left-chiral field ${\ensuremath {\nu_{b}}}= \frac{1}{2} (1-\gamma^5) \nu_b$ with charge $q_b = \pm 1$ and gauge coupling $g_l>0$. Leptons are neutral under U(1)$_B$ but all quarks $q=Q_L,\,u_R,\,d_R$ carry baryonic charge $1/3$ with gauge coupling $g_b > 0$. The SM Lagrangian is supplemented by the following terms $$\begin{aligned}
\mathcal{L}_{B} = {\ensuremath {\overline\nu_{b}}}\gamma^{\mu} (i \partial_{\mu} - g_l q_{b}
V_{\mu}) {\ensuremath {\nu_{b}}}- \frac{1}{3} g_b \sum_q \bar q \gamma^{\mu} q V_{\mu}
-\frac{1}{4} V_{\mu\nu}V^{\mu\nu} + \frac{1}{2} m_V^2 V_{\mu}V^{\mu}
+ \mathcal{L}_{m} .\end{aligned}$$ We have assumed that the new gauge field $V_{\mu}$ with field strength $V_{\mu\nu}$ has acquired a mass $m_V$ by the spontaneous breakdown of the U(1)$_B$ symmetry with Higgs$_b$ VEV $\langle \phi_b \rangle =
v_b/\sqrt{2}$; the sum in the second term runs over all SM quarks $q$. The part $\mathcal{L}_m$ is responsible for generating neutrino masses and the mixing between SM neutrinos and the new state [$\nu_{b}$]{}. A simple UV-completion of $\mathcal{L}_m$ is one where—once electroweak symmetry and U(1)$_b$ are broken—new right-handed neutrinos $\nu_R$ induce mass terms for SM neutrinos as well as for [$\nu_{b}$]{}: $$\begin{aligned}
\label{eq:Lm}
\mathcal{L}_m = \frac{1}{2} N_L^T \mathcal{C}^{\dag} M N_L + \mathrm{h.c.}
,\quad N_L =
\begin{pmatrix}
\nu_L' \\ \nu_R'^C \\ {\ensuremath {\nu_{b}}}'
\end{pmatrix},\quad
M =
\begin{pmatrix}
0 & m_D^T & 0\\
m_D & m_R & v_b b \\
0 & v_b b^T & 0
\end{pmatrix} .\end{aligned}$$ Here, $\nu'_L$ denote the three SM neutrinos with Dirac mass matrix $m_D$, $\nu_R'^C = \mathcal{C} \overline{\nu_R'}^T$ are the charge conjugate states of $\nu_R'$ with Majorana mass matrix $m_R$, and $b$ is a vector of Yukawa couplings generating mass for $\nu'_{b}$. In the simplest case the new right-handed neutrinos generate masses for $\nu'_L$ and $ {\ensuremath {\nu_{b}}}'$ simultaneously. Introduction of a right-handed partner for ${\ensuremath {\nu_{b}}}'$ will cancel a U(1)$_B^3$ triangle anomaly and remaining gauge anomalies can be cured by the introduction of a new family of heavy fermions with appropriate quantum numbers [@PhysRevLett.74.3122]. From the phenomenological viewpoint the details of this will not be of importance for this work.
The mass matrix $M$ in (\[eq:Lm\]) is diagonalized by $\mathcal{M}_{\mathrm{diag}} = (V_L^{\nu})^\dag M V_L^{\nu} $ where $V_L^{\nu}$ is a unitary matrix, defining the mass eigenstates $N_L =
(V_L^{\nu})^{\dag} N'_L$. Diagonalization of the charged lepton mass matrix by $3\times3$ unitary matrices $V_{L,R}^{l}$ with mass eigenstates $l_L =
(V_L^l)^{\dag} l'_L$, and $l_R = (V_R^l)^{\dag} l'_R$ where $l =
(e^{-}, \mu^{-},\tau^{-})$ then determines the mixing among the SM active neutrino states. Assuming that the eigenvalues of $m_R$ are much larger than any other values in $M$, the seesaw mechanism is operative and one can integrate out the heavy, right handed states. We are left with the $4\times4$ mixing matrix $U$ which connects “flavor” $\nu_{\alpha L}\,
(\alpha=e,\mu,\tau,b)$ and mass $n_{k L}\, (k=1,\,\dots,4)$ eigenstates: $$\begin{aligned}
\label{eq:fltoma}
n_{kL} = \sum_{\alpha } U^{*}_{k\alpha}
\nu_{\alpha L},
\quad U =
\begin{array}{cc}
\hphantom{b} & \begin{matrix}\!\!\!\! 1 & 2 & 3 & 4 \end{matrix} \\
\begin{matrix}e\\\mu\\\tau\\b\end{matrix} & \!\!
\begin{pmatrix} & & & \cdot\,\, \\ \multicolumn{3}{c}{U_{\mathrm{PMNS}}} & \cdot\,\,
\\ & & &\cdot\,\,\\ \cdot\,\, & \cdot\,\, & \cdot\,\, & \cdot\, \,
\end{pmatrix}\end{array} ,\end{aligned}$$ where $U_{\mathrm{PMNS}}$ is the usual $3\times3$ mixing matrix among the active flavors [@Nakamura:2010zzi]. The NCB current in the respective interaction and mass eigenbasis reads $$\begin{aligned}
j_{NCB}^{\mu} = {\ensuremath {\overline\nu_{b}}}\gamma^{\mu} {\ensuremath {\nu_{b}}}= \sum_{k,k'}\, U^{*}_{ 4 k}U_{4 k'}
\overline{n}_{kL} \gamma^{\mu} n_{k'L} .\end{aligned}$$
Neutrino oscillations and matter effects {#sec:matter-effects}
----------------------------------------
Aiming at a scenario in which the baryonic neutrino is coupled stronger than $G_F$, the question which immediately arises is whether new matter effects are to play a role. The index of refraction of [$\nu_{b}$]{}-propagation in matter is found by computing the forward scattering amplitude of ${\ensuremath {\nu_{b}}}$ on matter, described by the effective Lagrangian $$\begin{aligned}
\label{eq:Leff}
\mathcal{L}_{\mathrm{eff}} = - G_B j^{\mu}_{NCB} \sum_{N = n,p}
\overline N \gamma_{\mu} N ,\qquad G_B = q_b \frac{g_b g_l }{
m_V^2} = q_b {\ensuremath {\mathcal{N}}}G_F .\end{aligned}$$ In the last equality we have introduced the parameter ${\ensuremath {\mathcal{N}}}>0$ to measure $G_B$ in units of $G_F$ with the sign of the interaction determined by the charge $q_b$ of [$\nu_{b}$]{}. In an unpolarized medium, one obtains the following matter potential $$\begin{aligned}
V_{NCB} = \pm q_b {\ensuremath {\mathcal{N}}}G_F n_B \left( Y_N + 2Y_{\nu_b} \right), \qquad Y_f =
\frac{n_{f} - n_{\overline f}}{ n_B},\end{aligned}$$ where the plus sign is for [$\nu_{b}$]{} and the minus sign for [$\overline\nu_{b}$]{}; $Y_f$ is the particle-antiparticle asymmetry, normalized to the number density of baryons $n_B$. The first term in the first equation describes the potential in ordinary matter while the second term is the potential for $\stackrel{(-)}{\nu}_{\!\!bL}$ in a hypothetical sea of baryonic neutrinos.
In ordinary matter, with mass fraction $X_p$ in form of bound or unbound protons, the induced matter potentials (up to coherence factors) compare as follows $$\begin{aligned}
V_{NCB} : V_{CC} : V_{NC} = q_b {\ensuremath {\mathcal{N}}}: \sqrt{2} X_p : -\sqrt{2} (1-X_p)/2 ,\end{aligned}$$ where we have made use of the charge neutrality condition. As is evident, for ${\ensuremath {\mathcal{N}}}\gg1$ baryonic neutrinos experience the largest matter effect since $X_p$ is always of order unity.
The large strength of the NCB interaction may suggest that the flavor evolution in matter is dominated by $ V_{NCB}$. In a simplified two-neutrino case the Schrödinger-like equation describing the transition probabilities $P_{\alpha\to \beta}(x) = | \langle
\nu_{\beta} |\nu_{\alpha}(x) \rangle |^2 \equiv |\psi_{\alpha
\beta}(x)|^2$ from an initial state $| \nu_{\alpha} (0)\rangle$ to final state $|\nu_{\beta} \rangle$ can then be brought into the following form $$\begin{aligned}
i \frac{d}{dx}
\begin{pmatrix}
\psi_{\alpha\alpha}\\ \psi_{\alpha b}
\end{pmatrix}
\simeq
\frac{1}{4{\ensuremath {E_\nu}}}
\begin{pmatrix}
-\Delta m^2 \cos 2\theta-2{\ensuremath {E_\nu}}V_{NCB} & \Delta m^2 \sin 2\theta \\
\Delta m^2 \sin 2\theta & \Delta m^2 \cos 2\theta + 2{\ensuremath {E_\nu}}V_{NCB}
\end{pmatrix}
\begin{pmatrix}
\psi_{\alpha\alpha}\\ \psi_{\alpha b}
\end{pmatrix} .
$$ Here, ${\ensuremath {E_\nu}}$ is the neutrino energy and $\theta = \theta_{k 4}$ and $\Delta m^2 = \Delta m_{4k}^2$ with $k=1,2,3$ for $\alpha=e,\mu,\tau$ are the vacuum mixing angle and the mass squared difference between the new massive state $\nu_4$ and SM neutrinos $\nu_{1,2,3}$, respectively. The matter induced mixing angle $\theta_M$ reads $$\begin{aligned}
\label{eq:thetaM}
\tan 2\theta_M = \frac{\tan 2\theta}{1+ 2{\ensuremath {E_\nu}}V_{NCB} / (\Delta m^2 \cos
2 \theta ) },\end{aligned}$$ and, since $\mathrm{sign\,}(V_{NCB}) = q_b$, resonant flavor transitions for ${\ensuremath {\nu_{b}}}$ could be possible for $q_b = +1$ and $\theta >
\pi/4$ or for $q_b = -1$ and $\theta < \pi/4$ (and vice verse for [$\overline\nu_{b}$]{}.) We note, however, that the efficiency of a transition has a separate dependence on $\Delta m^2$, unrelated to Eq. (\[eq:thetaM\]), as the matter-induced oscillations seize to occur in the limit of $\Delta m^2\to 0$. From (\[eq:thetaM\]) we find the necessary condition for which NCB effects are least likely to play a role, $$\begin{aligned}
\label{eq:matter}
\Delta m^2 \cos{2\theta} \ll 10^{-4}\,{\ensuremath{\mathrm{eV}}}^2 \times \left(
\frac{E}{10\,{\ensuremath{\mathrm{MeV}}}} \right) \left( \frac{{\ensuremath {\mathcal{N}}}}{100} \right)
\left( \frac{\rho}{\mathrm{g}/{\ensuremath{\mathrm{cm}}}^3} \right) .\end{aligned}$$ In this work we focus on a parameter region which obeys this limit. A discussion for larger values of $ \Delta m^2$ is beyond the scope of this work.
Solar $\nu_b$ flux
------------------
Let us consider a scenario in which the baseline of ${\ensuremath {\nu_{b}}}$ oscillation $L_{\mathrm{osc}}$ is on the order of the earth-sun distance, $L_0 =
1\,\mathrm{AU} \simeq 1.5\times 10^8\,{\ensuremath{\mathrm{km}}}$. This “just-so” choice of parameters suggests a canonical mass squared difference, $$\begin{aligned}
\label{eq:Losc}
\frac{L_{\mathrm{osc}}}{L_0} \simeq 0.5 \times \left(
\frac{10^{-10}\,{\ensuremath{\mathrm{eV}}}^{2}}{\Delta m^2} \right) \left(
\frac{E_{\nu}}{10\,{\ensuremath{\mathrm{MeV}}}} \right) .\end{aligned}$$ “Flavor” eigenstates $\nu_{\alpha L}$ $(\alpha = e,\mu,\tau,b)$ are found from mass-eigenstates $n_{kL}$ by inversion of (\[eq:fltoma\]), $ \nu_{\alpha L } = \sum_{k} U_{\alpha k} n_{k
L}$, and their evolution is obtained by solving $$\begin{aligned}
\label{eq:schroedinger}
i \frac{d\Psi}{dx} = \mathcal{H} \Psi,\quad \mathcal{H} =
\frac{1}{2E} \left( U \mathcal{M}_d^2 U^{\dag} + \mathcal{A} \right).\end{aligned}$$ Here $\Psi$ is the vector of amplitudes for the flavor states, $\Psi=(\psi_e,\psi_{\mu},\psi_{\tau}, \psi_b)$, $\mathcal{M}_d^2 =
\mathrm{diag}(m_1^2,m_2^2,m_3^2,m_4^2)$ is the diagonalized mass matrix and the entries of $\mathcal{A} =
\mathrm{diag}(A_{CC}+A_{NC},A_{NC},A_{NC},A_{NCB})$ are related to the induced matter potentials via $A_x = 2E V_x$. In general, the baryonic neutrino flux at the Earth location is found upon numerical integration of (\[eq:schroedinger\]) from the production point $r_0$ of $\nu_e$ in the solar interior to earth at distance $L$ with initial condition $\Psi(r_0) = (1,0,0,0)$, This could be a complex problem when matter effects are involved, but fortunately not for the region of the parameter space we are interested in.
With a few simplifying assumptions the appearance probability at earth can be obtained analytically [@Pospelov:2011ha]. We seek access to the high energy end of the neutrino spectrum, ${\ensuremath {E_\nu}}\gtrsim 10\,{\ensuremath{\mathrm{MeV}}}$, because scatterings of ${\ensuremath {\nu_{b}}}$ will then more likely be picked up by a detector. The largest flux in combination with high endpoint energy comes from the neutrino emission in the decay of [${}^8\mathrm{B}$]{}. With [${}^4\mathrm{He}$]{}being the most tightly bound light nucleus, hep neutrinos have the highest endpoint in energy but come with a flux which is smaller by three orders of magnitude. The [${}^8\mathrm{B}$]{} and hep respective fluxes and endpoint energies are given by [@2005ApJ...621L..85B], $$\begin{aligned}
\Phi_{{\ensuremath{{}^8\mathrm{B}}}} = (5.69^{+0.173}_{-0.147})\times 10^6\,{\ensuremath{\mathrm{cm}}}^{-2}\,{\ensuremath{\mathrm{s}}}^{-1},\quad E_{\mathrm{max},
{\ensuremath{{}^8\mathrm{B}}}} = 16.36\,{\ensuremath{\mathrm{MeV}}}, \nonumber \\
\Phi_{\mathrm{hep}} = (7.93\pm0.155 )\times 10^3\,{\ensuremath{\mathrm{cm}}}^{-2}\,{\ensuremath{\mathrm{s}}}^{-1},\quad E_{\mathrm{max},
\mathrm{hep}} = 1.88\,{\ensuremath{\mathrm{MeV}}}.
\label{eq:fluxes}\end{aligned}$$ In the solution to the solar neutrino problem the MSW effect dominates the flavor evolution of the highly energetic part of the neutrino spectrum and neutrinos exit the sun mainly as $\nu_2$. Therefore, we assume a preferential mixing of the new neutrino to $\nu_2$, $ \theta_{24} \neq 0$, and neglect other mixings of $\nu_4$ to the active states. Choosing $q_b>0$, no resonant flavor transitions to ${\ensuremath {\nu_{b}}}$ inside the sun appear and with a ballpark of $\Delta m^2$ suggested in (\[eq:Losc\]) the standard solar MSW solution remains in place.
With these assumptions and using a tri-bimaximal mixing ansatz for the active states the following ${\ensuremath {\nu_{b}}}$ appearance probability at earth for the high energy part of the ${\ensuremath{{}^8\mathrm{B}}}$ (and hep) fluxes has been obtained in [@Pospelov:2011ha], $$\begin{aligned}
\label{eq:Peb}
P_{eb}(L,{\ensuremath {E_\nu}}) \simeq \sin^2(2\theta_b) \sin^2{\left[ \frac{{\ensuremath {\Delta m_b^2}}L(t)}{4 {\ensuremath {E_\nu}}} \right]} . \end{aligned}$$ It is assumed that mass mixings among active components are larger than mixings with ${\ensuremath {\nu_{b}}}$ so that one can address the diagonalization of the neutrino mass matrix sequentially. In this procedure, ${\ensuremath {\Delta m_b^2}}$ and $\theta_b$ denote the associated effective mass-squared difference and mixing angle between $\nu_2$ and ${\ensuremath {\nu_{b}}}$, respectively. The true vacuum mass eigenstates are $\nu_I = \cos{\theta_b}\nu_2 +
\sin{\theta_b}\nu_b$ and $\nu_{II} = -\sin{\theta_b}\nu_2 +
\cos{\theta_b}\nu_b$ and a phase builds up between $\nu_2$ exit from the sun and propagation to the detector at distance $L$, $$\begin{aligned}
L(t) = L_0 \left\{ 1-\epsilon \cos{\left[ \frac{2\pi(t-t_0)}{T}
\right]} \right\} ,\end{aligned}$$ where $\epsilon = 0.0167$ is the ellipticity of the earth’s orbit; the perihelion is reached at $t_0 \sim 3\,\mathrm{Jan}$. In what follows, it will be convenient to introduce an effective interaction parameter [$\mathcal{N}_{\mathrm{eff}}$]{}, $$\begin{aligned}
\label{eq:neff}
{\ensuremath {\mathcal{N}_{\mathrm{eff}}}}^2 = {\ensuremath {\mathcal{N}}}^2 \, \sin^2(2\theta_b) / 2 .\end{aligned}$$ In the limit of rapid oscillations this implies $P_{eb}G_B^2\to {\ensuremath {\mathcal{N}_{\mathrm{eff}}}}^2G_F^2$.
Direct Detection {#sec:dd}
================
In this section we provide a detailed investigation of current and future direct detection experiments. From the scaling (\[eq:scaling\]) we expect that elastic scattering off nuclei in direct detection experiments constitutes one of the most promising avenues in the search for a solar, long-baseline flux of ${\ensuremath {\nu_{b}}}$ particles.
The spin-independent elastic recoil cross section on nuclei obtained from (\[eq:Leff\]) essentially resembles the one from neutrino-nucleus coherent scattering [@PhysRevD.30.2295] with the replacement $G_F^2(N/2)^2\to G_B^2 A^2$ [@Pospelov:2011ha], $$\begin{aligned}
\label{eq:dsdcosth}
\frac{d\sigma_{\mathrm{el}}}{d\cos\theta_{*}} = \frac{G_B^2}{4\pi}
\,{\ensuremath {E_\nu}}^2 A^2 (1+\cos\theta_{*}) .\end{aligned}$$ Here $A$ is the atomic number of the nucleus and $\theta_{*}$ is the scattering angle in the CM frame. Equation (\[eq:dsdcosth\]) can be rewritten in terms of a recoil cross section, $$\begin{aligned}
\frac{d\sigma_{\mathrm{el}}}{d{\ensuremath {E_R}}} = \frac{G_B^2}{2\pi} \,A^2
m_N F^2(|\mathbf{q}|)\left[ 1 - \frac{(E_{\mathrm{min}})^2}{{\ensuremath {E_\nu}}^2} \right] , \quad
{\ensuremath {E_R}}= \frac{{\ensuremath {E_\nu}}^2}{m_N} \left( 1 - \cos{\theta_{*}} \right),\end{aligned}$$ where $E_{\mathrm{min}} = \sqrt{{\ensuremath {E_R}}m_N/2}$ is the minimum energy required to produce a recoiling nucleus of mass $m_N$ and kinetic energy ${\ensuremath {E_R}}$. Here we have now included the nuclear form factor suppression $F^2(|\mathbf{q}|)$ for scatterings with three-momentum transfer $\mathbf{q}$, and in our numerical evaluations we use the Helm form factor parametrization [@Helm:1956zz] with the nuclear skin thickness of 0.9[$\mathrm{fm}$]{}.
For the sake of comparison to the simplest case of spin-independent scattering of DM on nuclei, we can evaluate the total elastic scattering cross section of [$\nu_{b}$]{} on nuclei (at zero momentum transfer), $$\begin{aligned}
\label{eq:sigTotal}
\sigma_{\mathrm{el}} = \frac{G_B^2 }{\pi}A^2 {\ensuremath {E_\nu}}^2 \simeq 1.7\times
10^{-38}\,{\ensuremath{\mathrm{cm}}}^2 \times A^2 \left( \frac{{\ensuremath {\mathcal{N}}}}{100} \right)^2 \left(
\frac{{\ensuremath {E_\nu}}}{10\,{\ensuremath{\mathrm{MeV}}}} \right)^2 .\end{aligned}$$ The coefficient in front of the second relation serves as a figure of merit when compared to the DM-nucleon cross section $\sigma_n$. Given that direct detection experiments have put upper limits on $\sigma_n $ as low as $10^{-44}\,{\ensuremath{\mathrm{cm}}}^2$ [@Aprile:2011hi] the coefficient in (\[eq:sigTotal\]) is sizable. However, a much more stringent cut off in [$E_R$]{} makes it increasingly difficult for an essentially massless ${\ensuremath {\nu_{b}}}$ to scatter off heavier targets.
The recoil spectrum arising from the solar flux of [$\nu_{b}$]{} will have to include an average over the neutrino energy spectrum $df_i/d{\ensuremath {E_\nu}}$ of neutrino source $i$, weighted by the ${\ensuremath {\nu_{b}}}$ appearance probability and an overall flux modulation $[L_0/L(t)]^2$ due to the earth’s eccentric orbit, $$\begin{aligned}
\label{eq:rate}
\frac{dR}{dE_R} = N_T \left[ \frac{L_0}{L(t)} \right]^2 \sum_i\Phi_i
\int^{\mathrm{E_{\mathrm{max},i}}}_{\mathrm{E_{\mathrm{min}}}} dE_{\nu}\, P_{eb}(t,E_{\nu})\frac{df_i}{dE_{\nu}}
\frac{d\sigma_{\mathrm{el}}}{dE_R} . \end{aligned}$$ Note that $P_{eb}$ depends on ${\ensuremath {E_\nu}}$ so that it has to be included into the average; $\Phi_i$ is the integral flux given in (\[eq:fluxes\]) and $df_i/dE$ is normalized to unity, , and taken from [@PhysRevC.54.411; @PhysRevC.56.3391]. $N_T$ denotes the number of target nuclei per unit detector mass, and in our computations we take the fractional isotopic abundances of each element under consideration into account. The rate exhibits a non-trivial time-dependence. The maximum of the overall flux is attained at the perihelion in early January. However, the integral in Eq. (\[eq:rate\]) constitutes an additional source of modulation which depends on the neutrino energy. It will have its strongest effect on the differential rate in the “just-so” regime of Eq. (\[eq:Losc\]) where $L_{\mathrm{osc}}$ is on the order of the sun earth distance; we exploit this fact in the following section.
![*Left:* Solar neutrino fluxes as a function of energy as taken from [@PhysRevC.54.411; @PhysRevC.56.3391; @PhysRevD.49.3923]. *Right:* Associated recoil spectrum in a (perfect) germanium detector with a total exposure of 1 kg$\times$yr. Only [${}^8\mathrm{B}$]{} and hep neutrinos reach out to values of ${\ensuremath {E_R}}$ where direct detection experiments become sensitive. []{data-label="solnu"}](NUB_RecAllSolNu){width="\textwidth"}
The left panel of Fig. \[solnu\] shows the solar neutrino spectra of the various sources. In the right panel we compute the associated recoil spectra using (\[eq:rate\]) for a germanium detector without threshold, with perfect energy resolution, and an exposure of $1\,{\ensuremath{\mathrm{kg}}}\times$yr. For simplicity and since it does not affect the argument we use $P_{eb}$ as given in (\[eq:Peb\]) for all fluxes, while strictly speaking Eq. (\[eq:Peb\]) is only valid in the MSW regime. The deviation from MSW only affects the softest recoils, and can therefore be safely neglected in what follows. As can be seen, only [${}^8\mathrm{B}$]{} and hep neutrinos reach out to values of ${\ensuremath {E_R}}\gtrsim
\mathrm{few}\,{\ensuremath{\mathrm{keV}}}$ where direct detection experiments become sensitive. Furthermore, [${}^8\mathrm{B}$]{} neutrinos constitute the dominant part of the signal with hep giving a small correction only.
The spectrum in (\[eq:rate\]) is a theoretical one. To make contact with experiment we include effects from energy resolution, detector threshold and quenching of nuclear recoil energy. Details will be given when discussing the respective experiments. We start our discussion by considering the experiments with putative positive signal claims.
DAMA {#sec:dama}
----
The DAMA/NaI and its upgrade the DAMA/LIBRA experiment [@Bernabei:2008yh], situated in the northern hemisphere at the underground Gran Sasso National Laboratory (LNGS), were the first to report on a potential direct DM detection. The experiment uses large radiopure NaI(Tl) crystals to measure scintillation light resulting from nuclear recoils. Given that there is no other discriminating channel except requiring the candidate event to be a “single-hit”, a relatively large overall count-rate of $\sim 1\,
\mathrm{cpd/kg/keVee}$ is observed. The presence of a positive signal is inferred from the annual modulation of the residual count rate on the order of $\sim 0.02\, \mathrm{cpd/kg/keVee}$ over low-energy bins between 2 and 6[$\mathrm{keVee}$]{} once the average count rate per cycle is subtracted [@Bernabei:2008yi; @Bernabei:2010mq]; for recent discussions on potential modulating backgrounds see [@Ralston:2010bd; @Nygren:2011xu; @Blum:2011jf; @Chang:2011eb; @Bernabei:2012wp].
The modulation of the event rate has been observed over the course of more than a dozen annual cycles, collecting a cumulative exposure of 1.17ton$\times$yr [@Bernabei:2008yi; @Bernabei:2010mq]. The null-hypothesis, *i.e.* a rate constant in time, has been excluded at the $8.9\sigma$ level. The residuals of the DAMA/LIBRA runs in consecutive bins between (2-4)[$\mathrm{keV}$]{}, (2-5)[$\mathrm{keV}$]{} and (2-6)[$\mathrm{keV}$]{} are shown by the data points in Fig. \[residuals\]. The DAMA signal is usually decomposed as $$\begin{aligned}
S = S_0 + S_m \cos{\left[ \omega (t-t_0) \right]}\end{aligned}$$ where $S_0 \sim 1\,{\ensuremath {\mathrm{cpd}}}/{\ensuremath{\mathrm{kg}}}/{\ensuremath {\mathrm{keVee}}}$ is the baseline rate of single hit events and $S_m$ is the modulation amplitude, $$\begin{aligned}
\label{eq:mod-amp}
S_m = \frac{1}{2} \left( \left.\frac{dR}{d{\ensuremath {E_\mathrm{v}}}}\right|_{t_0}
- \left.\frac{dR}{d{\ensuremath {E_\mathrm{v}}}}\right|_{t_0+1/2\,\mathrm{yr}} \right) .\end{aligned}$$ The measured phase $t_0$ is reported to be compatible with the one expected from DM, $t_0 = 152.5\,{\ensuremath {\mathrm{days}}}$ (June 2nd) with a period of one year, *i.e.* $\omega = 2\pi/(1\,\mathrm{yr})$. The reported modulation amplitude is shown by the data points in Fig. \[dama\].
![The data points show the DAMA modulation amplitude as reported in [@Bernabei:2010mq] in units of counts per day (cpd) per kg detector material and recoil energy. The solid line is the best fit from [$\nu_{b}$]{} to the data. []{data-label="dama"}](NUB_Dama_revised){width="60.00000%"}
To see if [$\nu_{b}$]{} provides a viable explanation of the DAMA data one can either fit the modulation spectrum (\[eq:mod-amp\]) or directly the time series of the residual rate. Considering the solar ${\ensuremath {\nu_{b}}}$ origin for DAMA, Eq. (\[eq:rate\]) may not necessarily lead to a truly sinusoidal form of the signal as a function of time. In addition, $t_0$ is not expected to be identical with the DM value of 152.5days. At first sight, a direct fit of the time series seems therefore favorable. However, the reported residuals are binned in energy so that they only provide coarse-grained information on the recoil energy distribution. This, in contrast, calls for a fit of the modulation amplitude instead. We have implemented both approaches and discuss their results below. In addition, one can also attempt a joint fit of both data sets. This approach is complicated by the fact that the data sets are not independent.
We start by fitting the modulation amplitude. Observable scatterings of [$\nu_{b}$]{} occur on sodium only and no appreciable rate is expected for ${\ensuremath {E_\mathrm{v}}}\gtrsim 7\,{\ensuremath {\mathrm{keVee}}}$. The latter expectation is in accordance with what is seen in the data. Therefore, we only fit the first ten data points with ${\ensuremath {E_\mathrm{v}}}\lesssim 7\,{\ensuremath {\mathrm{keVee}}}$ in order not to bias the goodness-of-the-fit estimate. With the help of the usual $\chi^2$ function we obtain the following best fit values, $$\begin{aligned}
\label{eq:DAMA-bestFit}
\text{DAMA }S_m:\quad {\ensuremath {\Delta m_b^2}}= 2.43\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2,\quad
{\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 255,\quad \chi^2_{\mathrm{min}} /n_d = 9.5 / 8 .\end{aligned}$$ The minimum in $\chi^2$ is associated with a $p$-value of $p=0.3$; $n_d$ denotes the number of degrees of freedom. The result of this fit is shown by the solid line in Fig. \[dama\]. Confidence regions in [$\Delta m_b^2$]{} and [$\mathcal{N}_{\mathrm{eff}}$]{} are constructed by demanding, $$\begin{aligned}
\chi^2 ( {\ensuremath {\Delta m_b^2}}, {\ensuremath {\mathcal{N}_{\mathrm{eff}}}}) \leq \chi^2_{\mathrm{min}} + \Delta \chi^2 ,\end{aligned}$$ where $ \chi^2_{\mathrm{min}} $ is given in (\[eq:DAMA-bestFit\]). We choose $ \Delta \chi^2 = 9.21$ which corresponds to generous $99\%$ C.L. regions. The choice results in the two disjoint gray shaded regions shown in Fig. \[fit\].
![The data points show the DAMA/LIBRA reported residual event rate for various energy bins as a function of time. The red line is the residual event rate associated with the fit to the modulation amplitude in Fig. \[dama\]. The dotted line is a fit of the sinusoidal function $A\times \cos[\omega (t-t_0)]$ with $\omega$ corresponding to a period of one year and a phase $t_0$ as expected from DM. As can be seen the [$\nu_{b}$]{} signal is approximately out of phase by one month. For a quantitative statement see main text.[]{data-label="residuals"}](NUB_DAMAresiduals_revised){width="90.00000%"}
The above result looks promising. However, in contrast to the DM case one has to check how well the time dependence of the rate is met. The resulting signals in the various energy bins from the best fit values (\[eq:DAMA-bestFit\]) are shown by the solid (red) lines in Fig. \[residuals\]. For completeness we also show by the dotted lines fits of the data by the sinusoidal function $A_i\cos{[\omega(t-t_0)]}$ with period 1yr and $t_0=\mathrm{June}\,2^{\mathrm{nd}}$ as expected when the origin were due to DM scatterings. As can be seen by eye, the [$\nu_{b}$]{} signals seems to lag behind by approximately one month. Thus, the best fit corresponds to a phase inversion with a maximum rate at the aphelion with $t_0\sim \text{July
5th}$. For example, using the $(2-4)\,{\ensuremath {\mathrm{keVee}}}$ residuals, the best fit values of the modulation spectrum (\[eq:DAMA-bestFit\]), one finds $\chi^2/n_d = 101/41$ with $p= 5\times 10^{-7}$ for the residuals. This points towards a very poor description of the full data.
We can try to improve on the above situation by directly fitting the residual rate. This is an important check, since the time dependence of Eq. (\[eq:rate\]) is non-trivial. Can we find a corner in the considered parameter space in which one can alleviate the tension in the phase of DAMA and the [$\nu_{b}$]{} signal? From the $(2-4)\,{\ensuremath {\mathrm{keVee}}}$ data we obtain as best fit, $$\begin{aligned}
\text{DAMA residuals:}\quad {\ensuremath {\Delta m_b^2}}= 2.45\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2,\quad
{\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 183,\quad \chi^2_{\mathrm{min}} /n_d = 73.2 / 41 .\end{aligned}$$ Though the fit fares slightly better on the residuals with $p\simeq
10^{-3}$, this does not ameliorate the situation sufficiently. Moreover, the smaller value of [$\mathcal{N}_{\mathrm{eff}}$]{} now somewhat under-predicts the modulation amplitude. Finally, even when we perform a joint fit (neglecting potential covariances of the data sets) we do not find any substantial improvement. We conclude, that even though the DAMA modulation amplitude is fit rather nicely, the time series of events speaks against the [$\nu_{b}$]{} interpretation.
As a final remark, we comment on the sodium quenching factor. For our analysis above we used $Q=0.3$ in the conversion to electron equivalent recoil energy, ${\ensuremath {E_\mathrm{v}}}({\ensuremath {\mathrm{keVee}}}) = Q {\ensuremath {E_R}}({\ensuremath{\mathrm{keV}}})$. New measurements [@UCLAJuan] seem to indicate 1) lower values $Q\sim
0.15 $ and 2) a stronger energy dependence as previously thought. This has important implications for light DM as well as for the [$\nu_{b}$]{}hypothesis. We find that $Q=0.15$ moves the DAMA regions in Fig. \[fit\] towards larger values of [$\mathcal{N}_{\mathrm{eff}}$]{} which are already excluded by all the other considered null searches. The situation then becomes more similar to the one already witnessed for DM.
CoGeNT {#sec:cogent}
------
The CoGeNT experiment is a low-threshold nuclear recoil germanium detector situated in the Soudan Underground Laboratory. The latest data release is based on 442 live days taken with 0.33 kg target [@Aalseth:2010vx; @Aalseth:2011wp]. An unexplained exponential rise of the signal at lowest recoil energies 0.5–1[$\mathrm{keVee}$]{} is observed. The origin of it is unknown and has lead to the speculation that DM with a mass in the $\sim 8-10\,{\ensuremath{\mathrm{GeV}}}$ range may be the cause of it. For spin-independent DM-nucleus scattering, the excess requires a cross section $\sigma_{SI}\sim
10^{-40}\,{\ensuremath{\mathrm{cm}}}^2$. Such values are challenged by the null-result of XENON100 and by the low-threshold analysis of CDMS-II. However, more recently the collaboration has identified a source of surface-background events which may lead to a revision of the signal strength in the low-recoil bin 0.5–1[$\mathrm{keVee}$]{}. In the following we will investigate the possibility that the observed excess may be due to scattering of [$\nu_{b}$]{} in the detector. We will also account for the possibility that the collaboration’s results could be revised in the near future [@UCLAJuan].
In addition to the signal-rise below 1[$\mathrm{keVee}$]{} the data also appears to be annually modulated in the 0.5–3.2[$\mathrm{keVee}$]{} bracket. The observed event rate peaks in mid-to-late April (2010) with a modulation amplitude of $\sim 16\%$, most pronounced between 1.4–3.2[$\mathrm{keVee}$]{} [@Aalseth:2011wp]. The latter behavior is neither expected from DM scatterings nor could it be explained by [$\nu_{b}$]{}scatterings since the recoil spectrum arising from [${}^8\mathrm{B}$]{} neutrinos is cut off for ${\ensuremath {E_\mathrm{v}}}\gtrsim 1.4\,{\ensuremath {\mathrm{keVee}}}$. We also note that the modulation of nuclear recoil events in Ge in that energy regime has recently been challenged in a dedicated analysis by CDMS [@UCLAcdms]. We will therefore not further address the potential modulation of the CoGeNT signal and await further data.
![Recoil spectrum from the 442 live-day run of the CoGeNT experiment. The black (gray) data points show the signal after (before) subtraction of the cosmogenic radioactive background. The solid line is a fit to the black data points. It decomposes into the contribution from [$\nu_{b}$]{} (dashed line) and the contribution of a constant background (dotted line.)[]{data-label="cogent"}](NUB_Cogent){width="60.00000%"}
Cosmogenically induced radioactive background has to be subtracted from the CoGeNT data before fitting the exponential excess. The radioimpurities in the crystal have been identified by the collaboration, with the most prominent ones given by the electron capture decays of $^{68}$Ge and $^{65}$Zn centered at 1.3[$\mathrm{keVee}$]{} and 1.1[$\mathrm{keVee}$]{}, respectively. From a fit of observed K-shell electron capture peaks seen in the high energy data and from the expected ratio of L-to K-shell decays one can subtract the low-energy L-shell background in the 0.5–3.2[$\mathrm{keVee}$]{}window. We follow [@CogentWriteUp] in the subtraction and collect the time-stamped events in 0.1[$\mathrm{keVee}$]{} bins. The result of this procedure can be seen in Fig. \[cogent\] as the difference between gray (with peaks) and black (peaks subtracted) data points.
Nuclear recoil energies on germanium have to be converted into the measured ionization signal. We employ a Lindhard-type, energy dependent quenching factor, ${\ensuremath {E_\mathrm{v}}}({\ensuremath {\mathrm{keVee}}}) = Q \times
{\ensuremath {E_R}}({\ensuremath{\mathrm{keV}}})^{1.1204}$ with $Q = 0.19935$ [@CogentWriteUp] and account for a finite detector resolution by convolving the recoil signal with a Gaussian of width $\sigma^2 = (69.4\,{\ensuremath{\mathrm{eV}}})^2 +
0.858\,{\ensuremath{\mathrm{eV}}}\times{\ensuremath {E_\mathrm{v}}}({\ensuremath{\mathrm{eV}}})$ [@Aalseth:2008rx]. Finally, the efficiency of the detector is provided in Fig. 1 of [@Aalseth:2011wp].
When fitting the CoGeNT excess at low energies we follow two approaches: In the first case we seek an explanation of the excess exclusively in terms of [$\nu_{b}$]{} scatterings on Ge, allowing only for a constant background contribution. In the second case we relax the assumption on the background and allow, in addition, for an exponential background component, $A\times \exp(-B {\ensuremath {E_\mathrm{v}}})$, with coefficients $A$ and $B$ to be determined in the fit. Clearly, in the latter approach a [$\nu_{b}$]{}-induced contribution may not even be necessary as the excess resembles an exponential shape. This is therefore the most conservative way to treat the data in terms of new physics and it will show us the “compatibility” region in the ([$\mathcal{N}_{\mathrm{eff}}$]{},$\Delta m_b$) parameter space. As mentioned above, the CoGeNT excess is likely to be revised by the collaboration. In the second approach, the additional exponential background is here to mimic that possibility without quantifying its (yet unknown) concrete strength.
We first take the reported CoGeNT excess at face value and fit it by [$\nu_{b}$]{} scatterings on Ge together with a constant background rate. The best-fit values are $$\begin{aligned}
\label{eq:cogent-bestfit}
\text{CoGeNT:}\quad {\ensuremath {\Delta m_b^2}}= 1.76\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2,\quad
{\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 228,\quad \chi^2 /n_d = 33.6/24 ,\end{aligned}$$ which corresponds to $p\simeq 0.1$. The background rate is $c_0 =
3.36\,{\ensuremath {\mathrm{cpd}}}/{\ensuremath{\mathrm{kg}}}/{\ensuremath {\mathrm{keVee}}}$. A finer-grained bin size improves the goodness-of-fit to $\chi^2 /n_d = 47.6/47$ with $p=0.45$. We consider this a very good description of the data. One should keep in mind that the subtraction of the cosmogenic background has uncertainties itself which are not accounted for in the errorbar. Figure \[cogent\] shows the spectrum obtained from (\[eq:cogent-bestfit\]). The dashed line shows the signal from [$\nu_{b}$]{} only and the solid line includes the constant background. Below ${\ensuremath {E_\mathrm{v}}}\lesssim 0.5\,{\ensuremath {\mathrm{keVee}}}$ the detector efficiency decreases rapidly, which explains the turn-off of the scattering signal.
Figure \[fit\] shows the inferred 99% C.L. regions in the $({\ensuremath {\Delta m_b^2}},{\ensuremath {\mathcal{N}_{\mathrm{eff}}}})$ parameter space as labeled. We use $\Delta \chi^2=
9.21$, *i.e.* we treat the constant background rate as a nuisance parameter. This is equivalent of using a profile likelihood to infer the confidence regions. Two isolated islands are visible as labeled. The thin gray solid line labeled “CoGeNT hull” which touches the CoGeNT regions from above is obtained by allowing an additional exponential background in the fit (see discussion above.) Without further knowledge of the strength of this background, the full region below the line then becomes viable. Whenever the [$\nu_{b}$]{} signal becomes too weak, the background takes over in producing a viable fit. The general expectation is that once the collaboration revises their statements about the strength of the exponential rise, the CoGeNT favored regions will move vertically downwards, but at this point it is impossible to speculate by how much. In conclusion, we find that [$\nu_{b}$]{} can provide an excellent explanation to the CoGeNT data.
CRESST-II {#sec:cresst-ii}
---------
The CRESST-II experiment [@Angloher:2004tr], situated at LNGS, has recently presented their results from their DM “run32” with a total of 730 kg$\times$days effective exposure between 2009-2011 [@Angloher:2011uu]. The analysis has been carried out using data collected by eight CaWO$_4$ crystals which measure heat and scintillation light resulting from nuclear recoils. The calorimetric phonon channel allows for a precise determination of the recoil energy with a resolution better than 1keV. Nuclear recoils are again quenched in scintillation light. This is a virtue as it allows for a discrimination against $e^{-}$ and $\gamma$ induced events. Moreover, the quenching factors of Ca, W, and O differ. To a limited degree, recoils against the respective elements can therefore be distinguished.
The analysis [@Angloher:2011uu] finds an intriguing accumulation of a total of 67 events in their overall acceptance region between 10–40keV, shown by the solid line in Fig. \[cresst\]. The low-energy threshold of each detector-module is determined by the overlap between $e/\gamma$- and nuclear recoil band. Allowing for a leakage of one background $e/\gamma$-event per module distributes the individual detector thresholds between 10.2–19[$\mathrm{keV}$]{}. Whereas $e/\gamma$-events are a well controllable background, the experiment suffers from a number of less well-determined sources of spurious events. To assess how well the observed events can be explained in terms of new physics makes the modeling of such background—unfortunately—unavoidable.
In the following we briefly mention each of the known background sources and outline our treatment of them:
1. As alluded before, the thresholds of the detector modules are chosen such that a leakage of a total number of 8 $e/\gamma$-induced events into the nuclear bands are expected. We find the energy distribution of these events by digitizing and binning the corresponding line from Fig. 11 of [@Angloher:2011uu].
2. Degraded $\alpha$-particles from radioactive contamination in the clamping system holding the crystals can be misidentified as nuclear recoils once their energy falls below 40[$\mathrm{keV}$]{}. A sideband analysis above that energy indicates that the distribution in recoil energy is flat. This allows to estimate the number of $\alpha$-events in the acceptance region for each detector module and which is provided in Tab.2 of [@Angloher:2011uu]. We follow this prescription which yields a total of 9.2 events.
3. Related to the previous source, $^{210}$Pb $\alpha$-decays from radioactive lead on the clamps holding the crystal and with the $\alpha$-particle being absorbed by non-scintillating material constitutes another source of background. The peak at 103[$\mathrm{keV}$]{} full recoil energy in $^{206}$Po is clearly visible and a fit of it allows to infer the overall exponential tail distribution in the acceptance region below 40[$\mathrm{keV}$]{}. The low-energy tail is due to $^{206}$Po that is slowed down in the clamps before interacting in the crystal. We estimate the radioactive lead contamination of each detector module from the number of observed events in the reference region above 40[$\mathrm{keV}$]{}. This yields a nominal background of about 17 events.
4. Finally, low-energy neutron-nucleus scatterings in the crystals is a well known source of background. These neutrons can be produced by in-situ radioactive sources as well as by cosmogenic muons in- and out-side the detector housing. Since neutrons tend to scatter more than once, some information on the overall flux can be obtained from the amount of coincident events in different detectors. Such multiple scatters also tend to wash out the initial spectral information. The authors of [@Angloher:2011uu] parameterize the neutron spectrum by a simple exponential $dN_n/dE = A \times
\exp(-E/E_{\mathrm{dec}})$ where $E_{\mathrm{dec}}=(23.54\pm
0.92)\,{\ensuremath{\mathrm{keV}}}$ has been obtained from a neutron calibration run with an AmBe source. The best we can do is assuming a uniform neutron flux in all detector-modules. With $A=1$ one gets a total of about 9 events. When fitting [$\nu_{b}$]{} to the data we leave $A$ as a free parameter. We observe that $A$ is never too large once the goodness of the fit becomes acceptable. The reason is that the [$\nu_{b}$]{}-induced spectrum experiences are relatively sharp cutoff for energies in excess of $\sim 25\,{\ensuremath{\mathrm{keV}}}$. Thus, the high-energy part of the acceptance region has to be entirely explained by background for which $A =\mathcal{O}(1)$ provides the best fit. [$\nu_{b}$]{} scatters mainly on oxygen and calcium, so that there is no need to further dissect the neutron background as the latter also scatters to 90% on O [@Angloher:2011uu].
![CRESST-II recoil spectrum. The solid line is the histogram of reported events in the $730\,{\ensuremath{\mathrm{kg}}}\times{\ensuremath {\mathrm{days}}}$ run summing to a total of 67 events. The gray shaded (stacked) histograms show the best fit contribution from ${\ensuremath {\nu_{b}}}$ (darkest shading) and the modeled backgrounds as labeled and explained in the main text. The spiky dashed (red) solid line is the unbinned [$\nu_{b}$]{} signal. []{data-label="cresst"}](NUB_CRESST730){width="60.00000%"}
In [@Angloher:2011uu] the various nuclear recoil bands have not been resolved. Therefore, we compute rate predictions for CRESST summing up all events in Ca, O, and W. The fractional exposures of the individual detector modules lie within $\sim 20\%$ of a uniform one with value $1/8$. We use the accurate values as provided by [@Jens]. We account for a finite Gaussian energy resolution in the phonon channel with $\sigma =1\,{\ensuremath{\mathrm{keV}}}$. Since the number of observed events $n_i$ in each of the bins is small, we fit the data by minimizing the Poisson log-likelihood ratio $$\begin{aligned}
\label{eq:poisson-L}
\chi^2_P = 2 \sum_{i} \left[ y_i - n_i + n_i \ln{\left(
\frac{n_i}{y_i} \right)} \right] ,\end{aligned}$$ where the sum runs over all bins and $y_i$ is the sum of background and signal contributions; the last term is absent when $n_i=0$. Confidence regions are directly constructed from (\[eq:poisson-L\]).
Figure \[cresst\] shows the recoil spectrum induced by ${\ensuremath {\nu_{b}}}$ as the (magenta) continuous and falling line. Summing all background contributions to the [$\nu_{b}$]{} signal the best fit parameters read, $$\begin{aligned}
\label{eq:cresst-bestfit}
\text{CRESST-II:}\quad {\ensuremath {\Delta m_b^2}}= 3\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2,\quad
{\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 49,\quad \chi^2 /n_d = 27.7/27 ,\end{aligned}$$ with a $p$-value $p=0.48$ under the approximation that $\chi^2_P$ in (\[eq:poisson-L\]) follows a $\chi^2$ distribution with $n_d =
27$. The amplitude of the neutron background is found to be $A=1.23$. Fixing instead $A=1$ yields the same parameters (\[eq:cresst-bestfit\]) with negligible degradation in $\chi^2$. Discontinuous jumps in the count rate when going from lower to higher recoil energies are due to the onsets of the various detector modules with increasing energy thresholds. Also shown as a stacked histogram are the modeled sources of background as labeled.
Figure \[fit\] shows again the 99% confidence regions in the ([$\Delta m_b^2$]{},[$\mathcal{N}_{\mathrm{eff}}$]{}) parameter space. As can be seen, the favored region stretches across the plane and the trend for larger values of ${\ensuremath {\Delta m_b^2}}$ not shown in the plot can be easily be extrapolated. The CRESST region is compatible with the one inferred from the DAMA modulation amplitude. Once the CoGeNT excess is revised (see previous section), it is very likely that the resulting best fit region will overlap with CRESST too. CRESST spans a rather wide region in parameter space. The light yield distribution of the candidate events as a function of ${\ensuremath {E_\mathrm{v}}}$ is not published and has thus not been accounted for. With eventual better knowledge of this quantity, the region is expected to shrink in a joint fit. In the [$\nu_{b}$]{} scenario, most of the scatters are on oxygen which in turn yields most scintillation light among the CaWO$_4$ constituents.
It is also likely that the new CRESST data constrains larger values of ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}$. This is especially true given that the detector model with lowest threshold only observed one count between 10–12[$\mathrm{keV}$]{}. In order to put a constraint we use what has been termed “binned Poisson” technique in [@Savage:2008er]: for one bin, an average number of events $\nu = \nu_{\mathrm{s}} + \nu_{\mathrm{bg}}$ consisting of $ \nu_{\mathrm{s}} $ signal and $ \nu_{\mathrm{bg}} $ background events is excluded at a level $1-\alpha_{\mathrm{bin}}$ if the probability to see as few as $n_{\mathrm{obs}}$ observed events is $\alpha_{\mathrm{bin}}$. Since $n_{\mathrm{obs}}$ is a Poisson variable, $\alpha_{\mathrm{bin}}$ is given the lower tail of the Poisson distribution, $ \alpha_{\mathrm{bin}} =
\sum_{n=0}^{n_{\mathrm{obs}}} \nu^n \exp(-\nu)/n! $. When dealing with more than one bin the level of exclusion $1-\alpha$ is given by $$\begin{aligned}
\label{eq:confExcl}
1-\alpha = (1-\alpha_{\mathrm{bin}})^{N_{\mathrm{bin}}} ,\end{aligned}$$ where $N_{\mathrm{bin}}$ is the number of bins; $\alpha$ is the probability to see as few events as observed in at least one of the bins. For placing a constraint from CRESST we only use the bins from 10–25[$\mathrm{keV}$]{} as those are the ones for which ${\ensuremath {\nu_{b}}}$ can give a contribution. More conservative constraints are obtained when assuming that there is no background.
![Summary plot of direct detection favored regions and constraints in the parameters [$\Delta m_b^2$]{} and [$\mathcal{N}_{\mathrm{eff}}$]{} at 99% confidence. *Favored regions:* the broad light shaded gray band shows the CRESST-II region. The two darkest islands are the regions in which the CoGeNT excess is explained. In presence of an exponential background contamination (*e.g.* due to “surface events”), the region below the thin gray line labeled as “CoGeNT hull” becomes in principle viable (see main text for details.) The two medium gray shaded islands indicate the regions in which the DAMA modulation amplitude is fitted; these regions as well as any other parameter choices however exhibit a tension in timing when compared to the DAMA residuals. *Constraints:* [$\mathcal{N}_{\mathrm{eff}}$]{} values above the respective lines are excluded (or seriously challenged.) The top constraint is the one from Xenon100, the two degenerate ones below are obtained with the CRESST-II data and CDMS-II low-threshold data. The two dotted lines at the bottom show the constraints arising from the Xenon10 low-threshold analysis with two different assumptions on the ionization yield $Q_{y}$ (see main text for details.)[]{data-label="fit"}](NUB_ChiSq_revised){width="\textwidth"}
Null-searches {#sec:null-searches}
-------------
### CDMS-II low threshold analysis {#sec:cdms-ii-low}
The CDMS-II collaboration has published a low threshold analysis from the Soudan site using eight Ge detectors with a raw exposure of 241[$\mathrm{kg}$]{}[$\mathrm{days}$]{} [@Ahmed:2010wy]. At the expense of discriminating power of $e^{-}/\gamma$ against nuclear recoils, a threshold of $2\,{\ensuremath{\mathrm{keV}}}$ was reached. This is an interesting analysis because it uses the same target material as CoGeNT. Indeed, for the DM case the results indicate a serious conflict between the two experiments. Therefore, it is important to check if the CoGeNT explanation is also challenged in the baryonic neutrino scenario.
An exponential-like signal rise towards threshold with a maximum event rate of $\sim 1\,{\ensuremath {\mathrm{cpd}}}/{\ensuremath{\mathrm{kg}}}/{\ensuremath{\mathrm{keV}}}$ has been observed; see Fig. 1 in [@Ahmed:2010wy]. “Zero-charge” events from electron recoils near the edge of the detector are expected to yield a major contribution to the observed count rate. Given that the estimation of this background involves extrapolation and hard-to-control systematic errors, we obtain the most conservative limits on [$\mathcal{N}_{\mathrm{eff}}$]{} by not subtracting this background. This follows the approach taken by the collaboration itself.[^1] Only the first three data bins covering $2\,{\ensuremath{\mathrm{keV}}}\leq {\ensuremath {E_R}}\leq
3.5\,{\ensuremath{\mathrm{keV}}}$ are sensitive to [$\nu_{b}$]{} scattering. On those bins we perform a “binned Poisson” exclusion, similar to the one explained in Sec. \[sec:cresst-ii\]. We correct for efficiency according to Fig. 1 in [@Ahmed:2010wy] and use a Gaussian detector resolution of $0.2\,{\ensuremath{\mathrm{keV}}}$. In Fig. \[fit\] we show the resulting constraint at 99%C.L. Remarkably, CDMS-II does not challenge the CoGeNT-favored regions.
### SIMPLE {#sec:simple}
An interesting direct detection experiment in the current context is SIMPLE [@Felizardo:2011uw], operated in the Low Noise Underground Laboratory in southern France. It uses a dispersion of superheated liquid droplets made of C$_{12}$ClF$_{5}$ with an total active mass of 0.2[$\mathrm{kg}$]{}. Only nuclear scattering induces phase transition which results in bubble nucleation. The fact that the experiment contains mainly light elements makes it susceptible to [$\nu_{b}$]{}-scattering. Notably, fluorine with atomic number $A=19$ has a target mass fraction of $\sim 60\%$.
We use the results from Phase I of Stage II with 14.1[$\mathrm{kg}$]{}[$\mathrm{days}$]{}exposure [@Felizardo:2011uw]. A total of 8 events were observed with an expected neutron background of 12. We include this background in the derivation of an upper limit of [$\mathcal{N}_{\mathrm{eff}}$]{} as a function of [$\Delta m_b^2$]{}. We model bubble nucleation and heat transfer efficiency following [@Felizardo:2011uw] for which we use an energy threshold of $8\,{\ensuremath{\mathrm{keV}}}$. For better overview, we did not include the constraint in Fig. \[fit\]. It is superseded by the ones from CDMS-II and CRESST-II but more constraining than Xenon100 as will be discussed next.
### Xenon experiments {#sec:xenon}
In this work we consider the results from the Xenon10 and its upgrade, the Xenon100 experiment at LNGS [@Aprile:2010bt; @Aprile:2011hi]. Currently, the most stringent constraint for spin-inelastic scattering for DM masses in the $50\,{\ensuremath{\mathrm{GeV}}}$ ballpark is the one from the Xenon100 [@Aprile:2011hi]. The experiment also provides strong limits in the light-DM mass region $\mathcal{O}(10\,{\ensuremath{\mathrm{GeV}}})$. In the latter context, the low-threshold analysis [@Angle:2011th] from Xenon10 is of particular interest.
The experiments use a prompt scintillation light signal (S1) and a delayed one from ionization (S2) to detect the nature of the recoil event. Both signals are measured in units of photo-electrons (PEs). Nuclear recoil energies are obtained from the respective signals via $$\begin{aligned}
\label{eq:LeffEr}
S1:\: {\ensuremath {E_R}}= \frac{1}{\mathcal{L}_{\mathrm{eff}}} \frac{\mathrm{S}1}{L_y}
\frac{S_e}{S_n}, \qquad S2:\: {\ensuremath {E_R}}= \frac{\mathrm{S}2/\zeta}{Q_y} .\end{aligned}$$ Only a small fraction of the deposited recoil energy is emitted in form of scintillation light. The crucial quantity is the scintillation efficiency $\mathcal{L}_{\mathrm{eff}}$ which it determines the discrimination threshold of the experiment. Therefore, for Xenon100 we will mostly be interested in S1 since a low-threshold analysis has not yet been published. Conversely, for Xenon10 we focus on S2. For S1 we use the measurements of [@Plante:2011hw] with a conservative extrapolation of $\mathcal{L}_{\mathrm{eff}}$ to zero at $2\,{\ensuremath{\mathrm{keV}}}$ nuclear recoil. $L_y = 2.2 $ (Xenon100) is the light yield in PEs/keVee obtained from $\gamma$-calibration. $S_e$ and $S_n$ are quenching factors for scintillation light due to electron and nuclear recoils, respectively. They are experiment-specific and depend on the applied drift voltage; $S_e = 0.58$ and $S_n=0.95$ for Xenon100. $Q_y$ is the ionization yield per keV nuclear recoil and $\zeta$ is the measured number of PEs produced per ionized electron, $\zeta =
20\, (24)\,$PEs for Xenon100 (Xenon10); we will comment further on $Q_y$ below.
The S1 detector acceptance for Xenon100 is found from the lines presented in Fig. 2 of [@Aprile:2011hi] together with a low-energy threshold of 4PE which corresponds to $8.4\,{\ensuremath{\mathrm{keV}}}$ nuclear recoil energy. For S2, the trigger efficiency is effectively 100% in both experiments. We take the Poissonian nature on the expected number of scintillation photons/ionization electrons into account. For example, for S2 one computes $$\begin{aligned}
\frac{dR}{dn_e} = \int_{E_{\mathrm{min}}} d{\ensuremath {E_R}}\, \frac{dR}{d{\ensuremath {E_R}}} \times
\mathrm{Poiss}(n_e|\nu_e({\ensuremath {E_R}})) ,\end{aligned}$$ where $n_e = {\ensuremath {E_R}}Q_y$ is the number of ionized electrons; a PMT resolution of 0.5PE can be neglected when converting $n_e$ to S2; $\mathrm{S}2 = n_e\zeta$. We use a hard cut off at $E_{\mathrm{min}} = 1.4\,{\ensuremath{\mathrm{keV}}}$ following [@Angle:2011th].
We first discuss the constraint from Xenon100. Reference [@Aprile:2011hi] presents the results from a 100day run with a fiducial detector mass of 48[$\mathrm{kg}$]{}. Three events were observed in the acceptance region $8.4$–$44.6\,{\ensuremath{\mathrm{keV}}}$. We use Yellin’s maximum gap method [@Yellin:2002xd] to set an upper limit. The resulting constraint is again shown in Fig. \[fit\]. Given that Xe is a rather heavy target and that the scintillation threshold is $\mathcal{O}(10\,{\ensuremath{\mathrm{keV}}})$, the constraint on the parameter space is very week. Indeed, none of the favored regions is challenged.
In contrast, a more stringent limit can be expected from the Xenon10 low-threshold analysis in [@Angle:2011th]. Discarding the scintillation signal allows one to lower the threshold to ${\ensuremath {E_R}}=
\mathcal{O}(1\,{\ensuremath{\mathrm{keV}}})$. After all cuts are applied and with a resulting effective exposure of $\sim 6.15\,{\ensuremath{\mathrm{kg}}}\,{\ensuremath {\mathrm{days}}}$ a mere number of seven events in the region $
1.4\,{\ensuremath{\mathrm{keV}}}\lesssim {\ensuremath {E_R}}\lesssim 10\,{\ensuremath{\mathrm{keV}}}$ are observed. The calibration of the nuclear energy recoil scale now solely depends on $Q_y$ \[see Eq. (\[eq:LeffEr\])\]. The problem in the analysis is that one requires $Q_y$ in a regime where no data is available and the extrapolation of that quantity is not well supported by theoretical expectations. Indeed, the adopted values in [@Angle:2011th] by the Xenon10 collaboration have been repeatedly disputed [@Collar:2010ht; @Collar:2011wq]. Therefore, we adopt two different extrapolations of $Q_y$. First we use the solid line in Fig. 2 of [@Angle:2011th] and second we employ more conservative choice discussed in [@Collar:2011wq] in the estimation of the energy scale.
The resulting constraints are again presented in Fig. \[fit\]. In contrast to all other null searches, Xenon10 can challenge the entire region of interest with ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}\gtrsim 40$. We caution the reader that it is also the constraint with the largest uncertainty. We will further illustrate the sensitivity of the results on $Q_{y}$ below when considering a prospective Xenon100 low-threshold analysis.
Future sensitivity in Direct Detection {#sec:future-sens-direct}
--------------------------------------
### Xenon100 low threshold {#sec:xenon100-low-thresh}
In the previous section we have seen that a low-threshold analysis in Xenon10 can yield very stringent constraints. Therefore, it is conceivable that the collected 100 live days of data with the Xenon100 detector may offer another sensitive test for this model. Here we present projections for a Xenon100 ionization-only (S2) study.
![Projection for a Xenon100 low-threshold analysis for an exemplary parameter choice ${\ensuremath {\Delta m_b^2}}= 2.5\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2$ and ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 100$. The $x$-axis gives the ionization signal S2 in units of PEs. The horizontal and vertical dashed lines show the maximum rate from radioactive Kr decay and the S2 software threshold of the detector, respectively. The solid lines are the [$\nu_{b}$]{}-signals from [${}^8\mathrm{B}$]{} and hep neutrinos as labeled. The vertical arrow at 700 PEs indicates the current threshold of the S1 scintillation signal. The dotted line shows again [${}^8\mathrm{B}$]{} neutrinos for a calibration scale following [@Collar:2011wq] instead of [@Angle:2011th]. This highlights the severe sensitivity on the extrapolation of $Q_y$. []{data-label="xe100"}](NUB_XE100_S2){width="50.00000%"}
Once the prompt scintillation signal S1 is discarded, the goal lies in lowering the threshold in S2 as much as possible. In [@Aprile:2011hi] the Xenon100 quoted software threshold is 300PE which corresponds to 20 ionized electrons. This is a factor of five larger compared to the Xenon10 low-threshold analysis. The gain in exposure by about an order of magnitude somewhat compensates for this since the increase in detector mass does not seriously affect the extraction efficiency of ionized electrons. However, an air-leak during the run introduced unwanted Kr contamination at the ($700\pm 100$)ppt level. The associated rate in the electron recoil band reads, $$\begin{aligned}
\label{eq:kr}
R_{\mathrm{Kr}}< 22 \times 10^{-3}\: {\ensuremath {\mathrm{cpd}}}/ {\ensuremath{\mathrm{kg}}}/ {\ensuremath {\mathrm{keVee}}}, \end{aligned}$$ and is expected to be homogeneously distributed over recoil energy. Clearly, a putative [$\nu_{b}$]{} signal must overcome this background. We remark in passing that for future runs (\[eq:kr\]) will diminish since the $^{85}$Kr concentration is being continuously reduced by cryogenic distillation.
Figure \[xe100\] shows the sensitivity to [$\nu_{b}$]{} with the current dataset of 100 live days for ${\ensuremath {\Delta m_b^2}}= 2.5\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2$ and ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 100$. We present the recoil spectrum as a function of actual detected S2 in units of PE. The solid lines show the [${}^8\mathrm{B}$]{} and hep neutrino spectrum as labeled. The vertical dashed line indicates the Xenon100 threshold and the horizontal one is the rate (\[eq:kr\]) from Kr contamination. The dotted line is again a [${}^8\mathrm{B}$]{} spectrum but this time with a calibration scale following [@Collar:2011wq]. As a consequence the signal falls entirely below the threshold. Even a small change $\Delta {\ensuremath {E_R}}$ in the nuclear recoil calibration has a large effect since $\Delta \mathrm{S}2 \sim \zeta \Delta{\ensuremath {E_R}}$. This illustrates 1) that the Xenon10 constraints in the previous section should be viewed with care and 2) that without further experimental insight into $Q_y$ in (\[eq:LeffEr\]) a conclusive prediction for Xenon100 is not feasible.
### COUPP {#sec:coupp}
![Predictions for a COUPP 60 kg bubble chamber for ${\ensuremath {\Delta m_b^2}}= 3\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2$ and ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 60$ as a function of bubble nucleation threshold with one year of exposure. The solid lines are from top to bottom for the [${}^8\mathrm{B}$]{} and hep fluxes of [$\nu_{b}$]{}. For comparison, a signal from a 10[$\mathrm{GeV}$]{} DM particle with spin-independent nucleon cross section of $\sigma_n =
10^{-41}\,{\ensuremath{\mathrm{cm}}}^2$ is shown.[]{data-label="coupp"}](NUB_coupp){width="50.00000%"}
As already mentioned in Sec. \[sec:simple\], detectors which employ fluorocarbon compounds as target material are attractive because of their favorable kinematics in contrast to heavier targets. A large scale experiment of this type is the COUPP 60 kg bubble chamber currently in the progress of moving into the SNOLAB underground facility [@Ramberg:2010zz]. It uses a superheated CF$_3$I liquid with temperature and pressure adjusted such that only nuclear recoils set off bubble nucleation events. It is a counting experiment for events above adjustable threshold without *a priori* insight into the recoil energy distribution. We assume an exposure of 1 yr together with a detector efficiency $\varepsilon=0.7$.
Light target nuclei make COUPP particularly attractive for the searches of light WIMPs and [$\nu_{b}$]{}. In Fig. \[coupp\] the integral signal of [$\nu_{b}$]{} for ${\ensuremath {\Delta m_b^2}}= 3\times 10^{-10}\,{\ensuremath{\mathrm{eV}}}^2$ and ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 60$ as a function of detector threshold is shown. The two solid curves from top to bottom correspond to [${}^8\mathrm{B}$]{} and hep neutrinos, respectively. Already below a threshold energy of $\sim 20\,{\ensuremath{\mathrm{keV}}}$ the [${}^8\mathrm{B}$]{} flux induces a clear signal. Limited insight into the energy distribution should be possible by varying the rather “steplike” detector threshold. In particular, with a multi-year exposure (or larger values of [$\mathcal{N}_{\mathrm{eff}}$]{}) the crossover from the [${}^8\mathrm{B}$]{} to the hep neutrino spectrum may be observable. More importantly, the variation of the threshold provides discriminating power between a putative DM signal and [$\nu_{b}$]{}. The dashed line in Fig. \[coupp\] shows the integral event rate for a 10[$\mathrm{GeV}$]{} DM particle with spin-independent WIMP-nucleon cross section of $\sigma_n = 10^{-41}\,{\ensuremath{\mathrm{cm}}}^2$.
Neutrino searches {#sec:ns}
=================
Here we are going to consider the elastic scattering of [$\nu_{b}$]{} from the sun in solar neutrino experiments. The NC channels in those experiments are not necessarily sensitive to this class of new physics, given that the associated inelastic reactions exhibit the scaling (\[eq:scaling\]). As shown in [@Pospelov:2011ha], the NCB interaction does not yield an observable rate for the D-breakup at SNO. There is also the possibility of inelastic excitation of $^{12}$C with subsequent emission of a 4.44[$\mathrm{MeV}$]{} $\gamma$. The analysis of inelastic processes falls outside of the scope of the present study.
The experiments which are capable of detecting the elastic NCB signal employ hydrocarbon scintillators, namely, Borexino and KamLAND as well as the upcoming experiment SNO+. The dominant background at lowest energies comes from $^{14}$C contamination of the mineral oil which decays with a maximum $\beta $ energy of $Q = 156\,{\ensuremath{\mathrm{keV}}}$. Other, less prominent backgrounds by long-lived isotopes at lowest energies are the $\beta$-decays of $^{85}$Kr $(Q=687\,{\ensuremath{\mathrm{keV}}})$ and $^{210}$Bi $(Q=1.16\,{\ensuremath{\mathrm{MeV}}})$. Unfortunately, the decays from $^{14}$C prevent sensitivity to [$\nu_{b}$]{} in current detectors. For example, the Borexino experiment which uses dedicated mineral oil with low residual $^{14}$C content measured its concentration to be $^{14}\mathrm{C}/^{12}\mathrm{C}\simeq 2\times 10^{-18}$. Though this is a seemingly small ratio, it translates into a rate of approximately $\Gamma \simeq 6\times 10^4\, \mathrm{events}/{\ensuremath {\mathrm{day}}}/{\ensuremath {\mathrm{ton}}}$ below 0.2[$\mathrm{MeV}$]{}. Therefore, in the following we restrict our analysis to a future possibility when the level of $^{14}$C is much reduced so that one may hope to gain sensitivity to [$\nu_{b}$]{}.
The scattering of [${}^8\mathrm{B}$]{} neutrinos on protons can produce a recoil energy of ${\ensuremath {E_R}}\lesssim 0.5\,{\ensuremath{\mathrm{MeV}}}$. However, the proton recoil is quenched and the [$\nu_{b}$]{} signal may not show itself above the $^{14}$C peak. In organic scintillators Birk’s law provides a phenomenological description of the scintillation light yield per unit path length [@Birks:1964zz], $$\begin{aligned}
\label{eq:birk}
\frac{dL}{dx} = L_0 \frac{dE/dx}{1+k_B dE/dx} , \end{aligned}$$ where $k_B$ is Birk’s constant and $dE/dx$ is the ion stopping power in the material. The formula interpolates between the limiting cases of low energy losses (no quenching), $dE/dx \ll k_B^{-1}$ with linear dependence of the light output $dL/dx\sim L_0 dE/dx$ and high energy losses (quenching), $dE/dx \gg k_B^{-1}$ for which saturation occurs, $dL/dx\sim L_0/k_B$. Equation (\[eq:birk\]) suggests the (non-linear) relation between recoil and quenched energy, $$\begin{aligned}
\label{eq:quench}
{\ensuremath {E_\mathrm{v}}}= \int_0^{{\ensuremath {E_R}}} \frac{dE}{1+k_B\, dE/dx} .\end{aligned}$$
![Liquid scintillator neutrino detector filled with pseudocumine, C$_9$H$_{12}$, and modeled after Borexino. Shown are the signals for [$\nu_{b}$]{} for ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 100$ and ${\ensuremath {\Delta m_b^2}}= 2.5\times
10^{-10}\,{\ensuremath{\mathrm{eV}}}^{2}$. The spectrum is largely dominated by background decays of $^{14}$C for which a contamination in carbon of one part in $10^{-23}$ has been assumed. Solid (dotted) curves are with (without) detector-resolution. Proton recoil energies are quenched; the quenching of electrons at lowest recoil energies has been neglected.[]{data-label="borexino"}](NUB_Borexino){width="50.00000%"}
In the following we use Borexino, representative for other liquid scintillator experiments. It has a a fiducial detector mass of 0.278kton filled with the scintillator pseudocumine, C$_9$H$_{12}$, with a mass density of $\rho=0.88\,\mathrm{g}/{\ensuremath{\mathrm{cm}}}^3$. We employ the `SRIM` computer package to obtain the stopping power $dE/dx$ of protons in pseudocumine. Quenched energies are then obtained from (\[eq:quench\]) with $k_B = 0.01\,{\ensuremath{\mathrm{cm}}}/{\ensuremath{\mathrm{MeV}}}$. The scintillation light yield is approximately $500\,\mathrm{PE}/{\ensuremath{\mathrm{MeV}}}$ and, for simplicity, we assume a Gaussian energy resolution with $\sigma =
0.045\,{\ensuremath{\mathrm{MeV}}}\,\sqrt{{\ensuremath {E_\mathrm{v}}}}$ [@Dasgupta:2011wg] where ${\ensuremath {E_\mathrm{v}}}$ is in units of MeV.
We simulate the $^{14}$C background spectrum as follows. The decay rate from $^{14}$C decay in C$_9$H$_{12}$ is given by $$\begin{aligned}
\frac{d\Gamma_{14}}{dE} = \frac{R_{14}}{t_{1/2}\ln{2}} \frac{9
N_A}{M_{\mathrm{PC}}} \frac{df_{14}}{dE} = \frac{3.1\times
10^4}{{\ensuremath {\mathrm{day}}}\times {\ensuremath {\mathrm{ton}}}} \left( \frac{R_{14}}{10^{-18}}
\right)\times \frac{df_{14}}{dE}.\end{aligned}$$ Here, $R_{14}$ is the ratio of $^{14}$C/$^{12}$C, $t_{1/2} = 5730\,$yr is the half-life of $^{14}$C, $N_A$ is Avogadro’s number, and $M_{\mathrm{PC}} = 120.2~\mathrm{g/mol}$ is the molecular weight of pseudocumine. The $\beta$-spectrum is given by [@morita1973beta], $$\begin{aligned}
\label{eq:c14spec}
\frac{df_{14}}{dE} = \frac{1}{N}\times p_e E_e (E_0 - E_e)^2
F(Z,E_e) C(E_e) ,\end{aligned}$$ where $p_e$ and $E_e$ are the electron momentum and total energy, $E_0$ is the total endpoint energy and $F(Z,E_e)$ is the Fermi function for $^{14}$N; for the shape factor we use $C = 1 -
0.7\,E_e/{\ensuremath{\mathrm{MeV}}}$ [@Back:2002xz]. $N$ is chosen such that the spectrum is normalized to unity. Electron recoils are somewhat quenched by roughly $(10-40)\%$ for $(100-10)\,{\ensuremath{\mathrm{keV}}}$ [@Back:2002xz]. For simplicity we neglect this complication here as well as more serious resolution effects near the detector threshold. In a realistic scenario, the latter can affect the spectrum below 100[$\mathrm{keV}$]{} substantially, see *e.g.* [@Back:2002xz]. Here we are merely concerned with the question of how low the $^{14}$C content needs to be in order to gain sensitivity to [$\nu_{b}$]{} recoils.
In Fig. \[borexino\] we compare the $^{14}$C background to a solar [$\nu_{b}$]{} signal with parameters ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}= 100$ and ${\ensuremath {\Delta m_b^2}}= 2.5\times
10^{-10}\,{\ensuremath{\mathrm{eV}}}^{2}$. The contamination has been fixed to $R_{14} =
10^{-23}$ which is already 5 orders of magnitude below the level for the Borexino detector, $R_{14}(\mathrm{Borexino}) \simeq 2\times
10^{-18}$[@Alimonti:1998rc]. There is little hope that a signal would show its presence above the $^{14}$C-decay endpoint energy of $156\,{\ensuremath{\mathrm{keV}}}$. The detector resolution has the effect of smearing the cutoff out to larger values of energy, hence burying the neutrino signal. This can be seen by the difference between the dotted and solid lines. At lowest recoil energies $E_R\lesssim 60\,{\ensuremath{\mathrm{keV}}}$ the [${}^8\mathrm{B}$]{} signal dominates, but this may be a region which will not be explorable in large mass scintillator experiments due to their detector thresholds. Perhaps the only reasonable hope for using carbon-based scintillators is the isotopic purification of not excessively large, $O(100~{\rm kg})$, quantities of carbohydrates with the eventual setup similar to the prototype of the Borexino detector [@Alimonti:1998rc].
Conclusions {#sec:conclusions}
===========
We have performed an extensive analysis of the model of [$\nu_{b}$]{}neutrinos that are sourced by the sun, and elastically scatter on nuclei in underground dark matter experiments. The goal of this study was to assess the viability of this model as the explanation for the reported anomalies, that are often interpreted as a possible dark matter scattering signal. Our findings are summarized in the master plot of Figure 6, that is a direct analogue of the WIMP-mass vs. scattering cross section plot for DM scattering. We are now able to conclude the following:
- [*On the positive side for the model*]{}, the [$\nu_{b}$]{} scenario with an oscillation length comparable to the earth-sun distance and with an effective enhancement of the NCB current by $O(100)$ is currently not seriously challenged by any of the existing experiments. On the contrary, this corner of parameter space provides a natural fit to the CoGeNT excess, and to the CRESST anomaly. Given enough uncertainty in the current status of the CoGeNT excess, and in the background-contaminated CRESST events, it is not difficult to see that the same regions of the parameter space of [$\nu_{b}$]{} model can be responsible for these anomalies. The null results of many experiments that challenge WIMP explanations of these anomalies (such as CDMS and Xenon100), do not challenge [$\nu_{b}$]{} model.
- [*On the negative side for the model*]{}, the values ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}\simeq 200$ which are necessary to fit the DAMA modulation amplitude are challenged by the CDMS-II low threshold analysis and by recent CRESST-II results. It should be noted however, that in the presence of background [@Kudryavtsev:2010zza], smaller values for ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}$ become viable again. This requires larger modulation amplitudes of the signal, but this can presumably be achieved. Also the phase of the predicted modulation signal is deviant from the DAMA data. Even though one can achieve a phase reversal and have a maximum in the [$\nu_{b}$]{} scattering rate in early July, this is still far away from the DAMA phase, which corresponds to a modulation maximum in late May-early June. We do not know how to “correct” the model per se for this residual discrepancy. Further tension for the model may arise from the number of events predicted for the “ionization-only” signal at Xenon10. At face value, the parameter space of the model can potentially be constrained down to ${\ensuremath {\mathcal{N}_{\mathrm{eff}}}}\sim 40$, but the severity of constraint may well be mitigated by a poorly known energy calibration at those lowest recoils.
- [*Future prospects*]{} for probing [$\nu_{b}$]{}-scattering look reasonably bright. In contrast to many DM models, the [$\nu_{b}$]{} scattering pattern is fixed and we can make definite predictions as functions of only two parameters. For example, these predictions show that the COUPP experiment may be particularly sensitive to the [$\nu_{b}$]{} scattering signal. One can also conclude that a re-designed iteration of the CRESST-II experiment with reduced backgrounds will likely be very sensitive to the [$\nu_{b}$]{} model. Furthermore, a reduced electron-like background in the Xenon100 experiment will be able to probe the parameter space once the uncertain values for $Q_y$ and/or ${\cal L}_{\rm eff}$ are clarified.
Finally, as this paper was readied for the submission, a new preprint appeared [@Harnik:2012ni] that examines a similar set of ideas. It expands the set of interesting mediation mechanisms beyond NCB to [*e.g.*]{} $B-L$ and “massive photon” forces with extra-light mediators in the sub-MeV mass range. Unlike the [$\nu_{b}$]{} model, such modifications may already be under strong tension from astrophysical and cosmological constraints. We plan to return to [$\nu_{b}$]{}-related signatures in astrophysical and cosmological settings in future work.
[^1]: It has been speculated [@Collar:2011kf] that the CoGeNT and CDMS-II recoil spectra are indeed similar after correcting for a potential energy miscalibration. Given that the status of the CoGeNT recoil spectrum is uncertain itself, we do not follow up on that discussion in this work.
|
---
author:
- Oleksandr Shchur
- Stephan Günnemann
bibliography:
- 'bibliography.bib'
title: |
Overlapping Community Detection\
with Graph Neural Networks
---
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by the German Research Foundation, Emmy Noether grant GU 1409/2-1.
|
---
abstract: 'In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using space-time coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses space-time coarse cells. We construct space-time snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using randomized boundary conditions and oversampling (cf. [@randomized2014]). We construct the local spectral decomposition based on our analysis, as presented in the paper. We present numerical results to confirm our theoretical findings and to show that using our proposed approaches, we can obtain an accurate solution with low dimensional coarse spaces. We discuss using online basis functions constructed in the online stage and using the residual information. Online basis functions use global information via the residual and provide fast convergence to the exact solution provided a sufficient number of offline basis functions. We present numerical studies for our proposed online procedures. We remark that the proposed method is a significant extension compared to existing methods, which use coarse cells in space only because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space-time cells, and (3) local boundary conditions in space-time cells.'
author:
- 'Eric T. Chung[^1],'
- 'Yalchin Efendiev[^2],'
- 'Wing Tat Leung[^3],'
- 'Shuai Ye[^4]'
bibliography:
- 'references.bib'
title: 'Generalized multiscale finite element methods for space-time heterogeneous parabolic equations'
---
Introduction
============
Many multiscale processes vary over multiple space and time scales. These space and time scales are often tightly coupled. For example, flow processes in porous media can occur on multiple time scales over multiple spatial scales. Moreover, these scales can be non-separable. Reduced-order models for these problems require simultaneously treating spatial and temporal scales. Many previous approaches only handle spatial scales and spatial heterogeneities. These approaches have limitations when temporal heterogeneities arise. In this paper, we discuss a class of multiscale methods for handling space and time scales.
Some well-known approaches for handling [*separable*]{} spatial and temporal scales are homogenization techniques [@jikov2012homogenization; @pankov2013g; @pavliotis2008multiscale; @efendiev2005homogenization]. In these methods, one solves local problems in space and time. To give an example, we consider a well-known case of the parabolic equation $$\label{eq:parabolic_epsilon}
\begin{split}
\frac{\partial}{\partial t}u-\text{div}(\kappa(x,x/\epsilon^\alpha,t,t/\epsilon^\beta)\nabla u) = f,
\end{split}$$ subject to smooth initial and boundary conditions. Here, $\epsilon$ is a small scale, and the spatial scale is $\epsilon^\alpha$, and the temporal scale is $\epsilon^\beta$. One can show that (e.g., [@jikov2012homogenization; @pankov2013g]), the homogenized equation has the same form as (\[eq:parabolic\_epsilon\]), but with the smooth coefficients $\kappa^*(x,t)$. One can compute the coefficients using the solutions of local space-time parabolic equations in the periodic cell. This localization is possible thanks to the scale separation. The local problems may or may not include time-dependent derivatives depending on the interplay between $\alpha$ and $\beta$ since the cell problems are independent of $\epsilon$. One can extend this homogenization procedure to numerical homogenization type methods [@ming2007analysis; @abdulle2014finite; @efendiev2004numerical; @fish2004space], where one solves the local parabolic equations in each coarse block and in each coarse time step. To compute the effective property, one averages the solutions of the local problems. These approaches work well in the scale separation cases, but do not provide accurate approximations when there is no scale separation.
Previous researchers developed a number of multiscale methods for solving space-time multiscale problems in the absence of scale separation. These approaches use Multiscale Finite Element Methods [@hw97; @eh09; @kunze2012adaptive; @efendiev2004numerical], where one computes multiscale space-time basis functions, variational multiscale methods [@hughes1996space; @hughes1996space_1], and other approaches [@takizawa2011multiscale; @tezduyar1992computation; @nguyen1984space; @masud1997space] that are developed for stabilization. In [@owhadi2007homogenization], Owhadi and Zhang proposed a novel approach that uses global space-time information in computing multiscale basis functions. All these approaches use only a limited number of basis functions (one basis function) in each coarse block. We note that there has been a large body of works in space-time finite element methods. In this paper, our objective is to develop a general approach that can systematically construct multiscale basis functions, and provide analysis for multiscale high-contrast problems.
Our approaches use the Generalized Multiscale Finite Element Method (GMsFEM) Framework and develop a systematic approach for identifying multiscale basis functions. The GMsFEM is a generalization of MsFEM, proposed by Hou and Wu [@hw97]. The main idea of the GMsFEM is to construct multiscale basis functions by constructing snapshots spaces and performing local spectral decomposition in the snapshot spaces [@egh12; @Chung_adaptive14; @chan2015adaptive; @Efen_GVass_11; @ce09; @chung2015generalizedperforated; @Efen_GVass_11; @Efendiev_GLW_ESAIM_12; @Efen_GVass_11; @chung2013sub; @chung2015residual; @chung2014generalized; @chung2015online; @bush2014application]. The choice of the snapshot spaces and the local spectral decomposition is important for converging the resulting approach. We choose the snapshot spaces such that it can approximate the local solution space, while typically deriving local spectral decomposition from the analysis.
Previous approaches in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. The proposed method is a significant extension compared to existing methods, which use coarse cells in space only because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space-time cells, and (3) local boundary conditions in space-time cells. In our approach, we construct snapshot spaces in space-time local domains. We construct the snapshot solutions by solving local problems. We can construct a complete snapshot space by taking all possible boundary conditions; however, this can result to very high computational cost. For this reason, we use randomized boundary conditions for local snapshot vectors by solving parabolic equations subject to random boundary and initial conditions. We compute only a few more than the number of basis functions needed. Computing multiscale basis functions employs local spectral problems. These local spectral problems are in space-time domain. Using space-time eigenvalue problems controls the errors associated with $\partial u /\partial t$. We discuss several choices for local spectral problems and present a convergence analysis of the method.
In the paper, we present several numerical examples. We consider the numerical tests with the conductivities that contain high contrast and these high conductivity regions move in time. These are challenging examples since the high-conductivity heterogeneities vary significantly during one coarse-grid time interval. If only using spatial multiscale basis functions, one will need a very large dimensional coarse space. In our numerical results, we use oversampling and randomized snapshots. Our results show that one can achieve a small error by selecting a few multiscale basis functions. The numerical results confirm our convergence analysis.
In the paper, we also discuss online multiscale basis functions. In [@chung2015residual; @chung2015online], we present an online procedure for [*time-independent*]{} problems. The main idea of online multiscale basis functions is to use the residual information and construct new multiscale basis functions adaptively. We would like to choose a number of offline basis functions such that with only 1-2 online iterations, we can substantially reduce the error. This requires a sufficient number of online basis functions, with the online basis function construction typically derived by the analysis. In this paper, we present a possible online construction and show numerical results. Based on our previous results for [*time-independent*]{} problems, we show that one needs a sufficient number of offline basis functions to reduce the error substantially. In our numerical results, we observe a similar phenomena, i.e., the error decreases rapidly in 1-2 online iterations. We plan to investigate the convergence of the online procedure in our future work.
We organize the paper as follow. In Section \[sect:GMsFEM\], we present the underlying problem, the concepts of coarse and fine grids, the motivation of space-time approach, and the space-time GMsFEM framework. In Section \[sect:analysis\], we present the convergence analysis for our proposed method. In Section \[sect:online\], we present the new enrichment procedure of computing online multiscale basis functions. We present numerical results for offline GMsFEM and online GMsFEM in Section \[sect:NR1\] and Section \[sect:NR2\], separately. In Section \[sect:conclusion\], we draw conclusions.
Space-time GMsFEM {#sect:GMsFEM}
=================
Preliminaries and motivation
----------------------------
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with a Lipschitz boundary $\partial \Omega$, and $[0,T]$ $(T>0)$ be a time interval. In this paper, we consider the following parabolic differential equation $$\label{eq:para PDE}
\begin{split}
\frac{\partial}{\partial t}u-\text{div}(\kappa(x,t)\nabla u) & = f \quad\qquad \text{in }\Omega\times(0,T), \\
u & = 0 \quad\qquad \text{on }\partial\Omega\times(0,T),\\
u(x,0) & = \beta(x) \quad\;\; \text{in }\Omega,
\end{split}$$ where $\kappa(x,t)$ is a time dependent heterogeneous media (for example, a time dependent high-contrast permeability field), $f$ is a given source function, $\beta(x)$ is the initial condition. Our main objective is to develop space-time multiscale model reduction within GMsFEM and we use the time-dependent parabolic equation as an example. The proposed methods can be used for other models that require space-time multiscale model reduction.
We will introduce the space-time generalized multiscale finite element method in this section. The method follows the space-time finite element framework, where the time dependent multiscale basis functions are constructed on the coarse grid. Therefore, compared with the time independent basis structure, it gives a more efficient numerical solver for the parabolic problem in complicated media.
Before introducing our method, we need to define the mesh of the domain first. Let $\mathcal{T}^{h}$ be a partition of the domain $\Omega$ into fine finite elements where $h>0$ is the fine mesh size. Then we form a coarse partition $\mathcal{T}^{H}$ of the domain $\Omega$ such that every element in $\mathcal{T}^{H}$ is a union of connected fine-mesh grid blocks, that is, $\forall K_j\in\mathcal{T}^{H}$, $K_j=\cup_{F\in I_j}F$ for some $I_j\subset\mathcal{T}^{h}$. The set $\mathcal{T}^{H}$ is called the coarse grid and the elements of $\mathcal{T}^{H}$ are called coarse elements. Moreover, $H>0$ is the coarse mesh size. In this paper, we consider rectangular coarse elements for the ease of discussions and illustrations. The methodology presented can be easily extended to coarse elements with more general geometries. Let $\{x_i\}_{i=1}^{N_c}$ be the set of nodes in the coarse grid $\mathcal{T}^{H}$ (or coarse nodes for short), where $N_c$ is the number of coarse nodes. We denote the neighborhood of the node $x_i$ by $$\omega_i = \bigcup\{K_j\in\mathcal{T}^{H}: x_i\in\overline{K_j} \}.$$ Notice that $\omega_i$ is the union of all coarse elements $K_j\in\mathcal{T}^{H}$ sharing the coarse node $x_i$. An illustration of the above definition is shown in Figure \[Fig:plotnhood\]. Next, let $\mathcal{T}^{T}=\{(T_{n-1},T_{n})|1\leq n\leq N\}$ be a coarse partition of $(0,T)$ where $$0=T_{0}<T_{1}<T_{2}<\cdots<T_{N}=T$$ and we define a fine partition of $(0,T)$, $\mathcal{T}^{t}$ by refining the partition $\mathcal{T}^{T}$.
![Left: an illustration of fine and coarse grids. Right: an illustration of a coarse neighborhood and a coarse element.[]{data-label="Fig:plotnhood"}](plotschematic.pdf){width="\columnwidth"}
![Left: an illustration of fine and coarse grids. Right: an illustration of a coarse neighborhood and a coarse element.[]{data-label="Fig:plotnhood"}](plotnhood.pdf){width="\columnwidth"}
To fix the notations, we will use the standard conforming piecewise linear finite element method for the computation of the fine-scale solution. One can use discontinuous Galerkin coupling also [@eglmsMSDG; @cel14; @eric-2012]. Specifically, we define the finite element space $V_h$ with respect to $\mathcal{T}^{h}\times(0,T)$ as $$\begin{aligned}
V_{h} &=& \{v\in L^{2}((0,T);C^{0}(\Omega))\; | \; v=\phi(x)\psi(t) \text{ where }\phi|_{K}\in Q_{1}(K)\;\forall K\in\mathcal{T}^{h}, \; \psi|_{\tau}\in C^{0}(\tau)\;\forall \tau\in\mathcal{T}^{T} \\
& & \text{ and }\psi|_{\tau}\in P_{1}(\tau)\;\forall \tau\in\mathcal{T}^{t}\},\end{aligned}$$ then the fine-scale solution $u_h \in V_h$ is obtained by solving the following variational problem $$\label{eq:fine problem}
\int_{0}^{T}\int_{\Omega}\cfrac{\partial u_{h}}{\partial t}v+\int_{0}^{T}\int_{\Omega}\kappa\nabla u_{h}\cdot\nabla v+\sum_{n=0}^{N-1}\int_{\Omega}[u_{h}(x,T_{n})]v(x,T_{n}^{+})
=\int_{0}^{T}\int_{\Omega}fv+\int_{\Omega}\beta(x)v(x,T_{0}^{+}),\;\forall v\in V_{h},$$ where $[\cdot]$ is the jump operator such that $$\begin{cases}
[u_{h}(x,T_{n})]=u_{h}(x,T_{n}^{+})-u_{h}(x,T_{n}^{-}) & \text{ for }n\geq1,\\
{}[u_{h}(x,T_{n})]=u_{h}(x,T_{0}^{+}) & \text{ for }n=0.
\end{cases}$$ We assume that the fine mesh size $h$ is small enough so that the fine-scale solution $u_h$ is close enough to the exact solution. The purpose of this paper is to find a multiscale solution $u_{H}$ that is a good approximation of the fine-scale solution $u_h$.
Now we present the general idea of GMsFEM. We will use the space-time finite element method to solve problem (\[eq:para PDE\]) on the coarse grid. That is, we find $u_{H}\in V_{H}$ such that $$\label{eq: space-time FEM}
\int_{0}^{T}\int_{\Omega}\cfrac{\partial u_{H}}{\partial t}v+\int_{0}^{T}\int_{\Omega}\kappa\nabla u_{H}\cdot\nabla v+\sum_{n=0}^{N-1}\int_{\Omega}[u_{H}(x,T_{n})]v(x,T_{n}^{+})
=\int_{0}^{T}\int_{\Omega}fv+\int_{\Omega}\beta(x)v(x,T_{0}^{+}),\;\forall v\in V_{H},$$ where $V_{H}$ is the multiscale finite element space which will be introduced in the following subsections.
The computational cost for solving the equation (\[eq: space-time FEM\]) is huge since we need to compute the solution $u_{H}$ in the whole time interval $(0,T)$ at one time. In fact, if we assume the solution space $V_{H}$ is a direct sum of the spaces only containing the functions defined on one single coarse time interval $(T_{n-1},T_{n})$, we can decompose the problem (\[eq: space-time FEM\]) into a sequence of problems and find the solution $u_{H}$ in each time interval sequentially. Our coarse space will be constructed in each time interval and we will have $$V_{H}=\oplus_{n=1}^{N}V_{H}^{(n)},$$ where $V_{H}^{(n)}$ only contains the functions having zero values in the time interval $(0,T)$ except $(T_{n-1},T_{n})$, namely $\forall v\in V_{H}^{(n)},$ $$v(\cdot,t)=0\text{ for }t\in(0,T)\backslash(T_{n-1},T_{n}).$$ The equation (\[eq: space-time FEM\]) can be decomposed into the following problem: find $u_{H}^{(n)}\in V_{H}^{(n)}$ (where $V_{H}^{(n)}$ will be defined later) satisfying $$\begin{aligned}
\label{eq:space-time FEM coarse decoupled}
& \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial u_{H}^{(n)}}{\partial t}v+\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa\nabla u_{H}^{(n)}\cdot\nabla v+\int_{\Omega}u_{H}^{(n)}(x,T_{n-1}^{+})v(x,T_{n-1}^{+})\nonumber \\
= & \int_{T_{n-1}}^{T_{n}}\int_{\Omega}fv+\int_{\Omega}g_{H}^{(n)}(x)v(x,T_{n-1}^{+}),\;\forall v\in V_{H}^{(n)},\end{aligned}$$ where $$g_{H}^{(n)}(\cdot)=\begin{cases}
u_{H}^{(n-1)}(\cdot,T_{n-1}^{-}) & \text{ for }n\geq1,\\
\beta(\cdot) & \text{ for }n=0.
\end{cases}$$ Then, the solution $u_{H}$ of the problem (\[eq: space-time FEM\]) is the direct sum of all these $u_{H}^{(n)}$’s, that is $u_{H}=\oplus_{n=1}^{N}u_{H}^{(n)}$.
Next, we motivate the use of space-time multiscale basis functions by comparing it to space multiscale basis functions. In particular, we discuss the savings in the reduced models when space-time multiscale basis functions are used compared to space multiscale basis functions. We denote $\{t_{n1},\cdot\cdot\cdot,t_{np}\}$ are $p$ fine time steps in $(T_{n-1},T_n)$. When we construct space-time multiscale basis functions, the solution can be represented as $u_H^{(n)} = \sum_{l,i} c_{l,i} \psi_l^{\omega_i}(x,t)$ in the interval $(T_{n-1},T_n)$. In this case, the number of coefficients $c_{l,i}$ is related to the size of the reduced system in space-time interval. On the other hand, if we use only space multiscale basis functions, we need to construct these multiscale basis functions at each fine time instant $t_{nj}$, denoted by $\psi_{l}^{\omega_i}(x,t_{nj})$. The solution $u_H$ spanned by these basis functions will have a much larger dimension because each time instant is represented by multiscale basis functions. Thus, performing space-time multiscale model reduction can provide a substantial CPU savings.
In the next, we will discuss space-time multiscale basis functions. First, we will construct multiscale basis functions in the offline mode without using the residual. Next, in Section \[sect:online\], we will discuss online space-time multiscale basis construction.
Construction of offline basis functions
---------------------------------------
### Snapshot space
Let $\omega$ be a given coarse neighborhood in space. We omit the coarse node index to simplify the notations. The construction of the offline basis functions on coarse time interval $(T_{n-1},T_n)$ starts with a snapshot space $V_{\text{snap}}^{\omega}$ (or $V_{\text{snap}}^{\omega (n)}$). We also omit the coarse time index $(n)$ to simplify the notations. The snapshot space $V_{\text{snap}}^{\omega}$ is a set of functions defined on $\omega$ and contains all or most necessary components of the fine-scale solution restricted to $\omega$. A spectral problem is then solved in the snapshot space to extract the dominant modes in the snapshot space. These dominant modes are the offline basis functions and the resulting reduced space is called the offline space. There are two choices of $V_{\text{snap}}^{\omega}$ that are commonly used.
The first choice is to use all possible fine-grid functions in $\omega\times (T_{n-1},T_{n})$. This snapshot spaces provide accurate approximation for the solution space; however, this snapshot space can be very large. The second choice for the snapshot spaces consists of solving local problems for all possible boundary conditions. In particular, we define $\psi_{j}$ as the solution of $$\label{eq:locSnap}
\begin{split}
&\frac{\partial}{\partial t} \psi_{j} -\text{div} (\kappa(x,t) \nabla \psi_{j})=0\ \ \text{in}\ \omega\times (T_{n-1},T_{n}), \\
&\psi_{j}(x,t)=\delta_j(x,t)\ \ \text{on} \ \ \partial \left( \omega\times (T_{n-1},T_{n}) \right).
\end{split}$$ Here $\delta_j(x,t)$ is a fine-grid delta function and $\partial \left( \omega\times (T_{n-1},T_{n}) \right)$ denotes the boundaries $t=T_{n-1}$ and on $\partial \omega\times (T_{n-1},T_{n})$. In general, the computations of these snapshots are expensive since in each local coarse neighborhood $\omega$, $O(M_n^{\partial\omega})$ number of local problems are required to be solved. Here, $M_n^{\partial\omega}$ is the number of fine grids on the boundaries $t=T_{n-1}$ and on $\partial \omega\times (T_{n-1},T_{n})$. A smaller yet accurate snapshot space is needed to build a more efficient multiscale method. We can take an advantage of randomized oversampling concepts [@calo2014randomized] and compute only a few snapshot vectors, which will reduce the computational cost remarkably while keeping required accuracy. Next, we introduce randomized snapshots.
Firstly, we introduce the notation for oversampled regions. We denote by $\omega^{+}$ the oversampled space region of $\omega \subset\omega^{+}$, defined by adding several fine- or coarse-grid layers around $\omega$. Also, we define $(T_{n-1}^{*}, T_{n})$ as the left-side oversampled time region for $(T_{n-1},T_{n})$. In the following, we generate inexpensive snapshots using random boundary conditions on the oversampled space-time region $\omega^{+}\times(T_{n-1}^{*},T_{n})$. That is, instead of solving Equation (\[eq:locSnap\]) for each fine boundary node on $\partial \left( \omega\times (T_{n-1},T_{n}) \right)$, we solve a small number of local problems imposed with random boundary conditions $$\begin{split}
&\frac{\partial}{\partial t} \psi_{j}^{+} -\text{div} (\kappa(x,t) \nabla \psi_{j}^{+})=0\ \ \text{in}\ \omega^{+}\times (T_{n-1}^{*},T_{n}), \\
&\psi_{j}^{+}(x,t)= r_l\ \ \text{on} \ \ \partial \left( \omega^{+}\times (T_{n-1}^{*},T_{n}) \right),
\end{split}$$ where $r_l$ are independent identically distributed (i.i.d.) standard Gaussian random vectors on the fine-grid nodes of the boundaries $t=T_{n-1}^{*}$ and on $\partial \omega^{+}\times (T_{n-1}^{*},T_{n})$. Then the local snapshot space on $\omega^{+}\times (T_{n-1}^{*},T_{n})$ is $$V_{\text{snap}}^{\omega^{+}} = \text{span}\{\psi_{j}^{+}(x,t) | j=1,\cdot\cdot\cdot, L^{\omega}+p_{\text{bf}}^{\omega}\},$$ where $L^{\omega}$ is the number of local offline basis we want to construct in $\omega$ and $p_{\text{bf}}^{\omega}$ is the buffer number. Later on, we use the same buffer number for all $\omega$’s and simply use the notation $p_{\text{bf}}$. In the following sections, if we specify one special coarse neighborhood $\omega_i$, we use the notation $L_i$ to denote the number of local offline basis. With these snapshots, we follow the procedure in the following subsection to generate offline basis functions by using an auxiliary spectral decomposition.
### Offline space
To obtain the offline basis functions, we need to perform a space reduction by appropriate spectral problems. Motivated by our later convergence analysis, we adopt the following spectral problem on $\omega^{+}\times (T_{n-1},T_{n})$:
Find $(\phi,\lambda)\in V_{\text{snap}}^{\omega^{+}}\times\mathbb{R}$ such that $$\label{eq:eig-problem}
A_n(\phi,v) = \lambda S_n(\phi,v), \quad \forall v \in V_{\text{snap}}^{\omega^{+}},$$ where the bilinear operators $A_n(\phi,v)$ and $S_n(\phi,v)$ are defined by $$\begin{split}
A_n(\phi,v) &= \frac{1}{2} \left( \int_{\omega^{+}}\phi(x,T_{n})v(x,T_{n}) + \int_{\omega^{+}}\phi(x,T_{n-1})v(x,T_{n-1}) \right) + \int_{T_{n-1}}^{T_{n}}\int_{\omega^{+}}\kappa(x,t)\nabla\phi \cdot \nabla v, \\
S_n(\phi,v) &= \int_{\omega_{+}}\phi(x,T_{n-1})v(x,T_{n-1}) +
\int_{T_{n-1}}^{T_{n}}\int_{\omega^{+}}\widetilde{\kappa}^{+}(x,t)\phi v,
\end{split}$$ where the weighted function $\widetilde{\kappa}^{+}(x,t)$ is defined by $$\widetilde{\kappa}^{+}(x,t) = \kappa(x,t)\sum_{i=1}^{N_c}|\nabla\chi_i^{+}|^2,$$ $\{\chi_i^{+}\}_{i=1}^{N_c}$ is a partition of unity associated with the oversampled coarse neighborhoods $\{\omega_i^{+}\}_{i=1}^{N_c}$ and satisfies $|\nabla\chi_i^{+}|\geq|\nabla\chi_i|$ on $\omega_i$ where $\chi_i$ is the standard multiscale basis function for the coarse node $x_i$ (that is, with linear boundary conditions for cell problems). More precisely, $$\label{eq:POU}
\begin{split}
-\text{div}(\kappa(x,T_{n-1})\nabla\chi_i) &= 0, \quad \text{in }K\in\omega_i,\\
\chi_i &= g_i, \quad \text{on }\partial K,
\end{split}$$ for all $K\in\omega_i$, where $g_i$ is a continuous function on $\partial K$ and is linear on each edge of $\partial K$.
We arrange the eigenvalues $\{\lambda_j^{\omega^{+}}|j=1,2,\cdot\cdot\cdot\,L^{\omega}+p_{\text{bf}}^{\omega}\}$ from (\[eq:eig-problem\]) in the ascending order, and select the first $L^{\omega}$ eigenfunctions, which are corresponding to the first $L^{\omega}$ ordered eigenvalues, and denote them by $\{\Psi_1^{\omega^{+},\text{\text{off}}},\cdot\cdot\cdot, \Psi_{L^{\omega}}^{\omega^{+},\text{\text{off}}}\}$. Using these eigenfunctions, we can define $$\psi_j^{\omega^{+}}(x,t) = \sum_{k=1}^{L^{\omega}+p_{\text{bf}}^{\omega}} (\Psi_j^{\omega^{+},\text{off}})_k \psi_k^{+}(x,t), \qquad j=1,2,\cdot\cdot\cdot, L^{\omega},$$ where $(\Psi_j^{\omega^{+},\text{off}})_k$ denotes the $k$-th component of $\Psi_j^{\omega^{+},\text{off}}$, and $\psi_k^{+}(x,t)$ is the snapshot basis function computed on $\omega^{+}\times (T_{n-1}^{*},T_{n})$ as in the previous subsection. Then we can obtain the snapshots $\psi_{j}^{\omega}(x,t)$ on the target region $\omega\times (T_{n-1},T_{n})$ by restricting $\psi_j^{\omega^{+}}(x,t)$ onto $\omega\times (T_{n-1},T_{n})$. Finally, the offline basis functions on $\omega\times (T_{n-1},T_{n})$ are defined by $\phi_j^{\omega}(x,t) = \chi\psi_j^{\omega}(x,t)$, where $\chi$ is the standard multiscale basis function from (\[eq:POU\]) for a generic coarse neighborhood $\omega$. We also define the local offline space on $\omega\times (T_{n-1},T_{n})$ as $$V_{\text{off}}^{\omega} = \text{span}\{\phi_{j}^{\omega}(x,t) | j=1,\cdot\cdot\cdot, L^{\omega} \}.$$ Note that one can take $V_H^{(n)}$ in (\[eq:space-time FEM coarse decoupled\]) as $V_H^{(n)} = V_{\text{off}}^{(n)} = \text{span}\{\phi_{j}^{\omega_i}(x,t) | 1\leq i\leq N_c, 1\leq j\leq L_{i} \}$. As a result, $V_{H} = V_{\text{off}}= \oplus_{n=1}^{N}V_{H}^{(n)}$.
For the convenience of convergence analysis in Section \[sect:analysis\], we also denote by $\{\Psi_1^{\omega^{+},\text{\text{off}}},\cdot\cdot\cdot, \Psi_{L^{\omega}+p_{\text{bf}}^{\omega}}^{\omega^{+},\text{\text{off}}}\}$ all the eigenfunctions from (\[eq:eig-problem\]) corresponding to the ordered eigenvalues, and define $$\psi_j^{\omega^{+}}(x,t) = \sum_{k=1}^{L^{\omega}+p_{\text{bf}}^{\omega}} (\Psi_j^{\omega^{+},\text{off}})_k \psi_k^{+}(x,t), \qquad j=1,2,\cdot\cdot\cdot, L^{\omega}+p_{\text{bf}}^{\omega}.$$ We note that the snapshot space on $\omega^{+}\times (T_{n-1}^{*},T_{n})$ can be rewritten as $$V_{\text{snap}}^{\omega^{+}} = \text{span}\{\psi_j^{\omega^{+}}(x,t) | j=1,\cdot\cdot\cdot, L^{\omega}+p_{\text{bf}}^{\omega}\},$$ and the snapshot space on $\omega\times (T_{n-1},T_{n})$ can be written as $$V_{\text{snap}}^{\omega} = \text{span}\{\psi_j^{\omega}(x,t) | j=1,\cdot\cdot\cdot, L^{\omega}+p_{\text{bf}}^{\omega}\},$$ where each $\psi_j^{\omega}(x,t)$ is the restriction of $\psi_j^{\omega^{+}}(x,t)$ onto $\omega\times (T_{n-1},T_{n})$. By collecting all local snapshot spaces on each $\omega\times (T_{n-1},T_{n})$, we can obtain the snapshot space $V_{\text{snap}}^{(n)}$ on $\Omega\times (T_{n-1},T_{n})$.
The offline space can be rewritten as $$V_{\text{off}}^{\omega} = \text{span}\{\chi\psi_{j}^{\omega}(x,t) | j\leq L^{\omega} \}.$$
One can use a more general spectral problem in (\[eq:eig-problem\]) with $$\begin{split}
A(\phi,v) =& \frac{1}{2} \left( \int_{\omega}\phi(x,T_{n})v(x,T_{n}) + \int_{\omega}\phi(x,T_{n-1})v(x,T_{n-1}) \right) + \int_{T_{n-1}}^{T_{n}}\int_{\omega}\kappa(x,t)\nabla\phi \cdot \nabla v \\
& + \int_{T_{n-1}}^{T_{n}}\int_{\omega}\kappa(x,t)(z_{\phi}z_{v}+\nabla z_{\phi}\cdot\nabla z_{v}),\\
S(\phi,v) =& \int_{\omega}\phi(x,T_{n-1})v(x,T_{n-1}) +
\int_{T_{n-1}}^{T_{n}}\int_{\omega}\widetilde{\kappa}(x,t)\phi v
+\int_{T_{n-1}}^{T_{n}}\int_{\omega}\kappa|\nabla\chi|^{2}z_{\phi}z_{v},
\end{split}$$ where for any $w \in V_{\text{snap}}^{\omega}$, $z_w$ satisfies $$-z_{w}(x,t)+\nabla\cdot(\kappa(x,t)\nabla z_{w}(x,t))=\chi \frac{\partial w}{\partial t}, \;\quad \forall t\in(T_{n-1},T_{n}).$$ With this spectral problem, one can simplify the proof presented in Section \[sect:analysis\]. However, the numerical implementation of this local spectral problem is more complicated.
Convergence analysis {#sect:analysis}
====================
In this section, we will analyze the convergence of our proposed method. To start, we firstly define two norms that are used in the analysis. We define $\|\cdot\|_{V^{(n)}}^{2}$ and $\|\cdot\|_{W^{(n)}}^{2}$ by $$\begin{aligned}
\|u\|_{V^{(n)}}^{2} & =\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa|\nabla u|^{2}+\cfrac{1}{2}\int_{\Omega}u^{2}(x,T_{n}^{-})+\cfrac{1}{2}\int_{\Omega}u^{2}(x,T_{n-1}^{+}),\\
\|u\|_{W^{(n)}}^{2} & =\|u\|_{V^{(n)}}^{2}+\int_{T_{n-1}}^{T_{n}}\|u_{t}(\cdot,t)\|_{H^{-1}(\kappa,\Omega)}^{2},\end{aligned}$$ where $$\|u\|_{H^{-1}_{(\kappa,\Omega)}} = \sup_{v\in H^1_0(\Omega)}\cfrac{\int_{\Omega} u v}{(\int_\Omega \kappa |\nabla v|^2)^{\frac{1}{2}}}.$$
In the following, we will show the $V^{(n)}$-norm of the error $u_{h}-u_{H}$ can be bounded by the $W^{(n)}$-norm of the difference $u_{h}-w$ for any $w\in V_{H}^{(n)}$, where $u_{h}$ is the fine scale solution from Eqn.(\[eq:fine problem\]), $u_{H}$ is the multiscale solution from Eqn.(\[eq:space-time FEM coarse decoupled\]), and $V_{H}^{(n)}$ is the multiscale space defined in the previous section. The proof of this lemma will be presented in the Appendix.
\[lem: cea lemma\] Let $u_{h}$ be the fine scale solution from Equation (\[eq:fine problem\]), $u_{H}$ be the multiscale solution from Equation (\[eq:space-time FEM coarse decoupled\]). We have the following estimate $$\|u_{h}-u_{H}\|_{V^{(n)}}^{2}\leq
\begin{cases}
C\|u_{h}-w\|_{W^{(n)}}^{2} & \mbox{for n}=1,\\
C(\|u_{h}-w\|_{W^{(n)}}^{2}+\|u_{h}-u_{H}\|_{V^{(n-1)}}^{2}) & \mbox{for n}>1,
\end{cases}$$ for any $w\in V_{H}^{(n)}$. If we define the $V^{(0)}$-norm to be $0$, then we can write $$\|u_{h}-u_{H}\|_{V^{(n)}}^{2}\leq
C(\|u_{h}-w\|_{W^{(n)}}^{2}+\|u_{h}-u_{H}\|_{V^{(n-1)}}^{2}) \quad \mbox{for n}\geq 1,$$ for any $w\in V_{H}^{(n)}$.
Therefore, to estimate the error of our multiscale solution, we only need to find a function $w$ in $V_{H}^{(n)}$ such that $\|u_{h}-w\|_{W^{(n)}}$is small. Except for Lemma \[lem: cea lemma\], we still need the following lemma to estimate $\|u_{h}-w\|_{W^{(n)}}.$
\[lem: Caccioppoli\] For any $v$ satisfying $$\frac{\partial}{\partial t}v-\text{div}(\kappa(x,t)\nabla v)=0\ \ \text{in}\ \ \omega_i\times(T_{n-1},T_{n}),$$ we have $$\begin{split}\int_{\omega_i}\chi_i^{2}v^{2}(x,T_{n}) + \int_{T_{n-1}}^{T_{n}}\int_{\omega_i}\kappa|\chi_i^{2}||\nabla v|^{2} \preceq \int_{\omega_i}\chi_i^{2}v^{2}(x,T_{n-1}) + \int_{T_{n-1}}^{T_{n}}\int_{\omega_i}\kappa|\nabla\chi_i|^{2}v^{2},
\end{split}$$ where the notation $F \preceq G$ means $F \leq \mathcal{C}G$ with a constant $\mathcal{C}$ independent of the mesh, contrast and the functions involved.
Now, we are ready to prove our main result in this section.
\[main thm\]
Let $u_{h}$ be the fine scale solution from Equation (\[eq:fine problem\]), $u_{H}$ be the multiscale solution from Equation (\[eq:space-time FEM coarse decoupled\]). Let $\tilde{u}_{h}=\text{argmin}_{v\in V_{\text{snap}}^{(n)}}\{\|u_{h}-v\|_{W^{(n)}}\}$ and we denote $\tilde{u}_{h}=\sum_{i}\chi_{i}\tilde{u}_{h,i}$ with $\tilde{u}_{h,i}=\sum_{j}c_{i,j}\psi_{j}^{\omega_i}$. There holds $$\|u_{h}-u_{H}\|_{V^{(n)}}^{2}\preceq
M(DEF+1)\sum_{i}\left(\frac{1}{\lambda_{L_{i}+1}^{\omega_i^{+}}}\|\tilde{u}_{h,i}^{+}\|_{V^{(n)}(\omega_i^{+})}^{2}\right)
+\|u_{h}-\tilde{u}_{h}\|_{W^{(n)}}^{2}+\|u_{h}-u_{H}\|_{V^{(n-1)}}^{2},$$ where
$M=\max_{K}\{M_{K}\}$ with $M_{K}$ is the number of coarse neighborhoods $\omega_{i}$’s which have nonempty intersection with $K$,
$D=\max\{D_{i}\}$ with $D_{i}=\sup_{v\in H_{0}^{1}(\Omega)}\cfrac{\int_{\omega_{i}}\kappa|\nabla\chi_{i}|^{2}v^{2}+\int_{\omega_{i}}\kappa\chi_{i}^{2}|\nabla v|^{2}}{\int_{\omega_{i}}\kappa|\nabla v|^{2}+\int_{\omega_{i}}\kappa v{}^{2}}$,
$E=\sup_{w\in H_{0}^{1}(\Omega)}\cfrac{\int_{\Omega}\kappa|\nabla w|^{2}+\int_{\Omega}\kappa w{}^{2}}{\int_{\Omega}\kappa|\nabla w|^{2}}$,
$F=\max\{F_{i}\}$ with $F_i = \cfrac{1}{\min_{x\in\omega_i}\{|\chi_i^{+}(x)|^2\}}$,
$\tilde{u}_{h,i}^{+}=\sum_{j}c_{i,j}\psi_{j}^{\omega_i^{+}}$ and the local norm $\|\cdot\|_{V^{(n)}(\omega_i^{+})}$ is defined by $$\|v\|_{V^{(n)}(\omega_i^{+})}^{2}=\int_{T_{n-1}}^{T_{n}}\int_{\omega_i^{+}}\kappa|\nabla v|^{2}+\frac{1}{2}\int_{\omega_i^{+}}v^{2}(x,T_{n}^{-})+\frac{1}{2}\int_{\omega_i^{+}}v^{2}(x,T_{n-1}^{+}).$$
By Lemma \[lem: cea lemma\], $$\label{eq:general estimate}
\|u_{h}-u_{H}\|_{V^{(n)}}^2 \preceq
\inf_{w\in V_H^{(n)}}\|u_{h}-w\|_{W^{(n)}}^2+\|u_{h}-u_{H}\|_{V^{(n-1)}}^{2}.$$ Therefore, we need to estimate $\inf_{w\in V_{H}^{(n)}}\|u_{h}-w\|_{W^{(n)}}^2.$ Note that $\tilde{u}_{h}=\sum_{i}\chi_{i}\tilde{u}_{h,i}=\sum_{i}\sum_{j}c_{i,j}\chi_{i}\psi_{j}^{\omega_i}$. Using this expression, we can define a projection of $\tilde{u}_{h}$ into $V_{H}^{(n)}$ by $$P(\tilde{u}_{h})=\sum_{i}\sum_{j\leq L_{i}}c_{i,j}\chi_{i}\psi_{j}^{\omega_i}.\label{eq:expression of projection of u_h}$$ Then $$\begin{aligned}
\inf_{w\in V_{H}^{(n)}}\|u_{h}-w\|_{W^{(n)}}^2
&\leq \|u_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^2 \nonumber\\
&\leq \|u_{h}-\tilde{u}_{h}\|_{W^{(n)}}^2 + \|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^2. \label{eq:estimate inf}\end{aligned}$$ We will estimate $\|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^2$.\
By the definition of $\|\cdot\|_{W^{(n)}}$, we have $$\begin{aligned}
\|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^{2} & =\|\sum_{i}\chi_{i}(\tilde{u}_{h,i}-P(\tilde{u}_{h,i}))\|_{V^{(n)}}^{2}
+\int_{T_{n-1}}^{T_{n}}\|\cfrac{\partial(\sum_{i}\chi_{i}(\tilde{u}_{h,i}-P(\tilde{u}_{h,i})))}{\partial t}\|_{H^{-1}(\kappa,\Omega)}^{2}\;,\end{aligned}$$ where $\tilde{u}_{h,i}=\sum_{j}c_{i,j}\psi_{j}^{\omega_i}$ and $P(\tilde{u}_{h,i})=\sum_{j\leq L_{i}}c_{i,j}\psi_{j}^{\omega_i}$. Let $e_{i}=\tilde{u}_{h,i}-P(\tilde{u}_{h,i})$, then $\tilde{u}_{h}-P(\tilde{u}_{h})=\sum_{i}\chi_{i}e_{i}$. Therefore, $$\label{eq:W-norm estimate}
\|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^{2}=\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}}^{2}
+\int_{T_{n-1}}^{T_{n}}\|\cfrac{\partial(\sum_{i}\chi_{i}e_{i})}{\partial t}\|_{H^{-1}(\kappa,\Omega)}^{2}.$$ In the following, we will estimate the two terms on the right hand side of (\[eq:W-norm estimate\]), separately. Then the proof is done.\
First, we estimate the term $\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}}^{2}$. We define the local norm $\|\cdot\|_{V^{(n)}(K)}$ by $$\|v\|_{V^{(n)}(K)}^{2}=\int_{T_{n-1}}^{T_{n}}\int_{K}\kappa|\nabla v|^{2} +\frac{1}{2}\int_{K}v^{2}(x,T_{n}^{-})+\frac{1}{2}\int_{K}v^{2}(x,T_{n-1}^{+}).$$ Then we have $$\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}}^{2}\leq\sum_{K}\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}(K)}^{2}.$$ Moreover, $$\begin{aligned}
\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}(K)}^{2} & \leq M_{K}\sum_{i}\|\chi_{i}e_{i}\|_{V^{(n)}(K)}^{2},\end{aligned}$$ where $M_{K}$ is the number of coarse neighborhoods $\omega_{i}$’s which have nonempty intersection with $K$. Therefore, $$\begin{aligned}
\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}}^{2} & \leq\sum_{K}M_{K}\sum_{i}\|\chi_{i}e_{i}\|_{V^{(n)}(K)}^{2}\nonumber\\
& \leq M\sum_{i}\|\chi_{i}e_{i}\|_{V^{(n)}(\omega_{i})}^{2}, \label{eq:V-norm estimate 1}\end{aligned}$$ where $M=\max_{K}\{M_{K}\}$. Now, we need to estimate the term $\|\chi_{i}e_{i}\|_{V^{(n)}(\omega_{i})}^{2}$. Since $\nabla(\chi_{i}e_{i})=e_{i}\nabla\chi_{i}+\chi_{i}\nabla e_{i}$, we obtain $$\begin{aligned}
\|\chi_{i}e_{i}\|_{V^{(n)}(\omega_{i})}^{2} & \leq & 2\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla\chi_i|^{2}e_{i}^{2}+2\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa\chi_i^{2}|\nabla e_{i}|^{2}\\
& & +\cfrac{1}{2}\int_{\omega_{i}}\chi_i^{2}e_{i}^{2}(x,T_{n}^{-})+\cfrac{1}{2}\int_{\omega_{i}}\chi_i^{2}e_{i}^{2}(x,T_{n-1}^{+}).\nonumber\end{aligned}$$ Using Lemma \[lem: Caccioppoli\], we have $$\begin{aligned}
\|\chi_{i}e_{i}\|_{V^{(n)}(\omega_{i})}^{2}
& \preceq
\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla\chi_i|^{2}e_{i}^{2}
+\int_{\omega_{i}}\chi_i^{2}e_{i}^{2}(x,T_{n-1}^{+})\\
& \preceq
\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla\chi_i|^{2}e_{i}^{2}
+\int_{\omega_{i}}e_{i}^{2}(x,T_{n-1}^{+}).\end{aligned}$$ Now we introduce notations in $\omega_i^{+}$ and denote $e_{i}^{+}=\tilde{u}_{h,i}^{+}-P(\tilde{u}_{h,i}^{+})$, where $\tilde{u}_{h,i}^{+}=\sum_{j}c_{i,j}\psi_{j}^{\omega_i^{+}}$ and $P(\tilde{u}_{h,i}^{+})=\sum_{j\leq L_{i}}c_{i,j}\psi_{j}^{\omega_i^{+}}$. It is obvious that $\tilde{u}_{h,i}^{+}|_{\omega_i} = \tilde{u}_{h,i}$, $P(\tilde{u}_{h,i}^{+})|_{\omega_i} =P(\tilde{u}_{h,i})$ and $e_{i}^{+}|_{\omega_i} = e_{i}$. And there holds the following two inequalities, $$\label{eq:oversample estimate 1}
\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla\chi_i|^{2}e_{i}^{2} \leq \int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\nabla\chi_i^{+}|^{2}|e_{i}^{+}|^{2},$$ and $$\label{eq:oversample estimate 2}
\int_{\omega_{i}}e_{i}^{2}(x,T_{n-1}^{+}) \leq \int_{\omega_i^{+}}|e_i^{+}(x,T_{n-1}^{+})|^{2}.$$ Thus, $$\label{eq:V-norm estimate 2}
\|\chi_{i}e_{i}\|_{V^{(n)}(\omega_{i})}^{2}
\preceq
\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\nabla\chi_i^{+}|^{2}|e_{i}^{+}|^{2}
+\int_{\omega_i^{+}}|e_i^{+}(x,T_{n-1}^{+})|^{2}.$$ Substituting (\[eq:V-norm estimate 2\]) into (\[eq:V-norm estimate 1\]), we immediately obtain $$\label{eq:V-norm estimate 3}
\|\sum_{i}\chi_{i}e_{i}\|_{V^{(n)}}^{2} \preceq M\sum_{i}\left(\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\nabla\chi_i^{+}|^{2}|e_{i}^{+}|^{2}
+\int_{\omega_i^{+}}|e_i^{+}(x,T_{n-1}^{+})|^{2}\right).$$\
Next, we will estimate the term $\int_{T_{n-1}}^{T_{n}}\|\frac{\partial(\sum_{i}\chi_{i}e_{i})}{\partial t}\|_{H^{-1}(\kappa,\Omega)}^{2}$. By definition, we have $$\begin{aligned}
\int_{T_{n-1}}^{T_{n}}\|\cfrac{\partial(\sum_{i}\chi_{i}e_{i})}{\partial t}\|_{H^{-1}(\kappa,\Omega)}^{2}
& =\int_{T_{n-1}}^{T_{n}}\sup_{w\in H_{0}^{1}(\Omega)}\frac{\left(\int_{\Omega}\sum_{i}\chi_{i}\frac{\partial e_{i}}{\partial t}w\right)^{2}}{\int_{\Omega}\kappa|\nabla w|^{2}}\nonumber \\
& \leq\int_{T_{n-1}}^{T_{n}}\sup_{w\in H_{0}^{1}(\Omega)}\frac{\left(\sum_{i}|\int_{\omega_i}\chi_{i}\frac{\partial e_{i}}{\partial t}w|\right)^{2}}{\int_{\Omega}\kappa|\nabla w|^{2}}.\label{eq:H^=00007B-1=00007D norm global to local}\end{aligned}$$ Since $e_{i}$ satisfies the equation $$\frac{\partial}{\partial t}e_{i}-\text{div}(\kappa(x,t)\nabla e_{i})=0\text{ in }\omega_{i}\times(T_{n-1},T_{n}),\label{eq:loc pde}$$ we have $$\begin{aligned}
\int_{\omega_{i}}\chi_{i}\frac{\partial e_{i}}{\partial t}w & =-\int_{\omega_{i}}\kappa(x,t)\nabla e_{i}\cdot\nabla(\chi_{i}w)\\
& =-\int_{\omega_{i}}\kappa(x,t)w\nabla e_{i}\cdot\nabla\chi_{i}-\int_{\omega_{i}}\kappa(x,t)\chi_{i}\nabla e_{i}\cdot\nabla w.\end{aligned}$$ Moreover, $$\begin{aligned}
\left|\int_{\omega_{i}}\chi_{i}\frac{\partial e_{i}}{\partial t}w \right|
= & \left|-\int_{\omega_{i}}\kappa w\nabla e_{i}\cdot\nabla\chi_{i}-\int_{\omega_{i}}\kappa\chi_{i}\nabla e_{i}\cdot\nabla w\right|\nonumber \\
\leq & \left(\int_{\omega_{i}}\kappa w^{2}|\nabla\chi_{i}|^{2}\right)^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}} + \left(\int_{\omega_{i}}\kappa\chi_{i}^{2}|\nabla w|^{2}\right)^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}}\nonumber \\
\leq & 2\left(\int_{\omega_{i}}\kappa w^{2}|\nabla\chi_{i}|^{2}+\int_{\omega_{i}}\kappa\chi_{i}^{2}|\nabla w|^{2}\right)^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}}.\label{eq:estimate loc H^=00007B-1=00007D norm 1}\end{aligned}$$ Let $$D_{i}=\sup_{v\in H_{0}^{1}(\Omega)}\cfrac{\int_{\omega_{i}}\kappa|\nabla\chi_{i}|^{2}v^{2}+\int_{\omega_{i}}\kappa\chi_{i}^{2}|\nabla v|^{2}}{\int_{\omega_{i}}\kappa|\nabla v|^{2}+\int_{\omega_{i}}\kappa v{}^{2}}.$$ From (\[eq:estimate loc H\^=00007B-1=00007D norm 1\]), we obtain $$\left|\int_{\omega_{i}}\chi_{i}\cfrac{\partial e_{i}}{\partial t}w\right|
\leq
2D_{i}^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla w|^{2}+\int_{\omega_{i}}\kappa w^{2}\right)^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}}.$$ Therefore, $$\begin{aligned}
\sum_{i}\left|\int_{\omega_i}\chi_{i}\cfrac{\partial e_{i}}{\partial t}w\right|
& \leq
2\sum_{i}D_{i}^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla w|^{2}+\int_{\omega_{i}}\kappa w^{2}\right)^{\frac{1}{2}}\left(\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}}\nonumber \\
& \leq
2\left(\sum_{i}D_{i}(\int_{\omega_{i}}\kappa|\nabla w|^{2}+\int_{\omega_{i}}\kappa w{}^{2})\right)^{\frac{1}{2}}\left(\sum_{i}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}}\nonumber \\
& \leq
2D^{\frac{1}{2}}M^{\frac{1}{2}}\left(\int_{\Omega}\kappa|\nabla w|^{2}+\int_{\Omega}\kappa w{}^{2}\right)^{\frac{1}{2}}\left(\sum_{i}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right)^{\frac{1}{2}},\label{eq:estimate loc H^=00007B-1=00007D norm 2}\end{aligned}$$ where $D=\max\{D_{i}\}$. Combining (\[eq:H\^=00007B-1=00007D norm global to local\]) with (\[eq:estimate loc H\^=00007B-1=00007D norm 2\]), we have $$\int_{T_{n-1}}^{T_{n}}\|\cfrac{\partial(\sum_{i}\chi_{i}e_{i})}{\partial t}\|_{H^{-1}(\kappa,\Omega)}^{2}\leq4DM\sup_{w\in H^{1}(\Omega)}\cfrac{\left(\int_{\Omega}\kappa|\nabla w|^{2}+\int_{\Omega}\kappa w{}^{2}\right)}{\int_{\Omega}\kappa|\nabla w|^{2}}\left(\sum_{i}\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right).$$ Let $$E=\sup_{w\in H_{0}^{1}(\Omega)}\cfrac{\int_{\Omega}\kappa|\nabla w|^{2}+\int_{\Omega}\kappa w{}^{2}}{\int_{\Omega}\kappa|\nabla w|^{2}},$$ then we have $$\int_{T_{n-1}}^{T_{n}}\|\cfrac{\partial(\sum_{i}\chi_{i}e_{i})}{\partial t}\|_{H^{-1}(\kappa,\Omega)}^{2}\leq4DME\left(\sum_{i}\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}\right).\label{eq:H^=00007B-1=00007D estimate}$$
Now, we substitute (\[eq:H\^=00007B-1=00007D estimate\]) and (\[eq:V-norm estimate 3\]) into (\[eq:W-norm estimate\]), then we have $$\begin{aligned}
\|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^{2}
& \preceq
M\sum_{i}\left(DE\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}
+\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla\chi_i|^{2}e_{i}^{2}
+\int_{\omega_{i}}\chi_{i}^{2}e_{i}^{2}(x,T_{n-1}^{+})\right)\nonumber\\
& \preceq
M\sum_{i}\left(DE\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}
+\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla\chi_i|^{2}e_{i}^{2}
+\int_{\omega_{i}}e_{i}^{2}(x,T_{n-1}^{+})\right).\label{eq:W-norm estimate 2}\end{aligned}$$ Note that $$\begin{aligned}
\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}
& \leq \int_{T_{n-1}}^{T_{n}} \frac{1}{\min_{x\in\omega_i}\{|\chi_i^{+}(x)|^2\}} \int_{\omega_{i}}\kappa|\chi_i^{+}|^2|\nabla e_{i}|^{2}\\
& \leq \frac{1}{\min_{x\in\omega_i}\{|\chi_i^{+}(x)|^2\}} \int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\chi_i^{+}|^2|\nabla e_{i}^{+}|^{2}.\end{aligned}$$ Applying Lemma \[lem: Caccioppoli\] for $\omega_{i}^{+}$ then implies $$\begin{aligned}
\int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}}\kappa|\nabla e_{i}|^{2}
& \leq \frac{1}{\min_{x\in\omega_i}\{|\chi_i^{+}(x)|^2\}} \left( \int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\nabla\chi_i^{+}|^{2}|e_{i}^{+}|^{2}
+ \int_{\omega_i^{+}}|\chi_i^{+}|^{2}|e_i^{+}(x,T_{n-1}^{+})|^{2} \right) \nonumber\\
& \leq F_i \left( \int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\nabla\chi_i^{+}|^{2}|e_{i}^{+}|^{2}
+ \int_{\omega_i^{+}}|e_i^{+}(x,T_{n-1}^{+})|^{2} \right), \label{eq:W-norm estimate 3}\end{aligned}$$ where $F_i = \cfrac{1}{\min_{x\in\omega_i}\{|\chi_i^{+}(x)|^2\}}$. Substituting (\[eq:W-norm estimate 3\]), (\[eq:oversample estimate 1\]) and (\[eq:oversample estimate 2\]) into (\[eq:W-norm estimate 2\]) gives $$\begin{aligned}
\|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^{2}
& \preceq
M\sum_{i}(DEF_i + 1) \left( \int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\kappa|\nabla\chi_i^{+}|^{2}|e_{i}^{+}|^{2}
+ \int_{\omega_i^{+}}|e_i^{+}(x,T_{n-1}^{+})|^{2} \right) \nonumber\\
& \preceq
M(DEF+1)\sum_{i} \left( \int_{T_{n-1}}^{T_{n}}\int_{\omega_{i}^{+}}\widetilde{\kappa}^{+}(x,t)|e_{i}^{+}|^{2}
+ \int_{\omega_i^{+}}|e_i^{+}(x,T_{n-1}^{+})|^{2} \right),\label{eq:W-norm estimate 4}\end{aligned}$$ where $F=\max\{F_{i}\}$. Using the spectral problem, we have $$\label{eq:W-norm estimate 5}
\|\tilde{u}_{h}-P(\tilde{u}_{h})\|_{W^{(n)}}^{2}\preceq
M(DEF+1)\sum_{i}\left(\cfrac{1}{\lambda_{L_{i}+1}^{\omega_i^{+}}}\|\tilde{u}_{h,i}^{+}\|_{V^{(n)}(\omega_i^{+})}^{2}\right).$$ Combine (\[eq:general estimate\]), (\[eq:estimate inf\]) and (\[eq:W-norm estimate 5\]), and we finally obtain $$\|u_{h}-u_{H}\|_{V^{(n)}}^{2}\preceq
M(DEF+1)\sum_{i}\left(\frac{1}{\lambda_{L_{i}+1}^{\omega_i^{+}}}\|\tilde{u}_{h,i}^{+}\|_{V^{(n)}(\omega_i^{+})}^{2}\right)
+\|u_{h}-\tilde{u}_{h}\|_{W^{(n)}}^{2}+\|u_{h}-u_{H}\|_{V^{(n-1)}}^{2}.$$
Numerical results. Offline GMsFEM. {#sect:NR1}
==================================
In this section, we present a number of representative numerical examples to show the performance of the proposed method. In particular, we solve Equation (\[eq:para PDE\]) using the space-time GMsFEM to validate the effectiveness of the proposed approaches. The space domain $\Omega$ is taken as the unit square $[0,1]\times[0,1]$ and is divided into $10\times10$ coarse blocks consisting of uniform squares. Each coarse block is then divided into $10\times10$ fine blocks consisting of uniform squares. That is, $\Omega$ is partitioned by $100\times100$ square fine-grid blocks. The whole time interval is $[0, 1.6]$ (i.e., $T = 1.6$) and is divided into two uniform coarse time intervals and each coarse time interval is then divided into $8$ fine time intervals. We also use a source term $f = 1$ and impose a continuous initial condition $\beta(x,y)=\sin(\pi x)\sin(\pi y)$. We employ three different high-contrast permeability fields $\kappa(x,t)$’s to examine our method, which will be shown in the following three cases separately. In each case, we first solve for $u_h$ from Equation (\[eq:fine problem\]) to obtain the fine-grid solution. Then we solve for the multiscale solution $u_H$ using the space-time GMsFEM. To compare the accuracy, we will use the following error quantities: $$\label{err_formulus}
e_1 =\left( \frac{\int_0^{T} \|u_{H}(t)-u_{h}(t)\|^2_{L^2(\Omega)}}{\int_0^{T}\|u_{h}(t)\|^2_{L^2(\Omega)}} \right)^{1/2}, \qquad
e_2 =\left( \frac{\int_0^{T} \int_{\Omega} \kappa |\nabla(u_{H}(t)-u_{h}(t))|^2}{\int_0^{T} \int_{\Omega} \kappa |\nabla u_{h}(t)|^2} \right)^{1/2}.$$
Since we are using the technique of randomized oversampling in the computation of the snapshot space, we would like to introduce the concept of $\emph{snapshot ratio}$, which is calculated as the number of randomized snapshots divided by the number of the full snapshots on one coarse neighborhood $\omega_i$. Here, the number of the full snapshots refers to the number of functions $\delta_i(x,t)$ from Equation (\[eq:locSnap\]). In the following experiment with $100\times100$ fine-grid mesh, this number of the full snapshots on each coarse neighborhood is calculated by $n_{\text{total}}^{\text{snap}}=21\times21+40\times8 =761$.
High-contrast Permeability Field 1: High-contrast medium translated in time {#sect:media1}
---------------------------------------------------------------------------
We start with a high-contrast permeability field $\kappa(x,t)$, which is translated uniformly after every other fine time step. High-contrast permeability fields at the initial and final time steps are shown in Figure \[Figure:UniCoeff\]. Next, we consider applying the space-time GMsFEM to Equation (\[eq:para PDE\]) and solve for the multiscale solution $u_H$. Recall the procedures that are described in the Section \[sect:GMsFEM\], where we need to construct the snapshot spaces in the first place. The number of local offline basis that will be used in each $\omega_i$, denoted by $L_i$, and the buffer number $p_{\text{bf}}$ needs to be chosen in advance since they determine how many local snapshots are used. Then we can construct the lower dimensional offline space by performing space reduction on the snapshot space. In our experiments, we use the same buffer number and the same number of local offline basis for all coarse neighborhood $\omega_i$’s.
![High-contrast Permeability Field 1. Left: the permeability at the initial time. Right: the permeability at the final time.[]{data-label="Figure:UniCoeff"}](coeff1.png){width="\columnwidth"}
![High-contrast Permeability Field 1. Left: the permeability at the initial time. Right: the permeability at the final time.[]{data-label="Figure:UniCoeff"}](coeff17.png){width="\columnwidth"}
First, we fix $L_i = 11$ for all $\omega_i$’s and examine the influences of various buffer numbers on the solution errors $e_1$ and $e_2$. The results are displayed in the left table of Table \[Table:test bf Li 1\]. It is observed that when increasing the buffer numbers, one can get more accurate solutions, which is as expected. But the error decays very slowly, which indicates that using different buffer numbers doesn’t affect the convergence rate too much. Based on this observation, it is not necessary to choose a large buffer number in order to improve convergence rate. Then we consider the choice of $L_i$, the number of eigenbasis in a neighborhood. With the fixed buffer number $p_{\text{bf}}=8$, we examine the convergence behaviors of using different $L_i$’s. Relative errors of multiscale solutions are shown in the right table of Table \[Table:test bf Li 1\]. We observe that with a fixed buffer number, the relative errors are decreasing as using more offline basis. To see a more quantitative relationship between the relative errors and the values of $L_i$ as well as being inspired by the result in Theorem \[main thm\], we inspect the values of $1/\Lambda_{*}$ and the corresponding squared errors (see Table \[Table:minlambda\] and Figure \[Figure:LambdaVSError\]), where $\Lambda_{*}=\min_{\omega_i} \lambda_{L_i + 1}^{\omega_i}$ and $\{\lambda_j^{\omega_i}\}$ are the eigenvalues associated with the eigenbasis computed by spectral problem (\[eq:eig-problem\]) in each $\omega_i$. We note that when plotting Figure \[Figure:LambdaVSError\], we don’t use the values of case $L_i=2$, because in this case as in the case with one basis function per node, the method does not converge as we do not have sufficient number of basis functions. We note that the two curves in Figure \[Figure:LambdaVSError\] track each other somewhat closely. This indicates that $1/\Lambda_{*}$’s and $e_2^2$’s are correlated and we calculate for the correlation coefficient to be $corrcoef(1/\Lambda_{*}, e_2^2)= 0.9778.$ Observing the dimensions of the offline spaces $V_{\text{off}}$, one can see that compared with the traditional fine-scale finite element method, the proposed space-time GMsFEM uses much fewer degrees of freedom while achieving an accurate solution. Also, by inspecting the snapshot ratios, one can see that the use of randomization can reduce the dimension of snapshot spaces substantially. We would like to comment that oversampling technique is necessary for the randomization. For example, in the case $L_i=6$ and $p_{\text{bf}}=8$, if without oversampling the errors $e_1$ and $e_2$ are $11.19\%$ and $88.42\%$, respectively, which are worse than the errors obtained with oversampling.
$p_{\text{bf}}$ Snapshot ratio $e_1$ $e_2$
----------------- ---------------- ------- --------
1 0.0158 6.18% 53.90%
4 0.0197 5.66% 48.04%
8 0.0250 5.17% 45.86%
12 0.0302 5.16% 43.83%
20 0.0407 4.71% 41.14%
30 0.0539 4.35% 38.68%
40 0.0670 4.23% 37.60%
: First permeability field. Left: errors with the fixed number of offline basis $L_i=11$. Right: errors with the fixed buffer number $p_{\text{bf}}=8$. []{data-label="Table:test bf Li 1"}
$L_i$ $dim(V_{\text{off}})$ Snapshot ratio $e_1$ $e_2$
------- ----------------------- ---------------- -------- ---------
2 162 0.0131 17.03% 129.14%
6 486 0.0184 8.11% 62.59%
10 810 0.0237 6.97% 54.85%
20 1620 0.0368 4.81% 41.18%
30 2430 0.0499 3.29% 31.64%
40 3240 0.0631 2.28% 24.43%
50 4050 0.0762 1.54% 18.45%
: First permeability field. Left: errors with the fixed number of offline basis $L_i=11$. Right: errors with the fixed buffer number $p_{\text{bf}}=8$. []{data-label="Table:test bf Li 1"}
$L_i$ $1/\Lambda_{*}$ $e_1^2$ $e_2^2$
------- ----------------- --------- ---------
2 0.2734 2.90% 166.78%
6 0.0120 0.66% 39.17%
10 0.0085 0.49% 30.08%
20 0.0061 0.23% 16.96%
30 0.0053 0.11% 10.01%
40 0.0048 0.05% 5.97%
50 0.0042 0.02% 3.40%
: $1/\Lambda_{*}$ values and errors.[]{data-label="Table:minlambda"}
![Left: $1/\Lambda_{*}$ vs $L_i$; Right: $e_2^2$ vs $L_i$.[]{data-label="Figure:LambdaVSError"}](Lambda.png){width="\columnwidth"}
![Left: $1/\Lambda_{*}$ vs $L_i$; Right: $e_2^2$ vs $L_i$.[]{data-label="Figure:LambdaVSError"}](e2.png){width="\columnwidth"}
High-contrast Permeability Field 2: Four channels translated in time
--------------------------------------------------------------------
In this subsection, we consider a more structured high-contrast permeability field $\kappa(x,t)$, which has four channels inside and these four channels are translated uniformly in time. High-contrast permeability fields at the initial and final time steps are shown in Figure \[Figure:4chanlsTran\]. We repeat our steps from the previous example by fixing $L_i$ and $p_{\text{bf}}$, separately. The results are shown in Table \[Table:test bf Li 2\]. One can still observe that increasing the buffer numbers will slowly reduce the relative errors and with a fixed buffer number, the relative errors are decreasing as adding more offline basis. Using a similar approach, we can also get the cross-correlation coefficient between $e_2^2$ and $1/\Lambda_{*}$, which is $0.9863$. This suggests a linear relationship between $e_2^2$ and $1/\Lambda_{*}$ and verifies Theorem \[main thm\].
![High-contrast Permeability Field 2. Left: the permeability at the initial time. Right: the permeability at the final time.[]{data-label="Figure:4chanlsTran"}](coeff_4chanls_t0.png){width="\columnwidth"}
![High-contrast Permeability Field 2. Left: the permeability at the initial time. Right: the permeability at the final time.[]{data-label="Figure:4chanlsTran"}](coeff_4chanls_t17.png){width="\columnwidth"}
$p_{\text{bf}}$ Snapshot ratio $e_1$ $e_2$
----------------- ---------------- ------- --------
1 0.0158 7.42% 61.87%
4 0.0197 7.30% 58.95%
8 0.0250 7.14% 57.30%
12 0.0302 7.00% 54.01%
20 0.0407 6.81% 50.85%
30 0.0539 6.61% 49.30%
40 0.0670 6.43% 48.26%
: Second permeability field. Left: errors with the fixed number of offline basis $L_i=11$. Right: errors with the fixed buffer number $p_{\text{bf}}=8$. []{data-label="Table:test bf Li 2"}
$L_i$ $dim(V_{\text{off}})$ Snapshot ratio $e_1$ $e_2$
------- ----------------------- ---------------- -------- ---------
2 162 0.0131 11.91% 104.95%
6 486 0.0184 8.33% 70.82%
10 810 0.0237 7.25% 58.25%
20 1620 0.0368 5.67% 43.10%
30 2430 0.0499 3.90% 32.75%
40 3240 0.0631 2.73% 27.08%
50 4050 0.0762 1.86% 20.70%
: Second permeability field. Left: errors with the fixed number of offline basis $L_i=11$. Right: errors with the fixed buffer number $p_{\text{bf}}=8$. []{data-label="Table:test bf Li 2"}
High-contrast Permeability Field 3: Four channels rotated in time
-----------------------------------------------------------------
In the third example, we consider another structured high-contrast permeability field $\kappa(x,t)$ which has four channels inside and these four channels are rotated anticlockwise around the center by $11.25$ degrees after each fine time step. High contrast permeability fields at the initial time step is shown in Figure \[Figure:4chanlsRota\]. We repeat the same procedures as in the previous two examples. The results are shown in Table \[Table:test bf Li 3\] and one can draw similar conclusions as before. The cross-correlation coefficient between $e_2^2$ and $1/\Lambda_{*}$ is calculated as $0.9959$. This shows a linear relationship between $e_2^2$ and $1/\Lambda_{*}$ (see Theorem \[main thm\]).
![High-contrast Permeability Field 3 at the initial time.[]{data-label="Figure:4chanlsRota"}](coeff_4chanls_tt0.png){width="7cm"}
$p_{\text{bf}}$ Snapshot ratio $e_1$ $e_2$
----------------- ---------------- ------- --------
1 0.0158 8.68% 72.86%
4 0.0197 8.67% 71.67%
8 0.0250 8.56% 71.42%
12 0.0302 8.44% 68.87%
20 0.0407 8.18% 65.88%
30 0.0539 7.96% 61.56%
40 0.0670 7.58% 57.58%
: Third permeability field. Left: errors with the fixed number of offline basis $L_i=11$. Right: errors with the fixed buffer number $p_{\text{bf}}=8$. []{data-label="Table:test bf Li 3"}
$L_i$ $dim(V_{\text{off}})$ Snapshot ratio $e_1$ $e_2$
------- ----------------------- ---------------- -------- ---------
2 162 0.0131 10.41% 109.40%
6 486 0.0184 9.40% 83.60%
10 810 0.0237 8.63% 70.84%
20 1620 0.0368 7.42% 57.66%
30 2430 0.0499 6.14% 47.78%
40 3240 0.0631 4.75% 39.89%
50 4050 0.0762 3.29% 30.11%
: Third permeability field. Left: errors with the fixed number of offline basis $L_i=11$. Right: errors with the fixed buffer number $p_{\text{bf}}=8$. []{data-label="Table:test bf Li 3"}
Residual based online adaptive procedure {#sect:online}
========================================
As we observe in the previous examples, the offline errors do not decrease rapidly after several multiscale functions are selected. In these cases, online basis functions can help to reduce the error and obtain an accurate approximation of the fine-scale solution [@Chung:2015:ROG:2837849.2838155]. The use of online basis functions gives a rapid convergence. Next, we will derive a framework for the construction of online multiscale basis functions.
We use the index $m\geq1$ to represent the online enrichment level. At the enrichment level $m$, we use $V_{ms}^m$ to denote the corresponding space-time GMsFEM space and $u_{ms}^m$ the corresponding solution obtained in (\[eq:space-time FEM coarse decoupled\]). The sequence of functions $\{u_{ms}^m\}_{m\geq1}$ will converge to the fine-scale solution. We emphasize that the space $V_{ms}^m$ can contain both offline and online basis functions, and define $V_{ms}^0 = V_{\text{off}}$. We will construct a strategy for getting the space $V_{ms}^{m+1}$ from $V_{ms}^m$.
Next we present a framework for the construction of online basis functions. By online basis functions, we mean basis functions that are computed during the iterative process using the residual. This is the contrary to offline basis functions that are computed before the iterative process. The online basis functions for enrichment level $m+1$ are computed based on some local residuals for the multiscale solution $u_{ms}^m$. Thus, we see that some offline basis functions are necessary for the computations of online basis functions. In our numerical examples from the following section, we will also see how many offline basis functions are needed in order to obtain a rapidly converging sequence of solutions.
For brevity, we denote the left hand side of (\[eq:space-time FEM coarse decoupled\]) by $a(u_{ms}^{(n)},v)$ and the right hand side $F(v)$. That is, the solution $u_{ms} = \oplus_{n=1}^{N}u_{ms}^{(n)}$ where $u_{ms}^{(n)}$ satisfies $$a(u_{ms}^{(n)},v) = F(v), \quad \forall v\in V_{H}^{(n)}.$$ Consider a given coarse neighborhood $\omega_i$. Suppose that at the enrichment level $m$, we need to add an online basis function $\phi\in V_h$ in $\omega_i$. Then the required $\phi= \oplus_{n=1}^{N}\phi^{(n)}$ satisfies that $\phi^{(n)}$ is the solution of $$a(\phi^{(n)},v) = R^{(n)}(v), \quad \forall v\in V_{h},$$ where $R^{(n)}(v)= F(v) - a(u_{ms}^{m(n)},v)$ is the online residual at the coarse time interval $[T_{n-1},T_{n}]$.
In the following, we would like to form a residual based online algorithm in each coarse time interval $[T_{n-1},T_{n}]$, see Algorithm \[online\_algorithm\]. For simplicity, we will omit the time index $(n)$ on the spaces and solutions in this description. We consider enrichment on non-overlapping coarse neighborhoods. Thus, we divide the $\{\omega_i\}_{i=1}^{N_c}$ into $P$ non-overlapping groups and denote each group by $\{\omega_i\}_{i\in I_p}$, $p=1,...,P$. We denote by $M$ the number of online iterations.
**Initialization:** Offline space $V_{ms}^0 = V_{\text{off}}$, offline solution $u_{ms}^0 = u_{\text{H}}$.
\(1) On each $\omega_i (i\in I_p)$, compute residual $R^{m}(v) = a(u_{ms}^{m},v) - F(v),\quad v\in V_{h}$.
\(2) For each $i$, solve $a(\phi_i,v) = R^{m}(v), \quad \forall v\in V_{h}$.
\(3) Set $V_{ms}^m$ = $V_{ms}^m\cup\{\phi_i | i\in I_p\}$. \[online\_adding\]
\(4) Solve for a new $u_{ms}^m \in V_{ms}^m$ satisfying $a(u_{ms}^m,v) = F(v), \quad \forall v\in V_{ms}^m$. Set $V_{ms}^{m+1}$ = $V_{ms}^m$, and $u_{ms}^{m+1}$ = $u_{ms}^m$.
To further improve the convergence and efficiency of the online method, we can adopt an online adaptive procedure. In this adaptive approach, the online enrichment is performed for coarse neighborhoods that have a cumulative residual that is $\theta$ fraction of the total residual. More precisely, assume that the $V^{(n)}$ norm of local residuals on $\{\omega_i|i\in I_p\}$, denoted by $\{r_i|i\in I_p\}$, are arranged so that $$r_{p_1} \geq r_{p_2} \geq r_{p_3} \geq \cdot\cdot\cdot \geq r_{p_J},$$ where we suppose $I_p = \{p_1, p_2, p_3,\cdot\cdot\cdot, p_J\}$. Instead of adding $\{\phi_i | i\in I_p\}$ into $V_{ms}^m$ at step \[online\_adding\] in Algorithm \[online\_algorithm\], we only add the basis $\{\phi_1,\cdot\cdot\cdot,\phi_k\}$ for the corresponding coarse neighborhoods such that $k$ is the smallest integer satisfying $$\Sigma_{i=1}^{k}r_{p_i}^2 \geq \theta\Sigma_{i=1}^{J}r_{p_i}^2.$$ In the examples below, we will see that the proposed adaptive procedure gives a better convergence and is more efficient.
Numerical results. Online GMsFEM {#sect:NR2}
================================
In this section, we present numerical examples to demonstrate the performance of the proposed online method in solving Equation (\[eq:para PDE\]). To implement the space-time online GMsFEM, we will first choose a fixed number of offline basis functions for every coarse neighborhood, and calculate the resulting offline space $V_{\text{off}}$. Then we conduct the online process by following Algorithm \[online\_algorithm\]. In this experiment, we use the same space-time domain and mesh (coarse and fine), the same source term $f$ and initial condition $\beta(x,y)$, the same definitions of relative errors $e_1$ and $e_2$, as in Section \[sect:NR1\]. The permeability field $\kappa(x,t)$ is chosen as the high-contrast permeability field 1 from Section \[sect:media1\]. The buffer number in the computation of snapshot space is chosen to be 8.
First, we implement the space-time online GMsFEM by choosing different numbers of offline basis functions ($L_i = 1,2,3,4,5$) on every coarse neighborhood. The relative errors of online solutions are presented in Table \[L2ErrTable\_online\] and Table \[H1ErrTable\_online\]. Note that in the first column, we show the number of basis functions used for each coarse neighborhood $\omega_i$, and the degrees of freedom (DOF) of multiscale space on each coarse time interval which are the numbers in parentheses, after online enrichment. For example, $2(162)$ in the first column means that after online enrichment, $2$ multiscale basis are used on each $\omega_i$ and the DOF of multiscale space on each coarse time interval is $162$. And if we initially choose $L_i=1$, then it means $1$ online iteration is performed, which add $1$ online basis to each $\omega_i$. If $L_i=2$ initially, then it means we do not perform any online iteration and $2$ multiscale basis are offline basis functions. By observing each column, one can see that the errors decay fast with more online iterations being performed. This is observed for both $e_1$ and $e_2$ when $L_i\geq4$. This suggests that in this specific setting, we can get a fast online convergence with $4$ offline basis chosen on each $\omega_i$. After a small number of online iterations, the relative errors decrease to a significantly small level. We consider reducing the high contrast of the permeability field $\kappa(x,t)$ from $10^6$ to $100$. Then we look at the relative errors of online multiscale solutions (see Table \[L2ErrTable\_online2\] and Table \[H1ErrTable\_online2\]). The same phenomena can be observed except that the fast online convergence rate can be achieved for any choice of $L_i$. This implies that the number of offline basis functions used to guarantee a fast online convergence rate is related to the high contrast of the permeability field.
$DOF$ $e_1$($L_i=1$) $e_1$($L_i=2$) $e_1$($L_i=3$) $e_1$($L_i=4$) $e_1$($L_i=5$)
-------- ---------------- ---------------- ---------------- ---------------- ----------------
1(81) 97.57% - - - -
2(162) 93.20% 96.71% - - -
3(243) 44.24% 23.22% 21.27% - -
4(324) 15.37% 6.53% 7.17e-1% 10.20% -
5(405) 8.65% 3.69% 2.06e-1% 2.58e-1% 5.20%
6(486) 5.15% 1.71% 5.41e-2% 1.75e-2% 1.06e-1%
7(567) 2.58% 3.11e-1% 5.54e-3% 6.12e-4% 2.99e-3%
: Relative online errors $e_1$, with the different numbers of offline basis functions. High contrast = $10^6$.[]{data-label="L2ErrTable_online"}
$DOF$ $e_2$($L_i=1$) $e_2$($L_i=2$) $e_2$($L_i=3$) $e_2$($L_i=4$) $e_2$($L_i=5$)
-------- ---------------- ---------------- ---------------- ---------------- ----------------
1(81) 138% - - - -
2(162) 113% 114% - - -
3(243) 84.93% 139% 104% - -
4(324) 82.48% 82.08% 11.43% 73.50% -
5(405) 69.15% 51.13% 3.29% 4.78% 48.26%
6(486) 51.17% 34.00% 1.01% 3.53e-1% 1.86%
7(567) 37.93% 7.81% 1.05e-1% 9.89e-3% 4.75e-2%
: Relative online errors $e_2$, with the different numbers of offline basis functions. High contrast = $10^6$.[]{data-label="H1ErrTable_online"}
$DOF$ $e_1$(1 basis) $e_1$(2 basis) $e_1$(3 basis) $e_1$(4 basis) $e_1$(5 basis)
-------- ---------------- ---------------- ---------------- ---------------- ----------------
1(81) 19.28% - - - -
2(162) 1.97% 13.03% - - -
3(243) 2.81e-1% 9.81e-1% 9.27% - -
4(324) 3.48e-2% 1.24e-1% 2.23e-1% 8.34% -
5(405) 1.89e-3% 1.11e-2% 9.70e-2% 2.09e-1% 7.38%
6(486) 2.67e-5% 1.33e-4% 2.07e-4% 8.71e-3% 1.56e-1%
7(567) 2.51e-7% 9.32e-7% 1.45e-6% 1.16e-4% 8.62e-3%
: Relative online errors $e_1$, with the different numbers of offline basis functions. High contrast = $100$.[]{data-label="L2ErrTable_online2"}
$DOF$ $e_2$(1 basis) $e_2$(2 basis) $e_2$(3 basis) $e_2$(4 basis) $e_2$(5 basis)
-------- ---------------- ---------------- ---------------- ---------------- ----------------
1(81) 219% - - - -
2(162) 14.75% 123% - - -
3(243) 3.35% 8.37% 81.80% - -
4(324) 4.03e-1% 1.11% 2.63% 67.86% -
5(405) 2.11e-2% 1.01e-1% 1.68e-1% 2.29% 59.93%
6(486) 5.61e-4% 1.64e-3% 3.71e-3% 1.35e-1% 1.77%
7(567) 4.57e-6% 1.72e-5% 2.29e-5% 2.08e-3% 1.41e-1%
: Relative online errors $e_2$ with the different numbers of offline basis functions. High contrast = $100$.[]{data-label="H1ErrTable_online2"}
Next, we perform online adaptive basis construction procedure with $\theta = 0.7$. The numerical results for using $3$, $4$, and $5$ offline basis per coarse neighborhood are shown in Table \[Table:AdapUniMedia\]. Notice that “$M1+M2$” in the DOF columns means $M1$ degrees of freedom are used on the first coarse time interval and $M2$ degrees of freedom on the second coarse time interval. To compare the behaviors of online processes with and without adaptivity, we plot out the log values of $e_2$ against DOFs. See Figure \[Fig:AdapUniMedia\]. We observe that to achieve a certain error, fewer online basis functions are needed with adaptivity. This indicates that the proposed adaptive procedure gives us better convergence and is more efficient.
--------- ---------- --------- ---------- --------- ----------
DOF $e_2$ DOF $e_2$ DOF $e_2$
243+243 104% 324+324 73.50% 405+405 48.26%
323+322 10.57% 399+401 3.56% 471+473 1.95%
403+392 1.49% 468+466 2.13e-1% 533+536 1.21e-1%
480+465 9.81e-2% 541+529 1.03e-2% 599+603 6.81e-3%
552+533 4.24e-3% 611+601 5.00e-4% 670+669 3.41e-4%
--------- ---------- --------- ---------- --------- ----------
: Relative online adaptive errors $e_2$ with different numbers of offline basis functions.[]{data-label="Table:AdapUniMedia"}
![Adaptivity v.s. no adaptivity.[]{data-label="Fig:AdapUniMedia"}](adap3.png){width="\columnwidth"}
![Adaptivity v.s. no adaptivity.[]{data-label="Fig:AdapUniMedia"}](adap4.png){width="\columnwidth"}
![Adaptivity v.s. no adaptivity.[]{data-label="Fig:AdapUniMedia"}](adap5.png){width="\columnwidth"}
Conclusion {#sect:conclusion}
==========
In this paper, we consider the construction of the space-time GMsFEM to solve space-time heterogeneous parabolic equations. The main ingredients of our approach are (1) the construction of space-time snapshot vectors, (2) the local spectral decomposition in the snapshot space. To construct the snapshot vectors, we solve local problems in local space-time domains. A complete snapshot space will consist of the use of all possible boundary and initial conditions. However, this can result to very large computational cost and a high dimensional snapshot space. For this reason, we compute a number of randomized snapshot vectors. In fact, the number of snapshot vectors is slightly larger than that of the multiscale basis functions used in the simulations. To perform local spectral decomposition, we discuss a couple of choices for local eigenvalue problems motivated by the analysis. We present a convergence analysis of the proposed method. Several numerical examples are presented. In particular, we consider examples where the space-time permeability fields have high contrast and these high-conductivity regions move in the space. If only spatial multiscale basis functions are used, it will require a large dimensional space. Thanks to the space-time multiscale space, we can approximate the problem with a fewer degrees of freedom. Our numerical results show that one can obtain accurate solutions. We also discuss online procedures, where new multiscale basis functions are constructed using the residual. These basis functions are computed in each local space-time domain. Using online basis functions adaptively, one can reduce the error substantially at a cost of online computations.
In this paper, our main objective is to develop systematic multiscale model reduction techniques in space-time cells by constructing local (in space-time) multiscale basis functions. The proposed concepts can be used for other applications, where one needs space-time multiscale basis functions.
Appendix
========
Proof of Lemma \[lem: cea lemma\]
---------------------------------
By the definition of $\|\cdot\|_{V^{(n)}},$ $$\begin{aligned}
\|u_{h}-u_{H}\|_{V^{(n)}}^{2} & = & \cfrac{1}{2}\int_{\Omega}(u_{h}-u_{H})^{2}|_{t=T_{n}^{-}}+\cfrac{1}{2}\int_{\Omega}(u_{h}-u_{H})^{2}|_{t=T_{n-1}^{+}}+\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa|\nabla(u_{h}-u_{H})|^{2}\nonumber
\\
& = & \cfrac{1}{2}\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial}{\partial t}(u_{h}-u_{H})^{2}+\int_{\Omega}(u_{h}-u_{H})^{2}|_{t=T_{n-1}^{+}}+\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa|\nabla(u_{h}-u_{H})|^{2}\nonumber \\
& = & \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-u_{H})}{\partial t}(u_{h}-u_{H})+\int_{\Omega}(u_{h}-u_{H})^{2}|_{t=T_{n-1}^{+}} \nonumber \\
& & + \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa|\nabla(u_{h}-u_{H})|^{2}\nonumber\\
& = & \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-u_{H})}{\partial t}(u_{h}-w)
+ \int_{\Omega}(u_{h}-u_{H})(u_{h}-w)|_{t=T_{n-1}^{+}}\nonumber \\
& & + \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa\nabla(u_{h}-u_{H})\cdot\nabla(u_{h}-w)
+ \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-u_{H})}{\partial t}(w-u_{H})\nonumber \\
& & + \int_{\Omega}(u_{h}-u_{H})(w-u_{H})|_{t=T_{n-1}^{+}}
+ \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa\nabla(u_{h}-u_{H})\cdot\nabla(w-u_{H})\label{eq:V norm equality}.\end{aligned}$$ From (\[eq:space-time FEM coarse decoupled\]) and the similar formulation for fine scale solution $u_h$, we have $$\begin{aligned}
& \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-u_{H})}{\partial t}v+\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa\nabla(u_{h}-u_{H})\cdot\nabla v+\int_{\Omega}(u_{h}-u_{H})v|_{t=T_{n-1}^{+}}\nonumber \\
= & \int_{\Omega}\left(g_{h}^{(n)}-g_{H}^{(n)}\right)v(x,T_{n-1}^{+}),\;\forall v\in V_{H}^{(n)}.\label{eq:difference of u_h and u_H}\end{aligned}$$ Therefore, taking $v=w-u_{H}$ and combining the equation (\[eq:V norm equality\]) and (\[eq:difference of u\_h and u\_H\]), we obtain $$\begin{aligned}
\|u_{h}-u_{H}\|_{V^{(n)}}^{2} & = & \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-u_{H})}{\partial t}(u_{h}-w)+\int_{\Omega}(u_{h}-u_{H})(u_{h}-w)|_{t=T_{n-1}^{+}}\\
& & +\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa\nabla(u_{h}-u_{H})\cdot\nabla(u_{h}-w) +\int_{\Omega}\left(g_{h}^{(n)}-g_{H}^{(n)}\right)(w-u_{H})|_{t=T_{n-1}^{+}}.\end{aligned}$$ Using integration by parts, we have $$\begin{aligned}
& \int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-u_{H})}{\partial t}(u_{h}-w)+\int_{\Omega}(u_{h}-u_{H})(u_{h}-w)|_{t=T_{n-1}^{+}}\\
= & -\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-w)}{\partial t}(u_{h}-u_{H})+\int_{\Omega}(u_{h}-u_{H})(u_{h}-w)|_{t=T_{n}^{-}}.\end{aligned}$$ Thus, $$\begin{aligned}
\|u_{h}-u_{H}\|_{V^{(n)}}^{2}
& = & -\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\cfrac{\partial(u_{h}-w)}{\partial t}(u_{h}-u_{H})
+\int_{\Omega}(u_{h}-u_{H})(u_{h}-w)|_{t=T_{n}^{-}}\\
& & +\int_{T_{n-1}}^{T_{n}}\int_{\Omega}\kappa\nabla(u_{h}-u_{H})\cdot\nabla(u_{h}-w)
+\int_{\Omega}\left(g_{h}^{(n)}-g_{H}^{(n)}\right)(u_{h}-u_{H})|_{t=T_{n-1}^{+}}\\
& & +\int_{\Omega}\left(g_{h}^{(n)}-g_{H}^{(n)}\right)(w-u_{h})|_{t=T_{n-1}^{+}}\\
&\leq& C\|\cfrac{\partial(u_{h}-w)}{\partial t}\|_{L^{2}((T_{n-1},T_{n});H^{-1}(\kappa))}
\|u_{h}-u_{H}\|_{L^{2}((T_{n-1},T_{n});\kappa)}\\
& & +\|(u_{h}-u_{H})(\cdot,T_{n}^{-})\|_{L^{2}(\Omega)}\|(u_{h}-w)(\cdot,T_{n}^{-})\|_{L^{2}(\Omega)}\\
& & +\|u_{h}-w\|_{L^{2}((T_{n-1},T_{n});\kappa)}\|u_{h}-u_{H}\|_{L^{2}((T_{n-1},T_{n});\kappa)}\\
& & +\|g_{h}^{(n)}-g_{H}^{(n)}\|_{L^{2}(\Omega)}(\|(u_{h}-u_{H})(\cdot,T_{n-1}^{+})\|_{L^{2}(\Omega)}\\
& & +\|(u_{h}-w)(\cdot,T_{n-1}^{+})\|_{L^{2}(\Omega)}).\end{aligned}$$ Using Young’s inequality, we have $$\|u_{h}-u_{H}\|_{V^{(n)}}^{2} \leq
\cfrac{1}{2}\|u_{h}-u_{H}\|_{V^{(n)}}^{2} + 2\left( C\|u_{h}-w\|_{W^{(n)}}^{2}+\|g_{h}^{(n)}-g_{H}^{(n)}\|_{L^{2}(\Omega)}^{2}\right).$$ and $$\begin{aligned}
\|g_{h}^{(n)}-g_{H}^{(n)}\|_{L^{2}(\Omega)}^{2} & =\begin{cases}
0 & \text{ for }n=1\\
\|u_{h}^{(n-1)}(\cdot,T_{n-1}^{-})-u_{H}^{(n-1)}(\cdot,T_{n-1}^{-})\|_{L^{2}(\Omega)}^{2} & \text{ for }n>1
\end{cases}\\
& \leq\begin{cases}
0 & \text{ for }n=1\\
\|u_{h}-u_{H}\|_{V^{(n-1)}}^{2} & \text{ for }n>1
\end{cases}.\end{aligned}$$ Therefore, we proved the lemma.
[^1]: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR. This research is partially supported by the Hong Kong RGC General Research Fund (Project number: 400411).
[^2]: Department of Mathematics, Texas A&M University, College Station, TX; Numerical Porous Media SRI Center, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia
[^3]: Department of Mathematics, Texas A&M University, College Station, TX.
[^4]: Department of Mathematics, Texas A&M University, College Station, TX.
|
---
abstract: 'We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. Our proof has the significant advantage that it allows for a much stronger conclusion, namely that of a new dimensionless weighted $L^p$ estimate with optimal exponent. Other than previous arguments, only a small part of our proof is based on special auxiliary functions, the core of the argument is a weak type estimate and a sparse decomposition of the stochastic process by X.D. Li, whose projection is the Riesz vector.'
author:
- 'K. Dahmani, K. Domelevo, S. Petermichl'
title: Dimensionless $L^p$ estimates for the Riesz vector on manifolds
---
Introduction
============
In this paper, we are interested in dimensionless weighted and unweighted $L^p$ norm estimates of the Riesz vector on manifold. In the Euclidian setting, the $i$–th Riesz transfrom in ${\mathbb{R}}^n$ is defined as $$R_i = \frac{\partial}{\partial x_i} (-\Delta)^{-1/2},$$ where $\Delta = \sum_{i=1}^n \partial^2 / \partial_{x_i}^2$ is the usual Laplacian in ${\mathbb{R}}^n$. The vector Riesz transform $R$ is defined as the collection $R=(R_1,R_2,\ldots,R_n)$. In the one-dimensional setting, the Riesz transform is nothing but the Hilbert transform. The $L^p$ estimate of the Hilbert transform on the real line dates back to the work of Riesz [@Riesz1927] and Pichorides [@Pic1972]. Regarding the $L^p$ estimate of the Riesz vector in ${\mathbb{R}}^n$, see [@Stein1955; @Meyer1984; @Pisier1988; @BanWan1995; @IwaMar1996; @DraVol2006].
A corner stone in this line of results is the stochastic representation of Riesz transfroms in ${\mathbb{R}}^n$ by Gundy–Varopoulos [@GV1979]. To this end, these authors define the so-called background noise which are Brownian trajectories in the upper half space started at infinity and stopped when hitting the boundary. To a given function $f$ defined on ${\mathbb{R}}^n$ and its Poisson extension in the upper half space, these authors associate a natural martingale $M^f$. They prove that the Riesz transforms can be written as a suitable conditional expectation of martingale transforms of $M^f$. This representation was extended to the Riemannian manifold setting by X.-D. Li [@Li2008; @LiErratum; @LiArXiv] and has thus enabled the first dimensionless estimates with the growth proportional to $(p-1)^{-1}$ when $p>2$ and $p-1$ when $p>2$ in this setting [@CD], [@BO].
For early considerations of $L^p$ boundedness of Riesz transforms on manifolds, we refer to [@SteinTopics]. We mention also the works [@Meyer1984; @Bakry1985; @Gundy1986; @Bakry1987; @Pisier1988; @Arcozzi] among which the papers of Bakry provide estimates of Riesz transforms for complete Riemannian manifolds under the general condition that the Bakry–Emery curvature is bounded below (see [@Emery]). Using stochastic techniques, linear dimensionless estimates of the Bakry–Riesz vector on manifold were announced in [@Li2008] [@LiErratum]. Using deterministic techniques, such estimates were proved in [@CD]. See also [@BanBau2013] for second order Riesz transforms on manifolds and [@BO] for Riesz transforms on manifolds, correcting a previous gap in the probabilistic proof.
In this paper, we will only consider manifolds with non-negative curvature. We will use stochastic tools relying on the stochastic representation of Riesz transforms on manifolds by X.-D. Li [@Li2008; @LiErratum; @LiArXiv].
Our proof is very different from previous ones in that it does not rely on a Bellman function for the problem. Rather, it develops a sparse domination of the stochastic process of Li. See the elegant and short argument in [@Lacey2015] for the first probabilistic object, a discrete time martingale transform and also [@DP2] for the continuous time case. One can deduce, from such domination a dimensionless bound. The sparse operators are particularly well suited for working with weights, which is why this so obtained dimensionless estimate also holds in the weighted setting.
The stochastic process by Li is a specific semi-martingale, built using a pair of martingales that have differential subordination and solving a certain stochastic differential equation. As such, our argument required several new tools. One of them is a weak type estimate of the maximal operator of this process. This is the only part of our proof that uses a (simple) Bellman function. The explicit form of the function is essential and not just its convexity and size properties. The first derivative of said Bellman function is used to control a drift term that arises because the process we consider is not a martingale. Further, we then show that this process has a sparse domination, according to the definition of sparse operator in [@DP2]. The specific form of the defining stochastic equation is used.
The rest of the arguments, to deduce the estimates for the Riesz vector, are considered standard.
Bakry–Riesz transforms on manifolds.
------------------------------------
Let $(M, g)$ be a complete Riemannian manifold with metric $g$ and dimension $n$. Let $\Delta$ be the non-positive Laplace–Beltrami operator, $\nabla$ the gradient operator, and $\nu$ the volume measure on $(M, g)$ such that $d \nu (x) = \sqrt{\det g (x)} d x$. Let moreover $\mu_{\varphi}$ be a weighted volume measure on $M$ with $d \mu_{\varphi} (x) = e^{- \varphi (x)} d
\nu (x)$, where $\varphi (x) \in \mathcal{C}^2 (M)$. The weighted Laplacian $\Delta_{\varphi}$ with respect to $\mu_{\varphi}$ on $M$ is defined for any function $f$ by $$\Delta_{\varphi} f {:=}\Delta - \nabla \varphi \cdot \nabla f.$$
We assume that $\mu_{\varphi} (M) < \infty$ and by normalizing, we may assume without loss of generality that $\mu_{\varphi}$ is actually a probability measure. The Bakry–Emery curvature tensor associated with $\Delta_{\varphi}$ is defined by $${\ensuremath{\operatorname{Ric}}}_{\varphi} = {\ensuremath{\operatorname{Ric}}} + \nabla^2 \varphi,$$ where ${\ensuremath{\operatorname{Ric}}}$ denotes the Ricci curvature tensor on $M$ and $\nabla^2
\varphi$ the Hessian of $\varphi$ with respect to the Levi-Civita connection on $(M, g)$.
All over this paper, we assume a non-negative curvature $${\ensuremath{\operatorname{Ric}}}_{\varphi} \geq 0.$$ We denote by $R$ the Bakry-Riesz transform defined as $$R = d \circ (-\Delta _{\varphi} )^{-1/2}.$$ It was proved by Bakry [@Bakry1987] that for any $p>1$ there exists a universal constant $C_p$ such that for any function $f\in C_0^\infty(M)$, there holds $$\|R f \|_{L^p(\mu)} \leq C_p \| f\|_{L^p(\mu)}.$$
#### Probabilistic setting and notations.
Let $B^M_t$ the diffusion process on $M$ with generator $\Delta_{\varphi}$ obeying $$dB^M_t = U_t dW_t - \nabla\varphi(B^M_t) dt,$$ where $W_t$ is the Brownian motion on ${\mathbb{R}}^n$, $U_t \in {\ensuremath{\operatorname{End}}}(T_x M,T_x M)$ denotes the stochastic parallel transport on $M$ along the trajectory $\{ B^M_s , 0\leq s \leq t\}$. Let further $B_t$ the one-dimensional Brownian motion started at $y>0$ with the normalisation $E [(B^M_t)^2] = 2t$ such that its generator is $\partial^2/\partial y^2$. Following [@Meyer1984; @GV1979; @Gundy1986], there exists a diffusion process $Z_t=(B^M_t,B_t)$ on $(M,{\mathbb{R}}^+)$ – the so-called background radiation process – associated to the generator $\Delta_\varphi + \partial^2/\partial y^2$ and with initial distribution $\mu \otimes \delta_y$.
We recall that the martingale $Y$ is said differentially subordinate to the martingale $X$ if the process $(\langle X,X\rangle_t-\langle Y,Y\rangle _t)_{t\geq0}$ is non-negative and non-decreasing in $t$, where the bracket $\langle \cdot,\cdot \rangle$ denotes the usual quadratic covariance process for real or vector-valued processes.
#### Probabilistic representation of the Bakry-Riesz vector on manifolds.
Using a martingale approach, one can represent the Riesz vector $R$ via a probabilistic representation. In the literature, it first appeared in [@GV1979], where the Riesz transform was defined on $\mathbb{R}^n$. In [@Arcozzi] Arcozzi extended this formula to compact Lie groups and spheres. In [@Li2008], [@LiErratum] Li presented a new formula adapted to complete Riemannian manifolds. The representation formula of the Riesz vector in this setting for a complete manifold with $\operatorname{Ric}_{\varphi}\geq 0$ is as follows
$$\label{rep}
-\frac{1}{2}(R f)(x)=\lim_{y\rightarrow \infty} \mathbb{E}_y\left[M_{\tau}\int_0^{\tau} M_s^{-1} dQ(f)(B^M_s,B_s)dB_s | B^M_{\tau}=x \right],$$
where
- $Q(f)(x,y)=e^{-y\sqrt{-\Delta_{\varphi}}}f(x)$ is the Poisson semigroup;
- $\tau = \inf \{t>0:B_t=0\}$ is the stopping time upon hitting the boundary of the upper half space;
- $M_t$ is the solution to the matrix-valued stochastic differential equation $$dM_t=V_tM_tdt, \ \ \ M_0=Id,$$ for some adapted and continuous process $(V_t)_{t \geq 0}$ taking values in the set of symmetric and non-positive $d \times d$ matrices.
Equivalently, one can rewrite this fomula as $$-\frac{1}{2}(R f)(x)=\lim_{y\rightarrow \infty} \mathbb{E}_y\left[ Z_\tau |B^M_{\tau}=x \right],$$ where $Z_t$ is a semi-martingale defined thanks to the auxiliary martingales $X_t$ and $Y_t$ as follows $$X_t = Qf(B^M_t,B_t) = Qf(B^M_0,y ) + \int_0^t (\nabla,\partial_y) Qf(B^M_s,B_s) d(U_s dW_s, B_s),$$
$$Y_t = \int_0^t \nabla Qf(B^M_s,B_s) dB_s,$$
$$Z_t = M_t \int_0^t M_s^{-1} dY_s,$$ where $Y_t$ is by construction differentially subordinate to $X_t$.
Main results
------------
We prove in Theorem \[unweightedRiesz\] a dimensionless estimate in $L^p$ spaces for the Riesz vector on manifold with non-negative curvature. The first proofs of this result are recent [@CD], [@LiErratum], [@BO] and all based on a form of a Bellman function. Our proof is via a sparse domination with continuous index. All these cited Bellman proofs give a better numeric estimate than our proof, but as mentioned earlier, our proof extends (for free) to the weighted case, which the previous ones do not. Our estimate is linear in $p$, which means proportional to $(p-1)^{-1}$ when $p<2$ and to $p-1$ when $p>2$. We note that [@BO] have the best numeric constant in this case. We note also that the proof in [@CD] gives the linear estimate with $p$ also in the case where the curvature is merely bounded below (and possibly negative) with an appropriately defined Riesz vector involving a Laplacian with a modified spectrum. We do not pursue this here, although parts of our arguments clearly go through also in this case.
\[unweightedRiesz\] Suppose that $M$ is a complete Riemannian manifold without boundary and $\operatorname{Ric}_{\varphi}\geq0$. Then for all $f \in C_c^{\infty}(M)$, $p\in (1,\infty)$, we have the following dimension-free estimate
$$\label{E1}
\| R f \|_{L^p(T^*_xM)} \leq 32 \dfrac{p^2}{p-1}\|f\|_{L^p(M)}.$$
We prove also a dimensionless weighted estimate in $L^p$ spaces for the Riesz vector on manifold with non-negative curvature. In the Euclidean setting, see [@DPW]. For the case of manifolds, such an estimate was previously only known in the case $p=2$ see [@D]. A priori the weight has to be globally in $L^2$ so as to be able to define the flow characteristic. $$\tilde{Q}_p(w)=\sup_{x,y}(Q(w))(x,y)(Q(w^{-\frac1{p-1}}))^{p-1}(x,y).$$ The collection of weights for which this characteristic is finite is denoted $\tilde{A}_p$. There is also a natural way to extend the class of the weights to resemble more the classical case allowing local $L^1$ weights. In this case we require that constants are integrable in $M$ with the measure $d\mu_{\varphi}$ so as to prove the theorem for cut weights, such as in [@D], that are in $L^1 \cap L^{\infty} \cap L^2$ and then define the characteristic by a limiting procedure and deduce the theorem. See [@D] for detailed exposition in the case $p=2$.
\[Riesz\] Suppose that $M$ is a complete Riemannian manifold without boundary and $\operatorname{Ric}_{\varphi}\geq0$. Then for all $f \in C_c^{\infty}(M)$, $p\in (1,\infty)$ and $ w \in \widetilde{A}_p$, we have the following dimension-free estimate
$$\label{E1w}
\| R f \|_{L^p(T^*_xM, w)} \leq 32 \dfrac{p^2}{p-1}\widetilde{Q}_p(w)^{\max(1,\frac{1}{p-1})}\|f\|_{L^p(M, w)}.$$
The technique used in this paper resembles the sparse domination principle for discrete time martingale transforms which originally appeared in [@Lacey2015]. This technique has witnessed considerable efforts in the last several years and has been used to prove numerous new results in harmonic analysis, using sparse operators defined on cubes. These cannot give dimensionless estimates, nor are satisfactory results known in the non-doubling case. As in [@DP2] we use a sparse operator with continuous stopping times, dominating Li’s process $Z_t$ whose projection is the Riesz vector. This is what enables us to use the flow itself without cutting it into cubes, thus resulting in clean dimensionless estimates.
Following [@DP2], we say that the operator $X \mapsto S(X)$ is called sparse if there exists an increasing sequence of adapted stopping times $0=T^{-1}\leq T^0 \leq \cdots $ with nested sets $E_j = \{T^j < \infty\}$, $E_j \subset E_{j-1}$ so that $$\label{sparse1}
S(X) = \sum_{j=-1}^{\infty}X_{T^j} \chi_{E_j} {\text{ where }} X_{T^j} = \mathbb{E}(X|\mathcal{F}_{T^j} );$$ $$\label{sparse2}
\forall A_j \subset E_j, \ A_j \in \mathcal{F}_{T_j} {\text{ there holds }} \mathbb{P}(A_j \cap E_{j+1}) \leq \dfrac{1}{2}\mathbb{P}(A_j).$$
The estimate we aim to show will be a consequence of a sparse domination of the stochastic process $Z_t$ (see \[NL,L,DP\]). Other than in [@DP2] the object is not a martingale, so the sparse domination is different and the key of the proof relies on the weak-$L^1$ estimate for the maximal function of the studied stochastic operator. We do not aim at the fullest generality here, keeping our goal in mind, an estimate of the Riesz vector. Certain assumptions can certainly be weakened, as the attentive reader will observe.
\[L: weak type\] Let $X$ be a real valued continuous path martingale and $Y$ a vector valued continuous path martingale so that $Y$ is differentially subordinate with respect to $X$. Let further $Z$ a continuous path semi-martingale whose increments satisfy $dZ_t=V_tZ_tdt +dY_t$ with $V_t$ continuous adapted process with values in non-positive, symmetric $d \times d$ matrices. Let $\lambda >0$. We have $$\mathbb{P}\left( (|Z_t|+|X_t|)^* \geq \lambda \right) \leq 2 \lambda ^{-1} \|X\|_1.$$
\[T: sparse decomposition\] Let $X$ be a real valued non-negative continuous path martingale and $Y$ a vector valued continuous path martingale so that $Y$ is differentially subordinate with respect to $X$. Let further $Z$ a continuous path semi-martingale whose increments satisfy $dZ_t=V_tZ_tdt +dY_t$ with $V_t$ continuous adapted process with values in non-positive, symmetric $d \times d$ matrices. Then there exists a sparse domination such that $$Z^* \leq 8S(X).$$
where we recall that we denote by $Z^*=\sup_{t\geq 0}|Z_t|$ the maximal function associated with $Z$.
\[T: weighted estimate for Z\] Let $X$ be a real valued non-negative continuous path martingale and $Y$ a vector valued continuous path martingale so that $Y$ is differentially subordinate with respect to $X$. Let further $Z$ a continuous path semi-martingale whose increments satisfy $dZ_t=V_tZ_tdt +dY_t$ with $V_t$ continuous adapted process with values in non-positive, symmetric $d \times d$ matrices. Then there holds the weighted estimate $$\|Z^{*}\|_{L^p(w)} \lesssim \Phi_p(Q^{\mathcal{F}}_p(w)) \|X\|_{L^p(w)},$$ where $\Phi_p(x)=x^{\max\{1, \frac1{p-1}\}}$.
In general for filtered spaces, the $A_p$ characteristic of $w$ (identified with its closure) is $$Q_p^{\mathcal{F}}(w)=\sup_{\tau}\| \mathbb{E} ( (\frac{w_{\tau}}{w})^{\frac1{p-1}} \mid \mathcal{F}_{\tau} )^{p-1} \|_{\infty}.$$ In the case of interest to us, the characteristic that appears is the one that corresponds to the filtration used by Li at height $y$, denoted $\mathcal{F}^{(y)}$. It can be seen, similarly as is known to the Euclidean case, that these characteristic, in a limiting sense, is comparable to the Poisson flow characteristic.
The stochastic process $Z$
==========================
In this section, we prove Lemma \[L: weak type\] and Theorem \[T: sparse decomposition\].
(of Lemma \[L: weak type\]).
This proof is modelled after the exposition in Wang [@Wang]. We aim to show $$\label{weak}
\mathbb{P}\left( (|Z_t|+|X_t|)^* \geq \lambda \right) \leq 2 \lambda ^{-1} \|X\|_1.$$ Indeed, it suffices to show the inequality for $\lambda =1$. To do this, define functions $V,U:\mathbb{R}\times \mathbb{R}^n \to \mathbb{R}$ by $$V(x,y) = \left\{
\begin{array}{ll}
-2|x| & \mbox{when } |x|+|y|<1, \\
1-2|x| & \mbox{when } |x|+|y| \geq 1.
\end{array}
\right.$$
$$U(x,y) = \left\{
\begin{array}{ll}
|y|^2-|x|^2 & \mbox{when } |x|+|y|<1, \\
1-2|x| & \mbox{when } |x|+|y| \geq 1.
\end{array}
\right.$$ Let us first observe that everywhere $V\le U$. Define the stopping time $$T=\inf \{t \geq 0: |X_t|+|Z_t| \geq 1\}.$$ Then $ |X_T|+|Z_T| \geq 1$ and $ |X_t|+|Z_t| < 1$ for $t<T$.\
We aim to prove that $\mathbb{E}U(X_T,Z_T) \leq 0$, since $V\le U$ the result will follow (see the end of the argument, where we detail the step). We split $$\mathbb{E}U(X_T,Z_T) = \mathbb{E}(U(X_T,Z_T)\chi_{\{T>0\}})+\mathbb{E}(U(X_T,Z_T)\chi_{\{T=0\}})$$ and we show that these contributions are both non-positive.\
\
**Part 1:** $\{T=0\}$.\
For such $\omega$ where $T=0$ then by definition of $T$ we have $|X_0|+|Z_0| \geq 1$ and $U(X_0,Z_0)=1-2|X_0|$. Assuming that $|Z_0| \leq |X_0|$, then $$1 \leq |X_0|+|Z_0| \leq 2 |X_0|,$$ i.e. $1-2|X_0| \leq 0$ and hence $$\mathbb{E}(U(X_T,Z_T)\chi_{\{T=0\}})=\mathbb{E}(U(X_0,Z_0)\chi_{\{T=0\}}) \leq 0.$$\
**Part 2:** $\{T>0\}$.\
By simple calculations on the derivatives of $U$ we check that $$\begin{aligned}
\partial_{y_i} U(x,y) &=& 2y_i \label{signU}\\
\partial^2_{xx} U(x,y)&=& -2 \label{signU2},\\
\partial^2_{xy_j} U(x,y)&=& 0,\\
\partial^2_{y_iy_j} U(x,y)&=& 2\delta_{ij} \label{signU3},\end{aligned}$$ for $|x|+|y| < 1$ and where $\delta_{ij}$ is the Kronecker delta.\
On $\{T >0\}$, the process evolves in the set $\{(x,y): \ |x|+|y|<1\}$, in the interior of which the function $U$ is twice differentiable, which means that we have the following Itô formula $$U(X_T,Z_T)=U(X_0,Z_0)+I_1+\frac{1}{2}I_2,$$ with $$\begin{aligned}
I_1 &=& \int_0^T \partial_x U(X_s,Z_s) dX_s + \sum_i \int_0^T \partial_{y_i} U(X_s,Z^{i}_s), dZ^{i}_s \\
I_2 &=& \int_0^T \partial^2_{xx} U(X_s,Z_s), d\langle X,X \rangle _s + 2\sum_i \int_0^T\partial^2_{xy_i} U(X_s,Z^{i}_s), d\langle X, Z^{i} \rangle _s \\
&& + \sum_i\sum_j \int_0^T \partial^2_{y_iy_j} U(X_s,Z_s), d\langle Z^{i},Z^{j} \rangle _s .\end{aligned}$$ Let’s first study $I_1$:\
Recall that $Z_t$ satisfies the following stochastic differential equation $$\label{dZ}
dZ_t=V_tZ_tdt+dY_t.$$ Now if we replace this formula in the expression of $I_1$, we will obtain a local martingale part which is $$\int_0^T \partial_x U(X_s,Z_s) dX_s + \int_0^T\langle \partial_y U(X_s,Z_s), dY_s\rangle$$ and a process $$A_T = \int_0^T\langle \partial_y U(X_s,Z_s), V_sZ_s\rangle ds.$$ We may assume that the local martingale is a true martingale without loss of generality and hence its expectation is null. As for the process $A_T$, by (\[signU\]) we have $$A_T=2\int_0^T \langle Z_s,V_sZ_s\rangle ds\le 0.$$ The non-positivity holds because the integrand is non-positive as well, since $V$ takes values in the class of non-positive matrices. Notice that just like in [@D], the form of the partial derivative of $U$ in the variable $y$ is crucial.\
\
Now we deal with $I_2$:\
By the formulas (\[signU2\])-(\[signU3\]), we obtain that $$\frac12 I_2 = ( \langle Z,Z \rangle_T -|Z_0|^2- \langle X,X \rangle_T + |X_0|^2 )\chi_{\{T>0\}},$$ and hence it suffices to prove $$\label{wang}
( \langle Z,Z \rangle_T -|Z_0|^2- \langle X,X \rangle_T + |X_0|^2 )\chi_{\{T>0\}}\leq 0,$$ for any stopping time $T$. Recall that for all $t$ we have $dZ_t=V_tZ_tdt+dY_t$. Thus by integrating we have, $$Z_t - Z_0 =\int_0^tV_sZ_sds + Y_t - Y_0.$$ Taking the quadratic covariance on both sides we obtain $$\begin{aligned}
\langle Z,Z \rangle_t - |Z_0|^2 &=& \langle Y,Y \rangle_t - |Y_0|^2, \ \ \ \forall t\geq 0 \\
& \leq & \langle X,X \rangle_t - |X_0|^2 \ \text{ by differential subordination}\end{aligned}$$ which in turn implies that $\mathbb{E}(I_2) \leq 0$.\
Finally, $U(X_0,Z_0)=Z_0^2-X^2_0 \leq 0$.
It remains to show the weak estimate (\[weak\]):\
We have $V \le U$ everywhere and $\mathbb{E}U(X_T,Z_T)\le 0$. Therefore $$\begin{aligned}
0 & \geq & \mathbb{E}U(X_T,Z_T)\\
& \geq & \mathbb{E}V(X_T,Z_T)\\
& = & \mathbb{E}(V(X_T,Z_T)\chi_{\{|X_T|+|Z_T| \geq 1\}}) + \mathbb{E}(V(X_T,Z_T)\chi_{\{|X_T|+|Z_T| < 1\}}) \\
& = & \mathbb{E}(-2|X_T|\chi_{\{|X_T|+|Z_T| \geq 1\}}) + \mathbb{E}((1-2|X_T|)\chi_{\{|X_T|+|Z_T| < 1\}}) \\
& = & \mathbb{P}( |X_T|+|Z_T| \geq 1 ) -2 \mathbb{E}|X_T|,\end{aligned}$$ from which we deduce $$\mathbb{P}( (|X_t|+|Z_t|)^{*} \geq 1 )\le 2\| X \|_1$$ and so the lemma is proved.\
(of Theorem \[T: sparse decomposition\]).
Now that we have a weak type result by Lemma \[L: weak type\], we are able to use a sparse argument as in [@DP2]. Recall for convenience we assumed $X$ non-negative.\
Consider the processes $Z_t^0=\dfrac{Z_t}{X_0}$ and $Y_t^0=\dfrac{Y_t}{X_0}$ and $X_t^0=\dfrac{X_t}{X_0}$. Applying the result obtained in the first step, we know that the measure of the set $$E_0=\{ \omega \in \Omega : \max \{ Z^{0*}(\omega), X^{0*}(\omega)\} >4\}$$ is small. Indeed, $$|E_0| \leq \dfrac{2}{4} \|X^0\|_1\le \frac12.$$ We can associate $T^{-1}=0$ and a stopping time $$T^0(\omega)=\inf\{t>0:\max \{ |Z_t^{0}(\omega)|, X_t^{0}(\omega)\} >4\}$$ as the hitting time of the set $L=(4,\infty)$, which is finite in $E_0$, almost surely, by definition.\
The key of the proof, besides the weak type estimate, relies on recursivity in order to construct a sparse operator.\
To start, let us suppose we have chosen an increasing stopping time sequence $T^k$. Set for times $t\in I_k(\omega)=[T^{k-1}(\omega),T^k(\omega)[$ and let us recall that on $I_k$ $$Z_t=Z_{T^{k-1}}+\int_{T^{k-1}}^{t}dZ_s$$ so that for all times $$\label{Zsum}
Z_t=\sum_{k=0}^{\infty} Z_t \chi_{t\in I_k}=\sum_{k=0}^{\infty} (Z^{(k)}_t+(Z_t-Z^{(k)}_t)) \chi_{t\in I_k}$$ with the $Z^{(k)}_t$ constructed below from $Z_t$ for times in $I_k$ by changing the foot. In order to be more precise, let us first define the martingales $X^{(k)}_t$ and $Y^{(k)}_t$. $${\text{For }} k>0: \; X^{(k)}_t=X_{T^{k-1}}+\int_{T^{k-1}}^{\max\{T^{k-1},t\}}dX_s.$$ and $$Y^{(0)}_t=Y_0+\int_0^{t} dY_s {\text{ and if }} k>0: \; Y^{(k)}_t=\int_{T^{k-1}}^{\max\{T^{k-1},t\}}dY_s.$$ Observe that these are martingales in $\mathcal{F}$ for all $k$ and that $Y^{(k)}$ differentially subordinate to $X^{(k)}$. Notice that the processes $$X^k_t=\frac{X^{(k)}_t}{X_{T^{k-1}}} {\text{ and }} Y^k_t=\frac{Y^{(k)}_t}{X_{T^{k-1}}},$$ are also adapted in $\mathcal{F}=(\mathcal{F}_t)_{t\ge 0}$ since at times $t<T^{k-1}$ these processes are constant and hence adapted and at later times the denominator is measurable. Notice that the event $\{T^{k-1}<t\} \in \mathcal{F}_t$ since $T^{k-1}$ is a stopping time.
Now set $$Z^{(0)}_t=Z_0+\int_0^{t}dZ_s$$ and let for $k>0$ $Z^{(k)}_t$ be the process satisfying $Z^{(k)}_t=0$ if $t \le T^{k-1}$ and evolving for $t>T^{k-1}$ according to $$dZ^{(k)}_t=V_tZ^{(k)}_t dt + dY_t$$ with initial condition at time $T^{k-1}$ be set 0. Notice that the so defined process $Z^{(k)}_t$ is adapted to $(\mathcal{F}_t)_{t\ge 0}$ and solves $dZ^{(k)}_t=V_tZ^{(k)}_t dt + dY^{(k)}_t$ for all times with zero increments for $t<T^{k-1}$. For times $t\ge T^{k-1}$ we know that $W^{(k)}_t=(Z-Z^{(k)})_t$ solves the homogenous equation $$dW^{(k)}_t = V_tW^{(k)}_t dt$$ with initial condition $W^{(k)}_{T^{k-1}}=Z_{T^{k-1}}$. Now observe that $$d\langle W^{(k)}_t,W^{(k)}_t\rangle = 2\langle d W^{(k)}_t, W^{(k)}_t \rangle = \langle V_t W^{(k)}_t,W^{(k)}_t \rangle \le 0$$ because $V_t$ takes values in the non-positive matrices. So we have for $t\ge T^{k-1}$ that $$| Z_t-Z^{(k)}_t |= | W^{(k)}_t | \le |W^{(k)}_{T^{k-1}} | = |Z_{T^{k-1}} |$$ which will give us a control on the error term we induced in the sum (\[Zsum\]). Using similar arguments as above, we can consider $Z^k_t=\frac{Z^{(k)}_t}{X_{T^{k-1}}}$ and retain these properties, now with respect to martingales $X^k$ and $Y^k$.
We now explain how to choose the sequence of stopping times. Set $$E_k=\{ \omega \in E_{k-1} : \max \{ Z^{k*}(\omega), X^{k*}(\omega)\} >4\}$$ and its associated stopping time $T^k$ of hitting time. By the above, we know that processes $X^k$, $Y^k$ and $Z^k$ satisfy the assumptions of the weak type estimate and we thus control $|E_k|\le \frac12 \|X^k\chiÑ{E_{k-1}}\|_1\le \frac12|E_{k-1}|$. The second technical assumption of sparse operator in this setting can be shown similarly (see [@DP2]). By standard arguments we obtain the pointwise domination
$$\begin{aligned}
Z^*(\omega) &\leq & 8 \sum_{j=-1}^{\infty}X_{T^j}(\omega) \chi_{E_j}(\omega) \\
&=& 8 S(X)(\omega),\end{aligned}$$
by considering $E_{-1}=\Omega $.
(of Theorem \[T: weighted estimate for Z\]).
This follows from the sparse domination and the corresponding estimate for the sparse operator, see [@DP2].
The Riesz vector
================
The proof of the main result now follows standard arguments.
Following Li [@Li2008], recall that $$X_t = Qf(X_t,B_t)-Qf(X_0,y).$$ By taking the probabilistic representation of the Riesz transform (\[rep\]), one can write $$\begin{aligned}
\|R f\|_{L^p(w)}^p &\leq & \lim_{y \rightarrow \infty} 2^p \|Z_{\tau}\|_{L^p(w)}^p \\
& \leq & \lim_{y \rightarrow \infty} 2^p \|Z^*\|_{L^p(w)}^p \\
& \leq & \lim_{y \rightarrow \infty} (32 \dfrac{p^2}{p-1})^p \Phi_p(Q^{\mathcal{F}^{(y)}}_p(w)) \| X\|_{L^p(w)}^p \\
& \leq & \lim_{y \rightarrow \infty}(32 \dfrac{p^2}{p-1})^p \Phi_p(Q^{\mathcal{F}^{(y)}}_p(w))\left( \|Qf(B^M_{\tau},B_{\tau})\|_{L^p(w)}^p + \|Qf(B^M_0,y)\|_{L^p(w)}^p \right) \\
& \leq & (32 \dfrac{p^2}{p-1})^p \Phi_p(\tilde{Q}_p(w)) \|f(B^M_{\tau})\|_{L^p(w)}^p \\
& \leq & (32 \dfrac{p^2}{p-1})^p \Phi_p(\tilde{Q}_p(w)) \|f\|_{L^p(w)}^p ,\end{aligned}$$ Notice that sparse domination itself depends upon the used filtration (and hence $y$). Here the norm $\|X\|_{L^p(w)}$ is at $t=\infty$, which is $\tau$ in our stopped processes. We use that $\|Qf(B^M_0,y)\|_{\infty} \to 0$ as $y \to \infty$ and $w \in L^1$ by assumption. $Q^{\mathcal{F}^{(y)}}_p(w)$ is the $A_p$ characteristic that corresponds to the filtration when $B_0=y$ and $ \tilde{Q}_p(w)$ is the Poisson flow characteristic.
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abstract: |
We study the set of $k$-abelian critical exponents of all Sturmian words. It has been proven that in the case $k = 1$ this set coincides with the Lagrange spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense non-closed set. This is in contrast with the case $k = 1$, where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra. Our geometric approach allows the study of $k$-abelian powers in Sturmian words by means of continued fractions.
[: Sturmian word, $k$-abelian equivalence, Lagrange spectrum, continued fraction]{}
author:
- 'Jarkko Peltomäki[^1]'
- 'Markus A. Whiteland'
title: 'On $k$-abelian Equivalence and Generalized Lagrange Spectra'
---
Introduction
============
The critical exponent of an infinite word ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ is the supremum of exponents of fractional powers occurring in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. Famously Thue showed in 1906 [@1906:uber_unendliche_zeichenreihen] that the fixed point of the substitution $0 \mapsto 01$, $1 \mapsto 10$, now known as the Thue-Morse word for Morse’s independent contribution [@1921:recurrent_geodesics_on_a_surface_of_negative_curvature], has critical exponent $2$ meaning that it avoids powers with exponent at least $3$. The notion of critical exponent is central in the study of powers and their avoidance which have since Thue been a central theme in the area of combinatorics on words.
Another important subject in combinatorics on words is the theory of Sturmian words. Sturmian words comprise a large class of extensively studied words with strong connections to number theory, particularly to continued fractions (see, e.g., [@2007:sturmian_and_episturmian_words], [@2002:algebraic_combinatorics_on_words Chapter 2], [@2002:substitutions_in_dynamics_arithmetics_and_combinatorics Chapter 6] and the references therein). The powers occurring in Sturmian words are well-understood, and a formula for the critical exponent of a Sturmian word was determined by Damanik and Lenz [@2002:the_index_of_sturmian_sequences] and Justin and Pirillo [@2001:fractional_powers_in_sturmian_words]. For example, the critical exponent of the Fibonacci word, the fixed point of the substitution $0 \mapsto 01$, $1 \mapsto 0$, is $(5 + \sqrt{5})/2$ as was already derived in [@1992:repetitions_in_the_fibonacci_infinite_word]. The critical exponent of the Fibonacci word is minimal among all Sturmian words.
In recent years, there has been a substantial amount of research in generalizations of the concept of a power. A popular generalization is that of an abelian power; other generalizations are $k$-abelian powers (see below) and those based on $k$-binomial equivalence [@2015:another_generalization_of_abelian_equivalence_binomial]. Two words $u$ and $v$ are abelian equivalent, written $u \sim_1 v$, if one is obtained from the other by permuting letters. If $u_0$, $u_1$, $\ldots$, $u_{n-1}$ are abelian equivalent words of length $m$, then their concatenation $u_0 u_1 \cdots u_{n-1}$ is an abelian power of exponent $n$ and period $m$ (only integer exponents are considered). Thus an abelian power is a generalization of the usual notion of a power: the abelian equality relation is used in place of the usual equality relation. Questions regarding abelian powers were already raised by Erd[ő]{}s in 1957 [@1957:some_unsolved_problems]. More recently there has been a burst of activity on the subject starting, perhaps, with the 2011 paper [@2011:abelian_complexity_of_minimal_subshifts] by Richomme, Saari, and Zamboni. See, e.g., the references of [@2016:abelian_powers_and_repetitions_in_sturmian_words] and especially the papers [@2013:abelian_returns_in_sturmian_words; @2013:some_properties_of_abelian_return_words; @2016:abelian_powers_and_repetitions_in_sturmian_words; @2017:abelian-square-rich_words] related to Sturmian words.
The first author studied with Fici et al. the abelian critical exponents of Sturmian words in [@2016:abelian_powers_and_repetitions_in_sturmian_words], where it was shown that there are abelian powers of arbitrarily high exponent starting at each position of a Sturmian word, a result also obtained in [@2011:abelian_complexity_of_minimal_subshifts]. This means that directly generalizing the notion of a critical exponent to the abelian setting only in terms of the exponent does not produce a quantity of interest (at least for Sturmian words). Thus an alternative definition was adopted in [@2016:abelian_powers_and_repetitions_in_sturmian_words]. The abelian critical exponent of an infinite word ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ is defined as the quantity $$\label{eq:informal}
\limsup_{m \to \infty} \left\{ \frac{n}{m} : \text{$u$ is an abelian power of exponent $n$ and period $m$ occurring in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$} \right\}$$ measuring the maximal ratio between the exponents and periods of abelian powers in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. This alternative definition does lead to an interesting quantity. The abelian critical exponent of a Sturmian word can be finite or infinite, and again the Fibonacci word has minimal exponent; this time the value being $\sqrt{5}$.
Surprisingly, the set of abelian critical exponents of all Sturmian words turns out to coincide with the Lagrange spectrum. The Lagrange constant of an irrational $\alpha$ is the infimum of the real numbers $\lambda$ such that for every $c > \lambda$ the inequality ${\lvert\alpha - n/m\rvert} < 1/cm^2$ has only finitely many rational solutions $n/m$. The Lagrange constant $\lambda(\alpha)$ of $\alpha$ is computed as follows: $$\begin{aligned}
\lambda(\alpha) &= \limsup_{t \to \infty} (q_t {\lVertq_t\alpha\rVert})^{-1} \\
&= \limsup_{t \to \infty} ([a_{t+1}; a_{t+2}, \ldots] + [0; a_t, a_{t-1}, \ldots, a_1]),
\end{aligned}$$ where $[a_0; a_1, a_2, \ldots]$ is the continued fraction expansion of $\alpha$ and $(q_k)$ is the sequence of denominators of its convergents (${\lVertx\rVert}$ measures the distance of $x$ to the nearest integer). The connection here is that for a fixed Sturmian word, the number $n$ in , when maximal, equals the integer part of $1/{\lVertm\alpha\rVert}$ for a certain irrational $\alpha$ (for details, see and ).
The Lagrange spectrum is the set of finite Lagrange constants of irrational numbers. The Lagrange spectrum has been studied extensively, but many of its properties still remain a mystery. The spectrum has a curious structure: its initial part inside the interval $[\sqrt{5}, 3)$ is discrete as shown by Markov already in late 19th century [@1879:sur_les_formes_quadratiques_binaires_indefinies; @1880:sur_les_formes_quadratiques_binaires_indefinies_ii], but it contains a half-line as was famously proven by Hall in 1947 [@1947:on_the_sum_and_products_of_continued_fractions]. Good sources for information on the Lagrange spectrum are the monograph of Cusick and Flahive [@1989:the_markoff_and_lagrange_spectra] and Aigner’s book [@2013:markovs_theorem_and_100_years_of_the_uniqueness].
Another relatively recent development in combinatorics on words is the systematic study of a generalization of abelian equivalence called $k$-abelian equivalence initiated by Karhumäki, Saarela, and Zamboni in [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite]. This generalization originally appears in a 1980 paper of Karhumäki [@1980:generalized_parikh_mappings_and_homomorphisms]. Two words $u$ and $v$ are said to be $k$-abelian equivalent, written $u \sim_k v$, if ${\lvertu\rvert}_w = {\lvertv\rvert}_w$ for each nonempty word $w$ of length at most $k$ (here ${\lvertu\rvert}_w$ stands for the number of occurrences of $w$ as a factor of $u$). Thus $1$-abelian equivalence is simply the abelian equivalence discussed above. The $k$-abelian equivalence relation is clearly an equivalence relation, but it is also a congruence relation. For $k = 1, 2, \ldots$, the corresponding $k$-abelian equivalence relations can be seen as refinements of the abelian equivalence relation approaching the usual equality relation. The $k$-abelian equivalence has been studied especially from the points of view of factor complexity and power avoidance; for more information, see the recent paper [@2017:on_growth_and_fluctuation_of_k_abelian_complexity] and its references.
The purpose of the current paper is to generalize the research of [@2016:abelian_powers_and_repetitions_in_sturmian_words] on abelian critical exponents of Sturmian words to the $k$-abelian setting. That is, we use the general $k$-abelian equivalence in place of abelian equivalence to obtain the notion of $k$-abelian critical exponent and study the set ${\mathcal{L}_{k}}$ of $k$-abelian critical exponents of Sturmian words. As ${\mathcal{L}_{1}}$ is the Lagrange spectrum, the sets ${\mathcal{L}_{k}}$ for $k > 1$ can be seen as combinatorial generalizations of the Lagrange spectrum.
Our main contribution is the characterization of the $k$-Lagrange spectrum ${\mathcal{L}_{k}}$ in terms of the Lagrange spectrum ${\mathcal{L}_{1}}$. Our result, , states that the $k$-abelian critical exponent of a Sturmian word ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$ with abelian critical exponent $K$ equals $cK$ for a particular constant $c$, $0 < c < 1$, which depends on $k$ and ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$. The relation between ${\mathcal{L}_{1}}$ and ${\mathcal{L}_{k}}$ is thus quite simple. However, the sets ${\mathcal{L}_{k}}$ inherit the complicated structure of the Lagrange spectrum ${\mathcal{L}_{1}}$. We show that for $k > 1$ we have ${\mathcal{L}_{k}} \subseteq (\sqrt{5}/(2k-1), \infty)$, the number $\sqrt{5}/(2k-1)$ being the least accumulation point of ${\mathcal{L}_{k}}$ (). Moreover, we prove that the set ${\mathcal{L}_{k}}$ is dense in $(\sqrt{5}/(2k-1), \infty)$ (). This contrasts the case $k = 1$ where the initial part of ${\mathcal{L}_{1}}$ is discrete. The set ${\mathcal{L}_{1}}$ is known to contain a half-line. We do not know if ${\mathcal{L}_{k}}$ contains an analogous half-line for $k > 1$; we leave this problem open.
Our approach is to first give an arithmetical and geometric interpretation for what it means for two factors of a Sturmian word to be $k$-abelian equivalent and then to employ continued fractions to derive our results. This approach is similar to that of [@2016:abelian_powers_and_repetitions_in_sturmian_words] where the usage of continued fractions was crucial. The arithmetical interpretation complements the combinatorial methods of [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite]: we make some results of [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite] on Sturmian words more precise. Our approach also makes it possible to efficiently find the possible exponents and locations of $k$-abelian powers occurring in a given Sturmian word.
The paper is organized as follows. In , we give the necessary definitions and background information on Sturmian words and number theory. After this we present the main results and their proofs in . provides further discussion on some matters raised in . Finally, concludes the paper with open problems.
Preliminaries {#sec:preliminaries}
=============
We use standard terminology from combinatorics on words; we refer the reader to [@2002:algebraic_combinatorics_on_words] for any undefined terms. The words considered in this paper are finite or infinite binary words over the alphabet $\{0, 1\}$. We distinguish infinite words from finite words by referring to them with boldface symbols. By ${\lvertw\rvert}$ we mean the length of the finite word $w$. The $n^\text{th}$ *power* of a finite word $w$ is the word obtained by repeating it consecutively $n$ times, and it is denoted by $w^n$. For the infinite repetition of $w$, we use the notation $w^\omega$. An infinite word is *ultimately periodic* if it can be written in the form $uv^\omega$ for some finite words $u$ and $v$; otherwise it is *aperiodic*.
We denote by ${\lvertw\rvert}_u$ the number of occurrences of the nonempty word $u$ as a factor of $w$. If $u$ and $v$ are finite words over an alphabet $A$, then $u$ and $v$ are *abelian equivalent*, written $u \sim_1 v$, if ${\lvertu\rvert}_a = {\lvertv\rvert}_a$ for each letter $a$ of $A$. Let then $k$ be a fixed positive integer. We say that $u$ and $v$ are *$k$-abelian equivalent*, written $u \sim_k v$, if ${\lvertu\rvert}_w = {\lvertu\rvert}_w$ for each word $w$ of length at most $k$. Notice that if $k = 1$, then $k$-abelian equivalence is simply the abelian equivalence. For words of length at least $k - 1$ we can alternatively say that $u \sim_k v$ if and only if $u$ and $v$ have a common prefix and a common suffix of length $k - 1$ and ${\lvertu\rvert}_w = {\lvertu\rvert}_w$ for each word $w$ of length $k$ [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Lemma 2.3]. Thus, for words of length at most $2k-1$, the $k$-abelian equivalence is in fact the equality relation [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Lemma 2.4]. The $k$-abelian equivalence relation is a congruence relation. If $u_0$, $u_1$, $\ldots$, $u_{n-1}$ are $k$-abelian equivalent words of length $m$, then their concatenation $u_0 u_1 \cdots u_{n-1}$ is a *$k$-abelian power of exponent $n$ and period $m$*. In this paper, we consider only nondegenerate powers, that is, we assume that $n \geq 2$.
Recall that every irrational real number $\alpha$ has a unique infinite continued fraction expansion: $$\label{eq:cf}
\alpha = [a_0; a_1, a_2, a_3, \ldots] = a_0 + \dfrac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{a_3 + \ldots}}}$$ with $a_0 \in {\mathbb{Z}}$ and $a_t \in {\mathbb{Z}}_+$ for $t \geq 1$. The numbers $a_i$ are called the *partial quotients* of $\alpha$. By cutting the expansion after $t + 1$ terms, we obtain a rational number $[a_0; a_1, a_2, a_3, \ldots, a_t]$, which we denote by $p_t / q_t$. These rationals $p_t / q_t$ are the *convergents* of $\alpha$. The convergents of $\alpha$ satisfy the best approximation property, that is, $${\lVertq_t \alpha\rVert} = \min_{0 < m \leq q_{t+1}} {\lVertm\alpha\rVert}$$ for all $t \geq 1$. Here ${\lVertx\rVert}$ measures the distance of $x$ to the nearest integer. In other words, ${\lVertx\rVert} = \min\{ \{x\}, 1 - \{x\} \}$, where $\{x\}$ denotes the fractional part of $x$. Two numbers with continued fraction expansions $[a_0; a_1, \ldots]$ and $[b_0; b_1, \ldots]$ are *equivalent* if there exist integers $N$ and $M$ such that $a_{N+i} = b_{M+i}$ for all $i \geq 0$. As we shall see later, continued fractions are useful in studying Sturmian words (defined below). More details on the connection to Sturmian words can be found, e.g., in [@diss:jarkko_peltomaki Chapter 4].
Let $\alpha$ be an irrational number, and define the *Lagrange constant* $\lambda(\alpha)$ of $\alpha$ as the infimum of real numbers $\lambda$ such that for every $c > \lambda$ the inequality $$\label{eq:ls}
{\left\lvert\alpha - \frac{p}{q}\right\rvert} < \frac{1}{cq^2}$$ has only finitely many rational solutions $p/q$. Famously Hurwitz’s Theorem states that $\lambda(\alpha) \geq \sqrt{5}$ for any irrational $\alpha$, and there exists numbers with $\lambda(\alpha) = \sqrt{5}$. The numbers with finite Lagrange constant are often called *badly approximable numbers* in the literature. The Lagrange constant of $\alpha$ with continued fraction expansion as in is computed as follows: $$\label{eq:lagrange}
\lambda(\alpha) = \limsup_{t \to \infty} ([a_{t+1}; a_{t+2}, \ldots] + [0; a_t, a_{t-1}, \ldots, a_1]).$$ From this formula, it is clear that two equivalent numbers have the same Lagrange constant. The *Lagrange spectrum* is the set of finite Lagrange constants. This set has many curious properties, and we shall return to them later at the end of . For details on the Lagrange spectrum, see [@1989:the_markoff_and_lagrange_spectra] or [@2013:markovs_theorem_and_100_years_of_the_uniqueness].
Sturmian words are defined as the codings of orbits of points in an irrational circle rotation with two intervals. This understanding is sufficient for our purposes, but many other viewpoints exist; see, e.g., [@2002:substitutions_in_dynamics_arithmetics_and_combinatorics; @2002:algebraic_combinatorics_on_words]. Identify the unit interval $[0,1)$ with the unit circle ${\mathbb{T}}$, and let $\alpha$ be a fixed irrational. The mapping $R\colon {\mathbb{T}}\to {\mathbb{T}}, \, x \mapsto \{x + \alpha\}$ defines a rotation on ${\mathbb{T}}$. Partition the circle ${\mathbb{T}}$ into two intervals $I_0$ and $I_1$ defined by the points $0$ and $\{1-\alpha\}$. Let $\nu$ be the coding function defined by setting $\nu(x) = 0$ if $x \in I_0$ and $\nu(x) = 1$ if $x \in I_1$. Define ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ as the infinite word obtained by setting its $n^\text{th}, n \geq 0,$ letter to equal $\nu(R^n(x))$. The word ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ is called the *Sturmian word of slope $\alpha$ and intercept $x$*.
The above definition is not complete because we did not define how $\nu$ behaves in the endpoints $0$ and $\{1-\alpha\}$. There is some choice here, and we take either $I_0 = [0,\{1-\alpha\})$ and $I_1 = [\{1-\alpha\},1)$ or $I_0 = (0,\{1-\alpha\}]$ and $I_1 = (\{1-\alpha\},1]$. These options are determined by whether or not $0 \in I_0$. This little detail makes no difference to us: only interior points of intervals are considered. Let $x, y \in {\mathbb{T}}$ with $x < y$. Then by both $I(x, y)$ and $I(y, x)$ we mean the interval $[x, y)$ if $0 \in I_0$ and the interval $(x, y]$ if $0 \notin I_0$.
One particular example of a Sturmian word is the Fibonacci word ${ \ifcat\noexpandf\relax\bm{f} \else\mathbf{f}\fi}$. Its slope is $1/\varphi^2$, where $\varphi$ is the golden ratio, and its intercept equals its slope. We have $${ \ifcat\noexpandf\relax\bm{f} \else\mathbf{f}\fi} = 01001010010010100101001001010010 \cdots.$$ This word is also the fixed point of the substitution $0 \mapsto 01$, $1 \mapsto 0$.
The sequence $(\{n\alpha\})_{n\geq 0}$ is dense in $[0,1)$ by Kronecker’s Theorem, so Sturmian words of slope $\alpha$ have a common language $\mathcal{L}$ (the set of factors). Let $w$ denote a word $a_0 a_1 \cdots a_{n-1}$ of length $n$ in $\mathcal{L}$. Then there exists a unique subinterval $[w]$ of ${\mathbb{T}}$ such that the Sturmian word ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ begins with $w$ if and only if $x \in [w]$. Clearly $[w] = I_{a_0} \cap R^{-1}(I_{a_1}) \cap \ldots \cap R^{-(n-1)}(I_{a_{n-1}})$ (here the choice of endpoints matters, but we only consider interior points of these intervals). The points $0$, $\{-\alpha\}$, $\{-2\alpha\}$, $\ldots$, $\{-n\alpha\}$ partition the circle into $n+1$ subintervals which are exactly the intervals $[w]$ for factors of length $n$. We call these $n+1$ intervals the *level $n$ intervals*, and we denote the set containing them by $L(n)$. We abuse notation and write $\max L(n)$ (resp. $\min L(n)$) for the maximum (resp. minimum) length of a level $n$ interval.
In the rest of this paper, we keep the slope $\alpha$ with continued fraction expansion $[a_0; a_1, a_2, \ldots]$ fixed. Whenever we talk about the convergents $q_t$, the level $n$ intervals $L(n)$, the rotation $R$, etc., we implicitly understand that they relate to this fixed $\alpha$.
Main Results {#sec:results}
============
k-abelian Equivalence in Sturmian Words {#ssec:eq_classes}
---------------------------------------
Our first aim is to show that the $k$-abelian equivalence classes of factors of a Sturmian word correspond to certain intervals on the circle ${\mathbb{T}}$ and to characterize the endpoints of these intervals. We begin by recalling the following result of [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite] (specialized to Sturmian words).
\[prp:pref\_suff\_ab\] [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Proposition 2.8] Let $u$ and $v$ be two factors of the same length occurring in some Sturmian word. Then $u \sim_k v$ if and only if they share a common prefix and a common suffix of length $\min\{{\lvertu\rvert}, k-1\}$ and $u \sim_1 v$.
This result is interesting as it shows that rather weak conditions are enough for $k$-abelian equivalence in Sturmian words. This is not unique to Sturmian words: it holds for episturmian words [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Proposition 2.8], and in [@2018:on_the_k-abelian_complexity_of_the_cantor_sequence Theorem 1], it is shown that holds also for factors of the Cantor word, the fixed point of the substitution $0 \mapsto 000$, $1 \mapsto 101$. We will return to this matter in .
Let us then recall the following result which gives an arithmetical characterization of abelian equivalence in Sturmian words.
\[prp:abelian\_characterization\] [@2016:abelian_powers_and_repetitions_in_sturmian_words Proposition 3.3], [@2013:some_properties_of_abelian_return_words Theorem 19] Let $u$ and $v$ be two factors of the same length occurring in a Sturmian word of slope $\alpha$. Then $u \sim_1 v$ if and only if $[u], [v] \subseteq I(0, \{-{\lvertu\rvert}\alpha\})$ or $[u], [v] \subseteq I(\{-{\lvertu\rvert}\alpha\}, 1)$.
In other words, the two possible abelian equivalence classes for factors of length $m$ correspond to two intervals on the circle marked by the points $0$ and $\{-m\alpha\}$. Next we generalize for $k$-abelian equivalence.
By , we need to at least consider the prefixes and suffixes of length up to $k - 1$. Let $m \geq 1$, and define $\mathcal{D}_{k,m} = \{0, \{-\alpha\}, \{-2\alpha\}, \ldots, \{-\min\{m, k-1\}\alpha\}\}$. These points divide the circle into $\min\{m + 1, k\}$ intervals (which are the level $\min\{m, k-1\}$ intervals), and if points $x$ and $y$ belong to the same interval, then the prefixes of ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ and ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{y,\alpha}$ of length $\min\{m, k-1\}$ are equal. Now if $m \geq k - 1$, then $$R^{-(m - (k - 1))}(\mathcal{D}_{k,m}) = \{\{-(m - (k - 1))\alpha\}, \ldots, \{-m\alpha\}\},$$ and these points also divide the circle into $k$ intervals. If $x$ and $y$ belong to the same interval, then the prefixes of ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ and ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{y,\alpha}$ of length $m$ have a common suffix of length $k - 1$. Set $\mathcal{P}_{k,m} = \smash[t]{\mathcal{D}_{k,m} \cup R^{-(m - (k - 1))}(\mathcal{D}_{k,m})}$ if $m \geq k - 1$; otherwise set $\mathcal{P}_{k,m} = \mathcal{D}_{k,m}$.
$\mathcal{I}_{k,m}$ is the set of subintervals of ${\mathbb{T}}$ determined by the points of $\mathcal{P}_{k,m}$.
What me mean by this precisely is that, to define the intervals $I_i$ of $\mathcal{I}_{k,m}$, we order the points $x_i$ of $\mathcal{P}_{k,m}$: $0 = x_0 < x_1 < \ldots < x_{\ell - 1} < x_\ell = 1$, $\ell = {\lvert\mathcal{P}_{k,m}\rvert}$, and set $I_i = [x_i, x_{i+1})$ if $0 \in I_0$ and $I_i = (x_i, x_{i+1}]$ if $0 \notin I_0$ for $0 \leq i < \ell$. Observe that for $m < k-1$, the intervals $\mathcal I_{k,m}$ coincide with the level $m$ intervals.
As before for the level $m$ intervals $L(m)$, by writing $\max \mathcal{I}_{k,m}$ we mean the maximum length of an interval in $\mathcal{I}_{k,m}$. We claim that the intervals $\mathcal{I}_{k,m}$ determine the $k$-abelian equivalence classes.
\[thm:eq\_class\_intervals\] Let $u$ and $v$ be two factors of length $m$ occurring in a Sturmian word of slope $\alpha$. Then $u \sim_k v$ if and only if there exists $J \in \mathcal{I}_{k,m}$ such that $[u], [v] \subseteq J$.
Assume that $m < k - 1$. Then $u \sim_k v$ if and only if $u = v$. This means that $[u]$ and $[v]$ equal one of the level $m$ intervals. When $m < k - 1$, the intervals $\mathcal{I}_{k,m}$ are precisely the level $m$ intervals, so we are done. We may thus assume that $m \geq k - 1$.
Suppose first that there exists $J \in \mathcal{I}_{k,m}$ such that $[u], [v] \subseteq J$. By the definition of the intervals $\mathcal{I}_{k,m}$, the words $u$ and $v$ share a common prefix and a common suffix of length $k - 1$. Moreover they are abelian equivalent by because the point $\{-m\alpha\}$ separating the two abelian equivalence classes is among the points $\mathcal{P}_{k,m}$. Therefore implies that $u \sim_k v$.
Suppose that $u \sim_k v$. Then $u$ and $v$ share a common prefix and a common suffix of length $k - 1$. Assume for a contradiction that $[u]$ and $[v]$ are contained in distinct intervals of $\mathcal{I}_{k,m}$. Without loss of generality, we assume that $\sup[u] \leq \inf[v]$. Let $K$ be the interval containing exactly the points $z$ for which $\sup[u] \leq z \leq \inf[v]$. (If $\sup[u] = \inf[v]$, then we let $K$ to be the set containing the common endpoint of $[u]$ and $[v]$.) Since $[u]$ and $[v]$ are contained in distinct intervals of $\mathcal{I}_{k,m}$, there exists a point $x$ in $\mathcal{P}_{k,m}$ such that $x \in K$. Denote the set $\smash[t]{R^{-(m - (k - 1))}(\mathcal{D}_{k,m})}$ by $\mathcal{S}$. The point $x$ cannot be in $\mathcal{D}_{k,m}$ because $u$ and $v$ share a common prefix of length $k - 1$. Therefore we must have $x \in \mathcal{S}$. Let $y$ be an arbitrary point in $\mathcal{S}$. If $y \in \mathbb{T} \setminus ([u] \cup [v] \cup K)$, then either $[u] \subseteq I(x,y)$ and $[v] \cap I(x, y) = \emptyset$ or symmetrically $[v] \subseteq I(x,y)$ and $[u] \cap I(x, y) = \emptyset$. Then, by the definition of the points $\mathcal{S}$, we see that $u$ and $v$ have distinct suffixes of length $k - 1$, which is impossible. We conclude that $\mathcal{S} \subseteq K$ (see for this situation). Since $\{-m\alpha\} \in \mathcal{S}$, it follows by that $u$ and $v$ are not abelian equivalent. This is a contradiction.
Notice that $\mathcal{I}_{k,m}$ contains $2k$ intervals when $m \geq 2k - 1$ and $m + 1$ intervals when $0 \leq m \leq 2k - 2$. This number of abelian equivalence classes for factors of length $m$ characterizes Sturmian words; see [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Theorem 4.1].
\[ex:fibonacci\] Let us consider the $2$-abelian equivalence classes of length $5$ of the Fibonacci word; its slope $\alpha$ is $1/\varphi^2$. On the left in , there are two concentric circles. The outer circle represents the level $5$ intervals separated by the points $0$, $\{-\alpha\} (\approx 0.62)$, $\{-2\alpha\}$ ($\approx 0.24$), $\{-3\alpha\}$ ($\approx 0.85$), $\{-4\alpha\}$ ($\approx 0.47$), and $\{-5\alpha\}$ ($\approx 0.09$). The inner circle shows the endpoints of the $2$-abelian equivalence classes. The points $0$ and $\{-\alpha\}$ of $\mathcal{D}_{2,5}$ are shown in black while the points $\{-4\alpha\}$ and $\{-5\alpha\}$ of $R^{-4}(\mathcal{D}_{2,5})$ are represented by circles filled with white. The concentric circles on the right in give the corresponding intervals and points when $m = 7$.
We have $4$ $2$-abelian equivalence classes for length $5$: $\{00100\}$, $\{00101, 01001\}$, $\{01010\}$, and $\{10010, 10100\}$. The singleton classes are special. At the end of the proof of , we had to take some extra steps because factors corresponding to two distinct intervals of $\mathcal{I}_{k,m}$ could share prefixes and suffixes of length $k - 1$. Indeed here $00100$ and $01010$ have common prefixes and suffixes of length $1$, but this does not guarantee abelian equivalence.
(0,0) circle ([1.8]{}); in [0,...,5]{}[ ([(- floor()) \* 360]{}:[1.8]{}) circle (1pt); ]{}
(0,0) circle ([1.0]{}); /in [0/black, 1/black, 4/white, 5/white]{}[ ([(- floor()) \* 360]{}:[1.0]{}) circle (1pt); ]{}
at (2.3,0.5) ; at (1.5,1.7) ; at (-1.7,1.5) ; at (-2.25,-0.5) ; at (-0.2,-2.05) ; at (2.1,-1.0) ;
at (1.6,0) ; at (-1.05,-1.1) ; at (0.15,1.5) ; at (0.8,-1.2) ; at (-1.4,0.3) ; at (1.1,0.9) ;
(0,0) circle ([1.8]{}); in [0,...,7]{}[ ([(- floor()) \* 360]{}:[1.8]{}) circle (1pt); ]{}
(0,0) circle ([1.0]{}); /in [0/black, 1/black, 6/white, 7/white]{}[ ([(- floor()) \* 360]{}:[1.0]{}) circle (1pt); ]{}
at (2.4,0.5) ; at (1.5,1.8) ; at (-0.8,2.05) ; at (-2.1,1.2) ; at (-2.3,-0.7) ; at (-1.4,-1.8) ; at (0.9,-2.0) ; at (2.2,-1.0) ;
at (1.6,0) ; at (-1.05,-1.1) ; at (0.15,1.5) ; at (0.8,-1.2) ; at (-1.4,0.3) ; at (1.1,0.9) ; at (-0.4,-1.5) ; at (-0.7,1.35) ;
We make an observation regarding the part of the proof of showing that if two level $m$ intervals $[u]$ and $[v]$ are included in distinct intervals of $\mathcal{I}_{k,m}$ then $u \not\sim_k v$. The proof shows that if $u$ and $v$ have common prefixes and suffixes of length $k-1$, the only way that $u \not\sim_k v$ is when all the points $\mathcal{S}$ (this is the set $R^{-(m - (k - 1))}(\mathcal{D}_{m,k})$) are contained in one level $(k - 1)$ interval $J$. We claim that this phenomenon cannot occur if $k \geq 2$ and ${\lVert\alpha\rVert} > 1/(2(k-1))$. Notice that in the case $k = 2$ this may happen since ${\lVert\alpha\rVert} < 1/2$ always.
Assume that $k \geq 2$. There exist at least two points at distance ${\lVert\alpha\rVert}$ in $\mathcal{S}$ (e.g., $\{-(m-1)\alpha\}$ and $\{-m\alpha\}$) which implies that the length of $J$ is greater than ${\lVert\alpha\rVert}$. If $k - 1 \geq {\lfloor1/{\lVert\alpha\rVert}\rfloor}$, then each interval determined by the points $0$, $\{-\alpha\}$, $\ldots$, $\{-(k-1)\alpha\}$ has length at most ${\lVert\alpha\rVert}$, so we conclude that $k - 1 < {\lfloor1/{\lVert\alpha\rVert}\rfloor}$. The intervals determined by the points $0$, $\{-\alpha\}$, $\ldots$, $\{-(k-1)\alpha\}$ are now the same as those determined by the points $0$, $1 - {\lVert\alpha\rVert}$, $1 - 2{\lVert\alpha\rVert}$, $\ldots$, $1 - (k-1){\lVert\alpha\rVert}$, so all of them have length ${\lVert\alpha\rVert}$ except one that has length $1 - (k-1){\lVert\alpha\rVert}$. Thus $J$ has length $1 - (k-1){\lVert\alpha\rVert}$. Since $R$ is an isometry, $J$ contains $(k - 1)$ intervals of length ${\lVert\alpha\rVert}$ (defined by the points of $\mathcal{S}$), and we must have $(k - 1){\lVert\alpha\rVert} < 1 - (k - 1){\lVert\alpha\rVert}$, that is, ${\lVert\alpha\rVert} < 1/(2(k-1))$. Thus we obtain the following strengthening of .
\[thm:pref\_suff\_ab\_improved\] Let $u$ and $v$ be two factors of the same length occurring in a Sturmian word of slope $\alpha$. Then $u \sim_k v$ if and only if they share a common prefix and a common suffix of length $\min\{{\lvertu\rvert}, k-1\}$ and $u \sim_1 v$. Moreover, the condition $u \sim_1 v$ may be omitted if $2(k - 1){\lVert\alpha\rVert} > 1$.
The slope of the Fibonacci word is approximately $0.38$, so says that the condition $u \sim_1 v$ can then be omitted when $k \geq 3$. It is rather surprising that such a weak condition is sufficient to establish $k$-abelian equivalence. This raises the question if it is possible to improve on the Fibonacci word and have an infinite word for which the condition is redundant even when $k = 2$. We study this question in .
The k-Lagrange Spectrum {#ssec:k-lagrange_spectrum}
-----------------------
Let ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}$ be the maximum exponent of $k$-abelian powers of period $m$ occurring in a Sturmian word of slope $\alpha$. We define the *$k$-abelian critical exponent of slope $\alpha$* to be the quantity $$\limsup_{m\to\infty} \frac{{\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}}{m},$$ and we denote it by ${\mathpzc{A\mkern-3mu c}_{k}(\alpha)}$. It measures the maximal ratio between the exponent and period of a $k$-abelian power in a Sturmian word of slope $\alpha$; it was introduced in the case $k = 1$ in [@2016:abelian_powers_and_repetitions_in_sturmian_words] (in the current paper we follow the notation of the dissertation [@diss:jarkko_peltomaki] instead of the article [@2016:abelian_powers_and_repetitions_in_sturmian_words]). As mentioned in the introduction, the set of finite values of ${\mathpzc{A\mkern-3mu c}_{1}(\alpha)}$ is the Lagrange spectrum [@2016:abelian_powers_and_repetitions_in_sturmian_words Theorem 5.10], so the finite values of ${\mathpzc{A\mkern-3mu c}_{k}(\alpha)}$ can be viewed as a combinatorial generalization of the Lagrange spectrum. Thus we give the following definition.
The *$k$-Lagrange spectrum* ${\mathcal{L}_{k}}$ is the set $\{{\mathpzc{A\mkern-3mu c}_{k}(\alpha)} : \text{$\alpha$ is irrational}\} \cap {\mathbb{R}}$.
In order to study ${\mathcal{L}_{k}}$, we begin by showing how to compute ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}$ especially when $m$ is a denominator of a convergent of $\alpha$.
Say that a Sturmian word ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ of slope $\alpha$ and intercept $x$ begins with a $k$-abelian power of period $m$ and exponent $n$. The prefix of ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ of length $m$ and the factor of ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ of length $m$ starting after this prefix are $k$-abelian equivalent so, by , the points $x$ and $\{x + m\alpha\}$ lie in a common interval of $\mathcal{I}_{k,m}$. The distance of these points is ${\lVertm\alpha\rVert}$. Thus we see that the points $x$, $\{x + m\alpha\}$, $\ldots$, $\{x + (n - 1)m\alpha\}$ all lie in a common interval of $\mathcal{I}_{k,m}$, which must have length at least $(n - 1){\lVertm\alpha\rVert}$. Conversely, given such points, we see that the word ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}_{x,\alpha}$ begins with a $k$-abelian power of period $m$ and exponent $n$. Thus by considering the longest interval in $\mathcal{I}_{k,m}$, we obtain the following result (recall that $\max \mathcal{I}_{k,m}$ means the maximal length of an interval in $\mathcal{I}_{k,m}$).
\[lem:max\_exponent\] We have ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} = {\left\lfloor \frac{\max \mathcal{I}_{k,m}}{{\lVertm\alpha\rVert}} \right\rfloor} + \gamma$, where $\gamma$ is $1$ if $\max \mathcal{I}_{k,m} \neq {\lVertm\alpha\rVert}$ and $0$ otherwise.
( continued) The interval of the class $\{10010, 10100\}$ has length $\alpha$ which means by that using the words in the class a $2$-abelian power of period $5$ and exponent ${\lfloor\alpha/{\lVert5\alpha\rVert}\rfloor} + 1 = 5$ can be formed. Indeed, it is straightforward to check that $(10100)^2 (10010)^3$ is a factor of the Fibonacci word. Using words from the class $\{01010\}$ only $2$-abelian powers of exponent ${\lfloor{\lVert3\alpha\rVert}/{\lVert5\alpha\rVert}\rfloor} + 1 = 2$ can be formed. The word $(00100)^2$ is not a factor of the Fibonacci word since it contains $000$. Indeed, we see using that the exponent for this class is $1$.
Interestingly if $m = 7$, then the exponent for each equivalence class is $1$. The reason is that ${\lVert7\alpha\rVert}$ is large: we have ${\lVert7\alpha\rVert} \approx 0.33$ whereas ${\lVert5\alpha\rVert} \approx 0.09$. The $k$-abelian equivalence relation for $k > 1$ differs in this respect from abelian equivalence: it follows from [@2016:abelian_powers_and_repetitions_in_sturmian_words Theorem 4.7] that in any Sturmian word there exists an abelian square of period $m$ for each $m \geq 1$.
As the number $\max \mathcal{I}_{k,m}$ is generally difficult to find, let us argue next that when $m$ is chosen suitably then, in order to find ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}$, it is sufficient to study the level $2k - 2$ intervals. As in , the points $\mathcal{D}_{k,m} = \{0, \{-\alpha\}, \{-2\alpha\}, \ldots, \{-(k-1)\alpha\}\}$ together with the points $\mathcal{S} = R^{-(m - (k - 1))}(\mathcal{D}_{k,m}) = \{\{-(m - (k - 1))\alpha\}, \ldots, \{-m\alpha\}\}$ determine the intervals $\mathcal{I}_{k,m}$ of the $k$-abelian equivalence classes. Suppose now that ${\lVertm\alpha\rVert}$ is sufficiently small. Then the points $R^m(\mathcal{S}) = R^{k-1}(D_{k,m})$ are close to the points $\mathcal S$. In fact, when comparing the intervals $\mathcal{I}_{k,m}$ defined by the points $\mathcal{D}_{k,m} \cup \mathcal{S}$ to those intervals defined by the points $\mathcal{D}_{k,m} \cup R^{k - 1}(\mathcal{D}_{k,m})$, we see that some intervals are shortened by ${\lVertm\alpha\rVert}$ and some intervals are lengthened by ${\lVertm\alpha\rVert}$, but the order of the points is the same whenever ${\lVertm\alpha\rVert}$ is small enough. The points $\{-m\alpha\}$ and $0$ however merge, but this is irrelevant when considering $\max \mathcal{I}_{k,m}$ as we only lose a short interval of length ${\lVertm\alpha\rVert}$. Now $$\mathcal{D}_{k,m} \cup R^{k - 1}(\mathcal{D}_{k,m}) = \{\{-(k-1)\alpha\}, \ldots, \{-\alpha\}, 0, \alpha, \ldots, \{(k-1)\alpha\}\}.$$ Using the fact that $R$ is an isometry, we can study the set $R^{-(k - 1)}(\mathcal{D}_{k,m} \cup R^{k - 1}(\mathcal{D}_{k,m}))$ instead. This set is the set of endpoints of the level $2k - 2$ intervals. It is quite obvious from the preceding that ${\lVertm\alpha\rVert}$ is small enough whenever ${\lVertm\alpha\rVert} < \min L(2k - 2)$. We have thus argued that whenever ${\lVertm\alpha\rVert} < \min L(2k - 2)$, we have $${\lvert\max \mathcal{I}_{k,m} - \max L(2k - 2)\rvert} \leq {\lVertm\alpha\rVert}.$$ Therefore we have proved the following lemma.
\[lem:approximate\_exponent\] Let $m$ be a positive integer and suppose that ${\lVertm\alpha\rVert} < \min L(2k-2)$. Then $${\left\lvert{\left\lfloor\frac{\max L(2k-2)}{{\lVertm\alpha\rVert}}\right\rfloor} - {\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}\right\rvert} \leq 1.$$
This lemma shows that the exponents of $k$-abelian powers grow arbitrarily large (as we can make ${\lVertm\alpha\rVert}$ as small as desired). A more general result was obtained in [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Theorem 5.4].
With the results so far, we are able to show that for determining ${\mathpzc{A\mkern-3mu c}_{k}(\alpha)}$ it is sufficient to consider ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}$ only when $m$ is a denominator of a convergent. Recall that $q_t$ refers to the denominator of the $t^\text{th}$ convergent of $\alpha$.
\[prp:convergents\_enough\] For all large enough $t$, we have ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} \leq {\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)} + 2$ for all $1 \leq m < q_{t+1}$.
Let $t \geq 1$, and assume that $t$ be large enough so that ${\lVertq_t\alpha\rVert} < \min L(2k-2)$. Suppose that $m$ is an integer such that $1 \leq m < q_{t+1}$. By the best approximation property of the convergents, we have ${\lVertm\alpha\rVert} \geq {\lVertq_t\alpha\rVert}$. Suppose first that ${\lVertm\alpha\rVert} < \min L(2k-2)$. Then by , we have $${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} \leq \frac{\max L(2k-2)}{{\lVertm\alpha\rVert}} + 1 \leq \frac{\max L(2k-2)}{{\lVertq_t\alpha\rVert}} + 1,$$ so, by the same lemma, we have ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} \leq {\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)} + 2$. Suppose next that ${\lVertm\alpha\rVert} \geq \min L(2k-2)$. Then $$\frac{\max L(m)}{{\lVertm\alpha\rVert}} \leq \frac{\max L(m)}{\min L(2k-2)} \leq \frac{1}{\min L(2k-2)},$$ so ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}$ is bounded by a constant. Thus ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} < {\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)}$ for all large enough $t$. The sequence $({\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_i)})_i$ reaches arbitrarily high values due to .
can be improved: ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} \leq {\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)} + 1$ for all $1 \leq m < q_{t+1}$ and $t$ large enough. Proving this would complicate the argument significantly, and we do not need the improved statement in this paper. It is very well possible that ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)} > {\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_{t+1})}$. For example, if $k = 2$ and say $\alpha = [0; 3, 1, 1, 1, 100,
\overline{1}]$, then the sequence of denominators of convergents is $1$, $3$, $4$, $7$, $\ldots$, and it is readily computed that ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(4)} = 6 > 5 = {\mathpzc{A\mkern-3mu e}_{k,\alpha}(7)}$. On the other hand, if $k = 1$, then we have ${\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)} < {\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)}$ for all $t$ and $1 \leq m < q_t$ as can be readily observed from [@2016:abelian_powers_and_repetitions_in_sturmian_words Lemma 4.7].
For $t$ large enough, let $m$ be an integer such that $q_t \leq m < q_{t+1}$. It follows from that $$\frac{{\mathpzc{A\mkern-3mu e}_{k,\alpha}(m)}}{m} \leq \frac{{\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)} + 2}{q_t},$$ so we can conclude using that $${\mathpzc{A\mkern-3mu c}_{k}(\alpha)} = \limsup_{t \to \infty} \frac{{\mathpzc{A\mkern-3mu e}_{k,\alpha}(q_t)}}{q_t} = \limsup_{t \to \infty} \frac{\max L(2k-2)}{q_t{\lVertq_t\alpha\rVert}}.$$ When $k = 1$, we obtain $${\mathpzc{A\mkern-3mu c}_{1}(\alpha)} = \limsup_{t \to \infty} \frac{1}{q_t{\lVertq_t\alpha\rVert}},$$ so $${\mathpzc{A\mkern-3mu c}_{k}(\alpha)} = \max L(2k-2) \cdot {\mathpzc{A\mkern-3mu c}_{1}(\alpha)}.$$ Let us restate the result.
\[thm:main\_relation\] We have ${\mathpzc{A\mkern-3mu c}_{k}(\alpha)} = \max L(2k-2) \cdot {\mathpzc{A\mkern-3mu c}_{1}(\alpha)}$ for all $k \geq 1$.
Notice that ${\mathpzc{A\mkern-3mu c}_{1}(\alpha)}$ is finite if and only if $\alpha$ has bounded partial quotients; see . Therefore ${\mathpzc{A\mkern-3mu c}_{k}(\alpha)}$ is finite if and only if $\alpha$ has bounded partial quotients. As is well-known, numbers with bounded partial quotients comprise a set of measure zero.
As mentioned in , equivalent numbers have the same Lagrange constant. By , this is no longer true when $k > 1$ because $\max L(2k-2)$ depends on $\alpha$. It is not difficult to convince oneself that the points obtained in from a single class of equivalent numbers form a dense set. This is what we shall prove next. As a corollary we obtain , which states that the $k$-Lagrange spectrum ${\mathcal{L}_{k}}$ is itself dense when $k > 1$. In the statement of the following lemma, by $\max L_\beta(\ell)$ we mean the maximal length of a level $\ell$ interval of slope $\beta$.
\[lem:density\] Let $\alpha$ be irrational. The set $\{\max L_\beta(\ell) : \text{$\beta$ is equivalent to $\alpha$}\}$ is contained and dense in $(\tfrac{1}{\ell+1}, 1)$ for all $\ell > 1$.
Clearly $\max L_\beta(\ell) > \tfrac{1}{\ell + 1}$ since there are $\ell + 1$ level $\ell$ intervals. Let $\gamma \in \smash[b]{(\tfrac{1}{\ell+1}, 1)}$, and suppose without loss of generality that it is irrational. By cutting the continued fraction expansion of $1-\gamma$ after finitely many partial quotients, we obtain a fraction that is as close as $1-\gamma$ as we desire. Thus we can find a rational $\beta$ such that $\ell \beta$ is arbitrarily close to $1-\gamma$ (from either side). Now form an irrational $\beta'$ by continuing the continued fraction expansion of $\beta$ in such a way that it is equivalent to $\alpha$. By selecting the partial quotients appropriately, we find that $\ell \beta'$ is arbitrarily close to $1-\gamma$. Consider now the level $\ell$ intervals of slope $\beta'$. The longest such interval clearly has length $1-\ell\beta'$ since $\gamma > \tfrac{1}{\ell+1}$. As $1-\ell\beta'$ is as close to $\gamma$ as we like, the claim follows.
As the smallest element of the Lagrange spectrum is $\sqrt{5}$, and imply the following result.
\[thm:endpoints\] Let $k > 1$. Then ${\mathcal{L}_{k}} \subseteq (\tfrac{\sqrt{5}}{2k-1}, \infty)$ and $\tfrac{\sqrt{5}}{2k-1}$ is the least accumulation point of ${\mathcal{L}_{k}}$. In particular, the set ${\mathcal{L}_{k}}$ is not closed.
This proposition should be compared with the fact that ${\mathcal{L}_{1}}$ is closed; cf. [@1989:the_markoff_and_lagrange_spectra Theorem 2 of Chapter 3]. Notice that it also follows that when $k > 1$, the Fibonacci word no longer has minimal critical $k$-abelian exponent among all Sturmian words.
Let us then recall some remarkable facts about the Lagrange spectrum. *Hall’s ray* is the largest half-line contained in ${\mathcal{L}_{1}}$. It was proven by Hall that the half-line $[6, \infty)$ is contained in ${\mathcal{L}_{1}}$ [@1947:on_the_sum_and_products_of_continued_fractions]. By series of improvements by several researchers, it was finally determined by Freiman [@1975:diophantine_approximation_and_geometry_of_numbers] that Hall’s ray equals $[c_F, \infty)$, where $c_F$ is the *Freiman constant* $$c_F = \frac{2221564096+283748\sqrt{462}}{491993569} = 4.5278295661 \ldots$$ The detailed history and references can be found in [@1989:the_markoff_and_lagrange_spectra Chapter 4]. Hall’s result together with and imply the following theorem.
\[thm:dense\] The $k$-Lagrange spectrum ${\mathcal{L}_{k}}$ is dense in $(\tfrac{\sqrt{5}}{2k-1}, \infty)$ when $k > 1$.
By and Hall’s result, the intervals $(\tfrac{\sqrt{5}}{2k-1}, \sqrt{5})$ and $(\tfrac{c_F}{2k-1}, \infty)$ are dense with points of ${\mathcal{L}_{k}}$. Now $c_F$ is at most $6$, so $\smash[t]{\tfrac{c_F}{2k-1} \leq 2 < \sqrt{5}}$ meaning that these dense sets overlap.
We do not know if ${\mathcal{L}_{k}}$ contains a half-line when $k > 1$. If true, it is not a straightforward consequence of Hall’s and Freiman’s results: the union of the dense subsets obtained from each $\theta \in [c_F, \infty)$ by is not automatically a half-line. This poses an interesting open problem.
Does ${\mathcal{L}_{k}}$ contain a half-line when $k > 1$? If so, what is the largest such half-line? Is it $(\tfrac{c_F}{2k - 1}, \infty)$?
It is conceivable that a point in ${\mathcal{L}_{1}}$ below $c_F$ could map to $\tfrac{c_F}{2k-1}$. Moreover, ${\mathcal{L}_{1}}$ could contain an interval below $c_F$ (see below) that could produce an interval into ${\mathcal{L}_{k}}$.
The usual Lagrange spectrum is not dense between $\sqrt{5}$ and $c_F$. In fact, substantial amount of research has been done on maximal gaps occurring in this interval, see for instance [@1989:the_markoff_and_lagrange_spectra Chapter 5]. It is known for example that the set $[\sqrt{5}, 3] \cap {\mathcal{L}_{1}}$ is discrete and that the interior of the interval $[\sqrt{12}, \sqrt{13}]$ does not include any points of ${\mathcal{L}_{1}}$ while its endpoints are in ${\mathcal{L}_{1}}$. It is unknown if ${\mathcal{L}_{1}}$ contains an interval below $c_F$. The existence of such an interval could show that ${\mathcal{L}_{k}}$ also contains an interval below $\tfrac{c_F}{2k-1}$, but it is plausible that this could also happen for other reasons. For example, it is possible for uncountably many numbers to have the same Lagrange constant, so an interval could be produced by means of . One such example is the number $3$; it is the Lagrange constant of uncountably many numbers [@1992:continued_fractions Theorem 3, Chapter IV§6]. We do not believe that this particular example would provide an interval; we just mention it as a possibility. It is known that the part of ${\mathcal{L}_{1}}$ below $\sqrt{689}/8$ has measure zero [@1982:hausdorff_dimensions_of_cantor_sets]. It seems to us that studying intervals in ${\mathcal{L}_{k}}$ for $k > 1$ is of comparable difficulty as the study of intervals in ${\mathcal{L}_{1}}$.
Let us also point out that it is easy to come up with numbers greater than $\sqrt{5}/(2k-1)$ that are not in ${\mathcal{L}_{k}}$. The two smallest elements of ${\mathcal{L}_{1}}$ are $\sqrt{5}$ and $\sqrt{8}$, so any point in ${\mathcal{L}_{k}}$ between $\sqrt{5}/(2k-1)$ and $\sqrt{8}/(2k-1)$ is of the form $\max L_\alpha(2k-2) \cdot \sqrt{5}$ for some $\alpha$ equivalent to the golden ratio. The number $\max L_\alpha(2k-2)$ is always irrational, so rational multiples of $\sqrt{5}$ between $\sqrt{5}/(2k-1)$ and $\sqrt{8}/(2k-1)$ are not in ${\mathcal{L}_{k}}$.
The Spectrum L
--------------
As mentioned in the introduction, when the critical exponent is considered for the equality relation, it is typical to just measure the supremum of fractional exponents, not the ratio of the exponent and the period. In this final subsection, we briefly remark what happens if we look at the ratio instead.
Analogous to what we have done already, we set $${\mathpzc{A\mkern-3mu c}_{\infty}(\alpha)} = \limsup_{m\to\infty} \frac{{\mathpzc{A\mkern-3mu e}_{\infty,\alpha}(m)}}{m},$$ where ${\mathpzc{A\mkern-3mu e}_{\infty,\alpha}(m)}$ is the maximum integer exponent of a power of period $m$ occurring in a Sturmian word of slope $\alpha$. We further set ${\mathcal{L}_{\infty}} = \{{\mathpzc{A\mkern-3mu c}_{\infty}(\alpha)} : \text{$\alpha$ is irrational}\} \cap {\mathbb{R}}$. We show next that the set ${\mathcal{L}_{\infty}}$ contains every nonnegative real number.
We have ${\mathcal{L}_{\infty}} = {\mathbb{R}}_{\geq 0}$.
Consider powers occurring in a Sturmian word of slope $\alpha$ having continued fraction expansion $[0;a_1,a_2,\ldots]$ and sequence of convergents $(p_t/q_t)_t$. It is well-known that if $m$ is not a denominator of a convergent of $\alpha$, then any power of period $m$ has exponent at most $2$; see, e.g., [@2002:the_index_of_sturmian_sequences Lemma 3.6] or [@diss:jarkko_peltomaki Theorem 4.6.5]. Moreover, if $m = q_t$ with $t > 1$, then the highest integer exponent of a power of period $m$ is $a_{t+1} + 2$ [@2002:the_index_of_sturmian_sequences Lemma 3.4], [@diss:jarkko_peltomaki Theorem 4.6.5]. Given that we have chosen the partial quotients $a_1$, $a_2$, $\ldots$, $a_t$ and thus determined the convergent $q_t$, we have complete freedom to choose $a_{t+1}$ to make the ratio $(a_{t+1} + 2)/q_t$ to behave the way we like.
If the sequence $(a_t)_t$ of partial quotients is bounded, then we clearly have ${\mathpzc{A\mkern-3mu c}_{\infty}(\alpha)} = 0$ because the sequence $(q_t)_t$ is increasing. Hence $0 \in {\mathcal{L}_{\infty}}$. Let then $\lambda$ be a fixed positive real number, and let $k_1$ be the least integer such that $k_1 > 1$ and that there exist nonnegative integers $r_1$ and $s_1$ such that $0 \leq s_1 < q_{k_1}$ and $\lambda - (r_1 + s_1/q_{k_1}) < \tfrac12$. Set $a_{1,1} = a_1$, $a_{1,2} = a_2$, $\ldots$, $a_{1,k_1} = a_{k_1}$, $a_{1,k_1 + 1} = \max\{1, q_{k_1}(r_1 + s_1/q_{k_1}) - 2\}$, and let $a_{1,t} = 1$ for $t > k_1 + 1$ to obtain a new number $\alpha_1$ with continued fraction expansion $[0; a_{1,1}, a_{1,2}, \ldots]$. Analogously, select then $k_2$ to be the least positive integer such that $k_2 > k_1$ and that there exist nonnegative integers $r_2$ and $s_2$ such that $\lambda - (r_2 + s_2/q_{1,k_2}) < \tfrac14$ where $q_{1,k_2}$ is the denominator of the $\smash[t]{k_2^\text{th}}$ convergent of $\alpha_1$. Set $a_{2,1} = a_{1,1}$, $\ldots$, $a_{2,k_2} = a_{1,k_2}$, $a_{2,k_2 + 1} = \max\{1, q_{1,k_2}(r_2 + s_2/q_{1,k_2}) - 2\}$, and let $a_{2,t} = 1$ for $t > k_2 + 1$ to again obtain a number $\alpha_2$ with continued fraction expansion $[0; a_{2,1}, a_{2,2}, \ldots]$. Repeating this procedure yields sequences $(k_t)$, $(r_t)$, $(s_t)$ and a number $\beta$ with continued fraction expansion $[0; b_1, b_2, \ldots]$ and subsequence $(p'_t/q'_t)_t$ of its convergents such that $$\lambda - \frac{b_{k_t + 1} + 2}{q'_{k_t}} < \frac{1}{2^t}$$ for all $t \geq 1$ (the numbers $a_{t,k_t + 1}$ will grow arbitrarily large since $\lambda > 0$). We conclude that $$\limsup_{t \to \infty} \frac{{\mathpzc{A\mkern-3mu e}_{\infty,\beta}(q'_{k_t})}}{q'_{k_t}} = \lambda,$$ so ${\mathpzc{A\mkern-3mu c}_{\infty}(\beta)} \geq \lambda$. As we have constructed the sequence $(b_t)_t$ in such a way that $b_t = 1$ whenever $k_i < t < k_{i+1}$ for some $i$, it follows for such $i$ and $t$ large enough that $$\frac{b_t + 2}{q'_{t-1}} \leq \frac{b_{k_i} + 2}{q'_{t-1}} < \frac{b_{k_i} + 2}{q'_{k_i - 1}} \leq \lambda.$$ Therefore ${\mathpzc{A\mkern-3mu c}_{\infty}(\beta)} = \lambda$ and $\lambda \in {\mathcal{L}_{\infty}}$.
Additional Questions {#sec:examples}
====================
At the end of , we asked if there exists infinite words for which the condition of on abelian equivalence is redundant. The next proposition tells that such binary words exist but that they are rather uninteresting.
\[prp:no\_binary\] Let ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ be an infinite binary word such that for each of its factors $u$ and $v$ of equal length we have $u \sim_1 v$ if they share a common prefix and a common suffix of length $1$. Then ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ is ultimately periodic.
Suppose for a contradiction that ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ is aperiodic, so either $00$ or $11$ occurs in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. By symmetry, we assume that $00$ is a factor of ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. If $0011$ occurs also, then $001$ and $011$ occur. This is impossible as then by our assumption we should have $001 \sim_1 011$; this is clearly absurd. Thus $0010^n1$ occurs in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ for some $n \geq 1$. The factors $000$ and $010$ are also incompatible, so $000$ cannot occur in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. Hence $101$ and $1001$ are the only possible factors of the form $10^n 1$ with $n \geq 1$. Since $(100)^\omega$ is not a suffix of ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$, either $101$ occurs or $10011$ must occur. The latter case we already ruled out, so $101$ occurs meaning that $111$ is not a factor of ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. If $11$ is not a factor, then ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ has a suffix that is a concatenation of the words $10$ and $100$. Suppose then that $11$ is a factor. The only way this is possible is that we have an occurrence of $1011$. This means that we do not see the incompatible factor $1001$. Hence $00$ occurs only as a prefix of ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. We have concluded that ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ has a suffix that is a product of the words $01$ and $011$. Thus by mapping ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ with the coding $0 \mapsto 1$, $1 \mapsto 0$, we obtain a word satisfying the assumptions and which has a suffix that is a product of $10$ and $100$. Thus without loss of generality, we may assume that ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ has a suffix that is a product of $10$ and $100$.
If $100(10)^n100$ occurs in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ for two distinct values of $n$, then for some $m\geq 0$ both $00(10)^m100$ and $0(10)^{m+1}10$ are factors of ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$. By our assumption, we must have $00(10)^m100 \sim_1 0(10)^{m+1}10$, but this is false. Therefore $100(10)^n100$ can occur only for a single value $n$, and ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ must have either of the words $(10)^\omega$ or $(100(10)^n)^\omega$ as a suffix. This is a contradiction.
However, if we allow more than two letters, then aperiodicity is possible as is shown by the next proposition. Let $A$ and $B$ be alphabets. Recall that a substitution $f\colon A^* \to B^*$ is a mapping such that $f(uv) = f(u)f(v)$. The image of the infinite word $a_0 a_1 \cdots$ under $f$ is the infinite word $f(a_0) f(a_1) \cdots$. If $w = uv$, then by $wv^{-1}$ we mean the word $u$. In the next proof, we need to know some properties of Sturmian words; these can be found in [@2002:algebraic_combinatorics_on_words Chapter 2]. Firstly, Sturmian words are *balanced*. This means that for each two factors $u$ and $v$ of equal length occurring in some Sturmian word, we have ${\lvert{\lvertu\rvert}_0 - {\lvertv\rvert}_0\rvert} \leq 1$. Secondly in a Sturmian word, there exists exactly one right special factor of length $n$ for all $n \geq 0$. A factor $u$ of an infinite word ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ is *right special* if $ua$ and $ub$ occur in ${ \ifcat\noexpandw\relax\bm{w} \else\mathbf{w}\fi}$ for distinct letters $a$ and $b$.
Let $\sigma$ be the substitution defined by $\sigma(0) = 02$, $\sigma(1) = 1$. It is easy to see that the word $\sigma({ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi})$ is aperiodic for any Sturmian word ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$.
\[prp:ternary\_example\] Let $k \geq 2$ and ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$ be a Sturmian word containing $00$. Let $u$ and $v$ be two factors of the same length occurring in $\sigma({ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi})$. Then $u \sim_k v$ if and only if they share a common prefix and a common suffix of length $\min\{{\lvertu\rvert}, k-1\}$.
Suppose that $u$ and $v$ share a common prefix and a common suffix of length $\min\{{\lvertu\rvert}, k-1\}$. We proceed as in the proof of [@2013:on_a_generalization_of_abelian_equivalence_and_complexity_of_infinite Proposition 2.8] (this is the proof of ). In this proof it is assumed that $u \sim_1 v$ and a counting argument is used to show that $u \sim_{\ell+1} v$ if $u \sim_\ell v$ for $1 \leq \ell < k$. By a careful analysis, it can be seen that this counting argument only uses the fact that there exists at most one right special factor of length $n$ for each $n$. Let $w$ and $w'$ be two right special factors of equal length occurring in $\sigma({ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi})$. It is clear that both $w$ and $w'$ must end with $2$. By the form of the substitution $\sigma$, there exist words $a$ and $b$ and unique factors $x$ and $y$ of ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$ such that $a, b \in \{\varepsilon, 0\}$, ${\lvertx\rvert} \geq {\lverty\rvert}$, $aw = \sigma(x)$, and $bw' = \sigma(y)$. Since $w$ and $w'$ are right special, so are $x$ and $y$. It follows that $y$ is a suffix of $x$, so $w$ and $w'$ are suffixes of $\sigma(x)$. Since ${\lvertw\rvert} = {\lvertw'\rvert}$, they are equal. Thus we argued that $u \sim_k v$ if and only if they share a common prefix and a common suffix of length $\min\{{\lvertu\rvert}, k-1\}$ and $u \sim_1 v$. Thus it suffices to show that $u \sim_1 v$.
Like above, there exist words $a$ and $b$ and unique factors $x$ and $y$ of ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$ such that $a \in \{\varepsilon, 0\}$, $b \in \{\varepsilon, 2\}$, $aub = \sigma(x)$, and $avb = \sigma(y)$. Let us show next that $x$ and $y$ are abelian equivalent. The claim follows from this. Since $k \geq 2$, the words $x$ and $y$ end in a common letter $c$. Now $x \sim_1 y$ if and only if $xc^{-1} \sim_1 yc^{-1}$ so, by replacing $x$ with $xc^{-1}$ and $y$ with $yc^{-1}$ if necessary, we may assume that $x$ and $y$ end with the letter $0$ ($1$ is always preceded by $0$ since ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$ is balanced). For each binary word $w$, we have ${\lvert\sigma(w)\rvert} = {\lvertw\rvert} + {\lvertw\rvert}_0$. Since ${\lvertu\rvert} = {\lvertv\rvert}$ (if $x$ and $y$ were replaced, we must replace $u$ and $v$ respectively by $\sigma(xc^{-1})$ and $\sigma(yc^{-1})$), we have $$\label{eq:lengths}
{\lvertx\rvert} + {\lvertx\rvert}_0 = {\lverty\rvert} + {\lverty\rvert}_0.$$ Suppose without loss of generality that ${\lvertx\rvert} \geq {\lverty\rvert}$, and write $x = zt$ with ${\lvertz\rvert} = {\lverty\rvert}$. By plugging this into , we obtain that ${\lvertt\rvert} + {\lvertt\rvert}_0 = {\lverty\rvert}_0 - {\lvertz\rvert}_0$. Since ${ \ifcat\noexpands\relax\bm{s} \else\mathbf{s}\fi}$ is balanced, we see that ${\lvertt\rvert} + {\lvertt\rvert}_0 \leq 1$. Thus $t = \varepsilon$ or $t = 1$. The latter case is impossible as $x$ ends with $0$, so $t = \varepsilon$. Thus ${\lvertx\rvert} = {\lverty\rvert}$ and so ${\lvertx\rvert}_0 = {\lverty\rvert}_0$ by . This means that $x \sim_1 y$.
Sturmian and episturmian words satisfy the property of and it was shown in [@2018:on_the_k-abelian_complexity_of_the_cantor_sequence] that the Cantor word satisfies the property as well. The authors of [@2018:on_the_k-abelian_complexity_of_the_cantor_sequence] asked what sort of words satisfy this property. As we remarked above in the proof of , any infinite word having at most one right special factor of each length also satisfies this property. provides more examples of such words.
Further Open Problems {#sec:open_problems}
=====================
It would be nice if our combinatorial generalization of the Lagrange spectrum had some number-theoretic interpretation, perhaps in connection to rational approximations of irrational numbers. We are unaware of such a connection.
Is there an arithmetical characterization of the $k$-Lagrange spectrum ${\mathcal{L}_{k}}$?
An obvious open problem is to determine the $k$-abelian critical exponent of non-Sturmian infinite words. For example: what is the $k$-abelian critical exponent of the Tribonacci word, the fixed point of the substitution $0 \mapsto 012$, $1 \mapsto 02$, $2 \mapsto 0$? What about the Thue-Morse word? The case $k = 1$ is clear for the Thue-Morse word as the whole infinite word is an abelian power of infinite exponent and period $2$.
Instead of looking at particular words or classes of words, it would be interesting to determine the set of critical exponents of all infinite words. In [@2007:every_real_number_greater_than_1], Krieger and Shallit show that every real number greater than $1$ is a critical exponent of some infinite word. The result of Freiman shows that every real number greater than $c_F$ is the abelian critical exponent of some infinite word. Our result shows that a dense subset of $(\tfrac{c_F}{2k-1}, \infty)$ is attainable as $k$-abelian critical exponents when $k > 1$. We are thus led to ask the following question.[^2]
Is every nonnegative real number the $k$-abelian critical exponent of some infinite word?
In [@2016:abelian_powers_and_repetitions_in_sturmian_words], the abelian periods of factors of Sturmian words were studied (for definitions, see [@2016:abelian_powers_and_repetitions_in_sturmian_words]). It was proven for example that the abelian period of a factor of the Fibonacci word is always a Fibonacci number. Same sort of questions could be asked in the $k$-abelian setting for Sturmian words more generally. We have not attempted this study.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the referee for a careful reading of the paper, which improved the presentation.
[^1]: Corresponding author.\
E-mail addresses: <r@turambar.org> (J. Peltomäki), <mawhit@utu.fi> (M. A. Whiteland).
[^2]: The question has been solved in the positive; see [@2019:every_nonnegative_real_number_is_a_critical].
|
---
abstract: 'Single molecule mechanical unfolding experiments are beginning to provide profiles of the complex energy landscape of biomolecules. In order to obtain reliable estimates of the energy landscape characteristics it is necessary to combine the experimental measurements (the force extension curves, the mechanical unfolding trajectories, force or loading rate dependent unfolding rates) with sound theoretical models and simulations. Here, we show how by using temperature as a variable in mechanical unfolding of biomolecules in laser optical tweezer or AFM experiments the roughness of the energy landscape can be measured without making any assumptions about the underlying reaction coordinate. The efficacy of the formalism is illustrated by reviewing experimental results that have directly measured roughness in a protein-protein complex. The roughness model can also be used to interpret experiments on forced-unfolding of proteins in which temperature is varied. Estimates of other aspects of the energy landscape such as free energy barriers or the transition state (TS) locations could depend on the precise model used to analyze the experimental data. We illustrate the inherent difficulties in obtaining the transition state location from loading rate or force-dependent unfolding rates. Because the transition state moves as the force or the loading rate is varied it is in general difficult to invert the experimental data unless the curvature at the top of the one dimensional free energy profile is large, i.e the barrier is sharp. The independence of the TS location on force holds good only for brittle or hard biomolecules whereas the TS location changes considerably if the molecule is soft or plastic. We also comment on the usefulness of extension of the molecule as a surrogate reaction coordinate especially in the context of force-quench refolding of proteins and RNA.'
author:
- 'Changbong Hyeon$^1$ & D. Thirumalai$^{1,2}$'
title: Measuring the energy landscape roughness and the transition state location of biomolecules using single molecule mechanical unfolding experiments
---
= 22pt
Introduction
============
Representation of the large conformational space of RNA and proteins in terms of (low-dimensional) energy landscape has played an important role in visualizing their folding routes [@DillNSB97; @OnuchicCOSB04; @HyeonBC05]. It is suspected that the energy landscape of many evolved proteins is relatively smooth so that they can be navigated very efficiently. By smooth we mean that the gradient of the energy landscape $\Delta F(\chi,
R_g)$ towards the native basin of attraction (NBA) is “large” enough that the biomolecule does not get kinetically trapped in competing basins of attraction for long times during the folding process. Here, $F(\chi, R_g)$ is expressed in terms two non-unique variables, namely, the radius of gyration $R_g$ and $\chi$, (or equivalently $Q$ the fraction of native contacts), an order parameter that measures how similar a given conformation is to the native state. However, perfectly smooth with energy landscapes are difficult to realize because of energetic and topological frustration [@ThirumAccChemRes96; @ClementiJMB00]. In proteins, the hydrophobic residues prefer to be sequestered in the interior while polar and charged residues are better accommodated on the surfaces where they can interact with water. Often these conflicting requirements cannot be simultaneously satisfied and hence proteins can be energetically “frustrated”. It is clear from this description that only evolved or well designed sequences can minimize energetic frustration. Even if a particular foldable sequence minimizes energetic conflicts it is nearly impossible to eliminate topological frustration which arises due to chain connectivity [@GuoBP95; @ThirumalaiRNA00]. If the packing of locally formed structures is in conflict with the global fold then the polypeptide or polynucleotide chain is topologically frustrated. Both sources of frustration, energetic and topological, render the energy landscape rugged on length scales that are larger than those in which secondary structures ($\approx (1-2)$ $nm$) form even if folding can be globally described using only two-states (folded and unfolded).
An immediate consequence of frustration is that the free energy, projected along a one dimensional coordinate, is rough on certain length scale and may be globally smooth on larger scale. Let us assume that the characteristic roughness $\overline{\epsilon}$ has a Gaussian distribution. Following footnote 45 in [@ThirumalaiPRA89] the overall transit time from the unfolded basin may be written as $$\tau_{U\rightarrow F}\approx \tau(\beta)\int^{\infty}_{k_BT}d\overline{\epsilon}
e^{\beta\overline{\epsilon}}e^{-\overline{\epsilon}^2/2\epsilon^2}\approx
\tau(\beta)e^{\beta^2\epsilon^2/2}
\label{eqn:roughness}$$ where $\epsilon$ is the average value of ruggedness. The second part of the equation holds good at low temperatures. The additional slowing down in the folding time $\tau_{U\rightarrow F}$ arising from the second term in Eq.\[eqn:roughness\], was derived in an elegant paper by Zwanzig [@ZwanzigPNAS88] and was also obtained in [@Bryngelson89JPC] by analyzing the dynamics of Derrida’s random energy model [@Derrida81PRB]. If $\beta\epsilon$ is small then $\tau_{U\rightarrow F}\approx \tau_0e^{\beta\Delta F^{\ddagger}}$ where $\Delta F^{\ddagger}$ is the overall folding free energy barrier.
If folding takes place in a rough energy landscape then the characteristic scale-dependent time scale may be estimated as $$\tau(l)\approx\left\{ \begin{array}{ll}
\tau_{SS}\approx (10-100) ns& \mbox{$l\approx (1-2)$ $nm$}\\
\tau_{U\rightarrow F}& \mbox{$l\approx L$}\end{array}\right.
\label{eqn:fluctuation}$$ where $L$ is the effective contour length of the biomolecule. We have assumed that structures on $l\approx (1-2)$ $nm$ form in $\tau_{SS}\approx l^2/D$ where the diffusion constant is on the order of $(10^{-6}-10^{-7})$ $cm^2/sec$. The estimate of $\tau_{SS}$ is not inconsistent with the time needed to form $\alpha$-helices or $\beta$-hairpin especially given the crude physical picture.
With the possibility of manipulating biological molecules (Fig.\[RoughreviewFig1.fig\]), one molecule at a time, using force it is becoming possible to probe the features of their energy landscape (such as roughness and the transition state location) that are not easily possible using conventional experiments. Such experiments, performed using Laser Optical Tweezers (LOTs) [@TinocoARBBS04; @Ritort06JPHYS] or Atomic Force Microscopy (AFM) [@FernandezTIBS99], have made it possible to mechanically unfold proteins [@MarquseeScience05; @FernandezNature99; @RiefPNAS04; @GaubSCI97; @BustamantePNAS00; @Oberhauser98Nature; @BustamanteSCI94], RNA [@Bustamante01Science; @Bustamante03Science; @Woodside06PNAS; @Block06Science; @TinocoBJ06; @Tinoco06PNAS; @OnoaCOSB04], and their complexes [@Moy94Science; @Fritz98PNAS; @EvansNature99; @Schwesinger00PNAS; @ZhuNature03; @NevoNSB03], or initiate refolding of proteins [@FernandezSCI04] and RNA [@TinocoBJ06; @Tinoco06PNAS]. These remarkable experiments show how the initial conditions affect refolding and also enable us to examine the response of biological molecules over a range of forces and loading rates. In addition, fundamental aspects of statistical mechanics, including non-equilibrium work theorems [@JarzynskiPRL97; @Crooks99PRE], can be rigorously tested using the single molecule experiments [@Bustamante02Science; @Trepagnier04PNAS]. Here, we are concerned with using the data and theoretical models to extract key characteristics of the energy landscape of biological systems.
The crude physical picture of folding in a rough energy landscape (Fig.\[landscape\]) is not meaningful unless the ideas can be validated experimentally which requires direct measurement of the roughness energy scale $\epsilon$, barrier height, etc. In conventional experiments, in which folding is triggered by temperature, it is difficult to measure $\tau_0$ and $\Delta F^{\ddagger}$ even when $\beta\epsilon\equiv 0$ [@GruebeleNature03]. We proposed, using theoretical methods, that $\beta\epsilon$ can be directly measured using forced-unfolding of biomolecules and biomolecular complexes. The Hyeon-Thirumalai (HT) theory [@HyeonPNAS03], which was based on Zwanzig’s treatment [@ZwanzigPNAS88], showed that if unbinding or unfolding lifetime (or rates) are known as a function of the stretching force ($f$) and temperature ($T$) then $\epsilon$ can be inferred without explicit knowledge of $\tau_0$ or $\Delta F^{\ddagger}$. Recently, the loading-rate dependent unbinding times of a protein-protein complex using atomic force microscopy (AFM) at various temperatures have been used to obtain an estimate of $\epsilon$ [@ReichEMBOrep05]. Similarly, variations in the forced-unfolding rates as a function of temperature of *Dictyostelium discoideum filamin* (ddFLN4) were used to estimate $\epsilon$ [@RiefJMB05]. Although alternate interpretation of the data is proposed for temperature effect on ddFLN4 the variation in unbinding or unfolding rates of proteins as a function of $f$ and $T$ provides an opportunity to obtain quantitative estimates of the energy landscape characteristics.
Single molecule force spectroscopy can also be used to measure force-dependent unfolding rates from which the location of the transition state (TS) in terms of the spatial extension ($R$) can be computed. This procedure is a not straightforward because, as shown in a number of studies [@HyeonBJ06; @HyeonPNAS05; @Lacks05BJ; @West06BJ; @AjdariBJ04], the location of the transition state changes as $f$ changes unless the curvature of the free energy profile at the TS location is large, i.e, the barrier is sharp. The extent to which the TS changes depends on the load. We propose distinct scenarios for variation of the TS location as the external conditions change. By carefully considering the variations of force distributions it is possible to obtain reliable estimates of the TS location [@RiefJMB05]. Here, we review recent developments in single molecule force spectroscopy that have attempted to obtain the energy landscape characteristics of biological molecules [@BellSCI78; @Evans97BJ; @HyeonPNAS03; @Dudko03PNAS; @HummerBJ03; @Barsegov05PRL; @Barsegov06BJ; @AjdariBJ04]. Using theoretical models we also point out some of the ambiguities in interpreting the experimental data from dynamic force spectroscopy.
Theoretical background
======================
Single molecule mechanical unfolding experiments differ from conventional unfolding experiments in which unfolding (or folding) is triggered by varying temperature or concentration of denaturants or ions. In single molecule experiments folding or unfolding can be initiated by precisely manipulating the initial conditions. In both forced-unfolding and force-quench refolding the initial conformation, characterized by the extension of the biomolecules, are precisely known. On the other hand, the nature of the unfolded states, from which refolding is initiated, is hard to describe in ensemble experiments. For RNA and proteins, whose energy landscape is complex [@TreiberCOSB01; @HyeonBC05], details of the folding pathways can be directly monitored by probing the time dependent changes in the end-to-end distance $R(t)$ of individual molecules. Analysis of such mechanical folding and unfolding trajectories allows to explore regions of the energy landscape that are difficult to probe using ensemble experiments.
Force experiments can measure the extension of the molecule as a function of time. There are three modes in which the stretching experiments are performed. Most of the initial experiments were performed by unfolding biomolecules (especially proteins) by pulling on one of the molecule at constant velocity while keeping the other end fixed [@Bustamante01Science; @Bustamante03Science; @FernandezNature99; @Reif; @Fisher00NSB]. More recently, it has become possible to apply constant force on the molecule of interest using feed-back mechanism [@Visscher99Science; @Schlierf04PNAS; @FernandezSCI04; @TinocoBJ06; @Fernandez06NaturePhysics]. In addition, force-quench experiments have been reported in which the forces are decreased or increased linearly [@FernandezSCI04; @TinocoBJ06; @Tinoco06PNAS]. It is hoped that a combination of such experiments can provide a detailed picture of the complex energy landscape of proteins and RNA. In all the modes, the variable conjugate to $f$ is the natural coordinate that describes the progress of the reaction of interest (folding, unbinding or catalysis). If there is a energy barrier confining the molecular motion to a local minimum, whose height is greater than $k_BT$, then a sudden increase (decrease) of extension (force) signifies the transition of the molecule over the barrier. A cusp in the force-extension curve (FEC) is the signature of such a transition. Surprisingly, for proteins and RNA it has been found that the FECs can be quantitatively fit using the semi-flexible or worm-like chain model [@MarkoMacro96; @Bustamante97Science; @Bustamante01Science; @Bustamante03Science; @FernandezNature99; @Fisher00NSB]. From such fits, the global polymeric properties of the biomolecule, such as the contour length and the persistence length can be extracted [@MarkoMacro96; @BustamanteSCI94].
Single molecule experiments provide distributions of the unfolding times (or unfolding force) by varying external conditions. The objective is to construct the underlying energy landscape from such measurements and from mechanical folding or unfolding trajectories. However, it is difficult to construct from FEC or mechanical folding trajectories that report only changes at two points all the features of the energy landscape of biomolecules. For example, although signature of roughness in the energy landscape may be reflected as fluctuations in the dynamical trajectory it is difficult to estimate its value unless multiple pulling experiments are performed. We had proposed that the power of single molecules can be more fully realized if temperature ($T$) is also used as an additional variable [@Klimov01JPCB; @Klimov00PNAS; @HyeonPNAS03; @HyeonPNAS05]. By using $T$ and $f$ it is possible to obtain the phase diagram as a function of $T$ and $f$ that can be used to probe the nature of collapsed molten globules which are invariably populated but are hard to detect in conventional experiments. We also showed theoretically that the roughness energy scale ($\epsilon$) can be measured [@HyeonPNAS03] if both $f$ and temperature ($T$) are varied in single molecule experiments. The effect of $\epsilon$ manifests itself in the $1/T^2$ dependence of the rates of force-induced unbinding or unfolding kinetics. In the following subsections we first review the theoretical framework to describe the force-induced unfolding kinetics and show how the $1/T^2$ dependence emerges when the roughness is treated as a perturbation in the underlying energy profile.
[*Bell model:* ]{} Historically, a phenomenological description of the forced-unbinding of adhesive contacts was made by Bell [@BellSCI78] long before single molecule experiments were performed. In the context of ligand unbinding from binding pocket, Bell [@BellSCI78] conjectured that the kinetics of bond rupture can be described using a modified Eyring rate theory [@EyringJCP35], $$k=\kappa\frac{k_BT}{h}e^{-(E^{\ddagger}-\gamma f)/k_BT}
\label{eqn:Erying}$$ where $k_B$ is the Boltzmann constant, $T$ is the temperature, $h$ is the Planck constant, and $\kappa$ is the transmission coefficient. In the Bell description the activation barrier $E^{\ddagger}$ is reduced by a factor $\gamma\times f$ when the bond or the biomolecular complex is subject to external force $f$. The parameter $\gamma$ is a characteristic length of the system under load and specifies the distance at which the molecule unfolds or the ligand unbinds. The prefactor $\frac{k_BT}{h}$ is the vibrational frequency of a single bond. The Bell model shows that the unbinding rates increase when tension is applied to the molecule. Although Bell’s key conjecture, i.e., the reduction of activation barrier due to external force, is physically justified, the assumption that $\gamma$ does not depend on the load is in general not valid. In addition, because of the multidimensional nature of the energy landscape of biomolecules, there are multiple unfolding pathways which require modification of the Bell description of forced-unbinding. It is an oversimplification to restrict the molecular response to the force merely to a reduction in the free energy barrier. Nevertheless, in the experimentally accessible range of loads the Bell model in conjunction with Kramers’ theory of escape from a potential well have been remarkably successful in fitting much of the data on forced-unfolding of biological molecules.
[*Mean first passage times:*]{} In order to go beyond the popular Bell model many attempts have been made to describe unbinding process as an escape from a free energy surface in the presence of force [@Evans97BJ; @HummerBJ03; @HyeonPNAS03; @BarsegovPNAS05; @Dudko06PRL]. This is traditionally achieved by a formal procedure that adapts the Liouville equation that describes the time evolution of the probability density representing the molecular configuration on the phase space.
For the problem at hand, one can project the entire dynamics onto a single reaction coordinate provided the relaxation times of other degrees of freedom are shorter than the time scale associated with the presumed order parameter of interest [@Zwanzig60JCP; @ZwanzigBook]. In applications to force-spectroscopy, we assume that the the variable conjugate to $f$ is a reasonable approximation to the reaction coordinate. The probability density of the molecular configuration, $\rho(x,t|x_0)$, whose configuration is represented by order parameter $x$ at time $t$, obeys the Fokker Planck equation.
$$\frac{\partial\rho(x,t|x_0)}{\partial t}=\mathcal{L}_{FP}(x)\rho(x,t|x_0)=\frac{\partial}{\partial x}D(x)\left(\frac{\partial}{\partial x}+\frac{1}{k_BT}\frac{dF(x)}{dx}\right)\rho(x,t|x_0),
\label{eqn:FP}$$
where $D(x)$ is the position-dependent diffusion coefficient, and $F(x)$ is an effective one-dimensional free energy, $x_0$ is the position at time $t=0$. If the initial distribution is given by $\rho(x_0,t=0|x_0)=\delta(x-x_0)$ the formal solution of the above equation reads $\rho(x,t|x_0)=e^{t\mathcal{L}_{FP}}\delta(x-x_0)$. If we use absorbing boundary condition at a suitably defined location, the probability that the molecule remains bound (survival probability) at time $t$ is $$S(x_0,t)=\int dx\rho(x,t|x_0)=\int dxe^{t\mathcal{L}_{FP}}\delta(x-x_0).
\label{eq:S}$$ In terms of the the first passage time distribution, $p_{FP}(x_0,t)(=-dS(x_0,t)/dt)$, the mean first passage time can be computed using, $$\begin{aligned}
\tau(x_0)&=&\int_0^{\infty}dt \left[t p_{FP}(x_0,t)\right]\nonumber\\
&=&-\int^{\infty}_0dt\left[t\frac{dS(x_0,t)}{dt}\right]\nonumber\\
&=&\int^{\infty}_0dt\int dxe^{t\mathcal{L}_{FP}(x)}\delta(x-x_0)\nonumber\\
&=&\int^{\infty}_0dt\int dx\delta(x-x_0)e^{t\mathcal{L}^{\dagger}_{FP}(x)
}
\label{eq:detail}\end{aligned}$$ where $\mathcal{L}^{\dagger}_{FP}$ is the adjoint operator. In obtaining the above equation we used $S(x_0,t=\infty)=0$ and integrated by parts in going from the second to third third line. By operating on both sides of Eq.\[eq:detail\] with $\mathcal{L}_{FP}^{\dagger}(x_0)$ and exchanging the variable $x$ with $x_0$ we obtain $$\mathcal{L}^{\dagger}_{FP}(x)\tau(x)=e^{F(x)/k_BT}\frac{\partial}{\partial x}D(x)e^{-F(x)/k_BT}\frac{\partial}{\partial x}\tau(x)=-1.$$ The rate process with reflecting boundary $\partial_x\tau(a)=0$ and absorbing boundary condition $\tau(b)=0$ in the interval $a\leq x\leq b$, leads to the expression of mean first passage time, $$\tau(x)=\int^b_x dye^{F(y)/k_BT}\frac{1}{D(y)}\int^y_a dz e^{-F(z)/k_BT}.
\label{eqn:mfpt}$$
[*Diffusion in a rough potential:*]{} In the above analysis the one-dimensional free energy profile $F(x)$ that approximately describes the unfolding or unbinding event is arbitrary. In order to explicitly examine the role of the energy landscape ruggedness we follow Zwanzig and decompose $F(x)$ into $F(x)=F_0(x)+F_1(x)$ [@ZwanzigPNAS88]. where $F_0(x)$ is a smooth potential that determines the global shape of the energy landscape, and $F_1(x)$ is the periodic ruggedness that superimposes $F_1(x)$. By taking the spatial average over $F_1(x)$ using $\langle e^{\pm\beta F_1(x)}\rangle_l=\frac{1}{l}\int^l_0dx e^{\pm\beta F_1(x)}$, where $l$ is the ruggedness length scale, the associated mean first passage time can be written in terms of the effective diffusion constant $D^*(x)$ as, $$\begin{aligned}
D^*(x)&=&\frac{D(x)}{\langle e^{\beta F_1(x)}\rangle_l\langle e^{-\beta F_1(x)}\rangle_l}, \nonumber\\
\tau(x)&\approx&\int^b_x dye^{F_0(y)/k_BT}\frac{1}{D^*(y)}\int^y_a dz e^{-F_0(z)/k_BT}.
\label{eq:MFPT}\end{aligned}$$ An inversion of roughness barrier, i.e., $F_1\leftrightarrow -F_1$ does not alter $D^*(x)$. In the presence of roughness $D^*(x)\leq D(x)$. Depending on the distribution of roughness barrier, $D^*(x)$ can take various forms:
1. For $F_1(x)=\epsilon x/b$ $(0\leq x\leq b)$ and $F_1(x)=\epsilon(a-x)/(a-b)$ $(b\leq x\leq a)$ with $F_1(x)=F_1(x+a)$, $\langle e^{\beta F_1(x)}\rangle\langle e^{-\beta F_1(x)}\rangle=\left[\frac{\sinh{\beta\epsilon/2}}{\beta\epsilon/2}\right]^2$ [@Festa78PhysicaA].
2. For $F_1(x)=\epsilon\cos{qx}$, $\langle e^{\beta F_1(x)}\rangle\langle e^{-\beta F_1(x)}\rangle=[I_0(\beta\epsilon)]^2$ [@ZwanzigPNAS88].
3. If $P(F_1)$ is Gaussian with variance $\langle F_1^2\rangle=\epsilon^2$, $\langle e^{\beta F_1(x)}\rangle\langle e^{-\beta F_1(x)}\rangle=e^{\beta^2\epsilon^2}$ [@ZwanzigPNAS88].
4. If $P(F_1)=P(-F_1)$, then $\langle e^{\pm\beta F_1(x)}\rangle_l=\int dF_1P(F_1)\left[1+\frac{1}{2!}\beta^2\epsilon^2+\frac{1}{4!}\beta^4\epsilon^4+\cdots\right]$.
In all cases when $\beta\epsilon$ is small the effective diffusion coefficient can be approximated as $D_0^*\approx D\exp{\left(-\beta^2\epsilon^2\right)}$ where $D_0$ is the bare diffusion constant. If $P(F_1)$ is a Gaussian then this expression is exact.
From the recent experimental analysis on mechanical unfolding kinetics of the multi-ubiquitin construct, Fernandez and coworkers [@Fernandez06NaturePhysics] obtained a power-law distribution of unfolding times i.e., $P(\tau)\propto \tau^{-(1+a)}$, and showed that the distribution of energy barrier heights should be $P(F_1)\sim\exp(-|F_1|/\overline{F_1})$ where $\overline{F_1}=k_BT/a$ [@Fernandez06NaturePhysics]. This case belongs to class 4, thus one obtains $e^{-\epsilon^2/k_B^2T^2}$ behavior associated with the effective diffusion coefficient. These examples show that the coefficient associated with $1/T^2$ behavior is due to the energy landscape roughness provided the extension is a good reaction coordinate. The dependence of $e^{-\epsilon^2/k_B^2T^2}$ in $D^*$ suggests that the diffusion in rough potential can be substantially slowed even when the scale of roughness is not too large.
[*Barrier crossing dynamics in a tilted potential:*]{} In writing the one-dimensional Fokker-Planck equation (Eq.\[eqn:FP\]) we assumed that the order parameter $x$ is a slowly changing variable. This assumption is valid if the molecular extension, in the presence of $f$, describes accurately the conformational changes in the biomolecule.
Following the Bell’s conjecture we can replace $F(x)$ by $F(x)-f\cdot x$ in which $f$ “tilts” the free energy surface. Thus, in the presence of mechanical force Eq.\[eq:MFPT\] becomes $$k^{-1}(f)=\tau(f;x)\approx\int^b_x dye^{(F_0(y)-fy)/k_BT}\frac{1}{D^*(y)}\int^y_a dz e^{-(F_0(z)-fz)/k_BT}.
\label{eqn:MFPT_force}$$ As long as the energy barrier is large enough (see Fig.\[landscape\]) Eq.\[eqn:MFPT\_force\] can be further simplified using the saddle point approximation. The Taylor expansions of the free energy potential $F_0(x)-fx$ at the barrier top and the minimum result in the Kramers’ equation [@KramersPhysica40; @Hanggi90RMP], $$\begin{aligned}
k^{-1}(f)=\tau(f)&\approx&\frac{2\pi k_BT}{D^*m\omega_b(f)\omega_{ts}(f)}e^{\beta(\Delta F_0^{\ddagger}(f)-f\Delta x(f))}\nonumber\\
&=&\left(\frac{2\pi\zeta}{\omega_b(f)\omega_{ts}(f)}\langle e^{\beta F_1(x)}\rangle_l\langle e^{-\beta F_1(x)}\rangle_l\right)e^{\beta(\Delta F_0^{\ddagger}(f)-f\Delta x(f))}
\label{eqn:Kramers}\end{aligned}$$ where $\omega_b$ and $\omega_{ts}$ are the curvatures of the potential, $|\partial^2_xF_0(x)|$, at $x=x_b$ and $x_{ts}$, respectively, the free energy barrier $\Delta F_0^{\ddagger}=F_0(x_{ts})-F_0(x_b)$, $m$ is the effective mass of the biomolecule, $\zeta$ is the friction coefficient, and $\Delta x\equiv x_{ts}-x_b$.
In the presence of $f$, the positions of transition state $x_{ts}$ and bound state $x_b$ change because unbinding kinetics should be determined using $F_0(x)-fx$ and not $F_0(x)$ alone. Because $x_{ts}$ and $x_b$ satisfy the *force dependent condition* $F_0'(x)-f=0$, it follows that all the parameters, $\Delta x(f)$, $\omega_{ts}(f)$, and $\omega_b(f)$, are intrinsically $f$-dependent. Depending on the shape of free energy potential $F_0(x)$, the degree of force-dependence of $\Delta x$, $\omega_{ts}$, and $\omega_b$ can vary greatly. Previous theoretical studies [@HummerBJ03; @Dudko03PNAS; @HyeonBJ06; @RitortPRL06] have examined some of the consequences of the moving transition state. In addition, simulational studies [@HyeonPNAS05; @Lacks05BJ; @HyeonBJ06] in which the free energy profiles were explicitly computed from thermodynamic considerations alone clearly showed the change of $\Delta x$ when $f$ is varied. These authors also provided a structural basis for transition state movements in the case of unbinding of simple RNA hairpins. The nontrivial coupling of force and free energy profile makes it difficult to unambiguously extract free energy profiles from experimental data. In order to circumvent some of the problems Schlierf and Rief have used Eq.\[eqn:MFPT\_force\] to analyze the load-dependent experimental data on unfolding of ddFLN4 and extracted an effective one dimensional free energy surface $F(x)$ without making additional assumptions. The results showed that the effective free energy profile is highly anharmonic near the transition state region [@Schlierf06BJ].
[*Forced-unfolding dynamics at constant loading rate:* ]{} Many single molecule experiments are conducted by ramping force over time [@Bustamante02Science; @Bustamante03Science; @FernandezNature99; @FernandezTIBS99]. In this mode the load on the molecule or the complex increases with time. When the force increases beyond a threshold value, unbinding or bond-rupture occurs. Because of thermal fluctuations the unbinding events are stochastic and as a consequence one has to contend with the distribution of unbinding forces. The time-dependent nature of the force makes the barrier crossing rate also dependent on $t$. For a single barrier crossing event with a time-dependent rate $k(t)$, the probability of the barrier crossing event being observed at time $t$ is $P(t)=k(t)S(t)$ where the survival probability, that the molecule remains folded, is given as $S(t)=\exp{\left(-\int ^t_0d\tau k(\tau)\right)}$.
When the molecule or complex is pulled at a constant loading rate ($r_f$) the distribution ($P(f)$) of unfolding forces is asymmetric. The most probable $r_f$-dependent unfolding force ($f^*$) is often used to determine the TS location of the underlying energy landscape with the tacit assumption that the TS is stationary. When $r_f=df/dt$ is constant, the probability of observing an unfolding event at force $f$ is written as, $$P(f)=\frac{1}{r_f}k(f)\exp{\left[-\int^f_0df'\frac{1}{r_f}k(f')\right]}.
\label{eqn:force_distribution}$$ The most probable unfolding force is obtained from $dP(f)/df|_{f=f^*}=0$, which leads to $$\begin{aligned}
f^*&=&\frac{k_BT}{\Delta x(f^*)}\lbrace\log{\left(\frac{r_f\Delta x(f^*)}{\nu_D(f^*)e^{-\beta\Delta F^{\ddagger}_0(f^*)}k_BT}\right)}\nonumber\\
&+&\log{\left(1+f^*\frac{\Delta x'(f^*)}{\Delta x(f^*)}-\frac{\left(\Delta F^{\ddagger}_0\right)'(f^*)}{\Delta x(f^*)}
+\frac{\nu'_D(f^*)}{\nu_D(f^*)}\frac{k_BT}{\Delta x(f^*)}\right)}\nonumber\\
&+&\log{\langle e^{\beta F_1}\rangle_l\langle e^{-\beta F_1}\rangle_l}\rbrace,
\label{eqn:most_force}\end{aligned}$$ where $\Delta F^{\ddagger}_0\equiv F_0(x_{ts}(f))-F_0(x_0(f))$, $\prime$ denotes differentiation with respect to the argument, $\Delta x(f)\equiv x_{ts}(f)-x_0(f)$ is the distance between the transition state and the native state, and $\nu_D(f)\equiv \omega_o(f)\omega_{ts}(f)/2\pi\gamma$. Note that $\Delta F^{\ddagger}_0$, $\nu_D$, and $\Delta x$ depend on the value of $f$ [@HyeonPNAS03; @Lacks05BJ; @HyeonPNAS05]. Because $f^*$ changes with $r_f$, $\Delta x$ obtained from the data analysis should correspond to a value at a certain $f^*$, not a value that is extrapolated to $f^*=0$. Indeed, the pronounced curvature in the plot of $f^*$ as a function of $\log{r_f}$ makes it difficult to obtain the characteristics of the underlying energy landscape using data from dynamic force-spectroscopy without a reliable theory or a model. If $\Delta F^{\ddagger}_0$, $\nu_D$, and $\Delta x$ are relatively insensitive to variations in force, the second term on the right-hand side of Eq.\[eqn:most\_force\] would vanish, leading to $f^*\propto (k_BT/\Delta x)\log{r_f}$ [@Evans97BJ]. If the loading rate, however, spans a wide range so that the force-dependence of $\Delta F^{\ddagger}_0$, $\nu_D$, and $\Delta x$ are manifested, then the resulting $f^*$ can substantially deviate from the linear dependence to the $\log{r_f}$. Indeed, it has been shown that for a molecule or a complex known to have a single free energy barrier, the average rupture force $\overline{f}\approx (\log {r_f})^{\nu}$ where the effective exponent $\nu \le 1$. The precise value of $\nu$ depends on the nature of the underlying potential and is best treated as an adjustable parameter [@Dudko06PRL].
Provided $\Delta F^{\ddagger}_0$, $\nu_D$, and $\Delta x$ are assumed constant the applicability of Kramers’ equation can be checked. In principle, the use of Kramers’ equation is justified if the effective energy barrier is at least greater than the thermal energy $k_BT$ ($\Delta^{\ddagger}=\Delta F^{\ddagger}_0-f^*\Delta x>k_BT$) at the most probable unfolding force $f^*$. Substitution of $f^*$ from Eq.\[eqn:most\_force\] leads to $$\Delta^{\ddagger}=-k_BT\log{\frac{r_f\Delta x}{\nu_Dk_BT}}.$$ The condition that $\Delta^{\ddagger}>k_BT$ is satisfied as long as $r_f<\nu_D\frac{k_BT}{e\Delta x}=r_f^c$. For the set of parameters, $k_BT=4.14$ $pN\cdot nm$, $\Delta x\sim 1$ $nm$, and $\nu_D\sim 10^6$ $s^{-1}$, the critical loading rate $r_f^c\sim 10^6$ $pN/s$. The typical loading rate used in force experiments is several orders of magnitude smaller than this value. Therefore, it is legitimate to interpret the force experiments using the formalism based on the Kramers’ equation. If $r_f>r_f^c$, as is typically the case in steered molecular dynamics simulations [@SchultenBJ98], the forced-unfolding process can no longer be considered a thermally activated barrier crossing process. At such high loading rates average (not the same as $f^*$) rupture force ($\overline{f}$) grows (non-logarithmically) as $v^{1/2}$ where $v$ is the pulling speed [@HummerBJ03].
Measurement of energy landscape roughness
=========================================
In the presence of roughness we expect that the unfolding kinetics deviates substantially from an Arrhenius behavior. By either assuming a Gaussian distribution of the roughness contribution $F_1$ ($P(F_1)\propto e^{-F_1^2/2\epsilon^2}$) or simply assuming $\beta F_1\ll 1$ and $\langle F_1\rangle=0$, $\langle F_1^2\rangle=\epsilon^2$, one can further simplify Eq.\[eqn:Kramers\] to $$\log{k(f)/k_0}=-(\Delta F_0^{\ddagger}-f\cdot\Delta x)/k_BT-\epsilon^2/k_B^2T^2.
\label{eqn:k_mod}$$ This relationship suggests that roughness scale $\epsilon$ be extracted if $\log{k(f)}$ is measured over range of temperatures. Variations in temperature also result in changes in the viscosity, $\eta$, and because $k_0^{-1}\propto\eta$, corrections arising from the temperature-dependence of $\eta$ has to be taken into account in interpreting the experiments. It is known that $\eta$ for water varies as $\exp(A/T)$ over the experimentally relevant temperature range ($5^oC<T<50^oC$) [@CRC]. Thus, we expect $\log{k(f,T)}=a+b/T-\epsilon^2/T^2$. The coefficient of the $1/T^2$ term can be quantified by performing *force-clamp experiments* at several values of constant temperatures. In addition, the robustness of the HT theory can be confirmed by showing that $\epsilon^2$ is a constant even if the coefficients $a$ and $b$ change under different force conditions [@HyeonPNAS03]. The signature of the roughness of the underlying energy landscape is uniquely reflected in the non-Arrhenius temperature dependence of the unbinding rates. Although it is most straightforward to extract $\epsilon$ using Eq.\[eqn:k\_mod\] no roughness measurement, to the best of our knowledge, has been performed using force clamp experiments.
To extract the roughness scale, $\epsilon$, using dynamic force spectroscopy (DFS) in which the force increase gradually in time, an alternative but similar strategy as in force clamp experiments can be adopted. A series of dynamic force spectroscopy experiments should be performed as a function of $T$ and $r_f$ so that reliable unfolding force distributions are obtained. Since a straightforward application of Eq.\[eqn:most\_force\] is difficult due to the force-dependence of the variables in Eq.\[eqn:most\_force\], one should simplify the expression by assuming that the parameters $\Delta x(f)$, $\Delta F_0^{\ddagger}(f)$, and $\nu_D(f)$, depend only weakly on $f$. If this is the case then the second term of Eq.\[eqn:most\_force\] can be neglected and Eq.\[eqn:most\_force\] becomes $$f^*\approx\frac{k_BT}{\Delta x}\log{r_f}+\frac{k_BT}{\Delta x}\log{\frac{\Delta x}{\nu_De^{-\Delta F_0^{\ddagger}/k_BT} k_BT}}+\frac{\epsilon^2}{\Delta x k_BT}.
\label{eqn:most_force_approx}$$ One way of obtaining the roughness scale from experimental data is as follows [@ReichEMBOrep05]. From the $f^*$ vs $\log{r_f}$ curves at two different temperatures, $T_1$ and $T_2$, one can obtain $r_f(T_1)$ and $r_f(T_2)$ for which the $f^*$ values are identical. By equating the right-hand side of the expression in Eq.\[eqn:most\_force\] at $T_1$ and $T_2$ the scale $\epsilon$ can be estimated [@HyeonPNAS03; @ReichEMBOrep05] as
$$\begin{aligned}
\epsilon^2 &\approx&\frac{\Delta x(T_1)k_BT_1\times\Delta x(T_2)k_BT_2}{\Delta x(T_1)k_BT_1-\Delta x(T_2)k_BT_2}\nonumber\\
&\times& \left[\Delta F^{\ddagger}_0\left(\frac{1}{\Delta x(T_1)}-\frac{1}{\Delta x(T_2)}\right)+\frac{k_BT_1}{\Delta x(T_1)}\log{\frac{r_f(T_1)\Delta x(T_1)}{\nu_D(T_1)k_BT_1}}-\frac{k_BT_2}{\Delta x(T_2)}\log{\frac{r_f(T_2)\Delta x(T_2)}{\nu_D(T_2)k_BT_2}}\right].
\label{eqn:epsilon}\end{aligned}$$
In a recent study Nevo et. al. [@ReichEMBOrep05] used DFS to measure $\epsilon$ for a biomolecular protein complex consisting of nuclear import receptor importin-$\beta$ (imp-$\beta$) and the Ras-like GTPase Ran that is loaded with non-hydrolyzable GTP analogue (Fig.\[Nevo\_fig\]-C). The Ran-imp-$\beta$ complex was immobilized on a surface and the unbinding forces were measured using the AFM at three values of $r_f$ that varied by nearly three orders of magnitude. At high values of $r_f$ the values of $f^*$ increases as $T$ increases. At lower loading rates ($r_f\lesssim 2\times 10^3$ $pN/s$), however, $f^*$ decreases as $T$ increases (see Fig.\[Nevo\_fig\]-B). The data over distinct temperatures were used to extract, for the first time, an estimate of $\epsilon$. The values of $f^*$ at three temperatures (7, 20, 32$^oC$) and Eq.\[eqn:epsilon\] were used to obtain $\epsilon\approx 5-6k_BT$. Nevo et. al. explicitly showed that the value of $\epsilon$ was nearly the same from the nine pairs of data extracted from the $f^*$ vs $\log{r_f}$ curves. Interestingly, the estimated value of $\epsilon$ is about $0.2 \Delta F^{\ddagger}_0$ where $\Delta F^{\ddagger}_0$ is the major barrier for unbinding of the complex. This shows that, for this complex the free energy in terms of a one-dimensional coordinate, resembles the profile shown in Fig.\[landscape\]. It is worth remarking that the location of the transition state decreases from 0.44 nm at 7$^oC$ to 0.21 nm at 32$^oC$. The extracted TS movement using the roughness model is consistent with Hammond behavior (see below).
Extracting TS location ($\Delta x$) and unfolding rate ($\kappa$) using theory of DFS
=====================================================================================
The theory of DFS, $f^*\approx\frac{k_BT}{\Delta x}\log{r_f}+\frac{k_BT}{\Delta x}\log{\frac{\Delta x}{\kappa k_BT}}$, is used to identify the forces that destabilize the bound state of the complex or the folded state of a specific biomolecule. A linear regression provides the characteristic extension $\Delta x$ at which the molecule or complex ruptures (more precisely $\Delta x$ is the thermally averaged distance between the bound and the transition state along the direction of the applied force). It is tempting to obtain the zero force unfolding rate $\kappa$ from the intercept with the abscissa. Substantial errors can, however, arise in the extrapolated values of $\Delta x$ and $\kappa$ to the zero force if $f^*$ vs $\log{r_f}$ is not linear, as is often the case when $r_f$ is varied over several orders of magnitude. Nonlinearity of $[f^*,\log{r_f}]$ curve arises for two reasons. One is due to the complicated molecular response to the external load that results in dramatic variations in $\Delta x$. Like other soft matter, the extent of response (or their elasticity) depends on $r_f$ [@HyeonBJ06; @Lacks05BJ; @West06BJ]. The other is due to multiple energy barriers that are encountered in the unfolding or unbinding process [@EvansNature99].
If the TS ensemble is broadly distributed along the reaction coordinate then the molecule can adopt diverse structures along the energy barrier depending on the magnitude of the external load. Therefore, mechanical force should grasp the signature of the spectrum of the TS conformations for such a molecule. Mechanical unzipping dynamics of RNA hairpins whose stability is determined in terms of the number of intact base pairs is a good example. The conformation of RNA hairpins at the barrier top can gradually vary from an almost fully intact structure at small forces to an extended structure at large forces. Under these conditions the width of the TSE is large. The signature of diverse TS conformations manifests itself as a substantial curvature over the broad variations of forces or loading rates. Meanwhile, if the unfolding is a highly cooperative all-or-none process characterized by a narrow distribution of the TS, the nature of the TS may not change significantly.
The linear theory of DFS is not reliable if the TSE is plastic because it involves drastic approximations of the Eq.\[eqn:most\_force\]. From this perspective it is more prudent to fit the the experimental unbinding force distributions directly using analytical expressions derived from suitable models [@Dudko06PRL] (see also APPENDIX). If such a procedure can be reliably implemented then the extracted parameters are likely to be more accurate. Solving such an inverse problem does require assuming a reduced dimensional representation of the underlying energy landscape which cannot be *a priori* justified.
Direct Analysis of the unbinding force distribution
===================================================
Although the direct fit of the measured unbinding force distributions to the $P(f)$$\left(=\frac{1}{r_f}k(f)\exp{(-\int^f_0df'\frac{1}{r_f}k(f'))}\right)$ is numerically more complicated than using Eq.\[eqn:most\_force\_approx\], it avoids a potentially serious error due to the approximation in going from Eq.\[eqn:most\_force\] to Eq.\[eqn:most\_force\_approx\]. It is, however, difficult to unearth the energy landscape characteristics of the molecule from $P(f)$ alone because $P(f)$ cannot be expressed in a simple functional form. By approximating $P(f)$ by a Gaussian we can dissect the effect of molecular topology and sequence on $P(f)$. Note that in general the measured $P(f)$ is asymmetric so that the Gaussian approximation may be of limited utility. For purposes of illustrating the nuances in the data analysis, we approximate the measured $P(f)$ by a Gaussian using Taylor expansion, $$\begin{aligned}
P(f)&=&\exp\left[-\left(\int^f_0df\frac{k(f)}{r_f}-\log{\frac{k(f)}{r_f}}\right)\right]\nonumber\\
&=&\exp\left[\frac{\kappa}{r_f}\frac{1}{\beta\Delta x}(e^{\beta f\Delta x}-1)-\beta f\Delta x-\log{\frac{\kappa}{r_f}}\right]\nonumber\\
&\sim&\exp\left[-\frac{(f-\frac{r_f-\kappa k_BT/\Delta x}{\kappa})^2}{2\left(\frac{r_fk_BT}{\kappa\Delta x}\right)}\right]
=\exp{\left(-\frac{(f-\overline{f})^2}{2\sigma_f^2}\right)}
\label{eqn:Pf_Gaussian}\end{aligned}$$ where $k(f)=\kappa e^{\beta f\Delta x}$ and $\kappa\equiv k_0e^{-\beta \Delta F_0^{\ddagger}}e^{-\beta^2\epsilon^2}$. In going from the first to the second line we assume that $\Delta x$ and $\kappa$ are independent of $f$. Two conclusions can be drawn from Eq.\[eqn:Pf\_Gaussian\]. (i) Both $f^*\sim \overline{f}(=r_f/\kappa-k_BT/\Delta x)$ and the width of the force distribution $\sigma_f(=(\frac{r_fk_BT}{\kappa\Delta x})^{1/2})$ increase as $r_f$ increases. (ii) As $T$ increases, $f^*$ decreases while $\sigma_f$ increases, *provided $\Delta x$ and $\kappa$ are independent function of $T$ or $r_f$*.
Recently, Schlierf and Rief (SR) analyzed the unfolding force distributions (with $r_f$ fixed) of a single domain of *Dictyostelium discoideum* filamin (ddFLN4) at five different temperatures to infer the underlying one dimensional free energy surface [@RiefJMB05]. Going from the measured data ($P(f)$) to the underlying energy landscape is an inverse problem that requires a specific model. Using the Bell model ($k(f)=\kappa\exp{(f\Delta x/k_BT)}$) and an approximation that the Arrhenius pre-factor $k_0$ is a constant ($\kappa=k_0\exp{(-\Delta F^{\ddagger}/k_BT)}$ where $k_0=10^7s^{-1}$) that is independent of temperature (i.e., assuming zero roughness), Schlierf and Rief extracted the energy landscape parameters ($\Delta x$, $\Delta F^{\ddagger}$) for each force distribution. Somewhat surprisingly, the SR analysis indicated that the rate (with relatively large errors) for populating an intermediate at zero force ($\kappa$) increases as $T$ is lowered. Their fits indicate that the location of the TS *must increase* as $T$ increases. Therefore, they concluded that ddFLN4 protein exhibits an anti-Hammond behavior, i.e., the position of the TS moves towards the force-stabilized ensemble of states (unfolded states).
Visual inspection of $P(f)$ at different temperatures in Figure 2 of ref. [@RiefJMB05] by SR confirms that $f^*$ decreases but $\sigma_f$ decreases as $T$ increases. This means that $\kappa\times\Delta x$ must increase faster than the increase of $k_BT$ (Note that $\sigma_f=\left(\frac{r_fk_BT}{\kappa\Delta x}\right)^{1/2}$). Although the data were fit well by assuming that *$\Delta F^{\ddagger}$ is temperature dependent $\Delta F^{\ddagger}$ while $k_0$ is a constant*, an alternative explanation was also possible as recognized by SR. In the alternate model based on energy landscape roughness we assume that $\Delta F^{\ddagger}$ is constant but $k_0$ depends on temperature [@HyeonPNAS03]. The constancy of $\Delta F^{\ddagger}$ is *a posteriori* justified in light of the SR analysis. By adopting the the HT roughness model SR showed the data can be fit using $\epsilon = 4k_BT$ for ddFLN4 unfolding. We believe that SR’s main interpretation (temperature softening effect, or anti-Hammond behavior) still holds even within the framework of the roughness model because their conclusion is entirely based on the temperature dependence of $\Delta x$. The roughness model has the advantage that the data can be fit essentially with only one parameter. Moreover, experiments on protein folding have often been interpreted using changes in the prefactor due to $\epsilon$.
To further confirm SR’s conclusion we suggest an independent measurement of the precise values of $\Delta x$ by varying not only the temperature but also the loading rate, as is demonstrated by Nevo *et. al.* (Fig.\[Nevo\_fig\]-B) [@ReichEMBOrep05]. The slopes of Fig.\[Nevo\_fig\] at different temperatures succinctly show that imp-$\beta$-RanGppNHp complex exhibit the more common “Hammond”-behavior. Instead of using Eq.\[eqn:epsilon\], the energy landscape roughness $\epsilon$ can be extracted from the multiple sets of $P(f)$ at various $(T,r_f)$ conditions. Non-Arrhenius behavior characterized particularly by $1/T^2$ is the unique feature of the roughness scale of the energy landscape, and is not affected by TS movements. More explicitly, $P(f)$ should be modeled as $$P(f)=\frac{\kappa(T;r_f)}{r_f}e^{f\Delta x(T;r_f)/k_BT}\exp{\left[-\int^f_0df'\frac{\kappa(T;r_f)}{r_f}e^{f\Delta x(T;r_f)/k_BT}\right]}$$ with $\kappa(T;r_f)=\nu_D(T)e^{-\Delta F^{\ddagger}/k_BT}e^{-\epsilon^2/k_B^2T^2}$. The overall shape of $P(f)$ is determined by $\Delta x(T;r_f)$ and $\kappa(T;r_f)$. The multiple force distributions, generated at the same loading rate but at different temperatures, can be used to fit $\kappa(T;r_f)$ using $$\log{\kappa(T)}=a+b/T-\epsilon^2/T^2.$$ Such experiments would permit unambiguous extraction of roughness just as in the conceptually simpler force-clamp experiments (compare with Eq.\[eqn:k\_mod\]).
The SR results illustrate the potential difficulties in uniquely extracting model-free energy landscape parameters from dynamic force spectroscopy data. It is worth mentioning that Scalley and Baker have used similar arguments about the *anomalous temperature dependence* of refolding rates of proteins [@Scalley97PNAS]. They showed that the Arrhenius behavior of kinetics is retrieved when the protein stability is corrected for the temperature dependence.
Mechanical response of hard (brittle) versus soft (plastic) biomolecules
========================================================================
Regardless of the model used, it is obvious that the lifetimes of a complex decrease upon application of force. The compliance of the molecule is determined by the location of the TS, and hence it is important to understand the characteristics of the molecule that determine the TS. As we pointed out, many relevant paramters have strong dependence on $f$, $r_f$, or $T$. Thus, it is difficult to extract the energy landscape parameters without a suitable model. In this section we illustrate two extreme cases of mechanical response [@West06BJ; @HyeonBJ06; @RitortPRL06; @West06BJ] of a biomolecule using one-dimensional energy profiles. In one example the location of the TS does not move with force whereas in the other there is a dramatic movement of the TS. In the presence of force $f$, a given free energy profile $F_0(x)$ changes to $F(x)$ = $F_0(x)-fx$. The location of the TS at non-zero values of $f$ depends on the shape of barrier in the vicinity of the TS. Near the barrier ($x \approx x_{ts}$) we can approximate $F_0(x)$ as $$F_0(x) \approx F_0(x_{ts})-\frac{1}{2} F_0^{\prime\prime}(x_{ts}) (x - x_{ts})^2+\cdots.
\label{eqn:expansion}$$ In the presence of force the TS location becomes $x_{ts}(f)=x_{ts}-\frac{f}{F_0''(x_{ts})}$. If the transition barrier in $F_0(x)$ is sharp ($x_{ts}F_0''(x_{ts})\gg f$) then we expect very little force-induced movement in the TS. We refer to molecules that satisfy this criterion as hard or brittle. In the opposite limit the molecule is expected to be soft or plastic so that there can be dramatic movements in the TS. We illustrate these two cases by numerically computing $r_f$-dependent $P(f)$ using Eq.\[eqn:MFPT\_force\] and Eq.\[eqn:most\_force\] for two model free energy profiles.
[*Hard response:*]{} A nearly stationary TS position (independent of $f$) is realized if the energy barrier is sharp (Eq.\[eqn:expansion\]). We model $F_0(x)$ using $$\begin{array}{ll}
F_0(x)=-V_0|(x+1)^2-\xi^2| & \mbox{with $x\geq 0$}
\end{array}
\label{eqn:hard}$$ where $V_0=28$ $pN/nm$ and $\xi=4$ $nm$. The energy barrier forms at $x=1$ $nm$ and this position does not change much even in the presence of force as illustrated in Fig.\[hardplot.fig\]-A. In dynamic force spectroscopy the free energy profiles drawn at constant force may be viewed as snapshots at different times. The shape of the unbinding force distribution depends on $r_f$. We calculated $P(f)$ numerically using Eq.\[eqn:MFPT\_force\] and Eq.\[eqn:most\_force\] (see Fig.\[hardplot.fig\]-B). Interestingly, a plot of the the most probable force $f^*$ obtained from $P(f)$ does not exhibit any curvature when $r_f$ is varied over six orders of magnitude (Fig.\[hardplot.fig\]-C). Over the range of $r_f$ the $[f^*, \log{r_f}]$ plot is almost linear. The slight deviation from linearity is due to the force-dependent curvature near the bound state ($\omega_b(f)$). From the slope we find that $\Delta x\approx 1$ $nm$ which is expected from Eq.\[eqn:hard\]. In addition, we obtained from the intercept in Fig.\[hardplot.fig\]-C that $\kappa=1.58$ $s^{-1}$. The value of $\kappa[\equiv k(f=0)]$ directly computed using Eq.\[eqn:MFPT\_force\] is $\kappa=1.49$ $s^{-1}$. The two values agree quite well. Thus, for brittle response the Bell model is expected to be accurate.
[*Soft response:*]{} If the position of the TS *sensitively* moves with force the biomolecule or the complex is soft or plastic. To illustrate the behavior of soft molecules we model the free energy potential in the absence of force using $$\begin{array}{ll}
F_0(x)=-V_0\exp{(-\xi x)} & \mbox{with $x\geq 0$}
\end{array}
\label{eqn:soft}$$ where $V_0=82.8$ $pN\cdot nm$ and $\xi=4$ $(nm)^{-1}$. The numerically computed $P(f)$ and $[f^*, \log{r_f}]$ plots are shown Fig.\[softplot.fig\]. The slope of the $[f^*, \log{r_f}]$ plot is no longer constant but increases continuously as $r_f$ increases. The extrapolated value of $\kappa$ to zero $f$ varies greatly depending on the range of $r_f$ used. Even in the experimentally accessible range of $r_f$ there is curvature in the $[f^*,\log{r_f}]$ plot. Thus, unlike the parameters ($\Delta x$, $\kappa$) in the example of a brittle potential, all the extracted parameters from the force profile are strongly dependent on the loading rate. As a result, in soft molecules the extrapolation to zero force (or minimum loading rate) is not as meaningful as in hard molecules. Note how the extracted $\Delta x$ (see the inset of Fig.\[softplot.fig\]-A) changes as a function of $r_f$. For soft (plastic) molecules, the extracted parameters using the tangent at a certain $r_f^o$ are not the characteristics of the free energy profile in the absence of the load, but reflect the features for the modified free energy profile tilted by $(f^*)^o$ at $r_f^o$.
In practice, biomolecular systems lie between the two extreme cases (brittle and plastic). In many cases the $[f^*, \log{r_f}]$ appears to be linear over a narrow range of $r_f$. The linearity in narrow range of $\log{r_f}$, however, does not guarantee the linearity under broad variations of loading rates. In order to obtain energy landscape parameters it is important to perform experiments at $r_f$ as low as possible. The brittle nature of proteins (lack of change in $\Delta x$) inferred from AFM experiments may be the result of a relatively large $r_f$ ($\approx 1,000 pN/s$). On the other hand, only by varying $r_f$ over a wide range the molecular elasticity of proteins and RNA can be completely described. Indeed, we showed that even in simple RNA hairpins the transition from plastic to brittle behavior can be achieved by varying $r_f$ [@HyeonBJ06]. The load-dependent response may even have functional significance.
Hammond/anti-Hammond behavior under force and temperature variations
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The qualitative nature of the TS movement with increasing perturbations can often be anticipated using the Hammond postulate which has been successful in not only analyzing a large class of chemical reactions but also in rationalizing the observed behavior in protein and RNA foldings. The Hammond postulate states that the nature of TS resembles the least stable species along the reaction pathway. In the context of forced-unfolding it implies that the TS location should move closer to the native state as $f$ increases. In other words $\Delta x$ should decrease as $f$ is increased. Originally Hammond postulate was introduced to explain chemical reactions involving small organic molecules [@HammondJACS53; @LefflerSCI53]. Its validity in biomolecular folding is not obvious because there are multiple folding or unfolding pathways. As a result there is a large entropic component to the folding reaction. Surprisingly, many folding processes are apparently in accord with the Hammond postulate [@Fersht95Biochemistry; @Dalby98Biochemistry; @Kiefhaber00PNAS]. If the extension is an appropriate reaction coordinate for forced unfolding then deviations from Hammond postulate should be an exception than the rule. Indeed, anti-hammond behavior (movement of the TS closer to more stable unfolded state as $T$ increases) was suggested by SR based on a model used to analyze the AFM data. The simple free energy profiles used in the previous section (Eq.\[eqn:hard\] and Eq.\[eqn:soft\]) can be used to verify the Hammond postulate when the external perturbation is either force or temperature. First, for the case of hard response the TS is barely affected by force, thus the Hammond or anti-Hammond behavior is not a relevant issue when unbinding is induced by $f$. On the other hand, for the case of soft molecules $\Delta x$ always decreases with a larger force. The positive curvature in $[f^*,\log{r_f}]$ plot is the signature of the classical Hammond-behavior with respect to $f$.
As long as a one dimensional free energy profile suffices in describing forced-unfolding of proteins and RNA the TS location must satisfy Hammond postulate. In general, for a fixed force or $r_f$, $\Delta x$ can vary with $T$. The changes in $\Delta x$ with temperature can be modeled using $T$-dependent parameters in the potential. To evaluate the consequence of $T$-variations we set $$\xi=\xi_0+\alpha(T-300K)$$ for both free energies in Eq.\[eqn:hard\] and Eq.\[eqn:soft\]. Depending on the value of $\alpha$ the position of the TS can move towards or away from the native state. We set $\alpha=\pm 0.1$ for both the hard and soft cases. The numerically computed $[f^*, \log{r_f}]$ plots are shown in Fig.\[hard\_soft\_Hammond.fig\]. One interesting point is found in soft molecule that exhibits Hammond behavior. For wide range of $r_f$, $\Delta x$ decreases as $T$ increases. However, the most probable unbinding force $f^*$ at low temperatures can be larger or smaller than $f^*$ at high temperatures depending on the loading rate (see upper-right corner of Fig.\[hard\_soft\_Hammond.fig\]). A very similar behavior has been observed in the forced-unbinding of Ran-imp-$\beta$ complex [@ReichEMBOrep05] (see also Fig.\[Nevo\_fig\]-B). Although the model free energy profiles can produce a wide range of behavior depending on $T$, $f$, and $r_f$ the challenge is to provide a structural basis for the measurements on biomolecules.
Multidimensionality of energy landscape coupled to “memory” affect the force dynamics
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The natural one dimensional reaction coordinate in mechanical unfolding experiments is the extension $x$ of the molecule. However, local rupture events can couple to $x$, which would require a multidimensional description. For example, consider the case of forced-unfolding of a nucleic acid hairpin. The opening of a given base pair is dependent not only on its strength, which resists unfolding, but also on the increase in the available conformational space which favors unfolding. In this case fluctuations in the collective coordinates, which describe the local events, are coupled to the global coordinate $x$. Thus, suitable fluctuations in the local coordinate $r$ has to occur before an increase in the extension is observed. Such a coupling between local coordinates and global observable arises naturally in many physical situations [@Zwanzig90ACR]. In the context of ligand binding to myoglobin Zwanzig first proposed such a model by assuming that reaction (binding) takes place along $x$-coordinate and at the barrier top ($x=x_{ts}$) the reactivity is determined by the cross section of bottleneck described by $r$-coordinate [@Zwanzig92JCP]. We adopt a similar picture to describe the modifications when such a process is driven by force. First, consider the Zwanzig case i.e, $r_f = 0$. The equations of motion for $x$ and $r$ are, respectively, $$\begin{aligned}
m\frac{d^2x}{dt^2}&=&-\zeta\frac{dx}{dt}-\frac{dU(x)}{dx}+F_x(t)\nonumber\\
\frac{dr}{dt}&=&-\gamma r+F_r(t).
\label{eqn:eqofmotion}\end{aligned}$$ The Liouville theorem ($\frac{d\rho}{dt}=0$) describes the time evolution of probability density, $\rho(x,r,t)$, as $$\frac{d\rho}{dt}=\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x}\left(\frac{dx}{dt}\rho\right)+\frac{\partial}{\partial r}\left(\frac{dr}{dt}\rho\right)=0.
\label{eqn:Liouville}$$ By inserting of Eq.\[eqn:eqofmotion\] to Eq.\[eqn:Liouville\] and neglecting the inertial term, ($m\frac{d^2x}{dt^2}$), and averaging over the white-noise spectrum, and the fluctuation-dissipation theorem ($\langle F_x(t)F_x(t')\rangle=2\zeta k_BT\delta(t-t')$, $\langle F_r(t)F_r(t')\rangle=2\lambda\theta\delta(t-t')$ where $\langle r^2\rangle\equiv \theta$) leads to a Smoluchowski equation for $\rho(x,r,t)$ in the presence of a reaction sink. $$\frac{\partial\overline{\rho}}{\partial t}=\mathcal{L}_x\overline{\rho}+\mathcal{L}_r\overline{\rho}-k_rr^2\delta(x-x_{ts})\overline{\rho}
\label{eqn:Smol}$$ where $\mathcal{L}_x\equiv D\frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}+\frac{1}{k_BT}\frac{dU(x)}{dx}\right)$ and $\mathcal{L}_r\equiv \lambda\theta\frac{\partial}{\partial r}\left(\frac{\partial}{\partial r}+\frac{r}{\theta}\right)$. Integrating both sides of Eq.\[eqn:Smol\] using $\int dx\rho(x,r,t)\equiv \overline{C}(r,t)$ leads to $$\frac{\partial\overline{C}}{\partial t}=\mathcal{L}_r\overline{C}(r,t)-k_rr^2\overline{\rho}(x_{ts},r,t).
\label{eqn:step1}$$ By writing $\overline{\rho}(x_{ts},r,t)=\phi_x(x_{ts})\overline{C}(r,t)$ where $\phi(x_{ts})$ should be constant as $\phi(x_{ts})=e^{-U(x_{ts})/k_BT}/\int dx e^{-U(x)/k_BT}\approx \sqrt{\frac{U''(x_b)}{2\pi k_BT}}e^{-(U(x_{ts})-U(x_b))/k_BT}$, Eq.\[eqn:step1\] becomes $$\frac{\partial\overline{C}}{\partial t}=\mathcal{L}_r\overline{C}(r,t)-kr^2\overline{C}(r,t).
\label{eqn:step2}$$ where $k\equiv k_r\sqrt{\frac{U''(x_b)}{2\pi k_BT}}e^{-(U(x_{ts})-U(x_b))/k_BT}$. In all likelihood $k_r$ reflects the dynamics near the barrier, so we can write $k=\kappa \frac{\omega_{ts}\omega_b}{2\pi\gamma}e^{-\Delta U/k_BT}$ where $\kappa$ describes the geometrical information of the cross section of bottleneck. Now we have retrieved the equation in Zwanzig’s seminal paper where the survival probability ($\Sigma(t)=\int^{\infty}_0 dr\overline{C}(r,t)$) is given under a reflecting boundary condition at $r=0$ and Gaussian initial condition $\overline{C}(r,t=0)\sim e^{-r^2/2\theta}$. By setting $\overline{C}(r,t)=\exp{(\nu(t)-\mu(t)r^2)}$, Eq.\[eqn:step2\] can be solved exactly, leading to $$\begin{aligned}
\nu'(t)&=&-2\lambda\theta\mu(t)+\lambda\nonumber\\
\mu'(t)&=&-4\lambda\theta\mu^2(t)+2\lambda\mu(t)+k.\end{aligned}$$ The solution for $\mu(t)$ is obtained by solving $\frac{4\theta}{\lambda}\int^{\mu(t)-1/4\theta}_{1/4\theta}\frac{d\alpha}{\sigma^2-16\theta^2\alpha^2}=t$, and this leads to $$\begin{aligned}
\frac{\mu(t)}{\mu(0)}&=&\frac{1}{2}\left\{1+S\frac{(S+1)-(S-1)E}{(S+1)+(S-1)E}\right\}\nonumber\\
\nu(t)&=&-\frac{\lambda t}{2}(S-1)+\log{\left(\frac{(S+1)+(S-1)E}{2S}\right)^{-1/2}}\end{aligned}$$ with $\mu(0)=1/2\theta$. The survival probability, which was derived by Zwanzig, is $$\Sigma (t)=\exp{\left(-\frac{\lambda}{2}(S-1)t\right)}
\left(\frac{(S+1)^2-(S-1)^2E}{4S}\right)^{-1/2}
\label{eqn:solution}$$ where $S=\left(1+\frac{4k\theta}{\lambda}\right)^{1/2}$ and $E=e^{-2\lambda St}$. We wish to examine the consequences of coupling between local and global reaction coordinates under tension. In order to accomplish our goal we solve the Smoluchoski equation in the presence of constant load. In this case, $k$ in Eq.\[eqn:step2\] should be replaced with $ke^{t (r_f\Delta x/k_BT)}$. Eq.\[eqn:step2\]i, however, becomes hard to solve if the sink term depends on $t$. Nevertheless, analytical solutions can be obtained for special cases of $\lambda$. If $\lambda\rightarrow \infty$, $d\Sigma(t)/dt=-k\theta e^{t(r_f\Delta x/k_BT)}\Sigma(t)$, and hence, $$\Sigma(f)=\exp{\left(-k\theta\int^f_0df\frac{1}{r_f}e^{f\Delta x/k_BT}\right)}=\exp{\left(-\frac{k\theta k_BT}{r_f\Delta x} (e^{f\Delta x/k_BT}-1)\right)}$$ Using the rupture force distribution $P(f)=-\frac{d\Sigma(f)}{df}$ and $\frac{dP(f)}{df}|_{f=f^*}=0$, one can obtain the most probable force $$f^*=\frac{k_BT}{\Delta x}\log{\frac{r_f\Delta x}{(k\theta)k_BT}}.$$
If $\lambda$ is small ($\lambda\rightarrow 0$) then $\overline{C}(r,t)=\exp{[-\int_0^t dt kr^2\exp{(t\times r_f\Delta/k_BT)}]}=\exp{\left[-\frac{kr^2 k_BT}{r_f\Delta x}(e^{tr_f\Delta x/k_BT}-1)\right]}$ with the initial distribution of $e^{-r^2/2\theta}$. Thus, $$\begin{aligned}
\Sigma(f)&=& \int_0^{\infty}dr\exp{\left[-r^2\frac{kk_BT}{r_f\Delta x}(e^{f\Delta x/k_BT}-1)\right]}\sqrt{\frac{2}{\pi\theta}}\exp{\left[-r^2/2\theta\right]}\nonumber\\
&=& \sqrt{\frac{2}{\pi\theta}}\left(\frac{kk_BT}{r_f\Delta x}(e^{f\Delta x/k_BT}-1)+\frac{1}{2\theta}\right)^{-1/2}. \end{aligned}$$ Note that if $r_f\rightarrow 0$ we recover Zwanzig’s result $\Sigma(t)\sim (1+2k\theta t)^{-1/2}$. Using $P(f)=-d\Sigma(f)/df$ $$P(f)=\frac{1}{\sqrt{2\pi\theta}}\left(\frac{kk_BT}{r_f\Delta x}(e^{f\Delta x/k_BT}-1)+\frac{1}{2\theta}\right)^{-3/2}\frac{k}{r_f}e^{f\Delta x/k_BT}.$$ $dP(f)/df|_{f=f^*}=0$ gives $$f^*=\frac{k_BT}{\Delta x}\log{\left\{\left(\frac{r_f\Delta x}{k\theta k_BT}\right)\left(1-\frac{2k\theta k_BT}{r_f\Delta x}\right)\right\}},
\label{eqn:smalllambda}$$ in which $r_f\geq(1+2\theta)\frac{k\theta k_BT}{\Delta x}$ since $f^*\geq 0$. This shows that $f^*$ vs $r_f$ has a different form when $\lambda\rightarrow 0$ from the one when $\lambda\rightarrow\infty$. The deviation of Eq.\[eqn:smalllambda\] from the conventional relation is pronounced when $\left[r_f-(1+2\theta)\frac{kk_BT}{\Delta x}\right]\rightarrow 0^+$.
At present, experimental data have been interpreted using only one dimensional free energy profiles. The meaning and the validity of the extracted free energy profiles has not been established. At the least, this would require computing the force-dependent first passage times using the “experimental” free energy profile assuming that the extension is the only slowly relaxing variable. If the computed force-dependent rates (inversely proportional to first passage times) agree with the measured rates then the use of extension of the reaction coordinate would be justified. In the absence of good agreement with experiments other models, such as the one we have proposed here, must be considered. In the context of force-quench refolding we have shown (see below) that extension alone is not an adequate reaction coordinate. For refolding upon force-quench of RNA hairpins, the coupling between extension and local dihedral angles, which reports on the conformation of the RNA, needs to be taken into account to quantitatively describe the refolding rates.
Conclusions
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With the advent of single molecule experiments that can manipulate biomolecules using mechanical force it has become possible to get a detailed picture of their energy landscapes. Mechanical folding and unfolding trajectories of proteins and RNA show that there is great diversity in the explored routes [@HyeonSTRUCTURE06; @Bustamante03Science; @FernandezSCI04]. In certain well defined systems with simple native states, such as RNA and DNA hairpins, it has been shown using constant force unfolding that the hairpins undergo sharp bistable transitions from folded to unfolded states [@Bustamante02Science; @Block06Science; @Woodside06PNAS]. From the dynamics of the extension as a function of time measured over a long period the underlying force dependent profiles have been inferred. The force-dependent folding and unfolding rates and the unfolding trajectories can be used to construct the one-dimensional energy landscape. In a remarkable paper [@Block06Science], Block and coworkers have shown that the location of the TS can be moved, at will, by varying the hairpin sequence. The TS was obtained using the Bell model by assuming that the $\Delta x$ is independent of $f$. While this seems reasonable given the sharpness of the inferred free energy profiles near the barrier top it will be necessary to show the $\Delta x$ does not depend on force.
The fundamental assumption in inverting the force-clamp data is that the molecular extension is a suitable reaction coordinate. This may indeed be the case for force-spectroscopy in which the response of the molecule only depends on force that is coupled to the molecular extension, which may well represent the slow degrees of freedom. The approximation is more reasonable for forced-unbinding. It is less clear if it can be assumed that extension $x$ is the appropriate reaction coordinate when refolding is initiated by quenching the force to low enough values such that the folded state is preferentially populated. In this case the dynamic reduction in $x$ can be coupled to collective internal degrees of freedom. In a recent paper [@HyeonBJ06] we showed, in the context of force-quench refolding of a RNA hairpin, that the reduction in $x$ is largely determined by local conformational changes in the dihedral angle degrees in the loop region. Zipping by nucleation of the hairpin with concomitant reduction in $x$ does not occur until the transitions from *trans* to *gauche* state in a few of the loop dihedral angles take place. In this case, one has to consider at least a two dimensional free energy landscape. Fig.\[dih\_R\_2D\_map\_illust.fig\] clearly shows such a coupling between end-to-end distance ($R$) and the dihedral angle degrees of freedom. The “correctness” of the six dihedral angles representing the conformation of RNA hairpin loop region ($\phi_i$, $i=19,\ldots 24$) is quantified using $\langle 1-\cos{(\phi_i-\phi_i^o)}\rangle$, where $\phi^o_i$ is the angle value in the native state and $\langle\ldots\rangle$ is the average over the six dihedral angles. $\langle 1-\cos{(\phi-\phi^o)}\rangle=0$ signifies the correct dihedral conformation for the hairpin loop region. Once the “correct” conformation is attained in the loop region, the rest of the zipping process can easily proceed as we have shown in [@HyeonBJ06]. Before the correct loop conformation being attained, RNA spends substantial time in searching the conformational space related to the dihedral angle degree of freedom. The energy landscape ruggedness is manifested as in Fig.\[dih\_R\_2D\_map\_illust.fig\] when conformational space is represented using multidimensional order parameters. The proposed coupling between the local dihedral angle degrees of freedom and extension (global parameter) is fairly general. A similar structural slowing down, due to the cooperative link between local and global coordinates, should be observed in force-quench refolding of proteins as well.
The most exciting use of singe molecule experiments is in their ability to extract precise values of the energy landscape roughness $\epsilon$ by using temperature as a variable in addition to $f$. In this case a straightforward measurement of the unbinding rates as a function of $f$ or $r_f$ can be used to obtain $\epsilon$ without having to make *any assumptions* about the underlying mechanisms of unbinding. Of course, this involves performing a number of experiments. In doing so one can also be rewarded with diagram of states in terms of $f$ and $T$ [@HyeonPNAS05]. The theoretical calculations and arguments given here also show that the power of single molecule experiments can be fully realized only by using the data in conjunction with carefully designed theoretical and computational models. The latter can provide the structures that are sampled in the process of forced-unfolding and force-quench refolding as was illustrated for ribozymes and GFP [@HyeonSTRUCTURE06]. It is likely that the promise of measuring the energy landscapes of biomolecules, almost one molecule at a time, will be fully realized using a combination of single molecule measurements, theory, and simulations. Recent studies have already given us a glimpse of that promise with more to come shortly.
Appendix: Dynamic Force spectroscopy in a Cubic potential
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Assuming that the Bell model gives a correct description of forced-unbinding it was shown by Evans and Ritchie that the most probable unbinding force $f^* \approx \frac{k_BT}{\Delta x_{ts}}\log{r_f}$ assuming that the TS location is independent of $r_f$ [@Evans97BJ]. Deviations from logarithmic dependence of $f^*$ on $r_f$ occurs if the assumptions of the Bell model is relaxed as was first shown by Dudko et. al. [@Dudko03PNAS]. In this Appendix we give a simple derivation of the result for the cubic potential, the simplest one that has a potential well (bound state) and a barrier for which we can analytically formulate the dynamic force spectroscopy theory. The general form of the cubic potential is, $$\begin{array}{ll}
F_0(x)=ax^3+bx^2+cx+d & \mbox{with $a<0$}.
\end{array}$$ In the presence of constant force $f$, $F(x)=F_0(x)-fx$ can have a finite free energy barrier if the two roots of $F'(x)= 3ax^2+2bx+c-f=0$, namely, $$r_{\pm}=\frac{-2b\pm\sqrt{4b^2-12a(c-f)}}{6a}\nonumber\\$$ are real. Provided $b^2-3a(c-f)>0$, $r_-$ and $r_+$ correspond to $x_{ts}$ and $x_b$, respectively. Note that both $x_{ts}$ and $x_b$ are functions of $f$ in the cubic potential and so is $\Delta x=x_{ts}-x_b=-\frac{\sqrt{4b^2-12a(c-f)}}{3a}$. The distance between $x_{ts}$ and $x_b$ decreases as $f$ grows and vanishes when the force reaches a critical force $f_c=c-\frac{b^2}{3a}$, where the free energy barrier also becomes zero. The $f$-dependent free energy barrier($\Delta F^{\ddagger}$) is calculated by $\Delta F^{\ddagger}(f)=F(x_{ts})-F(x_b)$ as, $$\begin{aligned}
\Delta F^{\ddagger}(f)&=&a(r_-^3-r_+^3)+b(r_-^2-r_+^2)+c(r_--r_+)-f(r_--r_+)\nonumber\\
&=&\frac{2\sqrt{-12a}}{(-9a)}f_c^{3/2}\left(1-\frac{f}{f_c}\right)^{3/2}\nonumber\\
&=&U_c\epsilon_c^{3/2}.
\label{eqn:cubic_barrier} \end{aligned}$$ where $\epsilon\equiv 1-f/f_c$. The curvatures at the barrier ($x=x_{ts}$) and the bottom ($x=x_b$) of the potential, that are used in the Kramers’ rate expression, are calculated using $F''(x)|_{x=x_{ts}, x_b}=(6ax+2b)|_{x=x_{ts}, x_b}=m\omega^2_{ts,b}$. $$\begin{aligned}
\sqrt{m}\omega_{ts}=\sqrt{m}\omega_b&=&\left[-12af_c\left(1-\frac{f}{f_c}\right)\right]^{1/4}\nonumber\\
&=&\Omega_c\epsilon^{1/4}
\label{eqn:cubic_omega}\end{aligned}$$ Note that (i) $\Delta x=m\omega^2/(-3a)$ is satisfied for all $0<f<f_c$, and (ii) $\omega_{ts}=\omega_b$, the *point symmetry* around the inflection point $x_c$ that satisfies $F'(x_c)=F''(x_c)=0$, are the unique properties of the cubic function for any parameter set $(a,b,c,d,f)$.
This property is no longer valid if the free energy function is modeled using a higher order polynomial or special functions. What is worse for the higher order polynomial is that roots of $F'(x)=0$ are not easily found, and that if order is higher than 6 there is no general solution. Thus, if higher polynomials are needed to fit the free energy profile, even with the assumption that the extension is an appropriate reaction coordinate, then analytically tractable solutions are not possible. It may be argued that any smooth potential may be locally approximated using a cubic potential, and hence can be used to analyze the experimental data.
Using Eqs.\[eqn:Kramers\], \[eqn:force\_distribution\], \[eqn:cubic\_barrier\], and \[eqn:cubic\_omega\] we get $$\begin{aligned}
k_{cubic}(\epsilon)=\frac{\Omega_c^2\epsilon^{1/2}}{2\pi\gamma}\exp{\left(-U_c\epsilon^{3/2}/k_BT\right)}\end{aligned}$$ and $$\begin{aligned}
P(\epsilon)&=&\frac{1}{r_f}\frac{\Omega_c^2\epsilon^{1/2}}{2\pi\gamma}\exp{\left(-U_c\epsilon^{3/2}/k_BT\right)}\exp{\left(\frac{f_c}{r_f}\frac{\Omega_c^2}{2\pi\gamma}\int^{\epsilon}_1d\epsilon\epsilon^{1/2}e^{-U_c\epsilon^{3/2}/k_BT}\right)}\nonumber\\
&=&\mathcal{N}\epsilon^{1/2}\exp{\left(-\frac{U_c\epsilon^{3/2}}{k_BT}-\frac{k_BT f_c\Omega_c^2}{3\pi U_c\gamma r_f}e^{-U_c\epsilon^{3/2}/k_BT}\right)}\end{aligned}$$ where $\mathcal{N}=\frac{\Omega_c^2}{2\pi\gamma r_f}\exp{\left(\frac{k_BT f_c\Omega_c^2}{3\pi U_c \gamma r_f}e^{-U_c/k_BT}\right)}$. The most probable force $f^*$ is obtained using $dP(f)/df|_{f=f^*}=0$, which establishes the relation with respect to $\epsilon^*(\equiv 1-f^*/f_c)$ as $$-\frac{U_c}{k_BT}(\epsilon^*)^{3/2}=\log{\left[\frac{\gamma}{\Omega^2_cf_c}\left(\frac{U_c}{k_BT}-(\epsilon^*)^{-2/3}\right)\times r_f\right]}.$$ If $f^*\ll f_c$ which also satisfies the condition $\frac{U_c}{k_BT}\gg(\epsilon^*)^{-2/3}$, $$f^*\approx f_c\left\{ 1-\left[\frac{k_BT}{U_c}\log{\left(\frac{\gamma}{\Omega^2_cf_c}\frac{U_c}{k_BT}\times r_f\right)}\right]^{2/3}\right\}.$$ By defining $r_f^{min}\equiv \frac{U_c\Omega_c^2f_c}{k_BT\gamma}e^{U_c/k_BT}$ as the minimum loading rate that gives $f^*>0$, one can rewrite the above expression as $$f^*\approx f_c\left[1-\left(1+\frac{k_BT}{U_c}\log{\frac{r_f}{r_f^{min}}}\right)^{2/3}\right].
\label{eqn:f*approx}$$ The importance of the result in Eq.\[eqn:f\*approx\] was first emphasized by Dudko et. al.
Two comments about the announced deviation from the usual logarithmic dependence of $f^*$ on $r_f$ are worth making. (1) Provided $\frac{k_BT}{U_c}\log{\frac{r_f}{r_f^{min}}}\ll 1$, Eq.\[eqn:f\*approx\] becomes $$f^*\approx \frac{k_BT}{\left(\frac{3U_c}{2f_c}\right)}\log{\frac{r_f}{r_f^{min}}},$$ which is the typical $f^*$ vs $\log{r_f}$ relation. It is of particular interest to see if the condition $\frac{k_BT}{U_c}\log{\frac{r_f}{r_f^{min}}}\ll 1$ is satisfied in proteins or RNA. (2) Because $f^* \approx {log r_f}^{\nu}$ in both the Evans-Ritchie ($\nu = 1$) [@Evans97BJ] and Dudko ($\nu = \frac{2}{3}$) formulations [@Dudko06PRL] it is unclear whether they can be really distinguished using experimental data in which $r_f$ cannot be easily varied by more than three (at best) orders of magnitude. Thus, the reliability of the parameters obtained using either formulation is difficult to assess independently. Nevertheless, the reexamination of the logarithmic dependence of $f^*$ on $r_f$ shows that one should be mindful of the assumption that the location of the TS *does not depend on $f$ or $r_f$*.\
[**Acknowledgments:**]{} We are grateful to Reinat Nevo and Ziv Reich for providing the data in Fig. (3). This work was supported in part by a grant from the National Science Foundation through grant number CHE 05-14056.
**Figure Caption** {#figure-caption .unnumbered}
==================
[**Figure \[RoughreviewFig1.fig\] :**]{} Single molecule experiments using mechanical force. On the left we show a schematic setup of LOT experiments in which a ribozyme is mechanically unfolded. The *Tetrahymena* ribozyme is sandwiched between two RNA/DNA hybrid handles that are linked to micron-sized beads. The structures of a few of the displayed intermediates were obtained using simulations of the Self-Organized Polymer (SOP) model representation [@HyeonSTRUCTURE06] of the ribozyme. The sketch on the right shows a typical AFM setup in which we illustrate unfolding of GFP. The intermediate structures, that are shown, were obtained using simulations of the SOP model for GFP.
[**Figure \[landscape\] :**]{} Caricature of the rough energy landscape of proteins and RNA that fold in an apparent ”two-state” manner using extension $x$, the coordinate that is conjugate to force $f$. Under force $f$, the zero-force free energy profile ($F_0(x)$) is tilted by $f\times x$ and gives rise to the free energy profile, $F(x)$. In order to clarify the derivation of Eq.\[eqn:mfpt\] we have explicitly indicated the average location of the relevant parameters.
[**Figure \[Nevo\_fig\] :**]{} Dynamic force spectroscopy measurements of single imp-$\beta$-RanGppNHp pairs at different temperatures. [**A.**]{} Distributions of measured unbinding forces using AFM for the lower-strength conformation of the complex at different loading rates at 7 and 32$^oC$. Roughness acts to increase the separation between the distributions recorded at different temperatures. The histograms are fit using Gaussian distributions. The width of the bins represents the thermal noise of the cantilever. [**B.**]{} Force spectra used in the analysis. The most probable unbinding forces $f^*$ are plotted as a function of $log(r_f)$. The maximal error is $\pm10$% because of uncertainities in determining the spring constant of the cantilevers. Statistical significance of the differences between the slopes of the spectra was confirmed using covariance test. (Images courtesy of Reinat Nevo and Ziv Reich [@ReichEMBOrep05]). [**C.**]{} Ran-importin$\beta$ complex crystal structures (PDB id: 1IBR [@Vetter99Cell]) in surface (left) and ribbon (right) representations. In AFM experiments, Ran (red) protein complexed to importin$\beta$ (yellow) is pulled until the dissociation of the complex takes place.
[**Figure \[hardplot.fig\] :**]{} Dynamic force spectroscopy analysis using a model free energy profile $F_0(x)=-V_0|(x+1)^2-\xi^2|$ with $V_0=20pN/nm$, $\xi=4nm$, and $x\geq 0$. The lack of change in $x_{ts}$ as $f$ changes shows a *hard response* under tension. [**A.**]{} Effective free energy profile ($F(x)$) at various values of $f$. [**B.**]{} Distributions of unbinding forces at different loading rates. [**C.**]{} Plot of the most probable unbinding force ($f^*$) versus $\log{r_f}$.
[**Figure \[softplot.fig\] :**]{} Dynamic force spectroscopy for soft response to $f$ using the $f = 0$ free energy profile $F_0(x)=-V_0\exp{(-\xi x)}$ with $V_0=82.8pN\cdot nm$, $\xi=4(nm)^{-1}$. [**A.**]{} Effective free energy profile ($F(x)$) as a function of $f$. For emphasis on the soft response of the potential, the position of TS at each force value is indicated with arrows. [**B.**]{} Distributions of unbinding forces at varying loading rates. [**C.**]{} Plot of most probable unbinding force ($f^*$) versus $\log{r_f}$. The slope of the tangent at each loading rate value varies substantially, which suggests the variation in the TS (inset) as $r_f$ changes.
[**Figure \[hard\_soft\_Hammond.fig\] :**]{} By tuning the value of $\xi$ (see Eq. (23)) as a function of temperature, Hammond and anti-Hammond behaviors emerge in the context of force spectra in the free energy profiles that show hard and soft responses. The condition for Hammond or anti-Hammond behavior depends on $\alpha$ (Eq. (23)).
[**Figure \[dih\_R\_2D\_map\_illust.fig\] :**]{} [**A.**]{} A sample refolding trajectory of a RNA hairpin starting from the stretched state. The hairpin was, at first, mechanically unfolded to a fully stretched state and the force was subsequently quenched to zero at $t\approx 20$ $\mu s$. The time-dependence of the end-to-end distance shows that force-quench refolding occurs in steps. [**B.**]{} The deviation of the dihedral angles from their values in the native state as a function of time shows large departures from native values of the dihedral angles in loop region (indicated by the red strip). Note that this strip disappears around $t\approx 300$ $\mu s$, which coincides with the formation of bonds shown in [**C**]{}. $f_B$ is the fraction of bonds in pink that indicates that the bond is fully formed. [**D.**]{} The histograms collected from the projections of twelve stretching and force-quench refolding trajectories on the two dimensional plane characterized by the end-to-end distance ($R$) and the average correctness of dihedral angles ($\langle 1-\cos{(\phi-\phi^0)}\rangle$) around the loop region ($i=19-24$). The scale on the right gives the density of points in the two dimensional projection. This panel shows that the local dihedral angles are coupled to the end-to-end distance $R$, and hence extension alone is not a good reaction coordinate especially in force-quench refolding. The molecular extension $x$ is related to $R$ by $x = R -R_N$ where $R_N$ is the distance in the folded state.
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Particle physics has for long awaited the experimental deviation from the Standard Model predictions that will guide the way to the physics beyond the Standard Model. Such a discovery may be just around the corner. Recently, the ALEPH collaboration has reported an excess in the four–jet event cross section which is several sigmas above the Standard Model prediction [@alephone; @alephtwo]. Perhaps even more intriguing is the fact that both at LEP 1.5 and LEP 2 runs ALEPH has observed a sharp peak at $106.1\pm0.8$ GeV, corresponding to 18 events with 3.1 expected from QCD background. The di–jet mass difference distribution of the selected 18 events is consistent with a value around 10 GeV. If interpreted as a particle pair production this together with the information on the di–jet mass sum suggests that the two particle produced have masses of about 58 and 48 GeV and production of same mass particles is disfavored. By extracting information on the primary parton [@alephthree], it is concluded that the pair produced particles have a sizable charge and that neutral particle production is disfavored [@alephone; @alephtwo]. Absence of $b$–quarks in the final states disfavors the hypothesis of Higgs–boson production.
Recently, it was proposed [@giudice] that the ALEPH excess of four jet events can be explained in supersymmetric models with R–parity violation [@rparityv]. According to this proposal left–handed and right–handed selectrons ${\tilde e}_L{\tilde e}_R$ are pair produced at LEP. The selectron pair then decay further by the R–parity violating operator $$\lambda_{ijk} L^iQ^j{d}^k
\label{rxloperator}$$ where the standard notation for lepton and quark superfields has been used and $i,j,k$ denote the generation indices, and absence of top or bottom quarks in the final states restricts only the $\lambda_{1jk}$ with $j,k=1,2$ to be nonzero [@giudice].
It is well known however that R–parity violation may induce rapid proton decay. If in addition to the operator in Eq. (\[rxloperator\]) one also has the operator $$\eta_{ijk}u^id^jd^k
\label{rxboperator}$$ with unsuppressed $\eta_{ijk}$ couplings then the proton decays rapidly. A quick estimate of the constraint from proton lifetime gives $$(\lambda\eta)<g^2\left({M_{\rm squark}\over{M_{\rm GUT}}}\right)^2
\label{constraint}$$ Therefore, if $\lambda_{1jk}\sim10^{-4}$ as proposed in ref. [@giudice], and taking $M_{\rm squark}\sim1$ TeV $M_{\rm GUT}\sim10^{16}$ GeV, naively requires $\eta<10^{-22}$. Thus, proton constraints essentially requires $\eta\equiv0$ if the R–parity violation is to explain the excess of ALEPH four jet events.
Therefore, the problem is to understand why the couplings in Eq. (\[rxloperator\]) are allowed while the couplings in Eq. (\[rxboperator\]) are forbidden. In this paper I discuss this problem in the context of realistic superstring derived models.
To study this problem I examine the superstring models which are constructed in the free fermionic formulation [@fff; @revamp; @fny; @alr; @slm; @eu; @gcu; @custodial]. This class of superstring models reproduce many of the properties of the Standard Model, like three chiral generations with the Standard Model gauge group and existence of Higgs doublets which can generate realistic fermion mass spectrum. Two of the important features in this class of models is the existence of a stringy doublet–triplet splitting mechanism which resolves the GUT hierarchy problem [@ps] and the fact that the chiral generations all fall into the 16 of $SO(10)$. This last property admits the standard embedding of the weak hypercharge in $SO(10)$ and is crucial for the agreement of these models with $\sin^2\theta_W(M_Z)$ and $\alpha_{\rm strong}(M_Z)$ [@gcu; @df].
The superstring models under consideration are constructed in two steps. In the first step the observable gauge symmetry is broken to $SO(10)\times SO(6)^3$. There are 48 generations in the chiral 16 representation of $SO(10)$ with $N=1$ space–time supersymmetry. In the second step the $SO(10)$ symmetry is broken to one of its subgroups, $SU(5)\times U(1)$, $SO(6)\times SO(4)$ or $SU(3)\times SU(2)\times U(1)^2$. The flavor $SO(6)^3$ symmetries are broken to $U(1)^n$, where $n$ may vary between 3–9, and the number of generations is reduced to three. The symmetry is then broken further in the effective field theory and the weak hypercharge is some linear combination of the Cartan subgenerators. For example, in the standard–like models the weak hypercharge is given by, $$U(1)_Y={1\over3}U(1)_C+{1\over2}U(1)_L
\label{forexample}$$ The chiral generations in these superstring models are obtained from the 16 multiplets of $SO(10)$ and carry charges under the flavor symmetries. These models typically contain an “anomalous” $U(1)$ symmetry which requires that some fields in the massless string spectrum obtain non–vanishing VEVs [@dsw]. Further details on the construction of the realistic free fermionic models are given in ref. [@slm].
In general in string models one expects the appearance of R–parity violating terms of the form of Eq. (\[rxloperator\]) and (\[rxboperator\]). If both are not suppressed then the proton decays much too fast. If the $B-L$ generator is gauged like in $SO(10)$ then these terms are forbidden at the cubic level by gauge invariance. However, they may still be generated from nonrenormalizable terms that contain the right–handed neutrino. $$\eta_1(uddN)\Phi+\eta_2(QLdN)\Phi.
\label{quartic}$$ where $\Phi$ is a combination of fields that fixes the string selection rules and gets a VEV of $O(M_{Pl})$ and $N$ is the Standard Model singlet in the 16 of $SO(10)$. Thus, the ratio $\langle N\rangle/M_{Pl}$ controls the rate of proton decay. In general, terms of the form of Eq. (\[quartic\]) are expected to appear in string models at different orders of nonrenormalizable terms. For example, in the model of ref. [@eu] such terms appear at order $N=6$ $$\begin{aligned}
&(u_3d_3+Q_3L_3)d_2N_2\Phi_{45}{\bar\Phi}_2^{-}\nonumber\\
+&(u_3d_3+Q_3L_3)d_1N_1\Phi_{45}\Phi_1^{+}\nonumber\\
+&u_3d_2d_2N_3\Phi_{45}{\bar\Phi}_2^{-}+
u_3d_1d_1N_3\Phi_{45}\Phi_1^{+}\nonumber\\
+&Q_3L_1d_3N_1\Phi_{45}\Phi_3^+
+Q_3L_1d_1N_3\Phi_{45}\Phi_3^+\nonumber\\
+&Q_3L_2d_3N_2\Phi_{45}{\bar\Phi}_3^-
+Q_3L_2d_2N_3\Phi_{45}{\bar\Phi}_3^-.
\label{ordersix}\end{aligned}$$ In this model the states from the sector $b_3$ are identified with the lightest generation. It is therefore seen that if any of $N_1$, $N_2$ or $N_3$ gets a Planck scale VEV, dimension four operators may be induced which would result in rapid proton decay. It is interesting to note that all the terms in Eq. (\[ordersix\]) contain the field $\Phi_{45}$. If the VEV of $\Phi_{45}$ vanishes then all the higher order terms are identically zero. In this specific model due to the anomalous $U(1)$ symmetry $\Phi_{45}$ must get a VEV and, in general, dimension four operators may be induced. Nevertheless, this observation suggests the possibility that slight variation of the model will result in a field appearing in these terms which is not required to get a VEV. However, even if such a possibility can work we see that both the desired terms of the form $QLd$ and the undesired terms of the form $udd$ are induced, or forbidden, simultaneously.
In the flipped $SU(5)$ model similar terms may arise from the terms $$FF{\bar f}H\Phi^n.
\label{sufive}$$ Here $F$ and $H$ are in the $(10,1/2)$ representation and ${\bar f}$ is in the $({\bar 5},-3/2)$ representation of $SU(5)\times U(1)$. The field $F$ contains the $Q$, $d$, $N$ fields and ${\bar f}$ contains the $u$ and $L$ fields. The Standard Model singlet, $N$ in the Higgs field $H$ obtains a VEV which breaks the $SU(5)\times U(1)$ symmetry to the Standard Model symmetry. Thus, terms of the form of Eq. (\[sufive\]) produce simultaneously the terms in Eq. (\[rxloperator\]) and Eq. (\[rxboperator\]). Terms of the form of Eq. (\[sufive\]) are in general found in the string models [@elnone]. Therefore, to produce only the terms of the form of Eq. (\[rxloperator\]) while preventing the terms in Eq. (\[rxboperator\]) requires a different mechanism.
In the case of the $SO(6)\times SO(4)$ superstring models the Standard Model fermions are embedded in the $$\begin{aligned}
&F_L\equiv(4,2,1)=Q+L\nonumber\\
&{\bar F}_R\equiv({\bar 4},1,2)=u+d+e+N
\label{psleft}\end{aligned}$$ representations of the $SU(4)\times SU(2)_L\times SU(2)_R$. Note that $F_L+{\bar F}_R$ make up the 16 spinorial representation of $SO(10)$. The dangerous dimension four operators are obtained in this case from the operator $$F_LF_L{\bar F}_R{\bar H}_R~~{\rm and}~~
{\bar F}_R{\bar F}_R{\bar F}_R{\bar H}_R
\label{d4so64}$$ where ${\bar H}_R$ is the Higgs representation which breaks the extended non–Abelian symmetry. We observe that in the $SO(6)\times SO(4)$ type models, like the $SU(3)\times SU(2)\times U(1)^2$ type models, the operator in Eqs. (\[rxloperator\]) and (\[rxboperator\]) arise from two distinct operators.
Next, I turn to the model of Ref. [@custodial]. The detailed spectrum of this model and the quantum numbers are given in Ref. [@custodial]. In this model the observable gauge group formed by the gauge bosons from the Neveu–Schwarz sector alone is $$SU(3)_C\times SU(2)_L\times U(1)_C\times U(1)_L\times U(1)_{1,2,3,4,5,6}
\label{nsgaugebosons}$$ However, in this model two additional gauge bosons appear from the twisted sector ${\bf 1}+\alpha+2\gamma$. These new gauge bosons are singlets of the non–Abelian gauge group but carry $U(1)$ charges. Referring to this generators as $T^{\pm}$, then together with the linear combination $$T^3\equiv{1\over4}\left[U(1)_C+U(1)_4+U(1)_5+U(1)_6+U(1)_7-U(1)_9\right]
\label{t3}$$ the three generators $\{T^3,T^{\pm}\}$ together form an enhanced $SU(2)_{\rm custodial}$ symmetry group. Thus, the original observable symmetry group is enhanced to $$SU(3)_C \times SU(2)_L \times SU(2)_{\rm cust} \times
U(1)_{C'} \times U(1)_L
\times U(1)_{1,2,3} \times U(1)_{4',5',7''}$$ The different combinations of the $U(1)$ generators are given in ref. [@custodial; @df]. The weak hypercharge is still defined as a combination of $U(1)_C$ and $U(1)_L$. However in the present model $U(1)_C$ is part of the extended $SU(2)_{\rm custodial}$ symmetry. We can express $U(1)_C$ in terms of the new orthogonal $U(1)$ combinations, $${1\over3}\,U(1)_{C}~=~{2\over5}\biggl\lbrace U(1)_{C^\prime}+
{5\over{16}}\,\biggl\lbrack T^3+{3\over5}\,U_{7^{\prime\prime}}
\biggr\rbrack
\biggr\rbrace~.
\label{U1Cin274}$$ and the weak hypercharge is given as before by the linear combination $$U(1)_Y={1\over3}U(1)_C+{1\over2}U(1)_L
\label{weakhypercharge}$$ The weak hypercharge depends on the diagonal generator of the custodial $SU(2)$ gauge group. We can therefore instead define the new linear combination with this term removed, $$\begin{aligned}
U(1)_{Y'} &\equiv& U(1)_Y - {1\over 8}\,T^3 \nonumber\\
&=& {1\over 2}\,U(1)_L + {5\over{24}}\,U(1)_C \nonumber\\
&&~~~~~-{1\over8}\,\biggl\lbrack
U(1)_4+U(1)_5+U(1)_6+U(1)_7-U(1)_9\biggr\rbrack~,
\label{U1pin274}\end{aligned}$$ so that the weak hypercharge is expressed in terms of $U(1)_{Y'}$ as $$U(1)_{Y} = U(1)_{Y'} + {1\over 2}\,T^3 ~~~~\Longrightarrow~~~~
Q_{\rm e.m.} = T^3_L + Y = T^3_L + Y' + {1\over 2}\,T^3_{\rm
cust}~.\label{Qemin274}$$ The final observable gauge group then takes the form $$SU(3)_C \times SU(2)_L \times SU(2)_{\rm cust}\times U(1)_{Y'} ~\times
~\biggl\lbrace ~{\rm seven~other~}U(1){\rm ~factors}~\biggr\rbrace~.
\label{finalgroup}$$ These remaining seven $U(1)$ factors must be chosen as linear combinations of the previous $U(1)$ factors so as to be orthogonal to the each of the other factors in (\[finalgroup\]).
The full massless spectrum of this model is given in Ref. [@custodial]. In this model the charged and neutral leptons transform as doublets of the $SU(2)_{\rm custodial}$ symmetry while the quarks are singlets. Therefore, because of the custodial $SU(2)$ symmetry the terms of the form $$QLdN
\label{qldn}$$ are invariant under the custodial $SU(2)$ symmetry, while the terms of the form $$uddN
\label{uddn}$$ are not invariant. We could contemplate tagging another $N$ field to Eq. (\[uddn\]) which will render it invariant under $SU(2)_{\rm custodial}$. However, this will spoil the invariance under $U(1)_L$. We therefore find that the baryon number violating operators, Eq. (\[uddn\]) vanish to all orders in the model of ref. [@custodial]. Therefore, this model admits the type of custodial symmetries which allow the R–parity lepton–number violating operators of Eq. (\[rxloperator\]) while they forbid the baryon–number violating operators of Eq. (\[rxboperator\]). This conclusion was verified by an explicit search of nonrenormalizable terms up to order $N=10$. On the other hand we find already at order $N=6$ the non–vanishing terms $$\begin{aligned}
&Q_1d_3L_3N_1\Phi_{45}\Phi_1~+~Q_1d_2L_2N_2\Phi_{45}{\bar\Phi}_{45}\nonumber\\
&Q_1d_1L_3N_3\Phi_{45}\Phi_1~+~Q_1d_1L_2N_2\Phi_{45}{\bar\Phi}_{45}\nonumber\\
&Q_2d_3L_3N_2{\Phi_{45}}{\bar\Phi}_2+
Q_2d_2L_1N_1\Phi_{45}{\bar\Phi}_{45}\nonumber\\
&Q_2d_2L_3N_3\Phi_{45}{\bar\Phi}_2~+~
Q_2d_1L_1N_2\Phi_{45}{\bar\Phi}_{45}
\label{model7}\end{aligned}$$ At higher orders additional terms will appear. It is therefore seen that while the R–parity baryon number violating operators are forbidden to all orders of nonrenormalizable terms the lepton number violating operators are allowed. This is precisely what is required if the R–parity violation interpretation of the excess of four jet events observed by the ALEPH collaboration is correct.
Let us note some further remarks with in regard to the model proposed in Ref. [@giudice]. As claimed there the R–parity interpretation prefers low values of $\tan\beta$ and therefore to allow perturbative unification requires some intermediate thresholds. This is precisely the scenario suggested by the class of superstring standard–like models [@top]. In this class of models the top–bottom quarks mass hierarchy arises due to the fact that only the top quark gets its mass from a cubic level term in the superpotential while the bottom quark gets its mass term from a higher order term. Thus, in this class of models the top–bottom mass splitting arises due to a hierarchy of the Yukawa couplings rather than a large value of $\tan\beta$. It has similarly been proposed in the context of this class of superstring models that intermediate matter thresholds are required for resolution of the string scale gauge coupling unification problem [@gcu; @df].
To conclude, it was shown in this paper that string models can give rise to dimension four R–parity lepton number violating operators while forbidding the baryon number violating operators. Thus, R–parity violation is allowed while proton decay is forbidden. It will be of further interest to examine whether similar mechanism can operate in other string models [@lykken; @ibanez]. For example, the $SO(6)\times SO(4)$ type models are of particular interest as they also can in principle differentiate between the lepton–number and baryon–number violating operators. It is of further interest to study whether the string models can actually give sizable R–parity violation which is not in conflict with any observation. Finally, we eagerly await the experimental resolution of the observed excess in the ALEPH four jet events.
It is a pleasure to thank G. Giudice for valuable discussions and the CERN theory group for hospitality. This work was supported in part by DOE Grant No. DE-FG-0586ER40272.
[99]{} D. Buskulic [*et al.*]{}, (ALEPH Coll.), [*Z. Phys.*]{} [**C71**]{} (1996) 179. F. Ragusa, for the ALEPH coll., talk at the LEPC Meeting, November 19, 1996. D. Buskulic [*et al.*]{}, . M. Carena, G.F. Giudice, S. Lola and C.E.M. Wagner, CERN–TH/96–352, hep–ph/9612334. There are numerous papers on this subject. A partial list includes:\
L.J. Hall and M. Suzuki, ;\
I.H. Lee, 120;\
S. Dawson, 297;\
R. Barbieri and A. Masiero, ;\
V. Barger, G.F. Giudice, and T. Han, ;\
S. Dimopoulos [*et al.*]{}, ;\
H. Dreiner and G.G. Ross, . H. Kawai, D.C. Lewellen, and S.-H.H. Tye, ;\
I. Antoniadis, C. Bachas, and C. Kounnas, . I. Antoniadis, J.Ellis, J. Hagelin and D.V. Nanopoulos, ;\
J.L. Lopez, D.V. Nanopoulos and K. Yuan, . A.E. Faraggi, D.V. Nanopoulos, and K. Yuan, . I. Antoniadis, G.K. Leontaris and J. Rizos, ;\
G.K. Leontaris, . A.E. Faraggi, . A.E. Faraggi, . A.E. Faraggi, . A.E. Faraggi, . A.E. Faraggi, . J.C. Pati, . K.R. Dienes and A.E. Faraggi, . M. Dine, N. Seiberg and E. Witten, . J. Ellis, J.L. Lopez and D.V. Nanopoulos, ;\
G. Leontaris and T. Tamvakis, . A.E. Faraggi, ; hep–ph/9601332, Nuclear Physics [**B**]{}, in press. S. Chaudhoury, G. Hockney and J. Lykken, . L.E. Iba[ñ]{}ez [*et al,*]{} ;\
D. Bailin, A. Love and S. Thomas, ;\
A. Font [*et al,*]{} .
=7.5in =-18mm =-5mm
$$\begin{aligned}
\begin{tabular}{|c|c|c|rrrrrrr|c|rr|}
\hline
$F$ & SEC & $SU(3)_C\times SU(2)_L\times SU(2)_c $&$Q_{C'}$ & $Q_L$ & $Q_1$ &
$Q_2$ & $Q_3$ & $Q_{4'}$ & $Q_{5'}$ & $SU(5)_H\times SU(3)_H$ &
$Q_{6'}$ & $Q_{8''}$ \\
\hline
$L_1$ & $b_1 \oplus$ & $(1,2,2)$&$-{1\over 2}$ & $0$ & ${1\over 2}$ &
$0$ & $0$ & $-{1\over 2}$ & $-{1\over 2}$ & $(1,1)$ & $-{2\over 3}$ &
$0$ \\
$Q_1$ & $1+\alpha+2\gamma$&$(3,2,1)$&${1\over 6}$&$0$&${1\over 2}$ &
$0$ & $0$ & $-{1\over 2}$ & $-{1\over 2}$ & $(1,1)$ & ${4\over 3}$ &
$0$ \\
$d_1$ & & $(\overline 3,1,1)$&$-{1\over 6}$ & $-1$ & ${1\over 2}$ &
$0$ & $0$ & ${1\over 2}$ & ${1\over 2}$ & $(1,1)$ & $-{4\over 3}$ &
$0$ \\
$N_1$ & & $(1,1,2)$&${1\over 2}$ & $-1$ & ${1\over 2}$ &
$0$ & $0$ & ${1\over 2}$ & ${1\over 2}$ & $(1,1)$ & ${2\over 3}$ &
$0$ \\
$e_1$ & & $(1,1,2)$&${1\over 2}$ & $1$ & ${1\over 2}$ &
$0$ & $0$ & ${1\over 2}$ & ${1\over 2}$ & $(1,1)$ & ${2\over 3}$ &
$0$ \\
$u_1$ & & $(\overline 3,1,1)$&$-{1\over 6}$ & $-1$ & ${1\over 2}$ &
$0$ & $0$ & ${1\over 2}$ & ${1\over 2}$ & $(1,1)$ & $-{4\over 3}$ &
$0$ \\
\hline
$L_2$ & $b_2 \oplus$ & $(1,2,2)$&$-{1\over 2}$ & $0$&0 & ${1\over 2}$ &
$0$ & ${1\over 2}$ & $-{1\over 2}$ & $(1,1)$ & $-{2\over 3}$ &
$0$ \\
$Q_2$ & $1+\alpha+2\gamma$&$(3,2,1)$&${1\over 6}$&$0$&0&${1\over 2}$ &
$0$ & ${1\over 2}$ & $-{1\over 2}$ & $(1,1)$ & ${4\over 3}$ &
$0$ \\
$d_2$ & & $(\overline 3,1,1)$&$-{1\over 6}$ & $1$&0 & ${1\over 2}$ &
$0$ & $-{1\over 2}$ & ${1\over 2}$ & $(1,1)$ & $-{4\over 3}$ &
$0$ \\
$N_2$ & & $(1,1,2)$&${1\over 2}$ & $-1$ & 0 & ${1\over 2}$ &
$0$ & $-{1\over 2}$ & ${1\over 2}$ & $(1,1)$ & ${2\over 3}$ &
$0$ \\
$e_2$ & & $(1,1,2)$&${1\over 2}$ & $1$&0 & ${1\over 2}$ &
$0$ & $-{1\over 2}$ & ${1\over 2}$ & $(1,1)$ & ${2\over 3}$ &
$0$ \\
$u_2$ & & $(\overline 3,1,1)$&$-{1\over 6}$ & $-1$&0 & ${1\over 2}$ &
$0$ & $-{1\over 2}$ & ${1\over 2}$ & $(1,1)$ & $-{4\over 3}$ &
$0$ \\
\hline
$L_3$ & $b_3 \oplus$ & $(1,2,2)$&$-{1\over 2}$ & $0$&0&0 & ${1\over 2}$ &
$0$ & ${1}$ & $(1,1)$ & $-{2\over 3}$ &
$0$ \\
$Q_3$ & $1+\alpha+2\gamma$&$(3,2,1)$&${1\over 6}$&$0$&0&0&${1\over 2}$ &
$0$ & ${1}$ & $(1,1)$ & ${4\over 3}$ &
$0$ \\
$d_3$ & & $(\overline 3,1,1)$&$-{1\over 6}$ & $1$&0&0 & ${1\over 2}$ &
$0$ & $-{1}$ & $(1,1)$ & $-{4\over 3}$ &
$0$ \\
$N_3$ & & $(1,1,2)$&${1\over 2}$ & $-1$&0&0 & ${1\over 2}$ &
$0$ & $-{1}$ & $(1,1)$ & ${2\over 3}$ &
$0$ \\
$e_3$ & & $(1,1,2)$&${1\over 2}$ & $1$&0&0 & ${1\over 2}$ &
$0$ & $-{1}$ & $(1,1)$ & ${2\over 3}$ &
$0$ \\
$u_3$ & & $(\overline 3,1,1)$&$-{1\over 6}$ & $-1$&0&0 & ${1\over 2}$ &
$0$ & $-{1}$ & $(1,1)$ & $-{4\over 3}$ &
$0$ \\
\hline
\end{tabular}
\label{matter1}\end{aligned}$$
|
---
abstract: 'Hyperfine interactions between electron and nuclear spins in the quantum Hall regime provide powerful means for manipulation and detection of nuclear spins. In this work we demonstrate that significant changes in nuclear spin polarization can be created by applying an electric current in a 2-dimensional electron system at Landau level filling factor $\nu=1/2$. Electron spin transitions at $\nu=2/3$ and $1/2$ are utilized for the measurement of the nuclear spin polarization. Consistent results are obtained from these two different methods of nuclear magnetometry. The finite thickness of the electron wavefunction is found to be important even for a narrow quantum well. The current induced effect on nuclear spins can be attributed to electron heating and the efficient coupling between the nuclear and electron spin systems at $\nu$=1/2. The electron temperature, elevated by the current, can be measured with a thermometer based on the measurement of the nuclear spin relaxation rate. The nuclear spin polarization follows a Curie law dependence on the electron temperature. This work also allows us to evaluate the electron $g$-factor in high magnetic fields as well as the polarization mass of composite fermions.'
author:
- 'Y. Q. Li'
- 'V. Umansky'
- 'K. von Klitzing'
- 'J. H. Smet'
title: 'Current Induced Nuclear Spin Depolarization at Landau Level Filling Factor $\nu$=1/2'
---
\[sec:intro\]Introduction
=========================
In GaAs and many other semiconductors, electron spins and nuclear spins interact with each other via the hyperfine interaction [@Dyakonov84]. This interaction forms the basis for many ingenious methods to detect and manipulate nuclear spins via the electronic states [@Li08; @Kalevich08; @Reilly08; @ZhangYJ11]. Conversely, nuclear spins can also be used to probe and study electronic states in molecules and various condensed matter systems. For instance, the electron spin polarization can be measured with the Knight shift in nuclear magnetic resonance (NMR) experiments [@Barrett95; @Stern04; @Dementyev99; @Melinte00; @Knumada07; @Tiemann12]. Nuclear spin relaxation measurements yield important information about the electronic systems [@Berg90; @Smet02; @Hashimoto02; @Spielman05; @Kumada05; @Tracy06; @Kumada06; @Tracy07; @Zhang07; @Li09].
An essential ingredient of a nuclear spin relaxation measurement is to first drive the nuclear spin system out of equilibrium. NMR provides the most direct way for eliciting changes in the degree of nuclear spin polarization. The RF radiation in NMR experiments, however, unavoidably raises the electron temperature in the sample, in particular when operating at dilution refrigerator temperatures. This is detrimental to the fragile states such as those formed as a result of electron correlations in the fractional quantum Hall regime. Fortunately, alternative methods for manipulating nuclear spin polarization are available. They are based on the spin flip-flop term of the hyperfine interaction: $$\label{eq:flip-flop}
H_\mathrm{flip-flop}=\frac{1}{2}A_\mathrm{HF}({\hat{I}_{+}}\cdot \hat{S}_{-}+{\hat{I}_{-}}\cdot \hat{S}_{+}),$$ where $A_\mathrm{HF}$ is the hyperfine constant, and $\hat{I}_+$($\hat{I}_-$) and $\hat{S}_+$($\hat{S}_-$) are raising (lowering) operators for the nuclear and electron spins, respectively. This term describes processes in which the flip of an electron spin simultaneously triggers the reversal of a nuclear spin. Driving the electron spin system out of thermal equilibrium by an external source (e.g. microwave, light, or electric current) will cause the electron spins to relax back. The electron relaxation is accompanied by polarization of the nuclear spins. This dynamic nuclear polarization (DNP) process has been realized in numerous experiments, including optical pumping by circularly polarized light, [@Barrett95] electron spin resonance (ESR) [@Dobers88], inter-edge channel scattering in the quantum Hall regime [@Dixon97], current induced scattering near Landau level filling factor $\nu=$2/3 [@Kronmueller98; @Kronmueller99; @Smet01], and other fractional fillings [@Kraus02; @Stern04; @Kou10]. It also occurs in the breakdown regime of the integer and fractional quantum Hall effects [@Kawamura07; @Dean09; @Kawamura09].
In this work we highlight a different method for manipulating the nuclear spin polarization. It does not rely on the aforementioned conventional dynamic nuclear polarization processes. It will be referred to as electrically controlled thermal depolarization. It is based on current induced heating in the two-dimensional electron system (2DES) when a partially polarized composite fermion liquid at half filling of the lowest Landau level forms. The strong hyperfine interaction transfers energy from electrons to nuclear spins and hence raises the entropy as well as the temperature of the nuclear spin system. Significant changes in nuclear spin polarization can be obtained with low current densities. In contrast to previously reported techniques for electrical control of the nuclear spin polarization, this $\nu=1/2$ based technique can in principle produce spatially homogeneous changes in the nuclear spin polarization across a large area. The changes in nuclear spin polarization are measured with two different methods. One is based on the spin transition in the $\nu=1/2$ state itself [@Tracy07; @Li09], and the other relies on the spin phase transition at $\nu=2/3$ [@Kronmueller98; @Smet04]. The nuclear spin depolarization induced by the current can be described by a Curie law. This paper is organized as follows. In Sec.\[sec:theory\], we give a brief description of the composite fermion picture and the spin transitions at fillings 1/2 and 2/3. Sec.\[sec:Methods\] is devoted to the experimental details, including the sample preparation, the measurement setup, and the principles of the nuclear magnetometry using the 1/2 and 2/3 spin transitions. Effects associated with the finite thickness of the two-dimensional electron system will also be discussed in this section. In Sec.\[sec:Result\] the main experimental results will be presented. They include the current induced effects on the nuclear spins and electron transport properties. The mechanism for electrical controlled nuclear spin depolarization will be discussed based on measurements of electron temperatures. Finally, concluding remarks will be given in Sec.\[sec:summary\].
\[sec:theory\]Theoretical background
====================================
For a 2D electron system with density $n_s$ subject to a perpendicular magnetic field $B$, all electrons reside in the lowest Landau level when $B$ exceeds $n_s h/e$ (i.e. Landau level filling factor $\nu\equiv\frac{n_s/h}{eB}<1$), where $n_s$ is the density of the 2DES, $h$ is the Planck constant, and $e$ is the electron’s charge. The orbital degree of freedom is no longer relevant and the physics is governed by electron-electron interactions. They give rise to a large number of fractional quantum Hall states when the disorder is sufficiently weak [@DasSarma97]. The many-body wavefunctions proposed by Laughlin provide a solid foundation for understanding the nature of these states [@Laughlin83]. It is however also possible to describe the appearance of these correlated fractional quantum Hall ground states in an intuitive, single particle picture by introducing quasi-particles referred to as composite fermions [@Jain07; @HLR93].
A composite fermion (CF) comprises one electron and an even number ($q=2,4$) of magnetic flux quanta. CFs no longer experience the external magnetic field, but a drastically reduced effective magnetic field which in a mean field approximation [@HLR93] is given by $B_\mathrm{eff}=B-q n_s h/e$ and vanishes exactly at even denominator filling $1/q$. At this filling composite fermions form a compressible Fermi sea which in many ways resembles the 2D electron Fermi liquid at zero magnetic field. At filling factors away from 1/q, the Landau quantization of composite fermions in a non-zero $B_\mathrm{eff}$ gives rise to the integer quantum Hall effect of composite fermions. It is equivalent to the fractional quantum Hall effect at fillings $p/(pq\pm 1)$, where $p$ is the number of filled CF Landau levels. For example, the $\nu=2/3$ state corresponds to the integer quantum Hall state of composite fermions with two attached flux quanta when two CF Landau levels are completely filled.
\[subsec:SpinTransition\]Spin transitions at $\nu=2/3$ and $1/2$
----------------------------------------------------------------
 Energy level diagram for filling factor 2/3. The spin transition field $B_\mathrm{tr}$ is marked with the dotted line. When $B<B_\mathrm{tr}$, the ground state is spin unpolarized with the composite fermion Landau levels (0,$\uparrow$) and (0,$\downarrow$) occupied, while for $B>B_\mathrm{tr}$ the two filled levels are (0,$\uparrow$) and (1,$\uparrow$), and the ground state becomes fully polarized; (b) Spin transition for a non-interacting and disorder free composite fermion system at $\nu$=1/2. The density of states solely depends on the composite fermion effective mass. Transition from partial to full spin polarization takes place when the Zeeman energy surpasses the Fermi energy.](Fig1){width="7.5"}
The cyclotron mass and Landau quantization energy of composite fermions are not related to the conduction band mass of the electrons [@Park98]. This has some important implications for the spin related physics in the fractional quantum Hall regime. In the following, we only discuss two cases which will be important for the nuclear magnetometry carried out in this work. For simplicity, we start our discussion with an ideal 2D electron system, whose wavefunction has zero spread in the growth direction. We also do not consider the effect of nuclear spin polarization. It will be treated in a subsequent subsection.
At filling factor 2/3, composite fermions experience an effective (perpendicular) magnetic field of $-B/3$. The energy spectrum is quantized into a ladder of CF Landau levels ($n=0,1,2,...$) with a spacing of $\hbar\omega_\mathrm{CF} = \hbar \frac{e}{m_\mathrm{CF}}\frac{B}{3}=\frac{\hbar e}{3\xi m_e}\sqrt{B}$, where $m_e$ is the free electron mass, and the composite fermion mass is written as $m_\mathrm{CF}=\xi\sqrt{B}m_e$ [@Park98]. Each of the CF Landau levels is split into two spin sub-levels ($s=\uparrow,\downarrow$) by the Zeeman energy $E_Z=g_e\mu_B B_\mathrm{tot}$, where $B_\mathrm{tot}$ is the total magnetic field. Note the effective field only controls the orbital degree of freedom of the composite fermions. As shown in Fig.1(a), energy levels (0,$\downarrow$) and (1,$\uparrow$) cross each other at perpendicular field $B=B_\mathrm{tr}$ due to the different field dependencies of $E_C$ and $E_Z$. When $B<B_\mathrm{tr}$, the two occupied levels, (0,$\uparrow$) and (0,$\downarrow$), have opposite spin orientation, so the ground state is spin unpolarized. In contrast, when $B>B_\mathrm{tr}$ the two filled levels, (0,$\uparrow$) and(1,$\uparrow$), have identical spin orientations in Fig. 1(a) and the ground state is fully spin polarized. The spin transition field $B_\mathrm{tr}$ satisfies $$\label{eq:Btr_two3rds}
B_{\mathrm{tr}}|_{\lambda=0, B_N=0}=\frac{4}{9}(\frac{1}{g_e\xi})^2 \cos^2\theta,$$ where $\theta$ is the tilt angle, i.e. the angle between the total field $B_\mathrm{tot}$ and the perpendicular field $B$. Here the indices $\lambda=0$ and $B_N=0$ indicate that we are dealing with a special case of zero thickness of the electron wavefunction and zero nuclear spin polarization. More general cases will be discussed in subsequent sections.
For the Fermi sea at $\nu=1/2$, the composite fermion spins are not always fully polarized as well [@Kukushkin99; @Dementyev99; @Melinte00]. A transition from partial to full spin polarization takes place when the Zeeman energy exceeds the Fermi energy of the composite fermions. In the simplest model, the composite fermions are treated as non-interacting particles and the disorder is ignored. Under these assumptions, the Fermi energy can be written as $\varepsilon_F=\hbar^2 k_F^2/(2m_\mathrm{CF})$, where $k_F$ is the CF Fermi wavevector. It is straightforward to obtain that when the (perpendicular) magnetic field $B \geq (\frac{1}{g_e\xi} )^2 \cos^2\theta$, the composite fermion sea becomes fully spin polarized (see Fig.1(b)). It is interesting to note that the spin transition field at $\nu=2/3$ only differs from that at $\nu=1/2$ by a factor of $4/9$.
A useful feature of the partially polarized Fermi sea at $\nu=1/2$ is that the energy spectrum is continuous near the Fermi energy for both spin populations. A spin flip of a composite fermion may require a change in orbital momentum, but costs no or very little energy just like a nuclear spin flop. Hence, the interaction between nuclear spins and electron spins is expected to be strong. The situation resembles the Korringa type of nuclear spin-electron spin interaction in simple metals [@Korringa50; @Tracy07]. Indeed, in experiments the nuclear spin relaxation time has been found to be as short as $\sim$100s even at temperatures below $30$mK [@Li09]. Therefore, the $\nu=1/2$ state lends itself to manipulate nuclear spins because of the efficient coupling between these two spin systems. This will be the central theme of this paper.
\[subsec:FiniteThickness\]Finite thickness effect
-------------------------------------------------
It should be pointed out that the transition fields given in the previous subsection are calculated for truly 2D systems. For a 2DES which forms in a GaAs/AlGaAs quantum well or heterostructure, the finite thickness of the electron wavefunction in the growth direction softens the Coulomb potential. When the magnetic length, $l_B=\sqrt{\hbar/eB}$, becomes comparable to the thickness of the electron wavefunction, the ratio $\eta=E_Z/E_C$, which determines the spin transition fields for both $\nu=2/3$ and $1/2$, is no longer proportional to $\sqrt{B}$. In the strong $B$ limit, the Coulomb potential scales with $\log B$ instead of $\sqrt{B}$. As will be shown in Sec.\[subsec:Two3rds\], the finite thickness effect significantly modifies the spin transition field even for a narrow quantum well.
A precise treatment of the finite thickness effect requires numerical calculations [@Davenport12], which are beyond the scope of this paper. Here we follow an approach introduced by Zhang and Das Sarma [@Zhang86]. According to their theoretical study, a modified Coulomb potential of the form $V(r)\propto (r^2+\lambda^2)^{-1/2}$ provides a reasonably good approximation to more rigorous numerical calculations. Here $\lambda$ is a length parameter that can be viewed as the effective thickness of the electron wavefunction. In this model the Coulomb energy follows $E_C\propto(l_B^2+\lambda^2)^{-1/2}$, instead of $E_C\propto l_B^{-1}$ in the zero thickness limit. For analyzing experiments, it is useful to define $B_\lambda=\hbar/(e\lambda^2)$, and rewrite the Coulomb energy as $$E_C=\frac{1}{4\pi\epsilon\epsilon_0}\frac{e^2}{l_B}\sqrt{\frac{B_\lambda}{B+B_\lambda}}\propto \sqrt{\frac{B_\lambda B}{B+B_\lambda}} .$$ If the tilt angle $\theta$ is small so that the orbital effect of the in-plane field is negligible, the spin transition field, taking into account the non-zero thickness of the 2DES, can be written as $$\label{eq:Btr_finite_thickness}
B_\mathrm{tr}|_{B_N=0}=\frac{1}{2}\left(-B_\lambda+\sqrt{B_\lambda^2+4B_\mathrm{tr}^{0} B_\lambda\cos^2\theta}\right).$$ Here $B_\mathrm{tr}^0$ denotes the spin transition field for the case of zero-thickness, zero tilt angle, and no nuclear spin polarization. It equals $\frac{4}{9}(\frac{1}{g_e\xi})^2$ and $(\frac{1}{g_e\xi})^2$ for $\nu=2/3$ and $1/2$, respectively. Eq.(\[eq:Btr\_finite\_thickness\]) is quite different from the $\cos^2\theta$ dependence expected for the zero-thickness 2DES (see Eq.\[eq:Btr\_two3rds\]). Hence, the measurements in tilted fields can be used to evaluate how significant the finite thickness effect is.
\[subsec:NucSpins\]Influence of nuclear spin polarization
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The Zeeman term in the Hamiltonian of the hyperfine interaction between an electron and the nuclei is $$\mathcal{H}_\mathrm{Z}=A_\mathrm{HF}\hat{\mathbf{I}}_Z\cdot\hat{\mathbf{S}}_Z.$$ The influence of the polarized nuclear spins on the electron Zeeman splitting can be described in terms of an effective field $B_N$ $$\mathcal{H}_\mathrm{Z}=g_e\mu_B B_N.$$ For fully polarized nuclear spins in bulk GaAs $B_N$ equals $-5.3$T. [@Paget77] The minus sign reflects that $\mathbf{B}_N$ is opposite to the external magnetic field $\mathbf{B}_\mathrm{tot}$. The nuclear spin polarizations of all three isotopes in GaAs (i.e. $^{69}$Ga, $^{71}$Ga and $^{75}$As) follow the Brillouin function. For small $B/T$ ($<0.5$Tesla/mK approximately), this function can be simplified to a Curie law form: $$\label{eq:CurieLaw} \mathcal{P}_N=\frac{\langle I
\rangle}{I}=\frac{\gamma_{n}\hbar (I+1)B_{\rm tot}}{3k_{B}T}.$$ For all nuclei in GaAs, $I$ equals $3/2$ and $\gamma_n$ is the nuclear gyromagnetic ratio. Summing up the contributions from all three types of nuclei yields the following expression $$\label{eq:BNthermal}
B_N=\sum\limits_{i=1}^{3}b_{N,i}\mathcal{P}_{N,i}\simeq\frac{0.87 \mathrm{mK}}{g_e}\frac{B_\mathrm{tot}}{T},$$ where $b_{N,i}$ is the maximum effective field of the nuclei of type $i$. Note that $B_N$ is inversely proportional to the electron $g$-factor $g_e$. For a 2D electron system confined in a narrow GaAs quantum well, $|g_e|$ can be considerably smaller than the bulk value $|-0.44|$ [@Malinowski00]. It has been demonstrated in electron spin resonance experiments that strong perpendicular magnetic fields can further decrease the magnitude of $g_e$ [@Dobers88b].
Polarized nuclear spins only modify the electron Zeeman energy, i.e. $E_Z=g_e\mu_B(B_\mathrm{tot}+B_N)$. They do not affect the orbital motion of the electrons. Since the Coulomb energy remains unaltered, the spin transition fields for the $\nu=2/3$ and the $\nu = 1/2$ state change. Let’s first consider the case of the $\nu=2/3$ spin transition when the sample is mounted perpendicular to the external applied magnetic field ($\theta=0$). The transition field is obtained from equation $$\frac{\hbar e}{3 \xi m_e}\sqrt{\frac{B_\lambda B}{B+B_\lambda}} = g_e\mu_B(B+B_N).$$ When $B_N$ is much smaller than $B$, one finds $$\label{eq:Btr_smallBN}
B_\mathrm{tr}|_{\theta=0} \simeq B_\mathrm{tr}|_{\theta=0, B_N=0}-\left(1+\sqrt{\frac{B_\lambda}{B_\lambda+4B_\mathrm{tr}^0}}\right)B_N,$$ where $B_\mathrm{tr}|_{\theta=0, B_N=0}=\frac{1}{2}\left(-B_\lambda+\sqrt{B_\lambda^2+4B_\mathrm{tr}^0}\right)$ is the transition field in the absence of nuclear spin polarization. In the limit of zero-thickness, $B_\lambda\rightarrow\infty$, Eq.(\[eq:Btr\_smallBN\]) reduces to $$B_\mathrm{tr}|_{\lambda=0, \theta=0} \simeq B_\mathrm{tr}^0-2B_N.$$
When the effect of $B_N$, the influence of finite thickness as well as a sample tilt are all included, the calculation of the transition field is more tedious. Nevertheless, a simplified treatment is possible when $B_N$ follows the Curie law. The transition field is obtained by solving $$\frac{\hbar e}{3 \xi m_e}\sqrt{\frac{B_\lambda B}{B+B_\lambda}}=g_e\mu_B(1+\delta)B_\mathrm{tot},$$ with $\delta=0.87\mathrm{mK}/(g_eT)$. This gives rise to a solution similar to Eq.(\[eq:Btr\_finite\_thickness\]): $$\label{eq:Btr_two3rds_full}
B_\mathrm{tr}=\frac{1}{2}\left(-B_\lambda+\sqrt{B_\lambda^2+4\frac{B_\mathrm{tr}^0}{(1+\delta)^2}B_\lambda\cos^2\theta}\right).$$ For many quantum wells of interest, $g_e$ is smaller than zero and the nuclear spins in thermal equilibrium increase the spin transition field. For a typical $g$-factor ($g_e\sim-$0.4), the correction due to nuclear spins is, however, rather small even at very low temperatures. For instance at $T=20$mK, $1+\delta\approx0.9$. Also noteworthy is that in the limit of zero-thickness, Eq.(\[eq:Btr\_two3rds\_full\]) reduces to $B_\mathrm{tr}|_{\lambda=0}=B_\mathrm{tr}^0\cos^2\theta/(1+\delta)^2$.
\[sec:Methods\]Experimental methods
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\[subsec:Measurement\]Sample preparation and measurement setup
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The samples studied in this work are either single- or double-sided doped $\mathrm{Al}_x\mathrm{Ga}_{1-x}\mathrm{As}$/GaAs quantum wells with thicknesses between 16 nm and 18 nm. Although qualitatively similar results were observed on all samples, the data presented here were recorded on a 16nm thick, single-sided doped quantum well with an in-situ grown $n^+$-GaAs backgate. The sample was patterned into 400${\rm \mu m}$ wide Hall bars with voltage probes along the top and bottom perimeter that are 400${\rm \mu m}$ apart. The electron density and mobility at zero backgate voltage are $1.77\times10^{11}$cm$^{-2}$ and $0.8\times10^6$cm$^{2}/$V$\cdot$s, respectively.
Electron transport measurements were carried out in dilution refrigerators with base temperatures less than $20$mK. For the tilted field measurements, the samples were mounted on a stage with a low friction rotation mechanism driven by a high precision dc motor. The tilt angle $\theta$ was calibrated with low field Hall measurements. Standard lock-in techniques were used for the transport measurements. In order to study the current induced effects, two currents are applied to the samples. The first is a small ac current which is typically 1nA, and the other is a dc current or an ac current with different frequency from the first one. The lock-in amplifiers are locked to the first ac current, and hence they measure the differential longitudinal resistance $dV_\mathrm{xx}/dI$ and the differential Hall resistance $dV_\mathrm{xy}/dI$, which are denoted as $R_\mathrm{xx}$ and $R_\mathrm{xy}$, respectively, for simplicity. The differential resistances could be considerably different from the longitudinal resistance $V_\mathrm{xx}/I$ and the Hall resistance $V_\mathrm{xy}/I$ under sufficiently large bias current, but the latter are irrelevant for most of the discussion in this paper.
\[subsec:Two3rds\]Nuclear magnetometry based on the $\nu=2/3$ spin transition
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![\[Fig2\_Two3rd\_Transition\] Top panel: 2D plot of $R_\mathrm{xx}$ in the ($\nu,B$) plane with $I_\mathrm{ac}$=1nA at the base temperature. Regions labeled with $\uparrow \downarrow$ and $\uparrow \uparrow$ correspond to the incompressible $\nu = 2/3$ fractional quantum Hall state with spin polarization $\mathcal{P}=$0 and 1, respectively. The two ground states of the $\nu = 3/5$ quantum Hall fluid with different spin polarizations are also shown; Bottom panel: traces of $R_\mathrm{xx}$ and $R_\mathrm{xy}$ as a function of $B$ with $\nu$ fixed at 2/3. Dissipative transport takes place in the phase transition region.](Fig2){width="7"}
As described in Sec.\[subsec:SpinTransition\], the crossing of the CF levels (0,$\downarrow$) and (1,$\uparrow$) is accompanied by a transition from an unpolarized ground state with $\mathcal{P}=0$ to a fully polarized ground state with $\mathcal{P}=1$. This spin transition as a result of the competition between the Coulomb energy and the Zeeman energy leaves a signature in a measurement of $R_\mathrm{xx}$, since longitudinal transport becomes dissipative near the transition [@Engel92; @Smet01; @Kraus02; @Hashimoto02; @Hashimoto04]. Fig.\[Fig2\_Two3rd\_Transition\] shows an example of a transport measurement. A color rendition of the longitudinal resistance in the filling factor versus $B$-field plane is depicted in the top panel. Arrows mark the spin orientation of the Landau levels which are occupied for states with fractional fillings 2/3 and 3/5. At filling factor 2/3 the resistance becomes non-zero near 9.5 T. This is more clearly seen in a plot of the resistance along the line of constant filling factor 2/3 displayed in the bottom panel. Below this field the ground state is unpolarized. Also displayed in this graph is the Hall resistance. At the transition the Hall resistance deviates from the quantized Hall resistance. At higher $B$-fields, the formation of the fully spin polarized ground state gives rise to reentrant behavior in $R_{\rm xx}$ and $R_{\rm xy}$. As described above, the transition field $B_\mathrm{tr}$ is determined by the relative strength of $E_C$ and $E_Z$ and can therefore be varied either by tilting the magnetic field [@Kraus02], or changing the nuclear spin polarization [@Smet02]. Conversely, a measurement of the 2/3 transition field can serve as a method for nuclear magnetometry, since the displacement of the 2/3 transition provides information on the degree of nuclear spin polarization [@Smet04]. It follows from Eq.(\[eq:Btr\_smallBN\]) that for small nuclear fields $B_N$, the change in $B_N$ is simply proportional to the shift of the transition along the magnetic field axis, or more specifically, $$\label{eq:DeltaBN}
\Delta B_N \simeq -\left(1+\sqrt{\frac{B_\lambda}{B_\lambda+4B_\mathrm{tr}^0}}\right)^{-1}\Delta B_\mathrm{tr}.$$
A very useful scheme to study the interaction physics between nuclear spins and electron spins at filling factors other than 2/3 was introduced in Ref [@Smet04]. The measurement sequence is illustrated in Fig.3(a). It allows measuring the nuclear spin polarization at filling factor $\nu_\mathrm{rest}$, the filling factor of interest. Throughout this work, $\nu_\mathrm{rest}$ equals $1/2$. The system is allowed to relax and reach a steady state during an extended period of time $t_\mathrm{rest}$ at filling $\nu_\mathrm{rest}$. In order to determine the degree of nuclear spin polarization at this filling factor, the magnetic field is swept in small steps in a range large enough to cover the 2/3 phase transition peak. Each time after changing the magnetic field slightly, the system is again allowed to relax during a time $t_\mathrm{rest}$. The gate voltage tracks the externally applied magnetic field to ensure that the electron system remains at $\nu_\mathrm{rest}$ even during the short $B$-field sweep. So the filling factor is at all times $\nu_\mathrm{rest}$ except during a short excursion period to $\nu_\mathrm{meas}$ (typically a value close to 2/3) where we perform the nuclear magnetometry. A small ac current ($I_\mathrm{ac}$ =1nA) is turned on for recording $R_\mathrm{xx}$ during this excursion time $t_\mathrm{meas}$. In this work, $t_\mathrm{rest}$ was chosen to be 120 or 180s and $t_\mathrm{meas}$=1.5s in order to minimize the effect of nuclear spin relaxation at $\nu_\mathrm{meas}$ itself. It was verified that longer $t_\mathrm{rest}$ did not bring noticeable changes in the results on the nuclear spin phenomena at $\nu=1/2$. From the magnetic field at which the spin transition peak appears the effective field $B_{\rm N}$ is extracted. An important prerequisite to be able to calculate $B_\mathrm{N}$ is the knowledge of the transition field in the absence of nuclear spin polarization. How this reference value is obtained will be discussed in more detail in Sec.\[subsec:acNucDepol\].
 Measurement sequence for nuclear magnetometry based on the spin transition at $\nu$=2/3. (b) 2/3 transition peaks for $\nu_\mathrm{rest}$=1/2 measured at $T$=17, 54, and 134mK. (c) Temperature dependence of the 2/3 spin transition field $B_\mathrm{tr}$ for $\nu_\mathrm{rest}$=1/2. Plotted in solid diamonds are $B_\mathrm{tr}$ values extracted from single peak fits. Open circles represent the corrected $B_\mathrm{tr}$ values in which the offsets from the effects irrelevant to nuclear spins are removed. The dotted line marks the expected Curie law dependence of the transition field. See Sec.\[subsec:acNucDepol\] for more details.](Fig3){width="8"}
Fig.3(b) displays $R_{\rm xx}$ traces recorded according to the sequence in panel a for three different temperatures. The spin transition at filling factor 2/3, signaled by the peak in $R_{\rm xx}$, moves to lower magnetic fields as $T$ increases. As discussed in the previous section, the partially polarized Fermi sea allows for efficient coupling between the nuclear and electron spin systems. At thermal equilibrium, the nuclear spin temperature is the same as the electron temperature. Cooling the electrons at $\nu=1/2$ increases the degree of nuclear spin polarization. The nuclear spin polarization acts back on the electron spins as a result of the reduced Zeeman energy, i.e. $E_Z=B_\mathrm{tot}+B_N$, with $B_N$ given in Eq.\[eq:BNthermal\]. For the temperatures encountered in this experiment, $|B_N|<1$T, which is about one order of magnitude smaller than $B_\mathrm{tr}$, so the assumption under which Eq.(\[eq:DeltaBN\]) has been derived is satisfied. One would expect that the shift on $B_\mathrm{tr}$ follows the Curie law and depends linearly on $1/T$. The data plotted in Fig.\[Fig3\_Magnetometry\_Two3rds\](c), however, clearly does not follow a $1/T$ behavior. We attribute this to two factors. The resistance maximum associated with the spin transition is broad and possesses an asymmetric background as a result of thermal activation at high temperatures. This precludes us to extract precise values for the transition fields. A second problem is the difficulty in determining the electron temperature. The actual electron temperature $T_e$ may deviate from the bath temperature $T$, even though the thermometer for measuring the bath temperature is mounted very close to the sample. The difference between $T_e$ and $T$ becomes non-negligible at temperatures lower than 45mK. This issue will be discussed in the following sections.
 Measurement of the spin transition position at $\nu$=2/3 for $\nu_\mathrm{rest}=$1/2 in titled magnetic fields; (b) 2/3 transition fields plotted as a function of $\cos^2\theta$ (squares) and its fit to Eq.(\[eq:Btr\_two3rds\_full\]) with finite thickness effect included (dotted lines). Also plotted for comparison is the expected angular dependence of $B_\mathrm{tr}$ for a zero-thickness 2DES (solid line); (c) $B_\mathrm{tr}$ as the function of $B_\mathrm{extra}$ (squares) and its linear fit (line), which can serve as a calibration curve for the nuclear magnetometry based on the 2/3 transition. The linear fit gives $\Delta B_\mathrm{tr} \approx -1.24\Delta B_N $ for the 16nm quantum well. This is considerably different from $\Delta B_\mathrm{tr} \simeq -2\Delta B_N $ expected for the 2DES with zero-thickness.](Fig4){width="7.5"}
The results above suggest that the dependence of the maximum in the spin transition peak on the bath temperature does not provide a reliable framework to extract the nuclear spin polarization based on the 2/3 spin transition. This difficulty can, however, be overcome by measuring the transition field $B_\mathrm{tr}$ in titled magnetic fields. In this work we limited the measurements to small tilt angles so that the orbital effect of the in-plane field can be ignored. As displayed in Fig.\[Fig4\_Finite\_thickness\](a), the peak of the 2/3 spin transition moves to lower perpendicular field as the tilt angle increases. The height of the peak increases considerably with tilt, but the width of the peak varies very little. This is in contrast with the temperature dependent behavior which is dominated by strong broadening at high temperatures. This feature is particularly helpful for a precise evaluation of $B_\mathrm{tr}$. The transition field is plotted in Fig.\[Fig4\_Finite\_thickness\](b) as a function of $\cos^2\theta$. The experimental values of $B_\mathrm{tr}$ deviate significantly from what one would expect for a zero-thickness 2D electron system. Fitting the data to Eq.(\[eq:Btr\_two3rds\_full\]) yields $B_\mathrm{tr}^0/(1+\delta)^2\approx 26$T and $B_\lambda\approx$5.4T. The latter corresponds to an effective width of $\lambda\approx$11nm. It is slightly larger than the half width of the 16nm thick quantum well. Taking $(1+\delta)^{-2}\approx0.8$ at $T=20$mK (see Sec.\[subsec:NucSpins\]), one obtains $B_\mathrm{tr}^0\approx$21T. It follows from Eq.(\[eq:Btr\_smallBN\]) that $\Delta B_\mathrm{tr} \approx -1.2\Delta B_N$, significantly different from the $\Delta B_\mathrm{tr}=-2B_N$ expected in the limit of zero-thickness.
The main effect of the small angle tilted field is that the extra Zeeman energy brought by the in-plane field lowers the transition field. This is very similar to the role of thermally depolarized nuclear spins. Therefore, the extra Zeeman field, defined as $B_\mathrm{extra}=B_\mathrm{tr}(1/\cos\theta-1)$, could provide a convenient route to determine $\Delta B_N$ without the need for estimating $B_\mathrm{tr}^0$. An example is plotted in Fig.\[Fig4\_Finite\_thickness\](c). The linear fit of the $B_\mathrm{tr}$-$B_\mathrm{extra}$ data gives $\Delta B_\mathrm{tr}\approx -1.24 B_\mathrm{extra}$, or equivalently $\Delta B_\mathrm{tr} \approx -1.24\Delta B_N $ in case that the nuclear spins are involved.
\[subsec:OneHalf\]Nuclear magnetometry based on the $\nu=1/2$ spin transition
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The nuclear magnetometry based on the $\nu=$2/3 spin transition requires a rather sophisticated measurement sequence. Its application is limited to a narrow magnetic field range in the vicinity of the $\nu = 2/3$ spin transition only. For a 2D electron system residing in a wider quantum well, which is desirable in order to benefit from higher electron mobilities, the 2/3 transition peak does not even show up in the transport measurement at sufficiently low $T$. This makes the $\nu=$2/3 detection scheme no longer useful. Moreover, the magnetic field sweeps during the measurement sequence raise concerns over whether the nuclear spin system is truly in thermal equilibrium. This becomes even more problematic for filling factors at which the coupling between nuclear and electron spins is weak.
Here we describe a new type of nuclear magnetometry. It is based on the Fermi sea at $\nu=1/2$. The degree of spin polarization of this Fermi sea is also determined by $E_Z/E_C$ and can therefore readily be tuned as well by either changing the nuclear spin polarization [@Tracy07], or by tilting the sample while keeping the perpendicular field $B$ fixed [@Li09]. For the 16nm thick sample used in this work, the electron system remains partially polarized up to at least 16T if the magnetic field is not tilted. In tilted field measurements with a fixed perpendicular field $B$, the longitudinal resistance $R_\mathrm{xx}$ increases with $B_\mathrm{tot}$ (or $E_Z$) until it reaches a maximum near full spin polarization. As demonstrated in our previous work [@Li09], $R_\mathrm{xx}$ no longer responds to the change in $E_Z$ in case full spin polarization has been reached. The $R_\mathrm{xx}$-$B_\mathrm{tot}$ curves in Fig.\[Fig5\_OneHalf\_Magnetometry\](a) can be interpreted as $R_\mathrm{xx}$ versus $E_\mathrm{Z}$ curves and hence changes in the resistance can be converted into changes of the nuclear field, i.e. the degree of nuclear spin polarization. The orbital effect of the in-plane field, which is responsible for the small negative slope at large $B_\mathrm{tot}$, should be subtracted for large tilt angles. Fig.\[Fig5\_OneHalf\_Magnetometry\](b) shows an example for $B=8$T.
 Longitudinal resistance $R_\mathrm{xx}$ at $\nu=1/2$ as a function of $B_\mathrm{tot}$ with the perpendicular field $B$ fixed at 6T, 7T, 8T, 10T, and 12T. (b) $R_\mathrm{xx}$ plotted as a function of the extra Zeeman field. Triangles are experimental points obtained by directly converting $B_\mathrm{tot}$ to $B_\mathrm{tot}-B$. The line is the result of subtracting the contribution from the orbital effect of the in-plane magnetic field. (c) An example of the time evolution of $R_\mathrm{xx}$ (open dots) as a consequence of nuclear spin relaxation at $\nu=1/2$ and the corresponding change in the Zeeman field $\Delta E_Z/(g_e\mu_B)$ (line) as a function of time. The conversion from $R_\mathrm{xx}$ to $\Delta E_Z/(g_e\mu_B)$ is based on the calibration data in (b). ](Fig5){width="7.5"}
Nuclear magnetometry using the properties of the CF Fermi sea at $\nu=1/2$ is performed with the following sequence of operations: First, the sample relaxes at $\nu=1/2$, usually with a small current (typically 1nA) applied for monitoring $R_\mathrm{xx}$. The relaxation time, $t_\mathrm{relax}$, is usually chosen about one order of magnitude longer than the nuclear spin relaxation time $T_1$. In this experiment, $t_\mathrm{relax}$=900s unless otherwise specified. The nuclear spin system is expected to be close to equilibrium with the electron spin system at the end of this time period. The sample is then brought to the state of interest at filling factor $\nu_{pol}$ for a certain time. This may be the same or a different filling factor and RF radiation may be turned on in resonance with nuclear spins or the system may also be excited by a large current. Subsequently, the filling factor is set back to 1/2 (if it has been changed in the previous step) and all of the external excitation sources (if any were turned on) are shut off. Only the small measurement current ($I_\mathrm{ac}=$1nA) remains turned on in order to record the relaxation of $R_\mathrm{xx}$ at filling 1/2. After $R_\mathrm{xx}$ has saturated, the system is ready for a new set of measurements.
The time evolution of $R_\mathrm{xx}$ at filling 1/2 recorded after the excursion to filling factor $\nu_\mathrm{pol}$ can be fitted to the following exponential decay function: $$\label{eq:ExpDecayRxx}
R_\mathrm{xx}(t)=R_0+\Delta R\exp(-t/\tau).$$ Comparing $\Delta R$ with the $R_\mathrm{xx}$ curve recorded at a fixed perpendicular magnetic field as a function of tilt angle (for instance the one shown in Fig.\[Fig5\_OneHalf\_Magnetometry\](b)) enables to extract the time dependent change in the Zeeman energy, and hence the change in $B_N$. It should be noted that the time constant $\tau$ is equal to $T_1$ only when $R_\mathrm{xx}$ depends linearly on $E_Z$. A strong non-linearity would cause a large discrepancy between $\tau$ and $T_1$. A fitting procedure applicable even if $R_\mathrm{xx}$ depends in a non-linear fashion on $E_\mathrm{Z}$ can however be easily devised by converting the $R_\mathrm{xx}$ values to values of $\Delta E_Z/(g_e\mu_B)$ and then fitting the data to $$\label{eq:ExpDecayDeltaEz}
\Delta E_Z(t)=\Delta E_Z^0\exp(-t/T_1).$$ Fortunately for much of the region where the $\nu=1/2$ state is partially polarized, a linear approximation is justified if $\Delta E_Z$ remains small and consequently $T_1$ is usually close to $\tau$. For example, the relaxation shown in Fig.\[Fig5\_OneHalf\_Magnetometry\](c) yields a $\tau=250$s whereas the procedure using Eq.(\[eq:ExpDecayDeltaEz\]) gives $T_1=255$s.
\[sec:Result\]Results and Discussion
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\[subsec:NucDepol\]Current induced nuclear spin depolarization detected with the $\nu=2/3$ spin transition
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![\[Fig6\_NucSpinDepol\_dc\] Top panels: The phase transition diagrams of $R_\mathrm{xx}$ (defined as $\frac{dV_\mathrm{xx}}{dI}$) plotted in the ($\nu_\mathbf{meas}$, $B$) plane with $\nu_\mathrm{rest}$=1/2 and an applied $I_\mathrm{dc}$ of 0, 5, 20, 400nA from left to right. The detection filling factor $\nu_\mathrm{meas}$ is varied from 0.575-0.755 for each of the diagrams. Bottom panels: (a) The $\nu=2/3$ spin transition peaks for an applied $I_\mathrm{dc}$=0, 5, 20, 400nA during the time the system is left at $\nu_{\rm rest}$=1/2; (b) The dc current dependence of the phase transition field $B_\mathrm{tr}$ at $T$=18, 45, and 71mK. See text for details.](Fig6){width="7.5"}
Fig.\[Fig6\_NucSpinDepol\_dc\] illustrates measurements performed by using the sequence described in Sec.\[subsec:Two3rds\] involving the spin transition at filling 2/3. These experiments were carried out in order to investigate how the dc current influences the nuclear spin polarization at filling 1/2. For each filling factor $\nu_\mathrm{meas}$, the magnetic field was swept from 6.5T to 11.5T in steps of 0.1T. The field sweep rate was 0.1T/minute. The field was then set to remain constant for 2minutes before being ramped to the next value. The filling factor was fixed at 1/2 all the time except for a short excursion period of 1.5s during which the filling factor was changed to $\nu_\mathrm{meas}$ and a small ac current ($I_\mathrm{ac}$=1nA) was switched on to record $R_\mathrm{xx}$. The dc current $I_\mathrm{dc}$ was turned on only when $\nu_\mathrm{rest} = 1/2$. Repeating the above measurement for $\nu_\mathrm{meas}$=0.575 to 0.755 resulted in the color renditions of $R_\mathrm{xx}$ in the ($\nu_\mathrm{meas}$,$B$)-plane.
The four color plots in Fig.\[Fig6\_NucSpinDepol\_dc\] (top panels) reveal the spin phase transition for the $\nu = 2/3$ state at base temperature when a dc current excitation is applied during the time the system is kept at $\nu_\mathrm{rest} = 1/2$ with $I_\mathrm{dc}=$0, 5, 20 and 400nA (from left to right). The phase transition moves to lower field values as $I_\mathrm{dc}$ is increased. The transition field shifts by more than 1T when $I_\mathrm{dc}$ is varied from 0 to 400nA. Fig.\[Fig6\_NucSpinDepol\_dc\](b) shows the dependence of the transition field $B_\mathrm{tr}$ on $I_\mathrm{dc}$ for three different bath temperatures $T=$18, 45 and 71mK. As $T$ becomes higher, $B_\mathrm{tr}$ remains independent of the dc current up to high current values. At higher dc currents however, the curves nearly coincide.
![\[Fig7\_NucSpinDepol\_ac\] The ac current dependence of the phase transition field $B_\mathrm{tr}$ for $\nu_\mathrm{rest}$=1/2 at various bath temperatures. The values of $B_\mathrm{tr}$ are extracted from the spin transition peaks of $R_\mathrm{xx}$ with $I_\mathrm{ac}=1$nA. The lock-in amplifier is locked to $I_\mathrm{ac}$. The frequency of the second ac current $I_\mathrm{ac2}$ is chosen to be different from $I_\mathrm{ac}$ so that it does not interfere with the measurement of the differential resistance.](Fig7){width="7.5"}
Similar effects were also observed for ac current excitation. Fig.\[Fig7\_NucSpinDepol\_ac\] shows the results of a set of measurements carried out with a similar measurement sequence, except that $I_\mathrm{dc}$ is replaced with an ac current, denoted as $I_\mathrm{ac2}$ throughout this paper, with a frequency different from the one for measuring $R_\mathrm{xx}$, i.e. $I_\mathrm{ac}$. As previously for the dc-current, $I_\mathrm{ac2}$ was turned on during the time period when the filling is set to $\nu_\mathrm{rest} = 1/2$.
\[subsec:CIERxx\]Current induced effect on electron transport properties
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In Sec.\[subsec:Two3rds\], we have learned that raising the temperature lowers $B_\mathrm{tr}$ as a result of nuclear spin depolarization. The current induced decrease in $B_\mathrm{tr}$ is also attributed to changes in the nuclear spin polarization. A decrease in $B_\mathrm{tr}$ corresponds to a positive $\Delta B_N$ or a decrease of $|B_N|$. It is natural to suspect that the electron system gets heated as a consequence of weak electron-phonon coupling at ultra-low temperatures and that the entropy is transferred to the nuclei in case a strong interaction between the electron spins and nuclear spins exists. The current induced electron heating has been detected previously in many systems [@Dorlan79; @Roukes85; @Chow96], but to the best of our knowledge, the electron heating effects at $\nu=1/2$ have not been studied systematically. At $\nu=1/2$, the nuclear spin system near the 2DES has only one, but very effective way interacting with the environment, namely the spin flip-flop process via the Fermi contact hyperfine interaction. The elevated temperature of the electron system due to the applied current can therefore be transferred to the nuclear spin system.
![\[Fig8\_CIE\_transport\] The current induced effects on transport properties at $\nu$=1/2 measured at the various bath temperatures (less than 20 to 194mK). The differential resistance $R_\mathrm{xx}$ is plotted as a function of the second ac current $I_\mathrm{ac2}$. A perpendicular magnetic field of $B$=8T was applied in all of the measurements.](Fig8){width="7.5"}
The longitudinal resistance $R_\mathrm{xx}$ is subject to temperature dependent quantum corrections at low temperatures. This temperature dependence of $R_\mathrm{xx}$ seems the most obvious route to estimate the actual electron temperature under the influence of an externally imposed current in order to verify the above explanation for the current induced decrease of $B_\mathrm{tr}$ or $|B_N|$. A similar approach to estimate the electron temperature has been used in other systems such as for example copper thin films [@Roukes85]. The ac current dependence of $R_\mathrm{xx}$ plotted in Fig.8 was measured as follows. Two ac currents were applied. One is $I_\mathrm{ac}$=1nA with a frequency of 22.7Hz, which was locked to the detection electronics for recording $R_\mathrm{xx}$. The frequency of the other current, $I_\mathrm{ac2}$, was set to 83.9Hz. At $T>54$mK, the overall behavior of the current induced changes in the longitudinal resistance is similar to those observed previously in copper thin films [@Roukes85]. Yet, in our system there are some complications that limit the usefulness for extracting the actual electron temperature at lower temperatures. $R_\mathrm{xx}$ exhibits an anomaly at $\nu=$1/2 seen in Fig.8. The resistance at the base temperature with $I_\mathrm{ac2}\sim$10nA applied is lower than that at $T=$54mK. It is opposite to what one would expect from the electron heating picture. This is not completely surprising if one considers that the strongly interacting Fermi sea of electrons at filling 1/2 is not an ordinary Fermi liquid [@HLR93]. No theory is available yet to describe the transport properties of the partially polarized Fermi sea at 1/2. Sizeable nuclear spin polarization at low $T$ further complicates the interpretation of this nonlinear temperature dependence of $R_\mathrm{xx}$. Because of these complications, we did not further pursue the temperature dependence of $R_\mathrm{xx}$ to evaluate the amount of current induced electron heating in this work.
\[subsec:T1thermometry\]Using $T_1$ for sensing the electron temperature
------------------------------------------------------------------------
Here we describe a more attractive alternative to sense the actual electron temperature using the nuclear spin relaxation rate. At filling factor 1/2, efficient coupling between nuclear spins and electron spins is assured in case of partial spin polarization of the electronic system, because of the continuous energy spectrum near the Fermi energy for both electron spin directions. For a disorder free and non-interacting composite fermion system at $\nu=$1/2, the nuclear spin relaxation is described by the Korringa law. It states that the inverse of the relaxation time, $1/T_1$, is proportional to $D_\uparrow(\varepsilon_F) D_\downarrow(\varepsilon_F) T$. Here, $D_{\uparrow,\downarrow}(\varepsilon_F)$ are the density of states at the Fermi energy for up and down spins, respectively. If the spin polarization is close to 100%, disorder will result in an inhomogeneous spatial distribution of the minority spins and deviations from the Korringa law have been observed [@Li09]. Nevertheless, $T_1$ still depends monotonically on temperature, as discussed below and shown in Fig.\[Fig9\_T1\_Thermometry\](b).
 An example of the measurement sequence for extracting the nuclear spin relaxation rate $1/T_1$. A small ac current, $I_\mathrm{ac}$ is always turned on to measure the (differential) resistance $R_\mathrm{xx}$, and the second ac current $I_{\rm ac2}$ (usually much larger than $I_\mathrm{ac}$) is turned on and off alternately to drive the nuclear spin polarization into different values. An averaging over multiple runs and small changes in the nuclear spin polarization were used in the measurements in order to obtain reliable $1/T_1$-data. (b) Temperature dependence of $1/T_1$. The squares are the experimental data and the solid line is a linear fit down to $\sim 45$mK. In the lower inset $1/T_1$ at the base temperature is plotted as a function of $I_\mathrm{ac2}$. (c) $I_\mathrm{ac2}$ dependence of the electron temperature $T_e$ (diamonds). Also plotted for comparison are the fits to the square root of the current, i.e. $T_e\propto\sqrt{I_\mathrm{ac2}}$ (thicker line), as well as the predicted values of the parameter-free hydrodynamic model (thinner, red line).](Fig9){width="7.5"}
A measurement of $T_1$ in the presence of an extra ac current $I_{\rm ac2}$ was carried out as follows. An ac current of $I_\mathrm{ac}=$1-10nA was applied to the sample to monitor the time dependence of $R_\mathrm{xx}$. The system was left to rest for 900s before the second ac current, $I_\mathrm{ac2}$, was turned on. The resistance as a function of time is plotted in Fig. 9(a). It changes immediately after turning on $I_{\rm ac2}$ due to heating of the electronic system, The resistance would then change as a function of time as a result of the current induced agitation of the nuclear spins. This time dependent data after turning on $I_{\rm ac2}$ was fitted to the exponential decay function \[Eq.(\[eq:ExpDecayRxx\])\] to extract $T_1$. The result is displayed in the lower inset of Fig.\[Fig9\_T1\_Thermometry\](b). The nuclear spin relaxation rate $1/T_1$ increases as $I_{\rm ac2}$ becomes larger. This is presumably due to the electron heating. Being aware that the bath temperature may differ from the electron temperature at very low $T$, we only fit the $1/T_1$-$T$ data down to 45mK in order to obtain a reliable curve to extract the electron temperature $T_e$ from the $1/T_1$ values. The $1/T_1$-$I_\mathrm{ac2}$ data is then converted. $T_e$ is determined for every value of $I_\mathrm{ac2}$ by using the $1/T_1$ versus $T$ data. The outcome of this conversion is depicted in Fig.\[Fig9\_T1\_Thermometry\](c). The electron temperature approximately follows the power law $T_e\approx13.3\sqrt{I_\mathrm{ac2}/\mathrm{nA}}$mK. This is very close to the behavior predicted by the hydrodynamical model, which was developed long ago to describe electron heating in the plateau-to-plateau transition region in quantum Hall systems [@Chow96]. According to this model, $T_e\simeq 24.9(\sigma_\mathrm{xx}\rho_\mathrm{xx})^{1/4}[J/\mathrm{(A/m)}]^{1/2}$K, where $J$ is the current density. This would give $T_e\approx13.7\sqrt{I_\mathrm{ac2}/\mathrm{nA}}$mK. It is noteworthy that there is no free parameter in the hydrodynamic model [@Chow96]. Hence, from the nuclear spin relaxation rate we can determine the actual electron temperature.
\[subsec:acNucDepol\]Mechanism for current induced nuclear spin depolarization
------------------------------------------------------------------------------
![\[Fig10\_Btr\_Te\_Dep\] The $\nu=2/3$ transition field $B_\mathrm{tr}$ for $\nu_\mathrm{rest}$=1/2 plotted as a function of electron temperature $T_e$. The upper inset shows the raw data of the $I_\mathrm{ac2}$ dependence of $B_\mathrm{tr}$ measured at the base temperature. The electron temperature is converted from $I_\mathrm{ac2}$ using the thermometry based on the nuclear spin relaxation rates shown in Fig.\[Fig9\_T1\_Thermometry\]. In the lower inset the corresponding $|B_N|/B$, namely the ratio between the magnitude of the nuclear field and external magnetic field is plotted as function of $1/T_e$. It follows the Curie law $B_N= \frac{0.87\,\mathrm{mK}}{g_e}\frac{B}{T_e}$.](Fig10){width="7.5"}
The extraction of the electron temperature from the nuclear spin relaxation measurements is helpful to gain insight into the influence of the current on the nuclear spin polarization. We illustrate this with the data recorded in Fig.7 for the position of the spin phase transition at filling factor 2/3 at base temperature as a function of the applied current $I_\mathrm{ac2}$. In the top inset of Fig.\[Fig10\_Btr\_Te\_Dep\], this raw data has been replotted using $1/\sqrt{I_\mathrm{ac2}}$ as abscissa in view of the close connection between $1/\sqrt{I_\mathrm{ac2}}$ and $1/T_\mathrm{e}$. In the main graph, the current has been converted into the electron temperature using the nuclear spin relaxation data of Fig. 9. Data points are only shown for those points for which the electron temperature is no longer determined by the bath temperature, but predominantly controlled by the applied $I_\mathrm{ac2}$. The 2/3 spin transition field $B_\mathrm{tr}$ follows a linear dependence on $1/T_e$, namely $B_\mathrm{tr}=(8.38+0.0305/T_e$)T. For the smallest $I_{\rm ac2}$, $B_\mathrm{tr}$ reaches $9.44\,T$ (top inset). From the linear fit, we conclude that the electron temperature $T_e$ equals $28.7\,\mathrm{mK}$ in this case, which is considerably higher than the base temperature of the bath (15-18mK).
Based on the calibration of the finite thickness effect obtained in tilted field measurements, i.e. $\Delta B_\mathrm{tr}\simeq -1.24B_N$ (see Sec.\[subsec:Two3rds\]), the $B_\mathrm{tr}$ data can be converted into $B_N$ and are displayed in the lower inset of Fig.\[Fig10\_Btr\_Te\_Dep\]. Since $B_{\rm N}$ can be written as $0.87 \mathrm{mK} B / (g_\mathrm{e} T_\mathrm{e})$ according to Eq.(\[eq:BNthermal\]), it is possible to extract the electron $g$-factor $g_e\approx-0.34$ for this 16nm thick quantum well from this slope. This is consistent with a previous ESR experiment, in which the electron $g$-factor of a 15nm thick GaAs quantum well was determined to be $g_e=-(0.40-0.00575*B)$ for the lowest Landau level [@Dobers88b]. At $B=$9.44T, the ESR experiment would give $g_e=-0.35$. Considering there is about 5% uncertainty in the evaluation of $T_e$ from the nuclear relaxation time $T_1$, the agreement in the $g$-factor with ESR experiments is good. The linear fit of the $1/T_e$ dependence of the transition field also leads to $B_\mathrm{tr}=8.38$T in the limit $B_N\rightarrow 0$. An effective nuclear field $B_N=-(9.44-8.38)/1.24\approx-0.85$T at base temperature or an electron temperature $T_e = 28.7\ {\rm mK}$ can therefore be deduced from these measurements.
An independent confirmation of the validity of the nuclear magnetometry based on the 2/3 spin phase transition comes from the resistive detection method at $\nu=1/2$. The measurement sequence has been described in detail in Sec.\[subsec:OneHalf\], but is repeated here briefly for the sake of clarity. The system is allowed to equilibrate for $t_\mathrm{relax} = 900 s$ at half filling in the presence of a small current $I_\mathrm{ac} = 1nA$ used to monitor $R_\mathrm{xx}$, then $I_\mathrm{ac2}$ is turned on to depolarize the nuclear spins for 900 s. This depolarizing current is turned off and the time dependence of $R_\mathrm{xx}$ is recorded during a time period $t_\mathrm{relax}$. The procedure is then repeated for different values of $I_\mathrm{ac2}$. The $R_\mathrm{xx}$ relaxation data during timeperiod $t_\mathrm{relax}$ can then be fitted to Eq.(\[eq:ExpDecayRxx\]) in order to extract $\Delta B_N$ as described in Sec.\[subsec:OneHalf\]. The data are summarized in Fig.\[Fig11\_Compare\_OneHalf\_Two3rds\]. The change in the nuclear field, $\Delta B_N$, again has a linear dependence on $1/\sqrt{I_\mathrm{ac2}}$ (or $T_e$), similar to that observed in the $\nu$=2/3 detection experiment, in the regime where the electron temperature is controlled by the externally imposed current $I_\mathrm{ac2}$ and not by the bath temperature $T$. Extrapolating the data to the high current limit (corresponding to $T_e\rightarrow \infty$), one obtains $\Delta B_N\approx0.9$T for $B=$9T or a degree of nuclear spin polarization corresponding to $B_N = -0.9\ {\rm T}$ at base temperature. This is close to $B_N\approx-0.85$T extracted from the nuclear magnetometry method based on the 2/3 transition.
The Curie law dependence
------------------------
The Curie law dependence of $B_\mathrm{tr}$ on $T_e$ obtained from the current induced nuclear spin depolarization data is also consistent with the $B_\mathrm{tr}$ values measured with $I_\mathrm{ac2}$=0 in the experiment described in section III.B and summarized in Fig.\[Fig3\_Magnetometry\_Two3rds\], where the temperature of the bath is tuned in the intermediate temperature regime $T\sim$40-80mK. Deviations from the Curie law at higher temperatures are related to the shape of the 2/3 transition peak as a result of the thermally activated electron transport. It broadens the peak and makes it asymmetric which prevents a reliable extraction of $B_\mathrm{tr}$. This difficulty can be overcome, however, by using the information obtained from our studies on the influence of an additional ac-current $I_{\rm ac2}$. As shown in Fig.\[Fig8\_CIE\_transport\], for large enough $I_\mathrm{ac2}$, the longitudinal resistivity at $\nu=1/2$ is independent of the bath temperature. In this case, the electron temperature only depends on the current $I_{\rm ac2}$, even though it may not be the only parameter that controls the transport properties, as discussed in Sec.\[subsec:CIERxx\]. The small temperature dependence of $B_\mathrm{tr}$ observed at large $I_\mathrm{ac2}$ in Fig.\[Fig7\_NucSpinDepol\_ac\] can be attributed to artefacts related to the thermal broadening and asymmetric shape of the transition peak at high temperatures. Without these effects, all of the $B_\mathrm{tr}$-$I_\mathrm{ac2}$ data in Fig.\[Fig7\_NucSpinDepol\_ac\] would merge into a single curve at sufficiently high current densities. The offsets in $B_\mathrm{tr}$ observed in the large current limit can therefore be used to correct the $B_\mathrm{tr}$ values at high temperatures for $I_\mathrm{ac2}=0$. As shown in Fig.\[Fig3\_Magnetometry\_Two3rds\](c), the corrected $B_\mathrm{tr}$ values (open circles) agree very well with the Curie law dependence deduced from the current dependence of $B_\mathrm{tr}$ measured at the base temperature.
![\[Fig11\_Compare\_OneHalf\_Two3rds\] Comparison of the two methods of nuclear magnetometry, which are based on the spin transitions at $\nu$=2/3 (open symbols) and $\nu$=1/2 (solid squares). The ac current ($I_\mathrm{ac2}$) induced change in nuclear field, $\Delta B_N$, at $\nu$=1/2 is plotted as a function of $I_\mathrm{ac2}^{-1/2}$, which is proportional to $1/T_e$. See text for details.](Fig11){width="7.5"}
Now that all aspects of the experimental data are consistent with each other, we are in a position to extract the effective mass of the composite fermions with the help of Eq.(4-13). Taking $T_e=28.7$mK and $g_e=-0.34$, we obtain $1+\delta\approx0.87/|g_eT_e|=0.91$. Based on the fit of the tilted field data shown in Fig.\[Fig4\_Finite\_thickness\] to Eq.(\[eq:Btr\_two3rds\_full\]), one obtains $B_\mathrm{tr}^0/(1+\delta)^2\approx26$T. Using $B_\mathrm{tr}^0=(g_e\xi)^{-2}$ (see Eq.(\[eq:Btr\_finite\_thickness\]) and the related discussion), we get the effective mass parameter, $\xi\equiv\frac{m_\mathrm{CF}}{m_e}\frac{1}{\sqrt{B}}\approx0.4$Tesla$^{-1/2}$ (note that $m_\mathrm{CF}=\xi\sqrt{B}m_e$ with $B$ in units of Tesla), which is close to the predicted value ($\xi$=0.6Tesla$^{-1/2}$) for the polarization mass, but about 5 times larger than the activation mass [@Park98]. The latter is expected to be relevant in the thermally activated transport measurement in the incompressible regime. In contrast, the transport measurements in this work were carried out in the spin phase transition region where the energy gap is reduced and the displacement of the phase transition peak is used to obtain $m_\mathrm{CF}$ instead. As a result, the relevant composite fermion mass is the polarization mass, which was measured previously with optical experiments [@Kukushkin99] and NMR [@Dementyev99; @Melinte00].
\[sec:summary\]Summary
======================
It has been demonstrated in this work that the electric current applied to a 2D electron system at filling factor 1/2 can cause a large change in the degree of nuclear spin polarization. Much of the effect can be attributed to current induced electron heating. This can be described well by the hydrodynamic model. For the current densities applied in this work, the nuclear spin polarization follows a Curie law dependence on the electron temperature. The electron heating induced nuclear depolarization effect is mediated by the efficient coupling between the nuclear spin and the electron spin system.
Some advances in nuclear magnetometry have also been made in this work. The finite thickness effect has been included in the study of the spin transition at $\nu$=2/3. The finite thickness correction is found to be indispensable even for the 16nm thick quantum well sample. An alternative nuclear magnetometry technique based on the spin transition at $\nu=1/2$ has also been developed in this work. The results obtained from these two different methods of nuclear magnetometry are consistent with each other.
The capability of manipulating nuclear spin polarization with current as well as the detection of the change in the degree of nuclear spin polarization by electron transport provide a complete toolbox for *all-electrical* nuclear spin relaxation measurements. The advantage of this approach is that the measurement can be carried out under very weak external excitations (a small quasi-dc perturbation without the need for high frequency radiation) and hence at the lowest possible electron temperature. This is highly desirable for studying the spin properties of for instance fragile fractional quantum Hall states.
We are grateful for valuable discussions with N. R. Cooper, M. I. D’yakonov and J. P. Eisenstein. Y.Q.L. acknowledges technical assistance of T. Guan in recording the data for Fig.\[Fig2\_Two3rd\_Transition\]. We thank B. Frie[ß]{} for valuable comments on the manuscript. We acknowledge financial support from National Science Foundation of China (Project numbers: 10974240 and 91121003), National Basic Research Program of China (Project numbers: 2009CB929101 and 2012CB921703), the Hundred Talents Program of the Chinese Academy of Sciences, the German Ministry of Science and Education and German Israeli Foundation.
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abstract: 'We present results on the performance of the first prototype of the CASTOR quartz-tungsten sampling calorimeter, to be installed in the very forward region of the CMS experiment at the LHC. This study includes [geant]{} Monte Carlo simulations of the Čerenkov light transmission efficiency of different types of air-core light guides, as well as analysis of the calorimeter linearity and resolution as a function of energy and impact-point, obtained with 20-200 GeV electron beams from CERN/SPS tests in 2003. Several configurations of the calorimeter have been tested and compared, including different combinations of (i) structures for the active material of the calorimeter (quartz plates and fibres), (ii) various light-guide reflecting materials (glass and foil reflectors) and (iii) photodetector devices (photomultipliers and avalanche photodiodes).'
---
to 1 truecm
[**First performance studies of a prototype for the CASTOR** ]{}
3 truemm [**forward calorimeter at the CMS experiment**]{}\
1. truecm
[**X. Aslanoglou$^1$, A. Cyz$^2$, N. Davis$^3$, D. d’Enterria$^4$, E. Gladysz-Dziadus$^2$, C. Kalfas$^5$, Y. Musienko$^6$, A. Kuznetsov$^6$, A. D. Panagiotou$^3$**]{}\
8 truemm
[*$^1$ University of Ioannina, PO Box 1186, 45110 Ionnanina, Hellas*]{} 3 truemm [*$^2$ Institute of Nuclear Physics, Radzikowskiego 152, 31342 Krakow, Poland*]{} 3 truemm [*$^3$ University of Athens, Phys. Dept. 15701 Athens, Hellas*]{} 3 truemm [*$^4$ CERN, PH Dept., 1211 Geneva 23, Switzerland*]{} 3 truemm [*$^5$ NRC “Demokritos” INP, PO Box 60228, 15310 Ionnanina, Hellas*]{} 3 truemm [*$^6$ Northeastern University, Dept. of Physics, Boston, MA 02215, USA*]{} 3 truemm
KEYWORDS: CASTOR, CMS, LHC, forward, electromagnetic calorimeter, hadronic calorimeter, quartz, tungsten, sampling calorimeter, Čerenkov light.
Introduction
============
The CASTOR (Centauro And Strange Object Research) detector is a quartz-tungsten sampling calorimeter that has been proposed to study the very forward rapidity (baryon-rich) region in heavy ion collisions in the multi-TeV range at the LHC [@castor1] and thus to complement the heavy ion physics programme, focused mainly in the baryon-free midrapidity region [@ptdr]. CASTOR will be installed in the CMS experiment at 14.38 m from the interaction point, covering the pseudorapidity range 5.2 $< \eta <$ 6.6 and will, thus, contribute not only to the heavy ion program, but also to diffractive and low-$x$ physics in pp collisions [@castor_forw]. The CMS and TOTEM experiments supplemented by the CASTOR detector will constitute the largest acceptance system ever built at a hadron collider, having the possibility to measure the forward energy and particle flow up to $\eta$ = 6.6. With the design specifications for CASTOR, the total and the electromagnetic energies in its acceptance range ($E_{tot}\sim$180 TeV and $E_{em}\sim$50 TeV respectively according to [hijing]{} [@hijing] PbPb simulations at 5.5 TeV) can be measured with a resolution better than $\sim$1% and, therefore, “Centauro” and/or strangelets events with an unusual ratio of electromagnetic to total (hadronic) energies [@Gladysz-Dziadus:2001cq] can be well identified.\
A calorimeter prototype has been constructed and tested with electron beams at CERN/SPS in the summer 2003. The purpose of this beam test was to investigate and compare the performance of different component options (structure of the quartz active material, choice of the light guides/reflectors and photodetector devices), rather than to obtain precise quantitative results of the response of the final detector setup. The general view of the prototype is shown in Figure \[fig:castor\]. The different detector configurations considered in this work are shown schematically in Figure \[fig:proto\_scheme\]. Preliminary results of the analysis have been presented at different CMS meetings [@castor_meetgs]. Here we present a more quantitative analysis, including the beam profile data.
![CASTOR prototype I: frontal view (left picture) and lateral view (right picture, only one light guide is shown).[]{data-label="fig:castor"}](figures/Fig01_castor_photo.eps){width="12cm"}
![Configuration options investigated in the 2003 beam test: different quartz structures (fibres and plate) and reflectors (glass, foil). The points A-O and 4-8 are scan locations used in calorimeter response uniformity studies (see Section \[sec:area\_scan\]). $x-y$ units are mm.[]{data-label="fig:proto_scheme"}](figures/Fig02_prototype_scheme.eps){width="12cm" height="6cm"}
Technical description
=====================
The CASTOR detector is a Čerenkov-effect based calorimeter with tungsten (W) absorber and quartz (Q) as sensitive material. An incident high-energy particle will shower in the tungsten volume and produce relativistic charged particles that will emit Čerenkov light in the quartz plane. The Čerenkov light is then collected and transmitted to photodetector devices through air-core light-guides. The different instrumentation options, investigated in this work, are shown in Figure \[fig:proto\_scheme\]. In section \[sec:WQ\] we describe the various arrangements of the active (quartz) and passive (tungsten) materials of the calorimeter considered. Section \[sec:light\_guides\] discusses the light transmission efficiency of different light-guide geometries, section \[sec:light\_guide\_reflectors\] compares two different light-guide reflecting materials, and section \[sec:light\_read\] summarizes the characteristics of the photodetectors (photomultipliers and avalanche photodiodes) tested.
Tungsten - Quartz {#sec:WQ}
-----------------
The calorimeter prototype is azimuthally divided into 4 octants and longitudinally segmented into 10 W/Q layers (Fig. \[fig:castor\]). Each tungsten absorber layer is followed by a number of quartz planes. The tungsten/quartz planes are inclined at 45$^\circ$ with respect to the beam axis to maximize Čerenkov light output[^1]. The effective length of each W-plate is 7.07 mm, being inclined at 45$^\circ$. The total length is calculated to be 0.73$\lambda_{int}$ and 19.86$X_0$, taking a density for the used W-plates of $\sim$19.0 g/cm$^3$ and ignoring the contribution of the quartz material.
The calorimeter response and relative energy resolution were studied for quartz fibres (Q-F) and quartz plates (Q-P) (see Section \[sec:beamtests\]). We have tested four octant readout units of the calorimeter, arranged side-by-side in four azimuthal sectors. Each readout unit consisted of 10 sampling units. Each sampling unit for sectors J1, J2, and S2 (see Fig. \[fig:proto\_scheme\]) is comprised of a 5 mm thick tungsten plate and three planes of 640 $\mu$m thick quartz fibres. The quartz fibres were produced by Ceram Optec and have 600 $\mu$m pure fused silica core with a 40 $\mu$m polymer cladding and a corresponding numerical aperture NA = 0.37 (in general, an optical fibre consists of the core with index of refraction $n_{core}$, and the cladding with index $n_{clad}$, and NA = $\sqrt{n^2_{core}-n^2_{clad}}$). The sampling unit for sector S1 consisted of a 5 mm thick tungsten plate and one 1.8 mm thick quartz plate. Both types of quartz active material, fibre or plate, had about the same effective thickness. The filling ratio was 30% and 37% for the quartz fibres and quartz plates, respectively.
Air-core light guides {#sec:light_guides}
---------------------
The light guide constructed for the CASTOR prototype I is shown in Figure \[fig:light\_guide\]. It is an air-core light-guide made of Cu-plated 0.8 mm PVC (the internal walls are covered either with a glass reflector or with a reflector foil, which are compared in the next section). In this section the optimal design and dimensions of the light guide are obtained based on detailed [geant]{} Monte Carlo simulations.
In the simulations, the Čerenkov photons produced in the quartz of the calorimeter are collected and transmitted to the photodetectors by air-core light guides. The efficiency of light transmission and its dependence on the light-source position are crucial parameters characterizing the light guide and significantly affecting the performance of the calorimeter. We developed a [geant]{} 3.21-based code to simulate the transmission of Čerenkov photons produced in the quartz plane through a light guide [@mavro]. A photon is tracked until it is either absorbed by the walls or by the medium and is thus lost, or until it escapes from the light guide volume. In the latter case it is considered detected only if it escapes through the exit to the photodetector. If it is back-scattered towards the entry of the light guide it is also lost.
Inside the fibre core Čerenkov photons are practically produced isotropically. But those that are captured and propagate through the lightguide have an exit angle with respect to the fibre longitudinal axis up to a maximum value ($\theta_{core}$) which depends on the numerical aperture NA and the core refraction index ($n_{core}$). When traversing the core-air boundary at the entrance of the lightguide, the photons undergo refraction resulting in a larger angle ($\theta_{air}$). In the simulations, fibres of various numerical apertures (NA = 0.22 - 0.48) as well as light-guides of various shapes (fully square cross section or partially tapered) were used (see Fig. \[fig:light\_guide\_geom\]). The maximum values of core-exiting and air-entering angles ($\theta_{core}$, $\theta_{air}$) in degrees for various numerical apertures are given in Table \[tab:1\]. For the quartz plate, the air-entering angle, $\theta_{air}$, is larger than 30$^\circ$.
0.2cm \[tab:1\]
NA ($n_{core}$=1.46) $\theta_{core}$ $\theta_{air}$
---------------------- ----------------- ----------------
0.22 8.7 12.7
0.37 14.7 21.7
0.40 15.9 23.6
0.44 17.5 26.1
0.48 19.2 28.7
: Maximum values of the core-exiting ($\theta_{core}$) and air-exiting ($\theta_{air}$) angles, for various numerical apertures (NA) of the quartz fibres (index of refraction: $n_{core}$ = 1.46).
The walls of the [geant]{} light-guide have a reflection coefficient of 0.85 (simulating the transmittance of the reflecting internal mirror surface and the quantum efficiency of the photodetector devices, see next Section and Table \[tab:transmittance\]). The entrance plane of the light guide was uniformly scanned with the simulated light source. The percentage of photons escaping in the direction of the photodetector has been recorded as a function of the source position, giving, after integration over the complete surface, the light guide efficiency. The spatial uniformity of the light-guide performance can be quantified with the relative variation ($\sigma/$mean) of the efficiency across the entrance. Results for the light guides efficiency and uniformity studied are tabulated[^2] in Tables \[tab:1\]–\[tab:5\] and are plotted in Figures \[fig:effic\_lightguide\_0.37\] and \[fig:effic\_lightguide\_0.48\] for fibres with NA = 0.37 and 0.48, respectively. We studied air-core lightguides of square cross section (with entrance area 10$\times$10 cm$^2$), fully or partially tapered. The parameters $lg$ and $lm$ refer to the tapered and non-tapered sections of the light guide, as shown in Figure \[fig:light\_guide\_geom\], defined as [@mavro]:
$lg$ = ratio of the length of the tapered part over the width of the entrance plane, and
$lm$ = ratio of the length of non tapered part over the width of the entrance plane.
Thus, e.g. with a mean entrance length of 10 cm, a value $lg:lm$=1:2 indicates that the light-guide has a total length of 30 cm with 10 cm of tapering part, and a value $lg:lm$=2:0 indicates a fully tapered light-guide with length 20 cm, and so on. In tables \[tab:2\]–\[tab:5\], the row (column) indicates the magnitude of the parameters $lm$ ($lg$), respectively.
0.2cm \[tab:2\]
$_{\textstyle lg}\backslash^{\textstyle lm}$ 0 1 2
---------------------------------------------- ------ ------ ------
1 38.3 34.5 34.8
2 46.1 39.1 43.2
3 44.8 41.8 41.5
: Light-guide efficiency (%) for different values of the $lg$ and $lm$ parameters (see text) and quartz fibres with NA = 0.37.
0.2cm \[tab:3\]
$_{\textstyle lg}\backslash^{\textstyle lm}$ 0 1 2
---------------------------------------------- ------ ------ -----
1 39.3 35.5 3.6
2 8.9 38.3 3.4
3 3.3 22.8 3.2
: Relative variation of the light-guide efficiency across the entrance, $\sigma/$Mean (%), for different values of the $lg$ and $lm$ parameters (see text) and quartz fibres with NA = 0.37.
0.2cm \[tab:4\]
$_{\textstyle lg}\backslash^{\textstyle lm}$ 0 1 2
---------------------------------------------- ------ ------ ------
1 31.1 28.3 27.1
2 30.1 27.5 27.5
3 27.1 25.0 25.0
: Light-guide efficiency (%) for different values of the $lg$ and $lm$ parameters (see text) and quartz fibres with NA = 0.48.
0.2cm \[tab:5\]
$_{\textstyle lg}\backslash^{\textstyle lm}$ 0 1 2
---------------------------------------------- ------ ------ -----
1 20.4 23.8 4.1
2 3.9 28.4 4.6
3 3.8 23.2 3.7
: Relative variation of the light-guide efficiency across the entrance, $\sigma/$Mean (%), for different values of the $lg$ and $lm$ parameters (see text) and quartz fibres with NA = 0.48.
From the tables \[tab:1\]-\[tab:5\] and figures \[fig:effic\_lightguide\_0.37\] and \[fig:effic\_lightguide\_0.48\] we note that, as the NA of the fibre and hence the air-entering angle, $\theta_{air}$, increases, the transmission efficiency decreases. Also, the optimum length for the air-core light guide decreases, while the uniformity of the light exiting increases. In order to obtain an optimum efficiency and uniformity of light transmission within the realistically available space, the best option seems $lm$ = 0 and $lg$ = 2 for NA = 0.37 and 0.48. A more detailed study of the light guide performances – beyond the scope of our current paper – can be found in reference [@mavro].
Light guide reflecting material {#sec:light_guide_reflectors}
-------------------------------
The light transmittance in the light-guides was studied for two alternatives for the reflecting medium:
1. 0.5 mm thick float-glass with evaporations of AlO and MgFr (Fig. \[fig:reflectance\_vs\_wavelength\]a) and
2. Dupont polyester film reflector coated with AlO and reflection enhancing dielectric layer stack SiO$_2$+TiO$_2$, the so-called HF reflector foil (Fig. \[fig:reflectance\_vs\_wavelength\]b).
![Reflectance of two mirrors coated with (a) AlO+MgFr and (b) Dupont foil with AlO and SiO$_2$+TiO$_2$, as a function of the incident light wavelength.[]{data-label="fig:reflectance_vs_wavelength"}](figures/Fig07_reflectance_vs_wavelength.eps){width="14cm"}
To choose the most suitable reflector, we also have to take into account the quantum efficiency of the photodetector device (see Section \[sec:light\_read\]). In Table \[tab:transmittance\] we calculate the product of the light guide transmittance and Avalanche Photodiodes (APD) quantum efficiency for Q-fibres with NA = 0.37 and 3 internal reflections in the designed light guide. The light output is higher (lower) for the light-guides with reflector-foil (glass-reflector) for wavelengths above (below) $\lambda$ = 400 nm. We prefer the HF-reflector solution since the short wavelength Čerenkov light ($\lambda$ $<$ 400 nm) deteriorates fast with irradiation of the quartz material and thus a continuous compensation must be applied. The optimum combination of the HF-reflector and the Q-efficiency of the photodetector ensures that the total efficiency is maximized above 400 nm and falls sharply to zero below 400 nm.
0.2cm \[tab:transmittance\]
Wavelength Glass reflector (Al+MgF) Dupont + Layer stack
------------ -------------------------- ----------------------
650 nm 62% 64%
400 nm 53% 62%
350 nm 44% 7%
300 nm 10% $\sim$0%
: Light guide transmittance times the Avalanche Photodiode quantum efficiency at each wavelength (see Figure \[fig:quantum\_effic\]) for the two reflectors considered (in both cases the quartz fibres have NA = 0.37 and 3 internal reflections).
Photodetectors {#sec:light_read}
--------------
We instrumented the calorimeter prototype with two different types of light-sensing devices:
1. Two different kinds of Avalanche Photodiodes (APDs): Hamamatsu S8148 (APD1, developed for the CMS electromagnetic calorimeter [@apd1]) and Advanced Photonix Deep-UV (APD2), Fig. \[fig:apds\].
2. Two different types of photomultipliers (PMTs): Hamamatsu R374 and Philips XP2978.
We used 4 Hamamatsu APDs, each 5$\times$5 mm$^2$, in a 2$\times$2 matrix with total area of 1 cm$^2$. The Advanced Photonix DUV APD had an active area of 2 cm$^2$ (16 mm diameter). The Hamamatsu and Philips PMTs have both an active area of 3.1 cm$^2$. The Hamamatsu and Advanced Photonix APD quantum efficiencies are shown versus wavelength in Fig. \[fig:quantum\_effic\].
![The two types of APDs used in the beam test: Hamamatsu S8148 (left, 5$\times$5 mm$^2$, in a 2$\times$2 matrix with total 1 cm$^2$ active area) and Advanced Photonix DUV (right, active area of 2 cm$^2$).[]{data-label="fig:apds"}](figures/Fig08_apds.eps){width="11cm" height="5cm"}
![APDs quantum efficiencies versus wavelength: Hamamatsu S8148 (left) and Advanced Photonix (right, the curve labeled ’blue’ is relevant for this study).[]{data-label="fig:quantum_effic"}](figures/Fig09_quantum_effic_hamamatsuS8148.eps "fig:"){width="6.5cm"} ![APDs quantum efficiencies versus wavelength: Hamamatsu S8148 (left) and Advanced Photonix (right, the curve labeled ’blue’ is relevant for this study).[]{data-label="fig:quantum_effic"}](figures/Fig09_quantum_effic_AdvancedPhotonixDUV.eps "fig:"){width="7cm"}
Beam Test Results {#sec:beamtests}
=================
The beam test took place in summer 2003 at the H4 beam line of the CERN SPS. The calorimeter prototype was placed on a platform movable with respect to the electron beam in both horizontal and vertical (X,Y) directions. Telescopes of two wire chambers, as well as two crossed finger scintillator counters, positioned in front of the calorimeter, were used to determine the electron impact point. In the next two sections we present the measured calorimeter linearity and resolution as a function of energy and impact point for different prototype configurations.
Energy Linearity and Resolution
-------------------------------
To study the linearity of the calorimeter response and the relative energy resolution as a function of energy, the central points C (Fig. \[fig:proto\_scheme\]) in different azimuthal sectors have been exposed to electron beams of energy 20, 40, 80, 100, 150 and 200 GeV. The results of the energy scanning, analyzed for four calorimeter configurations, are shown in figures \[fig:adc\_signals1\]–\[fig:adc\_signals4\]. The distributions of signal amplitudes, after introducing the cuts accounting for the profile of the beam, are symmetric and well fitted by a Gaussian function.
For all configurations, the calorimeter response is found to be linear in the energy range explored (see Fig. \[fig:linearity\]). The average signal amplitude, expressed in units of ADC channels, can be satisfactorily fitted by the following formula:
$$\begin{aligned}
ADC & = & a + b \times E \end{aligned}$$
where the energy $E$ is in GeV. The fitted values of the parameters for each configuration are shown in Fig. \[fig:linearity\] and are tabulated in Table \[tab:linearity\_resol\]. The values of the intercept ’a’ are consistent with the position of the ADC pedestal values measured for the various configurations considered: 36.1 $\pm$ 0.3 (S1-Quartz Plate), 38.4 $\pm$ 1.8 (S2-Quartz Fibres), 35.3 $\pm$ 1.5 (J2-Quartz Fibres, glass reflector), 35.4 $\pm$ 0.6 (J1-Quartz Fibres, foil reflector).
The relative energy resolution of the calorimeter has been studied by plotting the normalized width of the Gaussian signal amplitudes (Figs. \[fig:adc\_signals1\]– \[fig:adc\_signals4\]), $\sigma/E$, with respect to the incident beam electron energy, E (GeV) and fitting the data points with two different functional forms [@resolution]:
$$\begin{aligned}
\sigma/E & = & p_0 + p_1/\sqrt{E} \label{eq:2} \\
\sigma/E & = & p_0 \oplus p_1/\sqrt{E} \oplus p_2/E \label{eq:3}\end{aligned}$$
where the $\oplus$ indicates that the terms have been added in quadrature. In expression (\[eq:3\]), three terms determine the energy resolution:
1. The constant term $p_0$, coming from the gain variation with changing voltage and temperature, limits the resolution at high energies.
2. The dominant stochastic term $p_1$, due to intrinsic shower photon statistics.
3. The noise $p_2$ term, which contains the noise contribution from capacitance and dark current.
![Energy resolution in sectors: (a) S1 (Philips PMT), (b) S2 (Philips PMT), (c) J2 (APD1), (d) S1 (APD2). Two fits are shown: $\sigma/E = p_0 + p_1/\sqrt{E}$ (solid); $\sigma/E = p_0 \oplus p_1/\sqrt{E} \oplus p_2/E$ (dashed), with $E$ given in GeV. The quoted $\sigma/E$ values are an average between both fits.[]{data-label="fig:energy_resol"}](figures/Fig18_Res_S1_Philips.eps "fig:"){width="6.8cm"} ![Energy resolution in sectors: (a) S1 (Philips PMT), (b) S2 (Philips PMT), (c) J2 (APD1), (d) S1 (APD2). Two fits are shown: $\sigma/E = p_0 + p_1/\sqrt{E}$ (solid); $\sigma/E = p_0 \oplus p_1/\sqrt{E} \oplus p_2/E$ (dashed), with $E$ given in GeV. The quoted $\sigma/E$ values are an average between both fits.[]{data-label="fig:energy_resol"}](figures/Fig19_Res_S2_Philips.eps "fig:"){width="6.8cm"}\
![Energy resolution in sectors: (a) S1 (Philips PMT), (b) S2 (Philips PMT), (c) J2 (APD1), (d) S1 (APD2). Two fits are shown: $\sigma/E = p_0 + p_1/\sqrt{E}$ (solid); $\sigma/E = p_0 \oplus p_1/\sqrt{E} \oplus p_2/E$ (dashed), with $E$ given in GeV. The quoted $\sigma/E$ values are an average between both fits.[]{data-label="fig:energy_resol"}](figures/Fig20_Res_J2_APD1.eps "fig:"){width="6.8cm"} ![Energy resolution in sectors: (a) S1 (Philips PMT), (b) S2 (Philips PMT), (c) J2 (APD1), (d) S1 (APD2). Two fits are shown: $\sigma/E = p_0 + p_1/\sqrt{E}$ (solid); $\sigma/E = p_0 \oplus p_1/\sqrt{E} \oplus p_2/E$ (dashed), with $E$ given in GeV. The quoted $\sigma/E$ values are an average between both fits.[]{data-label="fig:energy_resol"}](figures/Fig21_Res_S1_APD2.eps "fig:"){width="6.8cm"}\
Generally, both formulae satisfactorily fit the data (Fig. \[fig:energy\_resol\]). The fit parameters are shown in Table \[tab:linearity\_resol\]. The first thing to notice is that the constant term $p_0$ is close to 0 for all options. The average stochastic term $p_1$ is in the range $\sim$ 26% – 96% and indicates that we can measure the total Pb+Pb electromagnetic energy deposited in CASTOR at LHC energies ($\sim$ 40 TeV, according to [hijing]{} [@hijing]) with a resolution around 1%. The readout by avalanche photodiodes leads to the $p_2$ term, measured to be 1.25 GeV and 4.5 GeV for Advanced Photonix APD and Hamamatsu APD, respectively. It should be noted that the APDs are very sensitive to both voltage and temperature changes, but in this test there was no such stabilization. In Table \[tab:linearity\_resol\] we summarize the fit parameters for both parameterizations and for the four considered configurations.
0.2cm \[tab:linearity\_resol\]
Area scanning {#sec:area_scan}
-------------
The purpose of the area scanning was to check the uniformity of the calorimeter response, affected by electrons hitting points at different places on the sector area, as well as to assess the amount of “edge effects” and lateral leakage from the calorimeter, leading to cross-talk between neighbouring sectors.
For the area scanning of sector S2, connected to the Philips PMT, central points (A-E) as well as border points (I-O) have been exposed to electron beam of energy 100 GeV (see Fig. \[fig:proto\_scheme\]). The distributions are symmetric and well described by Gaussian fits for the majority of the points. Asymmetric distributions are seen only for points closer than $\sim$3 mm to the calorimeter outer edge or sector border.
Figure \[fig:energy\_impact\] shows the calorimeter response and relative resolution ($\sigma/E$) as a function of the distance $R$ from the calorimeter center, for both central and border points. The top plot shows the coordinates of the points, corrected for the beam impact point position. It can be seen that points E, F, J practically lie at the upper edge of the calorimeter. The rise of the signal amplitudes (bottom left), as well as of the distribution widths with R can be attributed to a lateral spread of the beam. For large $R$, a substantial part of the electron beam is outside of the calorimeter sector and falls directly onto the light guides. The bottom right plot shows that the energy resolution is $\sim$ 4.7% for 100 GeV electrons and is relatively independent of the position of the impact points.
![Dependence of signal amplitude on the distance $R$ from the calorimeter center in sector S2 (Philips PMT). Top: Coordinates of the scanned points. Bottom plots: Measured response to 100 GeV electrons on central (A-E, filled squares) and border (I-O, hollow squares) points.[]{data-label="fig:energy_impact"}](figures/Fig23_comparison_vs_R.eps){width="13cm"}
### S1 - S2 cross talk
Ten points, located at distances 2.5-32. mm from the S1/S2 sector border, have been exposed to the electron beam of energy 80 GeV. The simultaneous readout of both sectors has been done by Advanced Photonix APD and Hamamatsu PMT in S1 and S2, respectively. The upper left pad of Figure \[fig:impact\_point1\] shows the coordinates of the measured points in the calorimeter frame, corrected for the beam impact point position. The star symbol marks the coordinates of the border point between S1 and S2 sectors, found from the dependence of the signal amplitudes on X(Y) coordinates (lower pads).
The distributions of the signal amplitudes in S2 sector, for points distanced from the sector border more than $\sim$ 8 mm, are symmetric (Gaussian) and leakage to S1 sector is negligible. The relative energy resolution $\sigma/E$ is of the order $\sim$ 2.9% for 80 GeV electrons.
The dependence of the calorimeter response, leakage fraction and relative energy resolution, $\sigma$/response, on the distance $d$ from the sector border, for S1 and S2 sectors are shown in Figure \[fig:impact\_point2\].
Both the light output and energy resolution are a little better for S2 sector, connected to Hamamatsu PMT ($\sigma/E$ $\sim$ 2.9%), than for S1 sector, connected to Advanced Photonix APD ($\sigma/E$ $\sim$ 4.5%). This is expected since there is more light collected by the PMT as compared to the APD: area(PMT)/area(APD) = 1.55.
### Comparison of J1, J2 and S1 sectors
For comparison of the uniformity of calorimeter response, several points located at different places on the sectors have been exposed to the electron beam of 80 GeV energy. The points (A-E) at the middle of J1, J2 and S1 sectors and points (4-8) at the border of S1 sector have been studied (see Figure \[fig:proto\_scheme\]). All sectors have been connected to Hamamatsu PMT. Gaussian distributions of signal amplitudes in the middle of the sectors and asymmetric distributions close to the sector border (points 4-8) and sometimes also close to the inner (point A) and outer (point E) calorimeter edge in J1 sector are observed. The beam profile correction (aiming at selecting the central core of the impinging beam) reduces the asymmetry.
![Comparison of calorimeter response (left) and resolution (right) to 80 GeV electrons for several impact points (A-E) of J2, J1 and S1 sectors, readout with Hamamatsu PMTs.[]{data-label="fig:j1j2s1_comparison"}](figures/Fig26_Signals_vs_R_S1J2J1.eps "fig:"){width="6.2cm"} ![Comparison of calorimeter response (left) and resolution (right) to 80 GeV electrons for several impact points (A-E) of J2, J1 and S1 sectors, readout with Hamamatsu PMTs.[]{data-label="fig:j1j2s1_comparison"}](figures/Fig27_Res_vs_R_S1J2J1.eps "fig:"){width="6.5cm" height="4.75cm"}
Comparison of light output and relative energy resolution for all options studied is shown in Figure \[fig:j1j2s1\_comparison\]. Light output is highest in the S1 (QP-glass) sector and it is practically the same for the central and border points. It depends weakly on the distance $R$ of the impact point. For S1, a weak decrease and for J1 and J2 sectors a weak increase of the calorimeter response with distance R from the calorimeter center are observed. The relative energy resolution is almost independent of the position of the impact point and it is $\sim$ 1.5-2.5 % for S1 (QP-glass) and J2 (QF-glass) sectors and $\sim$ 3.5-4.0 % for J2 (QF-foil) for 80 GeV electrons.
Summary
=======
We have presented a comparative study of the performances of the first prototype of the CASTOR quartz-tungsten calorimeter of the CMS experiment using different detector configurations. [geant]{}-based MC simulations have been employed to determine the Čerenkov light efficiency of different types of air-core light guides and reflectors. Different sectors of the calorimeter have been setup with various quartz active materials and with different photodetector devices (PMTs, APDs). Electron beam tests, carried out at CERN SPS in 2003, have been used to analyze the calorimeter linearity and resolution as a function of energy and impact point. The main results obtained can be summarized as follows:
1\. Comparison between the calorimeter response using a single quartz plate or using a quartz-fibre bundle indicates that:
\(a) Good energy linearity is observed for both active medium options (Fig. \[fig:linearity\]).
\(b) The Q-plate gives more light output than equal thickness Q-fibres (Fig. \[fig:j1j2s1\_comparison\]).
\(c) The relative energy resolution is similar for quartz plates and quartz fibres (Fig. \[fig:energy\_resol\]). When readout with the same Hamamatsu PMT (S1, S2 sectors), we found $\sim$2% energy resolution for 80 GeV electrons (Fig. \[fig:j1j2s1\_comparison\]).
\(d) The constant term $p_0$ of the energy resolution, that limits performance at high energies, is less than 1% in both options for the same Philips PMT and glass reflector (Fig. \[fig:energy\_resol\]). The stochastic term $p_1$ is $\sim$36 % and $\sim$46% for quartz plates and quartz fibres, respectively (Table \[tab:linearity\_resol\]).
2\. Avalanche-photodiodes (APDs) appear to be a working option for the photodetectors, although they still need more investigation (radiation-hardness, cooling and voltage stabilization tests).
3\. The relative energy resolution is weakly dependent on the position of the impact point (Fig. \[fig:j1j2s1\_comparison\]). Leakage (cross-talk) between sectors is negligible for impact points separated more than 8 mm from the sector border. Only, electrons impinging less than 3 mm from the detector edge show a degraded energy response and worse resolution.
4\. The shape of the light guide is determined by tree parameters: (i) the type of quartz fiber (NA number), (ii) the maximum efficiency and uniformity of response, and (iii) the available space for the size of a calorimeter. The aim is to simultaneously achieve optimum efficiency and uniformity of light transmission within the realistically available space. From the analysis of the MC simulations we come to the conclusion that the above requirements are best satisfied with $lm$ = 0 and $lg$ = 2 for NA = 0.37 and 0.48.
5\. The light output is a little higher for the light-guides with glass reflector compared to those that use HF-foil, for the same photodetector (Hamamatsu PMT, Fig. \[fig:j1j2s1\_comparison\]). This is understood, since the HF reflecting foil is designed to cut Čerenkov light with $\lambda$ $<$ 400 nm, where the light output is greater. However, the HF-reflector foil has higher efficiency in the region $\lambda>$ 400 nm than the glass mirror (Table \[tab:transmittance\]).
In summary, this study suggests that equipping the CASTOR calorimeter with quartz-plates as active material, APDs as photodetector devices (with temperature and voltage stabilization), and light-guides with foil reflector is a promising option, although the final configuration would benefit from further (detailed) investigation to take into account the experimental conditions that will be encountered in the forward rapidity region of CMS. A beam test of the second prototype was carried out in 2004 and the results are reported elsewhere [@bt-2004].
Acknowledgments
===============
We wish to thank R. Wigmans and N. Akchurin for assistance in the early stage of the beam test. This work is supported in part by the Secretariat for Research of the University of Athens and the Polish State Committee for Scientific Research (KBN) SPUB-M nr. 620/E-77/SPB/CERN/P-03/DWM 51/2004-2006. D.d’E. acknowledges support from the 6th EU Framework Programme (contract MEIF-CT-2005-025073).
[999]{}
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[^1]: The index of refraction of quartz is $n = 1.46-1.55$ for wavelengths $\lambda$ = 600-200 nm. The corresponding Čerenkov threshold velocity is $\beta_c = 1/n = 0.65-0.69$, and therefore, for $\beta_c \approx$ 1 the angle of emission is $\theta_c = acos(1/n\beta) = 46^\circ-50^\circ$.
[^2]: Note, that only the points relevant for the actual light-guide construction are included in the table.
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abstract: 'The single-particle Green’s function (GF) of mesoscopic structures plays a central role in mesoscopic quantum transport. The recursive GF technique is a standard tool to compute this quantity numerically, but it lacks physical transparency and is limited to relatively small systems. Here we present a numerically efficient and physically transparent GF formalism for a general layered structure. In contrast to the recursive GF that directly calculates the GF through the Dyson equations, our approach converts the calculation of the GF to the generation and subsequent propagation of a scattering wave function emanating from a local excitation. This viewpoint not only allows us to reproduce existing results in a concise and physically intuitive manner, but also provides analytical expressions of the GF in terms of a generalized scattering matrix. This identifies the contributions from each individual scattering channel to the GF and hence allows this information to be extracted quantitatively from dual-probe STM experiments. The simplicity and physical transparency of the formalism further allows us to treat the multiple reflection analytically and derive an analytical rule to construct the GF of a general layered system. This could significantly reduce the computational time and enable quantum transport calculations for large samples. We apply this formalism to perform both analytical analysis and numerical simulation for the two-dimensional conductance map of a realistic graphene *p-n* junction. The results demonstrate the possibility of observing the spatially-resolved interference pattern caused by negative refraction and further reveal a few interesting features, such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene.'
author:
- 'Shu-Hui Zhang$^{1}$'
- 'Wen Yang$^{1}$'
- 'Kai Chang$^{2,3}$'
title: 'General Green’s function formalism for layered systems: Wave function approach'
---
Introduction
============
The single-particle retarded Green’s function (GF) is a key tool to calculate local and transport properties in mesoscopic systems [@DattaBook1995; @FerryBook1997], such as conductance, shot noise [@BlanterPhysRep2000], local density of states, and local currents [@CrestiPRB2003; @MetalidisPRB2005]. In the most popular scheme in which a scatterer is connected to two (or more) semi-infinite ballistic leads, the Landauer-Büttiker formula [@LandauerJRD1957; @LandauerPM1970; @BuettikerPRB1985] expresses the conductance $\sigma=(e^{2}/h)T(E_{F})$ in terms of the transmission probability $T(E_{F})$ across the scatterer on the Fermi surface. Typically, the electronic structure and transport properties of a mesoscopic system are described by a lattice model with a localized basis in real space, e.g., discretization of the continuous model [@KhomyakovPRB2004], empirical tight-binding [@HarrisonBook1989], or first-principles density-functional theory with a localized basis set [@KudrnovskyPRB1994; @ZellerPRB1995]. Then $T(E_{F})$ is constructed from the lattice GF $G(E_{F})$ across the scatterer through either an expression derived by Caroli *et al.* [@CaroliJPC1971] or the Fisher-Lee relations [@FisherPRB1981; @StoneIJRD1988; @BarangerPRB1989; @SolsAP1992] that express the scattering matrix $S(E_{F})$ of the scatterer in terms of the lattice GF $G(E_{F})$.
The recursive GF method (RGF) is a standard tool to compute the lattice GF of a scatterer connected to multiple leads [@FerryBook1997; @LewenkopfJCE2013]. This method is reliable, computationally efficient, and allows for a parallel implementation [@DrouvelisJCP2006]. It was pioneered by Thouless and Kirkpatrick [@ThoulessJPC1981] and by Lee and Fisher [@LeePRL1981]. Then MacKinnon presented a slice formulation for a general layered system, which is the form most used nowadays [@MacKinnonZPB1985]. Variations of the method were also introduced to treat multiple leads [@BarangerPRB1991], arbitrary geometries [@KazymyrenkoPRB2008; @WimmerThesis2009], and local scatterers inside an infinite periodic system [@ZhengJCP2010; @SettnesPRB2015]. With the development of many numerical algorithms, such as fast recursive or iterative schemes [@MacKinnonPRL1981; @SoukoulisPRB1982; @MacKinnonZPB1983; @SchweitzerJPC1984; @SanchoJPFMP1984; @SanchoJPF1985; @AokiPRL1985; @AndoPRB1989; @SolsJAP1989; @AndoPRB1990; @GodfrinJPCM1991; @NardelliPRB1999] and closed-form solutions [@UmerskiPRB1997; @SanvitoPRB1999; @Garcia-MolinerSS1994; @VasseurSSR2004; @RochaPRB2004], the development of RGF has culminated in many packages with different focus [@BrandbygePRB2002; @RochaPRB2006; @BirnerIEEE2007; @OzakiPRB2010; @FonsecaJCE2013] and is applicable to an arbitrary lattice Hamiltonian [@GrothNJP2014]. However, these techniques and our knowledge about the lattice GF still suffer from two drawbacks/limitations.
![Connection between two widely used approaches to mesoscopic quantum transport: the GF approach and wave function mode matching approach. The former calculates the GF $G(E)$, while the latter calculates the scattering matrix $S(E)$. The Fisher-Lee relations allow $S(E)$ to be constructed from $G(E)$, while the inverse relation remains absent for a general lattice model.[]{data-label="G_GFVSMODEMATCH"}](fig1GFVSModeMatch.eps){width="\columnwidth"}
First, there are two widely used approaches in mesoscopic quantum transport: the GF approach [@DattaBook1995; @Datta2000] that computes the GF $G(E)$ and the wave function mode matching approach [@AndoPRB1991; @NikolicPRB1994; @XiaPRB2006] that computes the unitary scattering matrix $S(E)$. However, the connection between the GF $G(E)$ and $S(E)$ and hence the connection between these two approaches remain incomplete. It is well known that $S(E)$ can be constructed from $G(E)$ through the Fisher-Lee relations (see Fig. \[G\_GFVSMODEMATCH\]), as first derived by Fisher and Lee [@FisherPRB1981] for a single parabolic band and two-terminal structures and later generalized to multiple leads [@StoneIJRD1988; @BarangerPRB1989; @SolsAP1992] and arbitrary lattice models [@SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009]. However, the inverse of this relation is nontrivial [^1]: explicit expressions of $G(E)$ in terms of $S(E)$ were derived [@StoneIJRD1988; @BarangerPRB1989; @SolsAP1992] only for a single parabolic band and in regions far from the scatterer. Generalization of this result to a general lattice model and over other regions would not only complete the equivalence [@KhomyakovPRB2005] between the GF approach and the wave function mode matching approach (Fig. \[G\_GFVSMODEMATCH\]), but also provides important tools to analyze the multi-probe scanning tunneling microscopy (STM), which has been applied to characterize a wide range of systems (see Refs. for recent reviews) in the past few years, including nanowires [@KuboAPL2006; @CherepanovRSI2012; @QinRSI2012], carbon nanotubes [@WatanabeAPL2001], graphene nanoribbons [@BaringhausNature2014; @BaringhausPRL2016], monolayer and bilayer graphene [@SutterNatMater2008; @JiNatMater2012; @EderNanoLett2013], and grain boundaries in graphene [@ClarkAcsNano2013; @ClarkPRX2014] and copper [@KimNanoLett2010]. With one STM probe at $\mathbf{R}_{1}$ and the other STM probe at $\mathbf{R}_{2}$, the Landauer-Büttiker formula expresses the conductance between the STM probes in terms of the GF $G(\mathbf{R}%
_{2},\mathbf{R}_{1},E_{F})$, which provides spatially resolved information about the sample; e.g., with an analytical expression for the GF of pristine graphene [@ZhengJCP2010; @SettnesPRB2015], Settnes *et al.* [@SettnesPRL2014] were able to identify the different scattering processes of local scatterers in graphene. However, this analysis is still qualitative. To go one step further to extract *quantitatively* the information about the scatterers, an explicit expression of the GF in terms of the scattering matrix is required.
Second, the time cost of RGF increases rapidly with the number of localized basis required to subtend the sample. This imposes a computational limit when addressing realistic experimental samples; e.g., many quantum transport studies on graphene consider narrow graphene nanoribbons rather than large-area graphene. Three methods have been proposed to lift this constraint. The modular RGF [@SolsJAP1989; @SolsAP1992; @RotterPRB2000] is limited to electrons in a single parabolic band and specific shape of the sample. The other two methods essentially reduce the number of transverse bases, either by projecting the system Hamiltonian onto a small number of transverse modes [@MaaoPRB1994; @ZozoulenkoPRB1996; @ZozoulenkoPRB1996a] or by assuming translational invariance [@LiuPRB2012] along the transverse direction. They could significantly reduce the time cost for wide samples, but the time cost still increases linearly with the length of the scatterer (along which transport occurs). To study a *long* sample, a more efficient method is desirable.
The origin of the above drawbacks/limitations is probably that the RGF treats the GF as a matrix and constructs the GF by a series of matrix recursion rules derived from the Dyson equation. Interestingly, although the rules for constructing the scattering matrix in terms of the GF (i.e., the Fisher-Lee relation) are concise and physically intuitive, their rigorous derivation (in which the GF is treated as a matrix) turns out to be rather tedious (see, e.g., Refs. ). This somewhat surprising fact suggests the possible existence of a very different way to represent and calculate the GF. This could not only enable a straightforward physical interpretation of the final results, but also shed light on some previous debates [@NikolicPRB1994; @KrsticPRB2002; @KhomyakovPRB2005] on the relationship between different calculation techniques in mesoscopic quantum transport.
In this work, we develop a numerically efficient and physically transparent GF formalism to address the above issues in layered systems, i.e., any system that is non-periodic along one direction, but is finite or periodic along the other directions. This includes a wide range of physical systems, such as interfaces and junctions, Hall bars, nanowires, multilayers, superlattices, carbon nanotubes, and graphene nanoribbons. Compared with the RGF that directly calculates the GF as a matrix through the Dyson equations, our approach converts the calculation of the GF to the generation and subsequent propagation of a scattering wave function emanating from a local excitation. This viewpoint provides several advantages. First, the procedures for calculating the GF $G(E)$ becomes physically transparent and existing results from the RGF (such as the Fisher-Lee relation) can be derived in a concise and physically intuitive manner. Second, the GF $G(E)$ can be readily expressed in terms of a few scattering wave functions with energy $E$. This provides an *on-shell* generalization of the standard spectral expansion in classic textbooks on quantum mechanics [@SakuraiBook1994; @GriffithsBook1995; @CohenBook2005], $G(\mathbf{R}%
_{2},\mathbf{R}_{1},E)=\sum_{\lambda}\langle\mathbf{R}_{1}|\Psi_{\lambda
}\rangle\langle\Psi_{\lambda}|\mathbf{R}_{2}\rangle/(E+i0^{+}-E_{\lambda})$, which involves *all* the eigenstates $\{|\Psi_{\lambda}\rangle\}$ and eigenenergies $\{E_{\lambda}\}$ of the system. In terms of a generalized scattering matrix $\mathcal{S}(E)$ that describes the scattering of both traveling and evanescent waves, we further establish a one-to-one correspondence between $G(E)$ and and $\mathcal{S}(E)$ (see Fig. 1). This identifies the contributions from each individual scattering channel (including evanescent channels) to the GF and hence allows this information to be extracted quantitatively from dual-probe STM. Third, the simplicity and physical transparency of the formalism further allows us to perform an infinite summation of the multiple reflection between different scatterers and arrive at an analytical construction rule for the GF of a general layered system containing an arbitrary number of scatterers. This could make the time cost independent of the length of the sample along the transport direction and hence significantly speed up the calculation. By further reducing the number of bases along the transverse direction [@MaaoPRB1994; @ZozoulenkoPRB1996; @ZozoulenkoPRB1996a; @LiuPRB2012], our formalism enables quantum transport calculations over macroscopic distances and on large samples.
Recently, the chiral tunneling [@KleinZP1929; @KatsnelsonNatPhys2006; @CheianovPRB2006] and negative refraction [@CheianovScience2007; @ParkNanoLett2008; @MoghaddamPRL2010] of graphene *p-n* junctions have received a lot of interest and the anomalous focusing effect was observed experimentally [@LeeNatPhys2015; @ChenScience2016], but previous theoretical studies are mostly based on the low-energy continuous model, whose validity is limited to the vicinity of the Dirac points. Here we apply our GF approach to perform an analytical analysis and numerical simulation for the two-dimensional conductance map of dual-probe STM experiments in a realistic graphene *p-n* junction described by the tight-binding model. The results demonstrate the possibility of observing the spatially resolved interference pattern caused by negative refraction and further reveals some interesting features (such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene) that may also be observed in dual-probe STM experiments.
This paper is organized as follows. In Sec. II, we introduce the model, review the commonly used RGF technique, and presents the key idea of our approach. In Sec. III, we derive the GF of an infinite system containing a single scatterer, as well as an analytical construction rule for the GF of a general layered system containing an arbitrary number of scatterers. In Sec. IV, we express the GF analytically in terms of a generalized scattering matrix or in terms of a few scattering states on the energy shell $E$. In Sec. V, we exemplify our results in a 1D chain and then apply it to analyze and simulate the two-dimensional conductance map of a realistic graphene *p-n* junction. Finally, a brief conclusion is given in Sec. VI.
Theoretical model and key ideas
===============================
![(a) A layered 2D structure consisting of multiple periodic slices (i.e., leads) and disordered slices (i.e., scatterers). (b) Regarding each slice as a unit cell (filled squares), the structure in (a) becomes a 1D lattice. (c) A semi-infinite lead connected to a scatterer. The unit cell Hamiltonian (filled squares) and nearest-neighbor hopping (double arrows) are $m$-independent inside the lead, but could be disordered inside the scatterer.[]{data-label="G_SETUP"}](fig2Setup.eps){width="\columnwidth"}
We consider a general layered system in the lattice representation. When each layer is an infinite, periodic repetition of a basic unit, we can make a Fourier transform to effectively reduce each layer to a single basic unit. Disorder inside each layer can also be introduced by using a sufficiently large unit cell and repeating it periodically. Then we can regard each layer as a finite-size unit cell, so the system becomes a 1D lattice, e.g., by taking each layer/slice of the structure in Fig. \[G\_SETUP\](a) as a unit cell, Fig. \[G\_SETUP\](a) becomes Fig. \[G\_SETUP\](b). A general 1D lattice can always be decomposed into a few nonperiodic regions (referred to as *scatterers*) consisting of different unit cells sandwiched between periodic regions (referred to as *leads*) consisting of identical unit cells; see Fig. \[G\_SETUP\](b) for an example.
Without losing generality, we consider nearest-neighbor hopping [^2] and use $M_{m}$ to denote the number of orthonormal local bases in the $m$th unit cell. In the representation of these bases, the lattice Hamiltonian is an infinite-dimensional block-tridiagonal matrix: $$\mathbf{H}=%
\begin{bmatrix}
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots\\
\cdots & \mathbf{H}_{-2,-2} & \mathbf{H}_{-2,-1} & 0 & 0 & 0 & \cdots\\
\cdots & \mathbf{H}_{-2,-1}^{\dagger} & \mathbf{H}_{-1,-1} & \mathbf{H}_{-1,0}
& 0 & 0 & \cdots\\
\cdots & 0 & \mathbf{H}_{-1,0}^{\dagger} & \mathbf{H}_{0,0} & \mathbf{H}_{0,1}
& 0 & \cdots\\
\cdots & 0 & 0 & \mathbf{H}_{0,1}^{\dagger} & \mathbf{H}_{1,1} &
\mathbf{H}_{1,2} & \cdots\\
\cdots & 0 & 0 & 0 & \mathbf{H}_{1,2}^{\dagger} & \mathbf{H}_{2,2} & \cdots\\
\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots
\end{bmatrix}
, \label{HAMIL}%$$ consisting of the $M_{m}\times M_{m+1}$ hopping matrix $\mathbf{H}_{m,m+1}$ between neighboring unit cells, its Hermitian conjugate $\mathbf{H}%
_{m+1,m}=\mathbf{H}_{m,m+1}^{\dagger}$, and the $M_{m}\times M_{m}$ Hamiltonian matrix $\mathbf{H}_{m,m}$ of the $m$th unit cell. In a lead, $\mathbf{H}_{m,m}=\mathbf{h}$ and $\mathbf{H}_{m,m+1}=\mathbf{t}$ are independent of $m$. In a scatterer, $\mathbf{H}_{m,m}$ and $\mathbf{H}%
_{m,m+1}$ could dependent on $m$ arbitrarily. Here, as a convention, the region of the scatterers is chosen such that the hopping between the lead and the surface of a scatterer is *the same* as that inside this lead, e.g., $\mathbf{H}_{m_{L}-1,m_{L}}=\mathbf{t}$ in Fig. \[G\_SETUP\](c), where $\mathbf{t}$ is the hopping inside the left lead. Except for the Hermiticity requirement $\mathbf{H}=\mathbf{H}^{\dagger}$, the lattice Hamiltonian is completely general. The (retarded) GF of the layered system is an infinite-dimensional matrix: $\mathbf{G}(E)\equiv(z-\mathbf{H})^{-1}$, where $z\equiv E+i0^{+}$. The GF from the unit cell $m_{0}$ to the unit cell $m$ is an $M_{m}\times M_{m_{0}}$ matrix and corresponds to the $(m,m_{0})$ block of $\mathbf{G}(E)$, i.e., $\mathbf{G}_{m,m_{0}}(E)\equiv\lbrack(z-\mathbf{H}%
)^{-1}]_{m,m_{0}}$. Hereafter we consider a fixed energy $E$ or $z\equiv
E+i0^{+}$ and omit this argument for brevity.
To highlight the distinguishing features of our approach and introduce relevant concepts, we first review the commonly used RGF method before presenting our idea.
Recursive Green’s function method
---------------------------------
The idea of the RGF is to build up the entire system out of disconnected subsystems by the Dyson equation. Let us start from two disconnected subsystems $A$ and $B$ characterized by the Hamiltonian $\mathbf{H}^{(A)}$ and $\mathbf{H}%
^{(B)}$, respectively. The (retarded) GFs of each subsystem are $\mathbf{G}%
^{(A)}\equiv(z-\mathbf{H}^{(A)})^{-1}$ and $\mathbf{G}^{(B)}\equiv
(z-\mathbf{H}^{(B)})^{-1}$. Next we connect the interface (denoted by $a$) of $A$ and the interface (denoted by $b$) of $B$ by local couplings $\mathbf{V}_{ab}$ and $\mathbf{V}_{ba}$. Thus the Dyson equation gives the GF $$\mathbf{G}=%
\begin{bmatrix}
\mathbf{G}_{AA} & \mathbf{G}_{AB}\\
\mathbf{G}_{BA} & \mathbf{G}_{BB}%
\end{bmatrix}$$ of the connected system in terms of the GFs of each subsystem [@VelevJPC2004]:
\[DSEQ1\]$$\begin{aligned}
\mathbf{G}_{AA} & =((\mathbf{G}^{(A)})^{-1}-\mathbf{V}_{ab}\mathbf{G}%
_{bb}^{(B)}\mathbf{V}_{ba})^{-1},\label{DSEQ1_AA}\\
\mathbf{G}_{BB} & =((\mathbf{G}^{(B)})^{-1}-\mathbf{V}_{ba}\mathbf{G}%
_{aa}^{(A)}\mathbf{V}_{ab})^{-1},\\
\mathbf{G}_{BA} & =\mathbf{G}_{Bb}^{(B)}\mathbf{V}_{ba}\mathbf{G}_{aA},\\
\mathbf{G}_{AB} & =\mathbf{G}_{Aa}^{(A)}\mathbf{V}_{ab}\mathbf{G}_{bB},\end{aligned}$$ or vice versa:
\[DSEQ2\]$$\begin{aligned}
\mathbf{G}^{(A)} & =\mathbf{G}_{AA}-\mathbf{G}_{Ab}\mathbf{V}_{ba}%
(1+\mathbf{G}_{ab}\mathbf{V}_{ba})^{-1}\mathbf{G}_{aA},\\
\mathbf{G}^{(B)} & =\mathbf{G}_{BB}-\mathbf{G}_{Ba}\mathbf{V}_{ab}%
(1+\mathbf{G}_{ba}\mathbf{V}_{ab})^{-1}\mathbf{G}_{bB}.\end{aligned}$$ Equation (\[DSEQ1\]) is the key to building up the entire system out of disconnected subsystems, while Eq. (\[DSEQ2\]) can be used to calculate the GF of each subsystem when the GF of the connected system is known (e.g., if the connected system is infinite and periodic [@SettnesPRB2015]). The first two equations of Eq. (\[DSEQ1\]) show that if we focus on one subsystem (say $A$), the presence of the other subsystem $B$ amounts to a self-energy correction to the interface of $A$: $\mathbf{H}_{a,a}%
^{(A)}\rightarrow\mathbf{H}_{a,a}^{(A)}+\mathbf{V}_{ab}\mathbf{G}_{bb}%
^{(B)}\mathbf{V}_{ba}$.
### General procedures of RGF
In RGF, to calculate the conductance of the general layered system as described earlier \[Eq. (\[HAMIL\])\], the system is first partitioned into the semi-infinite left lead $L$ (unit cells $m\leq0$), the central region $C$ (unit cells $1\leq m\leq N$), and the semi-infinite right lead $R$ (unit cells $m\geq N+1$). The *entire* central region $C$ is regarded as a scatterer \[see Fig. \[G\_SETUP\](a) for an example\], so the GF is
$$\mathbf{G}=%
\begin{bmatrix}
\mathbf{G}_{LL} & \mathbf{G}_{LC} & \mathbf{G}_{LR}\\
\mathbf{G}_{CL} & \mathbf{G}_{CC} & \mathbf{G}_{CR}\\
\mathbf{G}_{RL} & \mathbf{G}_{RC} & \mathbf{G}_{RR}%
\end{bmatrix}
.$$
Let us use $\mathbf{H}^{(C)}$ for the Hamiltonian of the central region, and $\mathbf{H}^{(p)}$ $(p=L,R)$ for the Hamiltonian of the lead $p$, as characterized by the unit cell Hamiltonian $\mathbf{h}_{p}$ and nearest-neighbor hopping $\mathbf{t}_{p}=\mathbf{H}_{m,m+1}^{(p)}$. Then the central region part of the GF is computed from $$\mathbf{G}_{CC}=(z-\mathbb{H})^{-1}, \label{GCC}%$$ where $\mathbb{H}$ is the effective central region Hamiltonian: it equals $\mathbf{H}^{(C)}$ in the interior of $C$, but differs from $\mathbf{H}^{(C)}$ at the two surface unit cells:
\[HCC\]$$\begin{aligned}
\mathbb{H}_{1,1} & =\mathbf{H}_{1,1}+\mathbf{\Sigma}^{(L)},\\
\mathbb{H}_{N,N} & =\mathbf{H}_{N,N}+\mathbf{\Sigma}^{(R)},\end{aligned}$$ due to self-energy corrections from the left and right leads:
$$\begin{aligned}
\mathbf{\Sigma}^{(L)} & =\mathbf{t}_{L}^{\dagger}\mathbf{G}_{\mathrm{s}%
}^{(L)}\mathbf{t}_{L},\\
\mathbf{\Sigma}^{(R)} & =\mathbf{t}_{R}\mathbf{G}_{\mathrm{s}}%
^{(R)}\mathbf{t}_{R}^{\dagger}\mathbf{.}%\end{aligned}$$
Here $\mathbf{G}_{\mathrm{s}}^{(L)}=[(z-\mathbf{H}^{(L)})^{-1}]_{0,0}$ is the GF of the left lead at the right surface, and $\mathbf{G}_{\mathrm{s}}%
^{(R)}=[(z-\mathbf{H}^{(R)})^{-1}]_{N+1,N+1}$ is the GF of the right lead at the left surface, so they are referred to as *surface GFs* in the literature. Finally, to compute the linear conductance, we need to set $E=E_{F}$ and use the Landauer-Buttiker formula [@LandauerJRD1957; @LandauerPM1970; @BuettikerPRB1985] $\sigma=(e^{2}%
/h)T(E_{F})$, where [@CaroliJPC1971] $$T(E_{F})=\mathrm{Tr}\mathbf{\Gamma}^{(R)}\mathbf{G}_{N,1}\mathbf{\Gamma}%
^{(L)}(\mathbf{G}_{N,1})^{\dagger} \label{TEF}%$$ and $\mathbf{\Gamma}^{(\alpha)}\equiv i(\mathbf{\Sigma}^{(\alpha)}-h.c.)$. Note that Eq. (\[TEF\]) only involves $\mathbf{G}_{N,1}$, the $(N,1)$ block of $\mathbf{G}_{CC}$. Instead of direct matrix inversion \[see Eq. (6)\], $\mathbf{G}_{N,1}$ can be computed by building up the central region layer by layer [@FerryBook1997] through Eq. (\[DSEQ1\]). Let us use $\mathbf{G}^{(n)}$ $(n=1,2,\cdots,N$) to denote the GF of the subsystem consisting of the unit cells $m=1,2,\cdots,n$. The RGF starts from $\mathbf{G}^{(1)}=(z-\mathbb{H}%
_{1,1})^{-1}$, first uses the iteration $$\mathbf{G}_{n,n}^{(n)}=(z-\mathbb{H}_{n,n}-\mathbf{H}_{n,n-1}\mathbf{G}%
_{n-1,n-1}^{(n-1)}\mathbf{H}_{n-1,n})^{-1} \label{RECURSIVE1}%$$ to obtain $\{\mathbf{G}_{n,n}^{(n)}\}$, and then uses the iteration $$\mathbf{G}_{1,n}^{(n)}=\mathbf{G}_{1,n-1}^{(n-1)}\mathbf{H}_{n-1,n}%
\mathbf{G}_{n,n}^{(n)} \label{RECURSIVE2}%$$ to obtain $\mathbf{G}_{1,N}=\mathbf{G}_{1,N}^{(N)}$. The number of iterations and hence the time cost of the above recursive algorithm scales linearly with the length of the scattering region.
Equation (\[TEF\]) gives the total transmission probability, i.e., the sum of the transmission probabilities of all channels. To identify the contributions from each individual transmission channels, it is necessary to use the GF to construct the transmission amplitude $S_{\beta,\alpha}^{(RL)}$ from the $\alpha$th traveling channel in the lead $L$ to the $\beta$th traveling channel in the lead $R$ through the Fisher-Lee relations [@FisherPRB1981; @StoneIJRD1988; @BarangerPRB1989; @SolsAP1992; @SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009] and then sum over all the traveling channels:$$T(E_{F})=\sum_{\alpha\beta\in\mathrm{traveling}}|S_{\beta,\alpha}^{(RL)}|^{2}.
\label{TEF2}%$$ Alternatively, it is also possible to calculate the transmission amplitudes $\{S_{\beta,\alpha}^{(RL)}\}$ (and more generally the entire scattering matrix) by directly calculating the scattering of an incident traveling wave through the wave function mode matching approach [@AndoPRB1991; @NikolicPRB1994; @XiaPRB2006]. The equivalence between Eqs. (\[TEF\]) and (\[TEF2\]), which establishes a connection between the GF approach and the wave function mode matching approach, is well known for a single parabolic band [@FisherPRB1981; @StoneIJRD1988; @BarangerPRB1989; @SolsAP1992; @DattaBook1995; @FerryBook1997]. For a general lattice model, there was suspicion [@KrsticPRB2002] that Eq. (\[TEF2\]) was incomplete since the GF in Eq. (\[TEF\]) includes both traveling waves and evanescent waves, while Eq. (\[TEF2\]) only includes the contributions from traveling waves. Later, a rigorous equivalence proof was provided by Khomyakov *et al.* [@KhomyakovPRB2005] and others [@WimmerThesis2009], but the presence of the evanescent states does suggest that the GF is not completely equivalent to the unitary scattering matrix.
There are still two remaining issues: the calculation of the self-energies $\mathbf{\Sigma}^{(L,R)}$ (or equivalently the surface GFs $\mathbf{G}%
_{\mathrm{s}}^{(L,R)}$) and a proper definition of the scattering channels and the transmission amplitudes $S_{\beta,\alpha}^{(RL)}$.
### Self-energies: recursive method and eigenmode method
The numerical algorithms for computing $\mathbf{\Sigma}^{(L,R)}$ or equivalently the surface GFs $\mathbf{G}_{\mathrm{s}}^{(L,R)}$ can be classified into two groups: recursive methods and eigenmode methods (see Ref. for a review). The former calculates an approximate surface GF through some recursive relations, while the latter provides exact—within the numerical precision—closed-form solutions to the surface GF.
The idea of the recursive methods is to split the left lead into the surface unit cell $m=0$ (subsystem $A$) and the remaining part (subsystem $B$); the Dyson equation \[Eq. (\[DSEQ1\_AA\])\] gives the recursive relation $$\begin{aligned}
\mathbf{G}_{\mathrm{s}}^{(L)}&=&(z-\mathbf{h}_{L}-\mathbf{t}_{L}^{\dagger
}\mathbf{G}_{\mathrm{s}}^{(L)}\mathbf{t}_{L})^{-1} \notag \\
\Leftrightarrow
\mathbf{\Sigma}^{(L)}&=&\mathbf{t}_{L}^{\dagger}(z-\mathbf{h}_{L}-\mathbf{\Sigma
}^{(L)})^{-1}\mathbf{t}_{L}.\end{aligned}$$ Similarly, by splitting the right lead into the surface unit cell $m=N+1$ (subsystem $A$) and the remaining part (subsystem $B$), Eq. (\[DSEQ1\_AA\]) gives the recursive relation$$\begin{aligned}
\mathbf{G}_{\mathrm{s}}^{(R)}&=&(z-\mathbf{h}_{R}-\mathbf{t}_{R}\mathbf{G}%
_{\mathrm{s}}^{(R)}\mathbf{t}_{R}^{\mathbf{\dagger}})^{-1} \notag
\\ \Leftrightarrow
\mathbf{\Sigma}^{(R)}&=&\mathbf{t}_{R}(z-\mathbf{h}_{R}-\mathbf{\Sigma}%
^{(R)})^{-1}\mathbf{t}_{R}^{\dagger}.\end{aligned}$$ Thus the surface GFs and self-energies can be obtained by simple or more efficient iteration techniques [@SanchoJPF1985; @SanchoJPFMP1984].
The eigenmode method has been derived independently several times [@LeePRB1981; @AndoPRB1991; @NikolicPRB1994; @UmerskiPRB1997; @SanvitoPRB1999; @KrsticPRB2002; @RochaPRB2006] and has been shown to be superior in accuracy and performance [@UmerskiPRB1997] compared to the recursive methods. The central results are explicit expressions for the self-energies:
\[SIGMA\]$$\begin{aligned}
\mathbf{\Sigma}^{(L)} & =\mathbf{t}_{L}^{\dagger}(\mathbf{P}_{-}^{(L)}%
)^{-1}\mathbf{,}\\
\mathbf{\Sigma}^{(R)} & =\mathbf{t}_{R}\mathbf{P}_{+}^{(R)},\end{aligned}$$ and surface GFs:
$$\begin{aligned}
\mathbf{G}_{\mathrm{s}}^{(L)} & =(z-\mathbf{h}_{L}-\mathbf{t}_{L}^{\dagger
}(\mathbf{P}_{-}^{(L)})^{-1})^{-1}=(\mathbf{t}_{L}\mathbf{P}_{-}^{(L)}%
)^{-1},\\
\mathbf{G}_{\mathrm{s}}^{(R)} & =(z-\mathbf{h}_{R}-\mathbf{t}_{R}%
\mathbf{P}_{+}^{(R)})^{-1}=\mathbf{P}_{+}^{(R)}(\mathbf{t}_{R}^{\dagger}%
)^{-1},\end{aligned}$$
in terms of the (retarded) *propagation matrices* $\mathbf{P}_{\pm
}^{(L,R)}$ (also referred to as Bloch matrices [@KhomyakovPRB2005] or amplitude transfer matrices [@VelevJPC2004] in the literature), which can be constructed from the (retarded) *eigenmodes* of each lead.
Now we introduce the propagation matrices and eigenmodes in some detail, since they will play a central role in our GF approach. Let us consider a lead characterized by the unit cell Hamiltonian $\mathbf{h}$ and nearest-neighbor hopping matrix $\mathbf{t}=\mathbf{H}_{m,m+1}$. The wave propagation in this lead is governed by the uniform Schrödinger equation$$-\mathbf{t}^{\dagger}|\Phi(m-1)\rangle+(z-\mathbf{h})|\Phi(m)\rangle
-\mathbf{t}|\Phi(m+1)\rangle=0, \label{SE}%$$ where $z\equiv E+i0^{+}$. Imposing the Bloch symmetry $|\Phi(m)\rangle
=e^{ikma}|\Phi\rangle$ ($a$ is the thickness of each unit cell) gives $$z|\Phi\rangle=(e^{-ika}\mathbf{t}^{\dagger}+\mathbf{h}+e^{ika}\mathbf{t}%
)|\Phi\rangle\equiv\mathbf{H}(k)|\Phi\rangle\label{MEQ1}%$$ for the eigenvector $|\Phi\rangle$. For an infinite lead, the wave function $|\Phi(m)\rangle$ must remain finite at $m\rightarrow\pm\infty$. This natural boundary condition dictates $k$ to be real, so that Eq. (\[MEQ1\]) gives $M$ real energy bands of the lead, where $M$ is the number of basis states in each unit cell of this lead. For certain complex $k$’s, the energies could still be real, which form the complex energy bands of the lead.
Conversely, given the energy $E$ and without imposing any boundary conditions, Eq. (\[SE\]) or (\[MEQ1\]) can be solved to yield $2M\ $(retarded) eigenmodes $\{k,|\Phi\rangle\}$ (see Appendix \[APPEND\_BULKMODE\]) [@AndoPRB1991; @NikolicPRB1994; @KhomyakovPRB2005; @XiaPRB2006], where the wave vector $k$ could be either real (i.e., *traveling* modes) or complex (i.e., *evanescent* modes). The eigenmodes are just the collection of eigenstates on the energy shell $E$ in the real and complex energy bands of the lead. As a convention, each eigenvector $|\Phi\rangle$ should be normalized to unity, but different eigenvectors are not necessarily orthogonal. For a traveling eigenmodes with wave vector $k$ and eigenvector $|\Phi\rangle$, its group velocity is $$v=\partial_{k}\langle\Phi|\mathbf{H}(k)|\Phi\rangle=-2a\operatorname*{Im}%
\langle\Phi|\mathbf{t}e^{ika}|\Phi\rangle, \label{VELOCITY}%$$ where $\langle\Phi|$ is the conjugate transpose of $|\Phi\rangle$, i.e., an $M$-component row vector. Then the $2M$ eigenmodes are classified into $M$ right-going ones and $M$ left-going ones: the former consist of traveling modes with a positive group velocity and evanescent modes decaying exponentially to the right (i.e., $\operatorname{Im}k>0$), while the latter consist of traveling modes with a negative group velocity and evanescent modes decaying exponentially to the left (i.e., $\operatorname{Im}k<0$). For clarity, we denote the $M$ right-going eigenmodes as $\{k_{+,\alpha}%
,|\Phi_{+,\alpha}\rangle\}$ and the $M$ left-going eigenmodes as $\{k_{-,\alpha},|\Phi_{-,\alpha}\rangle\}$, where $\alpha=1,2,\cdots,M$. For every right-going evanescent mode $(+,\alpha)$ with wave vector $k_{+,\alpha}$, there is always a left-going evanescent mode $(-,\alpha)$ with wave vector $k_{-,\alpha}$ $=k_{+,\alpha}^{\ast}$ [@MolinariJPA1997; @SanvitoPRB1999; @KhomyakovPRB2005].
The propagation matrix $\mathbf{P}_{s}$ for left-going $(s=-$) or right-going ($s=+$) waves is constructed as [@AndoPRB1991; @NikolicPRB1994; @KhomyakovPRB2005; @XiaPRB2006] $$\mathbf{P}_{s}\equiv\mathbf{U}_{s}%
\begin{bmatrix}
e^{ik_{s,1}a} & & \\
& \ddots & \\
& & e^{ik_{s,M}a}%
\end{bmatrix}
\mathbf{U}_{s}^{-1}, \label{PM}%$$ where $\mathbf{U}_{s}\equiv\lbrack|\Phi_{s,1}\rangle,\cdots,|\Phi_{s,M}%
\rangle]$ (i.e., its $\alpha$th column is $|\Phi_{s,\alpha}\rangle$). The propagation matrices are standard tools in the wave function mode matching approach [@AndoPRB1991; @NikolicPRB1994; @KhomyakovPRB2005; @XiaPRB2006] to describe wave propagation; e.g., a general right-going wave that obeys Eq. (\[SE\]) can be written as $|\Phi(m)\rangle=\mathbf{P}_{+}^{m}%
|\Phi(0)\rangle$, while a general left-going wave obeying Eq. (\[SE\]) can be written as $|\Phi(m)\rangle=\mathbf{P}_{-}^{m}|\Phi(0)\rangle$.
### Scattering channels and Fisher-Lee relations
Since the GF $\mathbf{G}(E)$ describes the scattering of both traveling and evanescent eigenmodes by the central region, while the unitary scattering matrix $\mathbf{S}(E)$ describes the scattering of traveling eigenmodes only, it is possible to construct $\mathbf{S}(E)$ in terms of $\mathbf{G}(E)$, i.e., the Fisher-Lee-type relations. Compared with the Caroli’s expression \[Eq. (\[TEF\])\] that gives the total transmission probability, the Fisher-Lee relations further provide information about the scattering of each individual traveling eigenmode. For a general lattice model, different eigenmodes $\{|\Phi_{s,\alpha}\rangle\}$ are not orthogonal; then for each lead, it is necessary to introduce $2M$ (retarded) dual vectors $\{|\phi_{s,\alpha}\rangle\}$ through [@SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009]$$%
\begin{bmatrix}
\langle\phi_{s,1}|\\
\vdots\\
\langle\phi_{s,M}|
\end{bmatrix}
\equiv\mathbf{U}_{s}^{-1},$$ where $M$ is the number of bases in each unit cell of this lead. In general, different left-going (right-going) eigenvectors are not orthogonal, so $\mathbf{U}_{s}$ is not necessarily unitary and $|\phi_{s,\alpha}\rangle$ is not necessarily equal to $|\Phi_{s,\alpha}\rangle$, but we always have the orthonormalization and completeness relations $$\begin{aligned}
\langle\phi_{s,\alpha}|\Phi_{s,\beta}\rangle & =\langle\Phi_{s,\alpha}%
|\phi_{s,\beta}\rangle=\delta_{\alpha,\beta},\label{ORTHO}\\
\sum_{\alpha}|\Phi_{s,\alpha}\rangle\langle\phi_{s,\alpha}| & =\sum_{\alpha
}|\phi_{s,\alpha}\rangle\langle\Phi_{s,\alpha}|=\mathbf{I}, \label{COMPLETE}%\end{aligned}$$ which follow from $\mathbf{U}_{s}\mathbf{U}_{s}^{-1}=\mathbf{U}_{s}%
^{-1}\mathbf{U}_{s}=\mathbf{I}$ ($\mathbf{I}$ is the identity matrix). By inserting the completeness relation, any column vector $|\Phi\rangle$ can be expanded as a linear combination of either the $M$ left-going eigenvectors or the $M$ right-going eigenvectors as $|\Phi\rangle=\sum_{\alpha}c_{s,\alpha
}|\Phi_{s,\alpha}\rangle$ with $c_{s,\alpha}=\langle\phi_{s,\alpha}%
|\Phi\rangle$. In terms of the eigenmodes and their dual vectors, Eq. (\[PM\]) can be written as$$\mathbf{P}_{s}\equiv\sum_{\alpha}e^{ik_{s,\alpha}a}|\Phi_{s,\alpha}%
\rangle\langle\phi_{s,\alpha}|, \label{P_EXPAND}%$$ which has a clear physical interpretation. For example, a general right-going wave is$$|\Phi(m)\rangle=\mathbf{P}_{+}^{m}|\Phi(0)\rangle=\sum_{\alpha}e^{ik_{+,\alpha
}ma}|\Phi_{+,\alpha}\rangle\langle\phi_{+,\alpha}|\Phi(0)\rangle,$$ i.e., $|\Phi(0)\rangle$ is first expanded as a linear combination of right-going eigenmodes $|\Phi(0)\rangle=\sum_{\alpha}|\Phi_{+,\alpha}%
\rangle\langle\phi_{+,\alpha}|\Phi(0)\rangle$ and then each right-going eigenmode propagates as $|\Phi_{+,\alpha}\rangle\rightarrow e^{ik_{+,\alpha
}ma}|\Phi_{+,\alpha}\rangle$.
Now the scattering channel can be labeled by the eigenmodes, which could be either traveling or evanescent. The scattering matrix $\mathbf{S}(E)$ provides a complete description for the scattering from one traveling eigenmode into another traveling eigenmode. For a general lattice model, the Fisher-Lee relations allow us to construct the scattering matrix from the lattice GF, e.g., the transmission amplitude from the right-going eigenmode $|\Phi
_{+,\alpha}^{(L)}\rangle$ of the left lead into the right-going eigenmode $|\Phi_{+,\beta}^{(R)}\rangle$ of the right lead is [@AndoPRB1991; @SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009]:$$S_{\beta,\alpha}^{(R,L)}|_{\alpha,\beta\in\mathrm{traveling}}=\sqrt
{\frac{v_{+,\beta}^{(R)}/a_{R}}{v_{+,\alpha}^{(L)}/a_{L}}}\langle\phi
_{+,\beta}^{(R)}|\mathbf{G}_{N,1}(\mathbf{g}^{(L)})^{-1}|\Phi_{+,\alpha}%
^{(L)}\rangle, \label{SUNITARY_RL}%$$ while the transmission amplitude from the left-going eigenmode $|\Phi
_{-,\alpha}^{(R)}\rangle$ of the right lead into the left-going eigenmode $|\Phi_{-,\beta}^{(L)}\rangle$ of the left lead is [@AndoPRB1991; @SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009]$$S_{\beta,\alpha}^{(L,R)}|_{\alpha,\beta\in\mathrm{traveling}}=\sqrt
{\frac{v_{-,\beta}^{(L)}/a_{L}}{v_{-,\alpha}^{(R)}/a_{R}}}\langle\phi
_{-,\beta}^{(L)}|\mathbf{G}_{1,N}(\mathbf{g}^{(R)})^{-1}|\Phi_{-,\alpha}%
^{(R)}\rangle, \label{SUNITARY_LR}%$$ where $a_{p}$ and $\mathbf{g}^{(p)}$ are, respectively, the unit cell thickness and *free GF* of the lead $p$ \[see Eq. (\[G00\])\], and $v_{s,\alpha}^{(p)}$ is the group velocity \[see Eq. (\[VELOCITY\])\] of the traveling eigenmode $|\Phi_{s,\alpha}^{(p)}\rangle$ in the lead $p$.
Our GF approach: Key ideas
--------------------------
Let us assume that there is a local excitation at the unit cell $m_{0}$, as described by an $M_{m_{0}}$-dimensional column vector $|\Phi_{\mathrm{loc}%
}\rangle_{m_{0}}$. This excitation generates a casual scattering wave $|\Phi(m)\rangle$ that has an energy $E$ and obeys the Schrödinger equation with a local source at $m_{0}$:$$\begin{aligned}
-\mathbf{H}_{m,m-1}|\Phi(m-1)\rangle+(z-\mathbf{H}_{m,m})|\Phi
(m)\rangle\nonumber\\
-\mathbf{H}_{m,m+1}|\Phi(m+1)\rangle=\delta_{m,m_{0}}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}.\label{EOM}%\end{aligned}$$ The solution is given by $$|\Phi(m)\rangle=\mathbf{G}_{m,m_{0}}|\Phi_{\mathrm{loc}}\rangle_{m_{0}%
},\label{PHI_GF}%$$ e.g., for a unit excitation of the $\alpha$th basis state, as described by $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}=[0,\cdots,1,0,\cdots,0]^{T}$ (only the $\alpha$th element is nonzero), Eq. (\[PHI\_GF\]) gives $|\Phi(m)\rangle$ as the $\alpha$th column of $\mathbf{G}_{m,m_{0}}$.
![Scattering state emanating from a local excitation at $m_{0}$ in an infinite lead (a), a semi-infinite lead connected to a scatterer on its right (b), and a finite lead sandwiched between two scatterers. The zeroth-order, first-order, and second-order partial waves are denoted by black, blue, and orange arrows, respectively.[]{data-label="G_BASICPROCESS"}](fig3BasicProcess.eps){width="\columnwidth"}
Equation (\[PHI\_GF\]) shows that the GF can be immediately obtained once the scattering state is determined, e.g., based on physical considerations on how the local excitation evolves to the scattering state. For example, if the local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ occurs inside a lead, then it first generates an outgoing zeroth-order wave consisting of a left-going one at $m\leq m_{0}$ and a right-going one at $m\geq m_{0}$. For an infinite lead, there are no scatterers, so this outgoing wave is the total scattering state \[Fig. \[G\_BASICPROCESS\](a)\]. For a semi-infinite lead connected to a scatterer on its right \[Fig. \[G\_BASICPROCESS\](b)\], the zeroth-order right-going wave will produce a first-order reflection wave, so the total scattering state in the lead is the sum of the zeroth-order and first-order waves. More generally, for a finite lead sandwiched between two scatterers \[Fig. \[G\_BASICPROCESS\](c)\], the right-going (left-going) zeroth-order wave will propagate to the right (left) scatterer and produce a first-order reflection wave, which in turn will propagate to the left (right) scatterer and then produce high-order reflection waves. The total scattering state would be the sum of all these waves.
In contrast to the commonly used RGF that treats the *entire* central region (regarded as a large scatterer) numerically \[see Fig. \[G\_SETUP\](a) for an example\], our method need only regard each *truly disordered* region as a scatterer for numerical treatment, while all the periodic subregions \[such as the middle lead in Fig. \[G\_SETUP\](a)\] inside the central region can be treated semianalytically by fully utilizing the translational invariance of these subregions. Physically, the wave propagation inside these periodic subregions leads to complicated multiple reflection between different scatterers, which is difficult to handle analytically in the standard RGF technique in which the GF is treated as a matrix. By contrast, in our approach, it is straightforward to perform analytically an infinite summation over all the multiple reflection waves, so that the time cost can be significantly reduced. This approach also provides a physically transparent expansion of the GF $G(E)$ in terms of a generalized scattering matrix $\mathcal{S}(E)$, which can be regarded as a reverse of the well-known Fisher-Lee relations [@FisherPRB1981; @StoneIJRD1988; @BarangerPRB1989; @SolsAP1992; @SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009] (see Fig. 1). In the next section, we will establish the procedures for calculating the GF within this framework in a physically transparent way.
Our Green’s function approach
=============================
Our GF approach essentially consists of two steps: generation of the zeroth-order outgoing partial wave by the local excitation and its propagation in the leads and scattering by the scatterers. The wave function mode matching approach [@AndoPRB1991; @NikolicPRB1994; @KhomyakovPRB2005; @XiaPRB2006] has developed useful tools to describe the latter process. Below we begin with an infinite lead, then we consider an infinite system containing a single scatterer. Finally, we give the analytical construction rule for the GF of a general layered system containing an arbitrary number of scatterers.
Infinite lead
-------------
Suppose that the lead is characterized by the unit cell Hamiltonian $\mathbf{h}$ and hopping $\mathbf{t}$. The scattering state $|\Phi(m)\rangle$ emanating from a local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ is determined by the Schrödinger equation with a local source at $m_{0}$: $$\begin{aligned}
-\mathbf{t}^{\dagger}|\Phi(m-1)\rangle+(z-\mathbf{h})|\Phi(m)\rangle
-\mathbf{t}|\Phi(m+1)\rangle \notag
\\=\delta_{m,m_{0}}|\Phi_{\mathrm{loc}}%
\rangle_{m_{0}}. \label{EOM_0}%\end{aligned}$$ In either the left region $(m\leq m_{0}-1$) or the right region ($m\geq
m_{0}+1$), the local source vanishes, so the general solution would be a linear combination of left-going and right-going eigenmodes with energy $E$. However, by causality considerations (due to the infinitesimal imaginary part of the energy $z=E+i0^{+}$), the solution in the left (right) region should be a left-going (right-going) wave \[see Fig. \[G\_BASICPROCESS\](a)\]: $$\begin{aligned}
|\Phi(m)\rangle|_{m\leq m_{0}-1} & =(\mathbf{P}_{-})^{m-m_{0}}|\Phi
(m_{0})\rangle,\\
|\Phi(m)\rangle|_{m\geq m_{0}+1} & =(\mathbf{P}_{+})^{m-m_{0}}|\Phi
(m_{0})\rangle.\end{aligned}$$ Substituting into Eq. (\[EOM\_0\]) gives $|\Phi(m_0)\rangle=\mathbf{g}%
|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$, where
\[G00\]$$\begin{aligned}
\mathbf{g} & \equiv(z-\mathbf{h-t}^{\dagger}\mathbf{P}_{-}^{-1}%
-\mathbf{tP}_{+})^{-1}\label{G00A}\\
& =[\mathbf{t(P}_{-}-\mathbf{P}_{+})]^{-1}=[\mathbf{t}^{\dagger}%
(\mathbf{P}_{+}^{-1}-\mathbf{P}_{-}^{-1})]^{-1}. \label{G00B}%\end{aligned}$$ Here we have used the equality [@KhomyakovPRB2005]
$$E-\mathbf{h=t}^{\dagger}\mathbf{P}_{\pm}^{-1}+\mathbf{tP}_{\pm}
\label{EQUALITY}%$$
in arriving at Eq. (\[G00B\]). From the scattering wave function, we immediately identify the GF of an infinite lead as $$\mathbf{g}_{m,m_{0}}\equiv\left\{
\begin{array}
[c]{ll}%
\mathbf{P}_{+}^{m-m_{0}}\mathbf{g\ \ } & (m\geq m_{0}),\\
\mathbf{P}_{-}^{m-m_{0}}\mathbf{g} & (m\leq m_{0}).
\end{array}
\right. \label{G_INFINITE}%$$ This recovers the previous result [@SanvitoPRB1999; @KhomyakovPRB2005] obtained by directly solving the equations of motion of the GF. For convenience, hereafter we call $\mathbf{g}_{m,m_{0}}$ the *free GF* of the lead since it describes the generation of the zeroth-order outgoing wave $|\Phi(m)\rangle=\mathbf{g}_{m,m_{0}}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ from a local excitation inside this lead.
Single scatterer
----------------
![Scattering state emanating from a local excitation inside a scatterer (a) or a lead (b). The black arrows denote the zeroth-order partial wave and the blue arrows denote the first-order partial wave due to scattering.[]{data-label="G_SINGLE"}](fig4Single.eps){width="\columnwidth"}
Let us consider a scatterer $C$ connected to two semi-infinite leads $L$ and $R$ \[Fig. \[G\_SINGLE\](a)\]. The left (right) surface of the scatterer is $m_{L}$ ($m_{R}$). The unit cell Hamiltonian and hopping inside the left (right) lead are $\mathbf{h}_{L}$ and $\mathbf{t}_{L}$ ($\mathbf{h}_{R}$ and $\mathbf{t}_{R}$).
### Local excitation inside the scatterer
The zeroth-order outgoing wave $|\Phi(m)\rangle$ emanating from a local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ at $m_{0}\in C$ obeys Eq. (\[EOM\]) with $m_{L}\leq m\leq m_{R}$, i.e., inside the scatterer. Inside the left lead, $|\Phi(m)\rangle$ obeys $$\begin{aligned}
-\mathbf{t}_{L}^{\dagger}|\Phi(m-1)\rangle+(z-\mathbf{h}_{L})|\Phi
(m)\rangle-\mathbf{t}_{L}|\Phi(m+1)\rangle=0\end{aligned}$$ with $m\leq m_{L}-1$. Inside the right lead, $|\Phi(m)\rangle$ obeys$$-\mathbf{t}_{R}^{\dagger}|\Phi(m-1)\rangle+(z-\mathbf{h}_{R})|\Phi
(m)\rangle-\mathbf{t}_{R}|\Phi(m+1)\rangle=0$$ with $m\geq m_{R}+1$. By causality, the solution in the left (right) lead is a left-going (right-going) wave \[see Fig. \[G\_SINGLE\](a)\]: $$\begin{aligned}
|\Phi(m)\rangle|_{m\in L} & =(\mathbf{P}_{-}^{(L)})^{m-m_{L}}|\Phi
(m_{L})\rangle,\\
|\Phi(m)\rangle|_{m\in R} & =(\mathbf{P}_{+}^{(R)})^{m-m_{R}}|\Phi
(m_{R})\rangle,\end{aligned}$$ where $\mathbf{P}_{\pm}^{(p)}$ are propagation matrices of lead $p$ \[see Eq. (22)\]. Substituting $|\Phi(m_{L}-1)\rangle=(\mathbf{P}_{-}^{(L)})^{-1}|\Phi
(m_{L})\rangle$ and $|\Phi(m_{R}+1)\rangle=\mathbf{P}_{+}^{(R)}|\Phi
(m_{R})\rangle$ into Eq. (\[EOM\]) gives a closed set of equations for $|\Phi(m)\rangle$ inside the scatterer. The solution is$$|\Phi(m)\rangle|_{m\in C}=\mathbb{G}_{m,m_{0}}|\Phi_{\mathrm{loc}}%
\rangle_{m_{0}}, \label{PHI0_SCATTERER}%$$ where $$\mathbb{G}\equiv(z-\mathbb{H})^{-1} \label{GG}%$$ and $\mathbb{H}$ is the effective Hamiltonian for the scatterer: it is equal to the scatterer part of the system Hamiltonian $\mathbf{H}$, except for the two surface unit cells:
\[HH\] $$\begin{aligned}
\mathbb{H}_{m_{L},m_{L}} & =\mathbf{H}_{m_{L},m_{L}}+\mathbf{t}_{L}^{\dagger
}(\mathbf{P}_{-}^{(L)})^{-1},\\
\mathbb{H}_{m_{R},m_{R}} & =\mathbf{H}_{m_{R},m_{R}}+\mathbf{t}_{R}%
\mathbf{P}_{+}^{(R)}.\end{aligned}$$
Since $\mathbb{G}$ converts a local excitation inside the scatterer into a scattering state inside the scatterer, we call it the *conversion matrix* of the scatterer. Comparing Eqs. (\[GG\]) and (\[HH\]) to Eqs. (\[GCC\]), (\[HCC\]), and (\[SIGMA\]), we see that $\mathbb{G}$ is just the scatterer part of the GF. Actually, from the scattering wave function, we immediately identify the GF:
\[G\_ALL\_S\]$$\begin{aligned}
\mathbf{G}_{m\in C,m_{0}\in C} & =\mathbb{G}_{m,m_{0}},\label{GSS}\\
\mathbf{G}_{m\in L,m_{0}\in S} & =(\mathbf{P}_{-}^{(L)})^{m-m_{L}}%
\mathbb{G}_{m_{L},m_{0}},\label{GLS}\\
\mathbf{G}_{m\in R,m_{0}\in S} & =(\mathbf{P}_{+}^{(R)})^{m-m_{R}}%
\mathbb{G}_{m_{R},m_{0}}. \label{GRS}%\end{aligned}$$ Equation (\[GLS\]) shows that the local excitation first evolves to an outgoing wave $\mathbb{G}_{m_{L},m_{0}}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ at the left surface of the scatterer, and then propagates to the unit cell $m$ as $(\mathbf{P}_{-}^{(L)})^{m-m_{L}}\mathbb{G}_{m_{L},m_{0}}|\Phi_{\mathrm{loc}%
}\rangle_{m_{0}}$. Equation (\[GRS\]) has a similar physical interpretation.
### Local excitation in the lead
As shown in Fig. \[G\_SINGLE\](b), for $m_{0}\in L$, the local excitation first generates a zeroth-order outgoing wave in the left lead:$\ |\Phi
^{(0)}(m)\rangle=\mathbf{g}_{m,m_{0}}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{0}%
}$, where $\mathbf{g}_{m,m_{0}}^{(p)}$ is the free GF of the lead $p$ \[Eq. (\[G\_INFINITE\])\]. Next the right-going partial wave reaches the left surface of $C$ and evolves into a scattering state $|\Psi(m)\rangle$. For $m_{0}\in R$, the local excitation first generates a zeroth-order outgoing wave in the right lead: $|\Phi^{(0)}(m)\rangle=\mathbf{g}_{m,m_{0}}^{(R)}%
|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$. Next the left-going partial wave reaches the right surface of $C$ and evolves to a scattering state $|\Psi(m)\rangle$. For either case, the total scattering state $|\Phi(m)\rangle$ emanating from the local excitation is the sum of the unscattered zeroth-order partial wave and the scattering state $|\Psi(m)\rangle$. The central issue is to determine the scattering state emanating from a known incident wave, in a way similar to the wave function mode matching approach to mesoscopic quantum transport [@AndoPRB1991; @NikolicPRB1994; @KhomyakovPRB2005; @XiaPRB2006].
First we consider the scattering state $|\Psi(m)\rangle$ emanating from a right-going incident wave $|\Phi_{\mathrm{in}}(m)\rangle$ in the left lead. The key observation is that for arbitrary $m_{1}\leq m_{L}$, the local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{1}}\equiv(\mathbf{g}^{(L)}%
)^{-1}|\Phi_{\mathrm{in}}(m_{1})\rangle$ generates a right-going partial wave $|\tilde{\Phi}(m)\rangle|_{m\geq m_{1}}=(\mathbf{P}_{+}^{(L)})^{m-m_{1}}%
|\Phi_{\mathrm{in}}(m_{1})\rangle$ that is equal to $|\Phi_{\mathrm{in}%
}(m)\rangle|_{m\geq m_{1}}$. Therefore, in the region $m\geq m_{1}$, the scattering state emanating from $|\Phi_{\mathrm{in}}(m)\rangle$ is the same as the scattering state emanating from this local excitation (see Appendix \[APPEND\_EQUIVALENCE\] for a rigorous proof). Taking $m_{1}=m_{L}$ immediately gives
$$|\Psi(m)\rangle|_{m\in C}=\mathbb{G}_{m,m_{L}}(\mathbf{g}^{(L)})^{-1}%
|\Phi_{\mathrm{in}}(m_{L})\rangle, \label{PHI_S1}%$$
i.e., first the incident wave amplitude $|\Phi_{\mathrm{in}}(m_{L})\rangle$ is converted back to a local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{L}%
}\equiv(\mathbf{g}^{(L)})^{-1}|\Phi_{\mathrm{in}}(m_{L})\rangle$, then the conversion matrix $\mathbb{G}$ of the scatterer further converts it to the total scattering state $|\Psi(m)\rangle|_{m\in C}$ according to Eq. (\[PHI0\_SCATTERER\]). Inside the left lead, $|\Psi(m)\rangle$ is the sum of the right-going incident wave and a left-going reflection partial wave $|\Phi_{\mathrm{r}}(m)\rangle|_{m\in L}=(\mathbf{P}_{-}^{(L)})^{m-m_{L}}%
|\Phi_{\mathrm{r}}(m_{L})\rangle$, where $$\begin{aligned}
|\Phi_{\mathrm{r}}(m_{L})\rangle & =|\Psi(m_{L})\rangle-|\Phi_{\mathrm{in}%
}(m_{L})\rangle\nonumber\\
& =[\mathbb{G}_{m_{L},m_{L}}(\mathbf{g}^{(L)})^{-1}-\mathbf{I}]|\Phi
_{\mathrm{in}}(m_{L})\rangle. \label{PHIR_ML}%\end{aligned}$$ Inside the right lead, $|\Psi(m)\rangle$ is the right-going transmission wave: $|\Psi(m)\rangle|_{m\in R}=(\mathbf{P}_{+}^{(R)})^{m-m_{R}}|\Psi(m_{R}%
)\rangle$, where$$|\Psi(m_{R})\rangle=\mathbb{G}_{m_{R},m_{L}}(\mathbf{g}^{(L)})^{-1}%
|\Phi_{\mathrm{in}}(m_{L})\rangle. \label{PHI_MR}%$$
Similarly, we can derive the scattering state $|\Psi(m)\rangle$ emanating from a left-going incident wave $|\Phi_{\mathrm{in}}(m)\rangle$ in the right lead. Inside the scatterer, the scattering state is $$|\Psi(m)\rangle|_{m\in S}=\mathbb{G}_{m,m_{R}}(\mathbf{g}^{(R)})^{-1}%
|\Phi_{\mathrm{in}}(m_{R})\rangle, \label{PHI_S2}%$$ as if it emanated from a local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{R}%
}\equiv(\mathbf{g}^{(R})^{-1}|\Phi_{\mathrm{in}}(m_{R})\rangle$ at the right surface of the scatterer \[cf. Eq. (\[PHI0\_SCATTERER\])\]. Inside the right lead, $|\Psi(m)\rangle$ is the sum of the left-going incident wave and a right-going reflection partial wave $|\Phi_{\mathrm{r}}(m)\rangle|_{m\in
R}=(\mathbf{P}_{+}^{(R)})^{m-m_{R}}|\Phi_{\mathrm{r}}(m_{R})\rangle$, where$$\begin{aligned}
|\Phi_{\mathrm{r}}(m_{R})\rangle & =|\Psi(m_{R})\rangle-|\Phi_{\mathrm{in}%
}(m_{R})\rangle\nonumber\\
& =[\mathbb{G}_{m_{R},m_{R}}(\mathbf{g}^{(R)})^{-1}-\mathbf{I}]|\Phi
_{\mathrm{in}}(m_{R})\rangle. \label{PHIR_MR}%\end{aligned}$$ Inside the left lead, $|\Psi(m)\rangle$ is the left-going transmission wave: $|\Psi(m)\rangle=(\mathbf{P}_{-}^{(L)})^{m-m_{L}}|\Psi(m_{L})\rangle$, where$$|\Psi(m_{L})\rangle=\mathbb{G}_{m_{L},m_{R}}(\mathbf{g}^{(R)})^{-1}%
|\Phi_{\mathrm{in}}(m_{R})\rangle. \label{PHI_ML}%$$
Using the above results, the scattering state emanating from the zeroth-order right-going partial wave in the left lead is given by Eqs. (\[PHI\_S1\])-(\[PHI\_MR\]) with $|\Phi_{\mathrm{in}}(m_{L})\rangle\equiv|\Phi
^{(0)}(m_{L})\rangle$. This allows us to identify the GF
\[G\_ALL\_L\]$$\begin{aligned}
\mathbf{G}_{m\in C,m_{0}\in L} & =\mathbb{G}_{m,m_{L}}(\mathbf{g}%
^{(L)})^{-1}\mathbf{g}_{m_{L},m_{0}}^{(L)},\label{GSL}\\
\mathbf{G}_{m\in R,m_{0}\in L} & =(\mathbf{P}_{+}^{(R)})^{m-m_{R}}%
\mathbb{G}_{m_{R},m_{L}}(\mathbf{g}^{(L)})^{-1}\mathbf{g}_{m_{L},m_{0}}%
^{(L)},\label{GRL}\\
\mathbf{G}_{m\in L,m_{0}\in L} & =\mathbf{g}_{m,m_{0}}^{(L)}+(\mathbf{P}%
_{-}^{(L)})^{m-m_{L}}[\mathbb{G}_{m_{L},m_{L}}(\mathbf{g}^{(L)})^{-1}%
-\mathbf{I}]\mathbf{g}_{m_{L},m_{0}}^{(L)}. \label{GLL}%\end{aligned}$$ These expressions have clear physical interpretations; e.g., Eq. (\[GSL\]) shows that the local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}\in L}$ first evolves to a right-going partial wave and propagates rightward to the left surface of the scatterer as $\mathbf{g}_{m_{L},m_{0}}^{(L)}%
|\Phi_{\mathrm{loc}}\rangle_{m_{0}\in L}$. There it is converted back to a local excitation by $(\mathbf{g}^{(L)})^{-1}$, and finally the conversion matrix of the scatterer $\mathbb{G}_{m,m_{L}}$ further converts it to the scattering state inside the scatterer. As another example, Eq. (\[GLL\]) shows that the total scattering wave inside the left lead is the sum of the zeroth-order partial wave $\mathbf{g}_{m,m_{0}}^{(L)}|\Phi_{\mathrm{loc}%
}\rangle_{m_{0}}$ and the reflection partial wave: first the local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}\in L}$ evolves to a zeroth-order partial wave and then propagates rightwards to the left surface of the scatterer as $\mathbf{g}_{m_{L},m_{0}}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{0}\in L}$, then $\mathbb{G}_{m_{L},m_{L}}(\mathbf{g}^{(L)})^{-1}-\mathbf{I}$ converts it to the reflection wave. Finally, $(\mathbf{P}_{-}^{(L)})^{m-m_{L}}$ propagates this reflection wave leftward to $m$.
Similarly, the scattering state emanating from the zeroth-order left-going partial wave in the right lead is given by Eqs. (\[PHI\_S2\])-(\[PHI\_ML\]) with $|\Phi_{\mathrm{in}}(m_{R})\rangle\equiv|\Phi^{(0)}(m_{R})\rangle$. This allows us to identify the GF
\[G\_ALL\_R\]$$\begin{aligned}
\mathbf{G}_{m\in C,m_{0}\in R} & =\mathbb{G}_{m,m_{R}}(\mathbf{g}%
^{(R)})^{-1}\mathbf{g}_{m_{R},m_{0}}^{(R)},\label{GSR}\\
\mathbf{G}_{m\in L,m_{0}\in R} & =(\mathbf{P}_{-}^{(L)})^{m-m_{L}}%
\mathbb{G}_{m_{L},m_{R}}(\mathbf{g}^{(R)})^{-1}\mathbf{g}_{m_{R},m_{0}}%
^{(R)},\label{GLR}\\
\mathbf{G}_{m\in R,m_{0}\in R} & =\mathbf{g}_{m,m_{0}}^{(R)}+(\mathbf{P}%
_{+}^{(R)})^{m-m_{R}}[\mathbb{G}_{m_{R},m_{R}}(\mathbf{g}^{(R)})^{-1}%
-\mathbf{I}]\mathbf{g}_{m_{R},m_{0}}^{(R)}. \label{GRR}%\end{aligned}$$ These can be interpreted in a similar way to Eqs. (\[G\_ALL\_S\]) and (\[G\_ALL\_L\]).
The above results cover previous results as special cases. For example, by directly solving the equation of motion, Sanvito *et al.* [@SanvitoPRB1999] and Krstić *et al.* [@KrsticPRB2002] obtain the GF of an infinite lead \[Eq. (\[G\_INFINITE\])\] and a semi-infinite lead consisting of the unit cells $m\leq0$ ($m\geq0$) \[Eq. (\[GL\_MM0\])\], which can be regarded as a single-unit-cell scatterer at $m=0$ connected to a semi-infinite left (right) lead. Khomyakov *et al.* [@KhomyakovPRB2005] further obtained the GF across a single scatterer \[Eq. (\[GRL\])\]. A sharp interface between a semi-infinite left lead and a semi-infinite right lead can also be regarded as a single-unit-cell scatterer connected to two semi-infinite leads. For reference, the explicit expressions of the GFs for these simple cases are given in Appendix \[APPEND\_EXAMPLE\].
Multiple scatterers
-------------------
A general layered system containing an arbitrary number of scatterers can be regarded as a *composite* scatterer connected to one or two semi-infinite leads, e.g., a scatterer $B$ connected to a finite lead $L$ and a semi-infinite lead $R$ can be regarded as a composite scatterer $C=(A+B)$ connected to one semi-infinite right lead \[Fig. \[G\_COMPOSITE\_SCATTERER\](a)\], while two scatterers sandwiched between three leads can be regarded a composite scatterer $C=(A+B)$ connected to two semi-infinite leads \[Fig. \[G\_COMPOSITE\_SCATTERER\](b)\]. Therefore, we can use Eqs. (\[G\_ALL\_S\]), (\[G\_ALL\_L\]), and (\[G\_ALL\_R\]) to obtain the GF of the entire system once the conversion matrix of the composite scatterer is known. In the RGF method, the conversion matrix (which coincides with the GF of the infinite system within the composite scatterer) is calculated by a numerical iteration algorithm that builds up the composite scatterer slice by slice, thus the time cost increases linearly with the total length of the composite scatterer. Here the physical transparency of our approach allows us to treat the multiple reflection between different scatterers analytically, so that the conversion matrix of a composite scatterer can be obtained by combining the conversion matrices of the constituent scatterers analytically with a significantly reduced time cost. The basic step is the combination of the conversion matrices $\mathbb{G}^{(A)}$ and $\mathbb{G}^{(B)}$ of two scatterers $A$ and $B$ into the conversion matrix $\mathbb{G}$ of a composite scatterer $C\equiv(A+B)$.
### Combining conversion matrices
![(a) A scatterer $B$ connected to a finite left lead $L$ and a semi-infinite right lead $R$ can be regarded as a composite scatterer $C$ connected to one semi-infinite right lead $R$. (b) Two scatterers sandwiched between three leads $L,M,R$ can be regarded as a composite scatterer $C$ connected to two semi-infinite leads $L$ and $R$.[]{data-label="G_COMPOSITE_SCATTERER"}](fig5CompositeScatterer.eps){width="\columnwidth"}
As shown in Fig. \[G\_COMPOSITE\_SCATTERER\](b), the left and right surfaces of the scatterer $A$ ($B$) are $a_{L}$ and $a_{R}$ ($b_{L}$ and $b_{R}$) and the three leads sandwiching the scatterers are the semi-infinite left lead $L,$ the middle lead $M,$ and the semi-infinite right lead $R$. For a local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ at $m_{0}\in C$, the total scattering state inside $C$ is $|\Phi(m)\rangle|_{m\in C}%
=\mathbb{G}_{m,m_{0}}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$, which allows us to identify $\mathbb{G}$ once the total scattering state has been obtained. For $m_{0}\in A$, the local excitation first produces a zeroth-order partial wave in $A$ and $M$. Next the right-going partial wave in $M$ undergoes multiple reflections between $B$ and $A$ and finally evolves to a scattering state. For $m_{0}\in B$, the local excitation first produces a zeroth-order partial wave inside $B$ and $M$. Next the left-going partial wave in $M$ undergoes multiple reflections and finally evolves to a scattering state. For $m_{0}\in M$, the local excitation first produces a zeroth-order outgoing partial wave in $M$. Next the left- and right-going partial waves each undergo multiple reflections and evolve to a scattering state. For each case, the total scattering state emanating from the local excitation is the sum of the unscattered zeroth-order partial wave and the scattering state(s) emanating from the scattered zeroth-order partial wave. Therefore, the key issue is to calculate the scattering state $|\Psi(m)\rangle$ emanating from a right- or left-going incident wave $|\Phi_{\mathrm{in}}(m)\rangle$ in the middle lead through multiple reflections between $A$ and $B$.
In the middle lead, $|\Psi(m)\rangle$ is the sum of the right-going part $|\Psi_{+}(m)\rangle=(\mathbf{P}_{+}^{(M)})^{m-b_{L}}|\Psi_{+}(b_{L})\rangle$ and the left-going part $|\Psi_{-}(m)\rangle=(\mathbf{P}_{-}^{(M)})^{m-a_{R}%
}|\Psi_{-}(a_{R})\rangle$. Inside the scatterer $A$, $|\Psi(m)\rangle$ is equal to the scattering state emanating from the total incident wave $|\Psi_{-}(a_{R})\rangle$ on $A$ \[cf. Eq. (\[PHI\_S2\])\]:
$$|\Psi(m)\rangle|_{m\in A}=\mathbb{G}_{m,a_{R}}^{(A)}(\mathbf{g}^{(M)}%
)^{-1}|\Psi_{-}(a_{R})\rangle,$$
Inside the scatterer $B$, $|\Psi(m)\rangle$ is equal to the scattering state emanating from the total incident wave $|\Psi_{+}(b_{L})\rangle$ on $B$ \[cf. Eq. (\[PHI\_S1\])\]:$$|\Psi(m)\rangle|_{m\in B}=\mathbb{G}_{m,b_{L}}^{(B)}(\mathbf{g}^{(M)}%
)^{-1}|\Psi_{+}(b_{L})\rangle.$$ Therefore, the scattering state inside $C$ is completely determined by $|\Psi_{+}(b_{L})\rangle$ and $|\Psi_{-}(a_{R})\rangle$. For brevity, we introduce the reflection matrices $$\begin{aligned}
\mathbb{R}_{B} & \equiv\mathbb{G}_{b_{L},b_{L}}^{(B)}(\mathbf{g}^{(M)}%
)^{-1}-\mathbf{I},\\
\mathbb{R}_{A} & \equiv\mathbb{G}_{a_{R},a_{R}}^{(A)}(\mathbf{g}^{(M)}%
)^{-1}-\mathbf{I}.\end{aligned}$$ The former (latter) converts a right-going (left-going) incident wave on the left (right) surface of $B$ ($A$) to a left-going (right-going) reflection wave. To describe the multiple reflections between $A$ and $B$, we introduce the following renormalized propagation matrices that incorporate multiple reflections:$$\begin{aligned}
\mathbf{P}_{b_{L}\leftarrow a_{R}} & \equiv\lbrack1-(\mathbf{P}_{+}%
^{(M)})^{\Delta m}\mathbb{R}_{A}(\mathbf{P}_{-}^{(M)})^{-\Delta m}%
\mathbb{R}_{B}]^{-1}(\mathbf{P}_{+}^{(M)})^{\Delta m},\\
\mathbf{P}_{a_{R}\leftarrow b_{L}} & \equiv\lbrack1-(\mathbf{P}_{-}%
^{(M)})^{-\Delta m}\mathbb{R}_{B}(\mathbf{P}_{+}^{(M)})^{\Delta m}%
\mathbb{R}_{A}]^{-1}(\mathbf{P}_{-}^{(M)})^{-\Delta m},\\
\mathbf{P}_{a_{R}\leftarrow a_{R}} & \equiv(\mathbf{P}_{-}^{(M)})^{-\Delta
m}\mathbb{R}_{B}\mathbf{P}_{b_{L}\leftarrow a_{R}}=\mathbf{P}_{a_{R}\leftarrow
b_{L}}\mathbb{R}_{B}(\mathbf{P}_{+}^{(M)})^{\Delta m},\\
\mathbf{P}_{b_{L}\leftarrow b_{L}} & \equiv(\mathbf{P}_{+}^{(M)})^{\Delta
m}\mathbb{R}_{A}\mathbf{P}_{a_{R}\leftarrow b_{L}}=\mathbf{P}_{b_{L}\leftarrow
a_{R}}\mathbb{R}_{A}(\mathbf{P}_{-}^{(M)})^{-\Delta m},\end{aligned}$$ where $\Delta m\equiv b_{L}-a_{R}$ is the distance between $A$ and $B$. For example, the renormalized propagation matrix $\mathbf{P}_{b_{L}\leftarrow
a_{R}}$ from $a_{R}$ to $b_{L}$ is the sum of the free propagation term $(\mathbf{P}_{+}^{(M)})^{\Delta m}$, the propagation term with two reflections $(\mathbf{P}_{+}^{(M)})^{\Delta m}\mathbb{R}_{A}(\mathbf{P}_{-}^{(M)}%
)^{-\Delta m}\mathbb{R}_{B}(\mathbf{P}_{+}^{(M)})^{\Delta m}$, and so on.
Using the above notations, when $|\Phi_{\mathrm{in}}(m)\rangle$ is a right-going incident wave on $B$, we have$$\begin{aligned}
|\Psi_{+}(b_{L})\rangle & =(1+\mathbf{P}_{b_{L}\leftarrow b_{L}}%
\mathbb{R}_{B})|\Phi_{\mathrm{in}}(b_{L})\rangle,\\
|\Psi_{-}(a_{R})\rangle & =\mathbf{P}_{a_{R}\leftarrow b_{L}}\mathbb{R}%
_{B}|\Phi_{\mathrm{in}}(b_{L})\rangle.\end{aligned}$$ When $|\Phi_{\mathrm{in}}(m)\rangle$ is a left-going incident wave on $A$, we have $$\begin{aligned}
|\Psi_{+}(b_{L})\rangle & =\mathbf{P}_{b_{L}\leftarrow a_{R}}\mathbb{R}%
_{A}|\Phi_{\mathrm{in}}(a_{R})\rangle,\\
|\Psi_{-}(a_{R})\rangle & =(1+\mathbf{P}_{a_{R}\leftarrow a_{R}}%
\mathbb{R}_{A})|\Phi_{\mathrm{in}}(a_{R})\rangle.\end{aligned}$$
Using the above results together with $|\Phi(m)\rangle|_{m\in C}%
=\mathbb{G}_{m,m_{0}}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$, we identify
\[G\_ALL\]$$\begin{aligned}
\mathbb{G}_{m\in B,m_{0}\in A} & =\mathbb{G}_{m,b_{L}}^{(B)}(\mathbf{g}%
^{(M)})^{-1}\mathbf{P}_{b_{L}\leftarrow a_{R}}\mathbb{G}_{a_{R},m_{0}}%
^{(A)},\label{GBA}\\
\mathbb{G}_{m\in A,m_{0}\in B} & =\mathbb{G}_{m,a_{R}}^{(A)}(\mathbf{g}%
^{(M)})^{-1}\mathbf{P}_{a_{R}\leftarrow b_{L}}\mathbb{G}_{b_{L},m_{0}}%
^{(B)},\label{GAB}\\
\mathbb{G}_{m\in A,m_{0}\in A} & =\mathbb{G}_{m,m_{0}}^{(A)}+\mathbb{G}%
_{m,a_{R}}^{(A)}(\mathbf{g}^{(M)})^{-1}\mathbf{P}_{a_{R}\leftarrow a_{R}%
}\mathbb{G}_{a_{R},m_{0}}^{(A)},\label{GAA}\\
\mathbb{G}_{m\in B,m_{0}\in B} & =\mathbb{G}_{m,m_{0}}^{(B)}+\mathbb{G}%
_{m,b_{L}}^{(B)}(\mathbf{g}^{(M)})^{-1}\mathbf{P}_{b_{L}\leftarrow b_{L}%
}\mathbb{G}_{b_{L},m_{0}}^{(B)},\label{GBB}\\
\mathbb{G}_{m\in M,m_{0}\in A} & =[(\mathbf{P}_{+}^{(M)})^{m-a_{R}%
}(1+\mathbb{R}_{A}\mathbf{P}_{a_{R}\leftarrow a_{R}})+(\mathbf{P}_{-}%
^{(M)})^{m-b_{L}}\mathbb{R}_{B}\mathbf{P}_{b_{L}\leftarrow a_{R}}%
]\mathbb{G}_{a_{R},m_{0}}^{(A)},\label{GMA}\\
\mathbb{G}_{m\in A,m_{0}\in M} & =\mathbb{G}_{m,a_{R}}^{(A)}(\mathbf{g}%
^{(M)})^{-1}[(1+\mathbf{P}_{a_{R}\leftarrow a_{R}}\mathbb{R}_{A}%
)\mathbf{g}_{a_{R},m_{0}}^{(M)}+\mathbf{P}_{a_{R}\leftarrow b_{L}}%
\mathbb{R}_{B}\mathbf{g}_{b_{L},m_{0}}^{(M)}],\label{GAM}\\
\mathbb{G}_{m\in M,m_{0}\in B} & =[(\mathbf{P}_{-}^{(M)})^{m-b_{L}%
}(1+\mathbb{R}_{B}\mathbf{P}_{b_{L}\leftarrow b_{L}})+(\mathbf{P}_{+}%
^{(M)})^{m-a_{R}}\mathbb{R}_{A}\mathbf{P}_{a_{R}\leftarrow b_{L}}%
]\mathbb{G}_{b_{L},m_{0}}^{(B)},\label{GMB}\\
\mathbb{G}_{m\in B,m_{0}\in M} & =\mathbb{G}_{m,b_{L}}^{(B)}(\mathbf{g}%
^{(M)})^{-1}[(1+\mathbf{P}_{b_{L}\leftarrow b_{L}}\mathbb{R}_{B}%
)\mathbf{g}_{b_{L},m_{0}}^{(M)}+\mathbf{P}_{b_{L}\leftarrow a_{R}}%
\mathbb{R}_{A}\mathbf{g}_{a_{R},m_{0}}^{(M)}],\label{GBM}\\
\mathbb{G}_{m\in M,m_{0}\in M} & =\mathbf{g}_{m,m_{0}}^{(M)}+(\mathbf{P}%
_{+}^{(M)})^{m-a_{R}}\mathbb{R}_{A}\mathbf{P}_{a_{R}\leftarrow b_{L}%
}\mathbb{R}_{B}\mathbf{g}_{b_{L},m_{0}}^{(M)}+(\mathbf{P}_{-}^{(M)})^{m-b_{L}%
}\mathbb{R}_{B}\mathbf{P}_{b_{L}\leftarrow a_{R}}\mathbb{R}_{A}\mathbf{g}%
_{a_{R},m_{0}}^{(M)}\label{GMM}\\
& +(\mathbf{P}_{-}^{(M)})^{m-b_{L}}(\mathbb{R}_{B}+\mathbb{R}_{B}%
\mathbf{P}_{b_{L}\leftarrow b_{L}}\mathbb{R}_{B})\mathbf{g}_{b_{L},m_{0}%
}^{(M)}+(\mathbf{P}_{+}^{(M)})^{m-a_{R}}(\mathbb{R}_{A}+\mathbb{R}%
_{A}\mathbf{P}_{a_{R}\leftarrow a_{R}}\mathbb{R}_{A})\mathbf{g}_{a_{R},m_{0}%
}^{(M)}.\nonumber\end{aligned}$$
These equations can be interpreted in a physically transparent way. For example, Eq. (\[GBA\]) shows that the local excitation at $m_{0}\in A$ evolves to the scattering wave at $m\in B$ through the following steps: first it is converted by $\mathbb{G}_{a_{R},m_{0}}^{(A)}$ to a zeroth-order partial wave at $a_{R}$, next it undergoes renormalized propagation from $a_{R}$ to $b_{L}$, and finally it is converted back to a local excitation and then to the scattering wave at $m\in B$.
### Analytical construction rule for multiple scatterers
![Green’s function $\mathbf{G}_{m,m_{0}}$ of an infinite system containing eight scatterers, where $m_{0}$ and $m$ are both inside the leads or at the surfaces of the scatterers.[]{data-label="G_MULTI_EXAMPLE"}](fig6Multiple.eps){width="\columnwidth"}
By repeatedly using Eq. (\[G\_ALL\]), the conversion matrix of a composite scatterer can be obtained analytically as functions of the conversion matrices of the constituent scatterers. In particular, the four surface elements of the conversion matrix of the composite scatterer can be immediately obtained from those of the constituent scatterers:
\[G\_BOUNDARY\]$$\begin{aligned}
\mathbb{G}_{b_{R},a_{L}} & =\mathbb{G}_{b_{R},b_{L}}^{(B)}(\mathbf{g}%
^{(M)})^{-1}\mathbf{P}_{b_{L}\leftarrow a_{R}}\mathbb{G}_{a_{R},a_{L}}%
^{(A)},\\
\mathbb{G}_{a_{L},b_{R}} & =\mathbb{G}_{a_{L},a_{R}}^{(A)}(\mathbf{g}%
^{(M)})^{-1}\mathbf{P}_{a_{R}\leftarrow b_{L}}\mathbb{G}_{b_{L},b_{R}}%
^{(B)},\\
\mathbb{G}_{a_{L},a_{L}} & =\mathbb{G}_{a_{L},a_{L}}^{(A)}+\mathbb{G}%
_{a_{L},a_{R}}^{(A)}(\mathbf{g}^{(M)})^{-1}\mathbf{P}_{a_{R}\leftarrow a_{R}%
}\mathbb{G}_{a_{R},a_{L}}^{(A)},\\
\mathbb{G}_{b_{R},b_{R}} & =\mathbb{G}_{b_{R},b_{R}}^{(B)}+\mathbb{G}%
_{b_{R},b_{L}}^{(B)}(\mathbf{g}^{(M)})^{-1}\mathbf{P}_{b_{L}\leftarrow b_{L}%
}\mathbb{G}_{b_{L},b_{R}}^{(B)},\end{aligned}$$ while the latter can be calculated through recursive techniques \[Eqs. (\[RECURSIVE1\]) and (\[RECURSIVE2\])\].
For example, let us consider an infinite layered system with eight scatterers $S_{1},S_{2},\cdots,S_{8}$ and calculate its GF $\mathbf{G}_{m,m_{0}}$. For simplicity we assume that $m_{0}$ and $m$ are both inside the leads or at the surfaces of the scatterers (see Fig. 6), so that $\mathbf{G}_{m,m_{0}}$ is just the $(m,m_{0})$ element of the conversion matrix $\mathbb{G}$ of the composite scatterer $(S_{1}+S_{2}+\cdots+S_{8})$, i.e., $\mathbf{G}_{m,m_{0}}=\mathbb{G}_{m,m_{0}}$, and $\mathbb{G}_{m,m_{0}}$ is completely determined by the surface elements of $\mathbb{G}^{(S_{1})},\cdots
,\mathbb{G}^{(S_{8})}$, which are readily obtained through recursive techniques. First, we use Eq. (\[G\_BOUNDARY\]) to calculate the surface elements of $\mathbb{G}^{(A)}$, $\mathbb{G}^{(B_{1})}$, and $\mathbb{G}%
^{(B_{2})}$ for the three composite scatterers $A\equiv(S_{1}+S_{2}+S_{3})$, $B_{1}\equiv(S_{4}+S_{5}+S_{6})$, and $B_{2}\equiv(S_{7}+S_{8})$. Next we regard $(B_{1}+B_{2})$ as a composite scatterer $B$ and calculate $\mathbb{G}_{m,b_{L}}^{(B)}$ ($b_{L}$ is the left surface of $B$) from Eq. (\[GMA\]). Now the entire system contains two scatterers $A$ and $B$; thus $\mathbb{G}_{m,m_{0}}$ can be obtained from Eq. (\[GBM\]).
Inverse of Fisher-Lee relation
==============================
In the previous section, we have developed a physically transparent and numerically efficient way to calculate the GF of a general layered system. There the GF is expressed as a matrix, i.e., in terms of the propagation matrices $\mathbf{P}_{\pm}$ and conversion matrix $\mathbf{g}$ of the leads and the conversion matrices $\mathbb{G}$ of the scatterers. In this section, we give further physical insight into our GF approach by expressing the GF analytically in terms of a generalized scattering matrix, which describes the scattering of both traveling states and evanescent states. This could be regarded as the inverse of the well-known Fisher-Lee relations [@FisherPRB1981; @StoneIJRD1988; @BarangerPRB1989; @SolsAP1992; @SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009] (see Fig. 1). The key is to express the conversion matrix $\mathbb{G}$ in terms of a generalized scattering matrix.
Generalized scattering matrix
-----------------------------
Let us consider an infinite system consisting of a single scatterer (with the left surface at $m_{L}$ and the right surface at $m_{R}$) connected to two semi-infinite leads $L$ and $R$ (see Fig. \[G\_SINGLE\]). For a right-going eigenmode $|\Phi_{\mathrm{in}}(m)\rangle=e^{ik_{+,\alpha}^{(L)}(m-m_{L})a_{L}%
}|\Phi_{+,\alpha}^{(L)}\rangle$ incident on the scatterer from the left lead, the resulting scattering state $|\Psi(m)\rangle$ follows from Eqs. (\[PHI\_S1\])-(\[PHI\_MR\]). Using Eq. (\[P\_EXPAND\]) for the propagation matrix, we obtain
$$\begin{aligned}
|\Psi(m)\rangle|_{m\geq m_{R}} & =\sum_{\beta}e^{ik_{+,\beta}^{(R)}%
(m-m_{R})a_{R}}|\Phi_{+,\beta}^{(R)}\rangle\mathcal{S}_{\beta,\alpha}^{(RL)}\\
|\Psi(m)\rangle|_{m\leq m_{L}} & =|\Phi_{\mathrm{in}}(m)\rangle+\sum_{\beta
}e^{ik_{-,\beta}^{(L)}(m-m_{L})a_{L}}|\Phi_{-,\beta}^{(L)}\rangle
\mathcal{S}_{\beta,\alpha}^{(LL)},\end{aligned}$$
where $$\mathcal{S}_{\beta,\alpha}^{(RL)}\equiv\langle\phi_{+,\beta}^{(R)}%
|\mathbb{G}_{m_{R},m_{L}}(\mathbf{g}^{(L)})^{-1}|\Phi_{+,\alpha}^{(L)}%
\rangle\label{SRL}%$$ is a generalized transmission amplitude from $|\Phi_{+,\alpha}^{(L)}\rangle$ at the left surface of the scatterer into $|\Phi_{+,\beta}^{(R)}\rangle$ at the right surface of the scatterer, and $$\mathcal{S}_{\beta,\alpha}^{(LL)}\equiv\langle\phi_{-,\beta}^{(L)}%
|[\mathbb{G}_{m_{L},m_{L}}(\mathbf{g}^{(L)})^{-1}-\mathbf{I}]|\Phi_{+,\alpha
}^{(L)}\rangle\label{SLL}%$$ is a generalized reflection amplitude from $|\Phi_{+,\alpha}^{(L)}\rangle$ into $|\Phi_{-,\beta}^{(L)}\rangle$ at the left surface of the scatterer. Similarly, for a left-going incident wave $|\Phi_{\mathrm{in}}(m)\rangle
=e^{ik_{-,\alpha}^{(R)}(m-m_{R})a_{R}}|\Phi_{-,\alpha}^{(R)}\rangle$ in the right lead, the resulting scattering state $|\Psi(m)\rangle$ follows from Eqs. (\[PHI\_S2\])-(\[PHI\_ML\]) as $$\begin{aligned}
|\Psi(m)\rangle|_{m\leq m_{L}} & =\sum_{\beta}e^{ik_{-,\beta}^{(L)}%
(m-m_{L})a_{L}}|\Phi_{-,\beta}^{(L)}\rangle\mathcal{S}_{\beta,\alpha}%
^{(LR)},\\
|\Psi(m)\rangle|_{m\geq m_{R}} & =|\Phi_{\mathrm{in}}(m)\rangle+\sum_{\beta
}e^{ik_{+,\beta}^{(R)}(m-m_{R})a_{R}}\mathcal{S}_{\beta,\alpha}^{(RR)}%
|\Phi_{+,\beta}^{(R)}\rangle,\end{aligned}$$ where $$\mathcal{S}_{\beta,\alpha}^{(LR)}\equiv\langle\phi_{-,\beta}^{(L)}%
|\mathbb{G}_{m_{L},m_{R}}(\mathbf{g}^{(R)})^{-1}|\Phi_{-,\alpha}^{(R)}%
\rangle\label{SLR}%$$ is a generalized transmission amplitude from $|\Phi_{-,\alpha}^{(R)}\rangle$ at the right surface of the scatterer into $|\Phi_{-,\beta}^{(L)}\rangle$ at the left surface of the scatterer, and $$\mathcal{S}_{\beta,\alpha}^{(RR)}\equiv\langle\phi_{+,\beta}^{(R)}%
|[\mathbb{G}_{m_{R},m_{R}}(\mathbf{g}^{(R)})^{-1}-\mathbf{I}]|\Phi_{-,\alpha
}^{(R)}\rangle\label{SRR}%$$ is a generalized reflection amplitude from $|\Phi_{-,\alpha}^{(R)}\rangle$ into $|\Phi_{+,\beta}^{(R)}\rangle$ at the right surface of the scatterer.
Equations (\[SRL\])-(\[SRR\]) define a generalized scattering matrix $\mathcal{S}(E)$ and express it in terms of the surface elements of the conversion matrix. They were first derived in the wave function mode matching approach [@AndoPRB1991; @NikolicPRB1994; @XiaPRB2006] and its connection to the GF approach was established later [@KhomyakovPRB2005]. They are valid for both traveling modes and evanescent modes. In our GF approach, these expressions have clear physical meanings. Taking $\mathcal{S}_{\beta,\alpha}^{(RL)}$ as an example, $\mathbb{G}_{m_{R},m_{L}}(\mathbf{g}^{(L)})^{-1}$ converts the incident eigenmode $|\Phi_{+,\alpha}^{(L)}\rangle$ at the left surface of the scatterer back to a local excitation and then to the scattering wave at the right surface of the scatterer. Then the dual vector $\langle\phi_{+,\beta}^{(R)}|$ projects the scattering wave onto the eigenmode $|\Phi_{+,\beta}^{(R)}\rangle
$ \[see Eqs. (24) and (25)\]. The transmission and reflection amplitudes of the unitary scattering matrix connecting two traveling eigenmodes are obtained by normalizing with respect to the current [@KhomyakovPRB2005]: $$S_{\beta,\alpha}^{(q,p)}|_{\alpha,\beta\in\mathrm{traveling}}=\sqrt
{\frac{|v_{s_{\mathrm{out}},\beta}^{(q)}| /a_{q}}{|v_{s_{\mathrm{in}},\alpha}^{(p)}|/q_{p}}}\mathcal{S}%
_{\beta,\alpha}^{(q,p)},$$ where $s_{\mathrm{in}}=s_{\mathrm{out}}=+$ for $(q,p)=(R,L)$; $s_{\mathrm{in}}=s_{\mathrm{out}}=-$ for $(q,p)=(L,R)$; $s_{\mathrm{in}}=+$, $s_{\mathrm{out}}=-$ for $(q,p)=(L,L)$; and $s_{\mathrm{in}}=-$, $s_{\mathrm{out}}=+$ for $(q,p)=(R,R)$, thus Eqs. (\[SRL\]) and (\[SLR\]) lead to Eqs. (\[SUNITARY\_RL\]) and (\[SUNITARY\_LR\]), respectively.
Inverse of Fisher-Lee relations
-------------------------------
The inverse of Eqs. (\[SRL\])-(\[SRR\]) gives the surface elements of $\mathbb{G}(E)$ in terms of the generalized scattering matrix $\mathcal{S}%
(E)$:
\[G\_BOUNDARY\_EXP\]$$\begin{aligned}
\mathbb{G}_{m_{R},m_{L}} & =\sum_{\alpha}\left( \sum_{\beta}\mathcal{S}%
_{\beta,\alpha}^{(RL)}|\Phi_{+,\beta}^{(R)}\rangle\right) \langle
\phi_{+,\alpha}^{(L)}|\mathbf{g}^{(L)},\label{GB_RL_EXP}\\
\mathbb{G}_{m_{L},m_{R}} & =\sum_{\alpha}\left( \sum_{\beta}\mathcal{S}%
_{\beta,\alpha}^{(LR)}|\Phi_{-,\beta}^{(L)}\rangle\right) \langle
\phi_{-,\alpha}^{(R)}|\mathbf{g}^{(R)},\label{GB_LR_EXP}\\
\mathbb{G}_{m_{L},m_{L}} & =\sum_{\alpha}\left( |\Phi_{+,\alpha}%
^{(L)}\rangle+\sum_{\beta}\mathcal{S}_{\beta,\alpha}^{(LL)}|\Phi_{-,\beta
}^{(L)}\rangle\right) \langle\phi_{+,\alpha}^{(L)}|\mathbf{g}^{(L)}%
,\label{GB_LL_EXP}\\
\mathbb{G}_{m_{R},m_{R}} & =\sum_{\alpha}\left( |\Phi_{-,\alpha}%
^{(R)}\rangle+\sum_{\beta}\mathcal{S}_{\beta,\alpha}^{(RR)}|\Phi_{+,\beta
}^{(R)}\rangle\right) \langle\phi_{-,\alpha}^{(R)}|\mathbf{g}^{(R)}.
\label{GB_RR_EXP}%\end{aligned}$$ These expressions have very clear physical interpretations. For example, Eq. (\[GB\_RL\_EXP\]) shows that a local excitation $|\Phi_{\mathrm{loc}}%
\rangle_{m_{L}}$ at the left surface of the scatterer evolves to the scattering wave at the right surface of the scatterer through two steps. First, it evolves to a partial wave $\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}%
}\rangle_{m_{L}}$, then it is expanded into linear combinations of right-going eigenmodes $\sum_{\alpha}|\Phi_{+,\alpha}^{(L)}\rangle\langle\phi_{+,\alpha
}^{(L)}|\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{L}}$ and each eigenmode transmits across the scatterer to its right surface as
$$|\Phi_{+,\alpha}^{(L)}\rangle\rightarrow\sum_{\beta}\mathcal{S}_{\beta,\alpha
}^{(RL)}|\Phi_{+,\beta}^{(R)}\rangle.$$
Similarly, Eq. (\[GB\_LL\_EXP\]) shows that a local excitation at the left surface of the scatterer first evolves to a partial wave $\mathbf{g}%
^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{L}}$, then it is expanded as $\sum_{\alpha}|\Phi_{+,\alpha}^{(L)}\rangle\langle\phi_{+,\alpha}%
^{(L)}|\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{L}}$ and each eigenmode evolves to a scattering wave:$$|\Phi_{+,\alpha}^{(L)}\rangle\rightarrow|\Phi_{+,\alpha}^{(L)}\rangle
+\sum_{\beta}|\Phi_{-,\beta}^{(L)}\rangle\mathcal{S}_{\beta,\alpha}^{(LL)},$$ which consists of the incident wave and the reflection wave.
Next, we can express other blocks of the GF, i.e., Eqs. (\[G\_ALL\_S\])-(\[G\_ALL\_R\]) with $m,m_{0}$ inside the leads, in terms of the generalized scattering matrix:
\[G\_ALL\_EXP\]$$\begin{aligned}
\mathbf{G}_{m\in R,m_{0}\in L}&=&\sum_{\alpha}\left( \sum_{\beta
}e^{ik_{+,\beta}^{(R)}(m-m_{R})a}|\Phi_{+,\beta}^{(R)}\rangle\mathcal{S}%
_{\beta,\alpha}^{(RL)}\right) \notag
\\ &&\times ~ e^{ik_{+,\alpha}^{(L)}(m_{L}-m_{0})a}%
\langle\phi_{+,\alpha}^{(L)}|\mathbf{g}^{(L)},\label{GRL_EXP}\\
\mathbf{G}_{m\in L,m_{0}\in R} &=&\sum_{\alpha}\left( \sum_{\beta
}e^{ik_{-,\beta}^{(L)}(m-m_{L})a}|\Phi_{-,\beta}^{(L)}\rangle\mathcal{S}%
_{\beta,\alpha}^{(LR)}\right) \notag
\\&& \times ~ e^{ik_{-,\alpha}^{(R)}(m_{R}-m_{0})a}%
\langle\phi_{-,\alpha}^{(R)}|\mathbf{g}^{(R)},\label{GLR_EXP}\\
\mathbf{G}_{m\in L,m_{0}\in L} &=&\mathbf{g}_{m,m_{0}}^{(L)}+\sum_{\alpha
}\left( \sum_{\beta}e^{ik_{-,\beta}^{(L)}(m-m_{L})a}|\Phi_{-,\beta}%
^{(L)}\rangle\mathcal{S}_{\beta,\alpha}^{(LL)}\right) \notag
\\&& \times ~ e^{ik_{+,\alpha}%
^{(L)}(m_{L}-m_{0})a}\langle\phi_{+,\alpha}^{(L)}|\mathbf{g}^{(L)}%
,\label{GLL_EXP}\\
\mathbf{G}_{m\in R,m_{0}\in R} &=&\mathbf{g}_{m,m_{0}}^{(R)}+\sum_{\alpha
}\left( \sum_{\beta}e^{ik_{+,\beta}^{(R)}(m-m_{R})a}|\Phi_{+,\beta}%
^{(R)}\rangle\mathcal{S}_{\beta,\alpha}^{(RR)}\right) \notag
\\&& \times ~ e^{ik_{-,\alpha}%
^{(R)}(m_{R}-m_{0})a}\langle\phi_{-,\alpha}^{(R)}|\mathbf{g}^{(R)}%
.\label{GRR_EXP}%\end{aligned}$$
The above expressions have clear physical meanings. For example, Eq. (\[GRL\_EXP\]) shows that a local excitation $|\Phi_{\mathrm{loc}}%
\rangle_{m_{0}}$ in the left lead first evolves to a partial wave $\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ and then propagates freely to the left surface of the scatterer as $\sum_{\alpha}e^{ik_{+,\alpha
}^{(L)}(m_{L}-m_{0})a}|\Phi_{+,\alpha}^{(L)}\rangle\langle\phi_{+,\alpha
}^{(L)}|\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$. Finally each eigenmode $|\Phi_{+,\alpha}^{(L)}\rangle$ evolves to a transmission wave: $$|\Phi_{+,\alpha}^{(L)}\rangle\rightarrow\sum_{\beta}e^{ik_{+,\beta}%
^{(R)}(m-m_{R})a_{R}}|\Phi_{+,\beta}^{(R)}\rangle\mathcal{S}_{\beta,\alpha
}^{(RL)}.$$ As another example, Eq. (\[GLL\_EXP\]) shows that $\mathbf{G}_{m\in
L,m_{0}\in L}$ is the sum of the free GF $\mathbf{g}_{m,m_{0}}^{(L)}$ and the reflection wave contribution, which emerges as follows: the local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ in the left lead first evolves to a partial wave $\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$ and then propagates freely to the left surface of the scatterer as $\sum_{\alpha
}e^{ik_{+,\alpha}^{(L)}(m_{L}-m_{0})a}|\Phi_{+,\alpha}^{(L)}\rangle\langle
\phi_{+,\alpha}^{(L)}|\mathbf{g}^{(L)}|\Phi_{\mathrm{loc}}\rangle_{m_{0}}$. Finally, each mode evolves to a reflection wave: $$|\Phi_{+,\alpha}^{(L)}\rangle\rightarrow\sum_{\beta}e^{ik_{-,\beta}%
^{(L)}(m-m_{L})a_{L}}|\Phi_{-,\beta}^{(L)}\rangle\mathcal{S}_{\beta,\alpha
}^{(LL)}.$$ Equation (\[G\_ALL\_EXP\]) not only allows us to construct the GF from the generalized scattering matrix, but also reveals the contribution of each individual scattering channels to the GF.
On-shell spectral expansion
---------------------------
The Fisher-Lee relations [@FisherPRB1981; @StoneIJRD1988; @BarangerPRB1989; @SolsAP1992; @SanvitoPRB1999; @KhomyakovPRB2005; @WimmerThesis2009] and its inverse \[Eq. (\[G\_ALL\_EXP\])\] provide a complete one-to-one correspondence between the lattice GF approach and the wave function mode matching approach [@AndoPRB1991; @NikolicPRB1994; @KhomyakovPRB2005; @XiaPRB2006] to mesoscopic quantum transport (see Fig. 1). Next we show that we can construct the GF $\mathbf{G}(E)$ analytically in terms of a few scattering states on the energy shell $E$. Since the latter can be readily obtained from standard textbook technique and approximation methods (such as the WKB approximation), this provides a convenient way to obtain the GF.
Let us introduce $2M$ *advanced* eigenmodes $\{\tilde{k}_{s,\alpha
},|\tilde{\Phi}_{s,\alpha}\rangle\}$ of each lead [@KhomyakovPRB2005],$$%
\begin{array}
[c]{ccc}%
\tilde{k}_{s,\alpha}\equiv k_{s,\alpha}, & |\tilde{\Phi}_{s,\alpha}%
\rangle\equiv|\Phi_{s,\alpha}\rangle\ & (\alpha\in\text{\textrm{evanescent}%
}),\\
\tilde{k}_{s,\alpha}\equiv k_{-s,\alpha}, & |\tilde{\Phi}_{s,\alpha}%
\rangle\equiv|\Phi_{-s,\alpha}\rangle & (\alpha\in\text{\textrm{traveling}}),
\end{array}$$ and their dual vectors:$$%
\begin{bmatrix}
\langle\tilde{\phi}_{s,1}|\\
\vdots\\
\langle\tilde{\phi}_{s,M}|
\end{bmatrix}
\equiv\mathbf{\tilde{U}}_{s}^{-1},$$ where $\mathbf{\tilde{U}}_{s}\equiv\lbrack|\tilde{\Phi}_{s,1}\rangle
,\cdots,|\tilde{\Phi}_{s,M}\rangle]$. The advanced eigenvectors $\{|\tilde
{\Phi}_{s,\alpha}\rangle\}$ and dual vectors $\{|\tilde{\phi}_{s,\alpha
}\rangle\}$ obey exactly the same orthonormal and completeness relations as their retarded counterpart \[Eqs. (\[ORTHO\]) and (\[COMPLETE\])\]. Using [@KhomyakovPRB2005]$$\mathbf{g}^{-1}=\sum_{\alpha}|\tilde{\phi}_{-,\alpha}\rangle\frac
{iv_{+,\alpha}}{a}\langle\phi_{+,\alpha}|=\sum_{\alpha}|\tilde{\phi}%
_{+,\alpha}\rangle\frac{-iv_{-,\alpha}}{a}\langle\phi_{-,\alpha}|,
\label{GINV_EXPANSION}%$$ and the completeness relations Eq. (\[COMPLETE\]), we obtain$$\begin{aligned}
\mathbf{g} & =\sum_{\alpha}|\Phi_{+,\alpha}\rangle\frac{a}{iv_{+,\alpha}%
}\langle\tilde{\Phi}_{-,\alpha}|=\sum_{\alpha}|\Phi_{-,\alpha}\rangle\frac
{a}{-iv_{-,\alpha}}\langle\tilde{\Phi}_{+,\alpha}|\label{G_EXPANSION}\\
& \longrightarrow\sum_{\alpha}|\Phi_{+,\alpha}\rangle\frac{a}{iv_{+,\alpha}%
}\langle\Phi_{+,\alpha}|=\sum_{\alpha}|\Phi_{-,\alpha}\rangle\frac
{a}{-iv_{-,\alpha}}\langle\Phi_{-,\alpha}|,\end{aligned}$$ where the second line holds when all the eigenmodes are traveling modes. Here $a$ is the unit cell spacing of the lead and $v_{s,\alpha}$ is the generalized group velocity of the eigenmode $(s,\alpha)$: for a traveling mode, it equals the group velocity \[Eq. (\[VELOCITY\])\]; for an evanescent mode, it is defined as $$v_{+,\alpha}=v_{-,\alpha}^{\ast}=-ia\langle\Phi_{-,\alpha}|(e^{ik_{-,\alpha}%
a})^{\ast}\mathbf{t}^{\dagger}-e^{ik_{+,\alpha}a}\mathbf{t}|\Phi_{+,\alpha
}\rangle. \label{VSA}%$$ Note that for an evanescent (traveling) mode, $v_{s,\alpha}$ depends (does not depend) on the choice of the phases of the eigenvectors $\{|\Phi_{s,\alpha
}\rangle\}$.
To express the GF in terms of on-shell scattering states, we introduce the eigenmode wave functions
\[BULK\_MODE\_DEF\]$$\begin{aligned}
|\Phi_{+,\alpha}(m)\rangle & \equiv\left\{
\begin{array}
[c]{ll}%
e^{ik_{+,\alpha}(m-m_{0})a}|\Phi_{+,\alpha}\rangle & (m\geq m_{0}),\\
0 & (m<m_{0}),
\end{array}
\right. \\
|\Phi_{-,\alpha}(m)\rangle & \equiv\left\{
\begin{array}
[c]{ll}%
0 & (m\geq m_{0}),\\
e^{ik_{-,\alpha}(m-m_{0})a}|\Phi_{-,\alpha}\rangle & (m<m_{0}),
\end{array}
\right.\end{aligned}$$ for the lead in which $m_{0}$ locates. Note that so-defined $|\Phi_{s,\alpha
}(m)\rangle$ depends on $m_{0}$, which is regarded as fixed and hence omitted for brevity. If there were no scatterers, then $\mathbf{G}_{m,m_{0}}$ would coincide with the free GF of this lead \[Eq. (\[G\_INFINITE\])\], which can be written as
$$\mathbf{g}_{m,m_{0}}=a%
%TCIMACRO{\dsum \limits_{\alpha}}%
%BeginExpansion
{\displaystyle\sum\limits_{\alpha}}
%EndExpansion
\left( \frac{|\Phi_{+,\alpha}(m)\rangle\langle\tilde{\Phi}_{-,\alpha}%
|}{iv_{+,\alpha}}+\frac{|\Phi_{-,\alpha}(m)\rangle\langle\tilde{\Phi
}_{+,\alpha}|}{-iv_{-,\alpha}}\right) . \label{GFREE_SPECTRAL_EXP}%$$
Due to the presence of scatterers, each eigenmode $|\Phi_{s,\alpha}(m)\rangle$ evolves to a corresponding scattering state $|\Psi_{s,\alpha}(m)\rangle$, so replacing $|\Phi_{s,\alpha}(m)\rangle$ in Eq. (\[GFREE\_SPECTRAL\_EXP\]) with $|\Psi_{s,\alpha}(m)\rangle$ gives the GF:$$\mathbf{G}_{m,m_{0}}=a%
%TCIMACRO{\dsum \limits_{\alpha}}%
%BeginExpansion
{\displaystyle\sum\limits_{\alpha}}
%EndExpansion
\left( \frac{|\Psi_{+,\alpha}(m)\rangle\langle\tilde{\Phi}_{-,\alpha}%
|}{iv_{+,\alpha}}+\frac{|\Psi_{-,\alpha}(m)\rangle\langle\tilde{\Phi
}_{+,\alpha}|}{-iv_{-,\alpha}}\right) . \label{G_SPECTRAL_EXP2}%$$ Since the total scattering state $|\Psi_{s,\alpha}(m)\rangle$ can be decomposed into the sum of the incident wave $|\Phi_{s,\alpha}(m)\rangle$ (which vanishes outside the lead in which $m_{0}$ locates) and the outgoing scattering wave $|\Psi_{s,\alpha}^{(\mathrm{out})}(m)\rangle\equiv
|\Psi_{s,\alpha}(m)\rangle-|\Phi_{s,\alpha}(m)\rangle$, Eq. (\[G\_SPECTRAL\_EXP2\]) can also be written as $$\mathbf{G}_{m,m_{0}}=\mathbf{g}_{m,m_{0}}+a%
%TCIMACRO{\dsum \limits_{\alpha}}%
%BeginExpansion
{\displaystyle\sum\limits_{\alpha}}
%EndExpansion
\left( \frac{|\Psi_{+,\alpha}^{(\mathrm{out})}(m)\rangle\langle\tilde{\Phi
}_{-,\alpha}|}{iv_{+,\alpha}}+\frac{|\Psi_{-,\alpha}^{(\mathrm{out}%
)}(m)\rangle\langle\tilde{\Phi}_{+,\alpha}|}{-iv_{-,\alpha}}\right) ,
\label{G_SPECTRAL_EXP3}%$$ where $\mathbf{g}_{m,m_{0}}$ is nonzero only for $m$ in the same lead as $m_{0}$.
Equation (\[G\_SPECTRAL\_EXP2\]) or (\[G\_SPECTRAL\_EXP3\]) shows that the GF $\mathbf{G}_{m,m_{0}}(E)$ is essentially certain scattering states $\{|\Psi_{s,\alpha}(m)\rangle\}$ (on the energy shell $E$) that obey outgoing boundary conditions; i.e., they emanate from the on-shell eigenmodes $|\Phi_{s,\alpha}(m)\rangle$ of the lead in which the local excitation occurs. Compared with the standard spectral expansion in classic textbook on quantum mechanics [@SakuraiBook1994; @GriffithsBook1995; @CohenBook2005] that expresses the GF in terms of all the eigenstates (both on-shell eigenstates and off-shell ones) of the system, Eq. (\[G\_SPECTRAL\_EXP2\]) or (\[G\_SPECTRAL\_EXP3\]) deepens our physical understanding about the GF and allows analytical reconstruction of the GF from a few scattering states of the system.
As an example, let us consider an infinite system containing a single scatterer (Fig. \[G\_SINGLE\]). For $m_{0}$ in the left lead, the left-going eigenmode $|\Phi_{-,\alpha}^{(L)}(m)\rangle$ is not scattered, so $|\Psi_{-,\alpha
}^{(L,\mathrm{out})}(m)\rangle=0$, while the right-going eigenmode $|\Phi_{+,\alpha}^{(L)}(m)\rangle$ produces an outgoing scattering wave$$\begin{aligned}
|\Psi_{+,\alpha}^{(L,\mathrm{out})}(m)\rangle|_{m\in L} & =\left(
\sum_{\beta}e^{ik_{-,\beta}^{(L)}(m-m_{L})a}|\Phi_{-,\beta}^{(L)}%
\rangle\mathcal{S}_{\beta,\alpha}^{(LL)}\right) e^{ik_{+,\alpha}^{(L)}%
(m_{L}-m_{0})a},\\
|\Psi_{+,\alpha}^{(L,\mathrm{out})}(m)\rangle|_{m\in R} & =\left(
\sum_{\beta}e^{ik_{+,\beta}^{(R)}(m-m_{R})a}|\Phi_{+,\beta}^{(R)}%
\rangle\mathcal{S}_{\beta,\alpha}^{(RL)}\right) e^{ik_{+,\alpha}^{(L)}%
(m_{L}-m_{0})a}.\end{aligned}$$ Substituting into Eq. (\[G\_SPECTRAL\_EXP3\]) gives Eqs. (\[GRL\_EXP\]) and (\[GLL\_EXP\]). Similarly, for $m_{0}$ in the right lead, the right-going eigenmode $|\Phi_{+,\alpha}^{(R)}(m)\rangle$ is not scattered, so $|\Psi_{+,\alpha}^{(R,\mathrm{out})}(m)\rangle=0$, while the left-going eigenmode $|\Phi_{-,\alpha}^{(R)}(m)\rangle$ produces the outgoing wave $$\begin{aligned}
|\Psi_{-,\alpha}^{(R,\mathrm{out})}(m)\rangle|_{m\in L} & =\left(
\sum_{\beta}e^{ik_{-,\beta}^{(L)}(m-m_{L})a}|\Phi_{-,\beta}^{(L)}%
\rangle\mathcal{S}_{\beta,\alpha}^{(LR)}\right) e^{ik_{-,\alpha}^{(R)}%
(m_{R}-m_{0})\ a},\\
|\Psi_{-,\alpha}^{(R,\mathrm{out})}(m)\rangle|_{m\in R} & =\left(
\sum_{\beta}e^{ik_{+,\beta}^{(R)}(m-m_{R})a}|\Phi_{+,\beta}^{(R)}%
\rangle\mathcal{S}_{\beta,\alpha}^{(RR)}\right) e^{ik_{-,\alpha}^{(R)}%
(m_{R}-m_{0})a}.\end{aligned}$$ Substituting them into Eq. (\[G\_SPECTRAL\_EXP3\]) gives Eqs. (\[GLR\_EXP\]) and (\[GRR\_EXP\]).
Example and applications
========================
Here we first exemplify our general results in a 1D chain and then apply the formalism to graphene *p-n* junctions described by the tight-binding model.
Simple example: 1D chain
------------------------
We consider a 1D chain characterized by one basis state in each unit cell, the unit cell Hamiltonian $\mathbf{h}=\varepsilon_{0}$, and the nearest-neighbor hopping $\mathbf{t}=-t<0$. For a given wave vector $k$, Eq. (\[MEQ1\]) can be solved to yield the energy $E(k)=\varepsilon_{0}-2t\cos(ka)$, which is real in two cases: (1) $k\in\mathbb{R}$; (2) $k=i\kappa$ or $k=\pi/a+i\kappa$ with $\kappa\in\mathbb{R}$. The former gives the real energy band$,$ while the latter gives the complex energy bands. For specificity we consider the energy $E\in\lbrack\varepsilon_{0}-2t,\varepsilon_{0}+2t]$, so there is one right-going traveling eigenmode $k_{+}=k$ with group velocity $v=2at\sin
(ka)>0$ and one left-going traveling eigenmode $k_{-}=-k$ with group velocity $-v$, where $k$ is the positive solution to $E=E(k)$. The eigenvectors of the eigenmodes and dual vectors are $\Phi_{\pm}=\phi_{\pm}=1$. The propagation matrices are $\mathbf{P}_{\pm}=e^{\pm ika}$, the conversion matrix of the lead is$$\mathbf{g}=\frac{1}{2it\sin(ka)}=\frac{1}{iv/a},$$ and the free GF of the lead is $$\mathbf{g}_{m,m_{0}}=\frac{e^{ik|m-m_{0}|a}}{iv/a}.$$
When the on-site energy of the unit cell $m_{1}$ of the infinite 1D chain is replaced by $\varepsilon_{0}+\delta$, the unit cell $m_{1}$ becomes a scatterer characterized the conversion matrix$$\mathbb{G}=\frac{1}{iv/a-\delta}, \label{CM_EXAMPLE}%$$ or equivalently the transmission amplitude $\mathcal{T}=\mathbb{G}%
\mathbf{g}^{-1}=(iv/a)/(iv/a-\delta)$ and reflection amplitude $\mathcal{R}%
=\mathbb{G}\mathbf{g}^{-1}-1=\delta/(iv/a-\delta)$. The GFs of the entire system are given by Eqs. (\[G\_ALL\_S\]), (\[G\_ALL\_L\]), and (\[G\_ALL\_R\]) as$$\begin{aligned}
\mathbf{G}_{m\geq m_{1},m_{0}\leq m_{1}} & =\frac{e^{ik(m-m_{1}%
)a}\mathcal{T}e^{ik(m_{1}-m_{0})a}}{iv/a}=\mathcal{T}\frac{e^{ik(m-m_{0})a}%
}{iv/a},\\
\mathbf{G}_{m\leq m_{1},m_{0}\geq m_{1}} & =\frac{e^{-ik(m-m_{1}%
)a}\mathcal{T}e^{-ik(m_{1}-m_{0})a}}{iv/a}=\mathcal{T}\frac{e^{-ik(m-m_{0})a}%
}{iv/a},\\
\mathbf{G}_{m\leq m_{1},m_{0}\leq m_{1}} & =\frac{e^{ika|m-m_{0}|}}%
{iv/a}+\frac{e^{-ik(m-m_{1})a}\mathcal{R}e^{ik(m_{1}-m_{0})a}}{iv/a},\\
\mathbf{G}_{m\geq m_{1},m_{0}\geq m_{1}} & =\frac{e^{ika|m-m_{0}|}}%
{iv/a}+\frac{e^{ik(m-m_{1})a}\mathcal{R}e^{-ik(m_{1}-m_{0})a}}{iv/a}.\end{aligned}$$ The above results are also consistent with Eq. (\[G\_ALL\_EXP\]).
Finally, when the on-site energies of unit cell $m_{1}$ and unit cell $m_{2}$ ($>m_{1}$) are both replaced by $\varepsilon_{0}+\delta$, then each unit cell becomes a scatterer characterized by the conversion matrix in Eq. (\[CM\_EXAMPLE\]). The conversion matrix $\mathbb{G}^{(\mathrm{C})}$ of the composite scatterer $(m_{1}+m_{2})$ is given by Eq. (\[G\_ALL\]). In particular, the surface elements are obtained from Eq. (\[G\_BOUNDARY\]) as$$\begin{aligned}
\mathbb{G}_{m_{2}m_{1}}^{(\mathrm{C})} & =\mathbb{G}_{m_{1}m_{2}%
}^{(\mathrm{C})}=e^{ik(m_{2}-m_{1})a}\frac{\mathcal{T}(1-e^{2ik(m_{2}-m_{1}%
)a}\mathcal{R}^{2})^{-1}\mathcal{T}}{iv/a},\\
\mathbb{G}_{m_{1}m_{1}}^{(\mathrm{C})} & =\mathbb{G}_{m_{2}m_{2}%
}^{(\mathrm{C})}=\mathbb{G}+e^{2ik(m_{2}-m_{1})a}\frac{\mathcal{TR}%
(1-e^{2ik(m_{2}-m_{1})a}\mathcal{R}^{2})^{-1}\mathcal{T}}{iv/a},\end{aligned}$$ from which we can obtain the generalized transmission amplitude across the composite scatterer as $$\mathcal{T}^{(\mathrm{C})}\equiv\mathbb{G}_{m_{2},m_{1}}^{(\mathrm{C}%
)}\mathbf{g}^{-1}=\mathcal{T}(1-e^{2ik(m_{2}-m_{1})a}\mathcal{R}^{2}%
)^{-1}e^{ik(m_{2}-m_{1})a}\mathcal{T},$$ and the generalized reflection amplitude as$$\mathcal{R}^{(\mathrm{C})}\equiv\mathbb{G}_{m_{1}m_{1}}^{(\mathrm{C}%
)}\mathbf{g}^{-1}-1=\mathcal{R}+e^{2ik(m_{2}-m_{1})a}\mathcal{TR}%
(1-e^{2ik(m_{2}-m_{1})a}\mathcal{R}^{2})^{-1}\mathcal{T}.$$
Chiral tunneling and anomalous focusing in graphene *p-n* junction
------------------------------------------------------------------
![(a) Sketch of graphene *p-n* junction with a smooth interface. Panels (b)-(d) show the choice of primitive vectors and $x,y$ axes when the interface is along the zigzag direction (b), the armchair direction (c), or a more general direction (d). The filled dots mark the Bravais lattice ($A$ sublattice) of graphene. The shaded regions mark the unit cells. In panel (b), the $X$ and $Y$ axes of the Cartesian coordinate system are also shown.[]{data-label="G_PNJ"}](fig7PNJ.eps){width="\columnwidth"}
Graphene is a single layer of carbon atoms in a honeycomb lattice that hosts massless Dirac fermions [@BeenakkerRMP2008; @CastroRMP2009; @PeresRMP2010; @DasSarmaRMP2011]. One of the unique properties of electrons in graphene is chiral tunneling [@KleinZP1929; @KatsnelsonNatPhys2006; @CheianovPRB2006]: an electron normally incident on a potential barrier will always be perfectly transmitted, independently of its kinetic energy and the height and width of the potential barrier (for oblique incidence, the transverse momentum serves as a gap-opening mass term, so the transmission is not perfect, and bound states may be created by a 1D potential well [@PhysRevB.74.045424; @PhysRevA.90.052116]). In recent years, the chiral tunneling in graphene *p-n* junctions has attracted a lot of attention (see Ref. for a review). Another interesting phenomenon for electrons in graphene *p-n* junctions is the anomalous focusing due to negative refraction, which was initially proposed by Veselago for electromagnetic waves [@VeselagoSPU1968; @PendryPRL2000; @ZhangNatMater2008; @PendryScience2012]: a spatially diverging pencil of rays is focused to a spatially converging one during the transition from a medium with positive refractive index across a sharp interface into a negative index medium. In 2007, Cheianov *et al*. [@CheianovScience2007] proposed that ballistic graphene *p-n* junctions can also exhibit negative refraction and hence focus the electron flow: in the electron-doped *n* (hole-doped *p*) region, the carrier group velocity is parallel (anti-parallel) to its momentum, in analogy to light propagation in a positive (negative) refractive index medium. Ever since then, there have been a lot of works on the negative refraction in graphene (see Refs. for examples) and on the surface of topological insulators [@ZhaoPRL2013]. Recently, the Veselago lens effect in graphene was observed experimentally [@LeeNatPhys2015; @ChenScience2016].
On the theoretical side, most of the previous studies focus on traveling states and are based on the low-energy continuous model, whose validity is limited to the vicinity of the Dirac points. A very recent work [@LogemannPRB2015], based on the tight-binding model, calculates numerically the propagation of a wave packet in a large but finite graphene flake to approximate the Klein tunneling and caustics of electron waves. Here we apply our general GF formalism to study the chiral tunneling in an infinite graphene *p-n* junction and pay special attention to the *evanescent* eigenmode. Our approach provides a clear physical picture and allows us to calculate the GF over long distances, so we further perform both analytical analysis and numerical simulations of dual-probe STM measurements. Our results demonstrate the possibility of observing the spatially resolved interference pattern caused by the negative refraction in graphene *p-n* junctions and further reveal a few interesting features, such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene.
We consider a graphene *p-n* junction with an interface that can be either sharp or smooth, as shown in Fig. \[G\_PNJ\](a). In the tight-binding model, the interface could align along different crystalline directions. To provide the simplest description, a *tilted* coordinate system is usually necessary: we choose one primitive vector $\mathbf{a}_{2}$ (defined as the $y$ axis) of the Bravais lattice of uniform graphene to be parallel to the interface, and choose the $x$ axis of our tilted coordinate system to be parallel to the other primitive vector $\mathbf{a}_{1}$; i.e., the two nonorthogonal unit vectors of the tilted coordinate are $\mathbf{e}_{x}%
\equiv\mathbf{a}_{1}/|\mathbf{a}_{1}|$ and $\mathbf{e}_{y}\equiv\mathbf{a}%
_{2}/|\mathbf{a}_{2}|$, as shown in Figs. \[G\_PNJ\](b)-\[G\_PNJ\](d) for the interface along the zigzag direction, armchair direction, and a more general direction. This choice of the primitive vectors and the tilted coordinate system ensures that the lattice Hamiltonian is invariant upon translation by $|\mathbf{a}_{2}|$ along the $y$ axis, so that the original 2D lattice model can be reduced to a 1D lattice model.
### Reduction from 2D to 1D
For specificity, we assume that the interface is along the zigzag direction \[Fig. \[G\_PNJ\](c)\], where $|\mathbf{a}_{1}|=|\mathbf{a}_{2}|=\sqrt
{3}a_{\mathrm{C-C}}\equiv a$ and $a_{\mathrm{C-C}}$ is the C-C bond length. The vanishingly small spin-orbit coupling in graphene makes the GF spin-independent, so we neglect the electron spin. In the tight-binding model, each unit cell of graphene consists of $M=2$ orbitals, i.e., $|A,m,n\rangle$ and $|B,m,n\rangle$ for the unit cell $(m,n)$, where $m$ ($n$) is the index along the $x$ ($y$) axis of the tilted coordinate and $A,B$ labels the sublattice \[see Fig. \[G\_PNJ\](c)\]. The Hamiltonian $\hat{H}=\hat{H}%
_{0}+\hat{V}$ is the sum of the uniform part $$\hat{H}_{0}=\sum_{m,n}t(\left\vert A,m+1,n\right\rangle +\left\vert
A,m,n-1\right\rangle +\left\vert A,m,n\right\rangle )\left\langle
B,m,n\right\vert +h.c.)$$ and the *p-n* junction potential $$\hat{V}=\sum_{m,n}V_{m}(\left\vert A,m,n\right\rangle \left\langle
A,m,n\right\vert +\left\vert B,m,n\right\rangle \left\langle B,m,n\right\vert
),$$ where $t\approx 3$ eV is the nearest-neighbor hopping constant[@CastroRMP2009]. The junction potential $V_{m}$ depends arbitrarily on $m$ inside the interface ($m_{L}\leq m\leq m_{R}$), but equals $V_{L}$ inside the left lead $L$ ($m\leq m_{L}-1$) and equals $V_{R}$ inside the right lead $R$ ($m\geq m_{R}+1$).
Due to the invariance of $\hat{H}$ upon translation by $|\mathbf{a}_{2}|$ along the $y$ axis, the problem can be reduced from 2D to 1D by a Fourier transform $$|k_{y}\rangle\equiv\frac{1}{\sqrt{N_{y}}}\sum_{n}e^{ik_{y}{n}a}|n\rangle
,\ \ \ |n\rangle\equiv\frac{1}{\sqrt{N_{y}}}\sum_{k_{y}}e^{-ik_{y}{n}a}%
|k_{y}\rangle, \label{FT}%$$ where $N_{y}$ is the number of unit cells along the $y$ axis. In the new basis, the Hamiltonian $\hat{H}=\sum_{k_{y}}\hat{H}_{\mathrm{1D}}(k_{y}%
)|k_{y}\rangle\langle k_{y}|$ is diagonal with respect to $k_{y}$, where $\hat{H}_{\mathrm{1D}}(k_{y})$ describes a 1D lattice with the unit cells labeled by $m$ and each unit cell containing $2$ basis states $|A\rangle
,|B\rangle$. Let us use $\mathbf{R}_{mn}\equiv{m}\mathbf{a}_{1}+n\mathbf{a}%
_{2}$ to denote the Bravais vector of the unit cell $(m,n)$ and use $\mathbf{G}(\mathbf{R}_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}},E)$ to denote the retarded GF from the unit cell $(m_{1},n_{1})$ to the unit cell $(m_{2}%
,n_{2})$ of the original 2D system, which is connected to the retarded GF $\mathbf{G}_{m_{2},m_{1}}(E,k_{y})$ of the 1D lattice from the unit cell $m_{1}$ to the unit cell $m_{2}$ via a Fourier transform $$\mathbf{G}(\mathbf{R}_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}},E)=\frac{1}{N_{y}%
}\sum_{k_{y}}{e^{ik_{y}(n_{2}-n_{1})a}\mathbf{G}_{m_{2},m_{1}}(E,}k_{y}{)},
\label{G2D_FT}%$$ where $\mathbf{G}(\mathbf{R}_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}},E)$ and $\mathbf{G}_{m_{2},m_{1}}(E,k_{y})$ are both $2\times2$ matrices. Below we consider fixed $E$ and $k_{y}$ and apply our general results to calculate the GF ${\mathbf{G}_{m_{2},m_{1}}}$ of the 1D lattice, with $E$ and $k_{y}$ omitted for brevity.
### Green’s function of 1D lattice
In the 1D lattice, the hopping is uniform: $$\mathbf{H}_{m,m+1}^{(\mathrm{1D})}=(\mathbf{H}_{m+1,m}^{(\mathrm{1D}%
)})^{\dagger}=\mathbf{t}=%
\begin{bmatrix}
0 & 0\\
t & 0
\end{bmatrix}
. \label{T}%$$ The unit cell Hamiltonian $\mathbf{H}_{m,m}^{(\mathrm{1D})}=V_{m}%
+\mathbf{h}_{0}$ is the sum of the unit cell Hamiltonian of pristine graphene,$$\mathbf{h}_{0}=%
\begin{bmatrix}
0 & t(1+e^{ik_{y}a})\\
t(1+e^{-ik_{y}a}) & 0
\end{bmatrix}
. \label{H0}%$$ and the *p-n* junction potential $V_{m}$. The entire infinite system consists of a single scatterer ($m_{L}\leq m\leq m_{R}$) connected to two semi-infinite leads $L$ and $R$ \[cf. Fig. \[G\_PNJ\](a)\], whose GFs can be constructed from the conversion and propagation matrices of the leads and the conversion matrix $\mathbb{G}$ of the *p-n* interface (see Sec. III and Sec. IV).
![Real energy band (black lines) and complex energy bands (orange lines) of pristine graphene at a fixed $k_{y}\approx0.14\times2\pi/a$.[]{data-label="G_SPECTRA"}](fig8Spectra.eps){width="\columnwidth"}
The remaining issue is to calculate the eigenmodes of each lead, as characterized by the hopping $\mathbf{t}$ and the unit cell Hamiltonian $\mathbf{h}=V+\mathbf{h}_{0}$, where $V=V_{L}$ (left lead) or $V_{R}$ (right lead). Given a complex wave vector $k$, we can solve the eigenvalue problem Eq. (\[MEQ1\]) and obtain the energy bands of the lead as $V\pm E_{0}(k)$, where $E_{0}(k)\equiv t\sqrt{f(k,k_{y})f(-k,-k_{y})}$ and $f(k,k_{y}%
)\equiv1+e^{ik_{y}a}+e^{-ika}$. Here $E_{0}(k)$ is real in two cases: (1) $k+k_{y}/2=\kappa$; (2) $k+k_{y}/2=i\kappa$ or $\pi/a+i\kappa$, where $\kappa\in\mathbb{R}$. The former gives the real energy bands, while the latter gives the complex energy bands, as shown in Fig. \[G\_SPECTRA\]. Conversely, given the energy $E$, we can solve Eq. (\[MEQ1\]) and obtain a right-going eigenmode $k_{+},|\Phi_{+}\rangle$ and a left-going eigenmode $k_{-},|\Phi_{-}\rangle$, where $k_{\pm}$ are the two solutions to $|E-V|=E_{0}(k)$ or equivalently the two intersection points of $E-V$ with the real and complex energy bands of pristine graphene. When $E-V$ lies in the range of the real energy bands (dashed black line in Fig. \[G\_SPECTRA\]), the wave vectors $k_{\pm}$ are both real and both eigenmodes are traveling modes. When $E-V$ lies outside the range of the real energy bands (dashed gray line in Fig. \[G\_SPECTRA\]), the wave vectors $k_{+}=k_{-}^{\ast}$ are complex and both eigenmodes are evanescent. A more convenient method to obtain the eigenmodes is to define $\lambda\equiv e^{ika}$ and rewrite Eq. (\[MEQ1\]) as $$\lambda^{2}bt+\lambda\lbrack|b|^{2}+t^{2}-(E-V)^{2}]+tb^{\ast}=0$$ with $b\equiv t(1+e^{ik_{y}a})$, so that $\lambda_{\pm}=e^{ik_{\pm}a}$ are obtained as the two solutions to this quadratic equation for $\lambda$. In addition to the two *normal* eigenmodes, there are also a pair of *ideally evanescent* eigenmodes, including a right-going one $\lambda_{+,0}=e^{ik_{+,0}a}=0$, $|\Phi_{+,0}\rangle=[1,0]^{T}$ and a left-going one $\lambda_{-,0}=e^{ik_{-,0}a}=\infty$, $|\Phi_{-,0}%
\rangle=[0,1]^{T}$ (see Appendix \[APPEND\_BULKMODE\]). The ideally evanescent eigenmodes do not propagate, so they do not directly contribute to the GF, but their existence does influence the generalized transmission and reflection amplitudes of the normal eigenmodes.
Using the eigenmodes, we can calculate the conversion matrix $\mathbb{G}$ of the *p-n* interface using Eq. (\[GG\]) and then obtain the generalized transmission and reflection amplitudes $\mathcal{S}^{(LL)}$, $\mathcal{S}%
^{(RL)}$, $\mathcal{S}^{(LR)}$, and $\mathcal{S}^{(RR)}$ of the normal eigenmode from Eqs. (\[SRL\])-(\[SRR\]). For $m\neq m_{0}$, the free GF of the left lead is essentially the sum of the left-going eigenmode $|\Phi
_{-}^{(L)}(m)\rangle$ and the right-going eigenmode $|\Phi_{+}^{(L)}%
(m)\rangle$ \[cf. Eq. (\[BULK\_MODE\_DEF\]) for their definitions\]:$$\mathbf{g}_{m,m_{0}}^{(L)}=\frac{a}{iv_{+}^{(L)}}|\Phi_{+}^{(L)}%
(m)\rangle\langle\tilde{\Phi}_{-}^{(L)}|+\frac{a}{-iv_{-}^{(L)}}|\Phi
_{-}^{(L)}(m)\rangle\langle\tilde{\Phi}_{+}^{(L)}|.$$ Due to the *p-n* interface, the eigenmode $|\Phi_{+}^{(L)}(m)\rangle$ produces a reflection wave in the left lead and a transmission wave in the right lead:$$|\Psi_{+}^{(L,\mathrm{out})}(m)\rangle=\left\{
\begin{array}
[c]{ll}%
e^{ik_{-}^{(L)}(m-m_{L})a}|\Phi_{-}^{(L)}\rangle\mathcal{S}^{(LL)}%
e^{ik_{+}^{(L)}(m_{L}-m_{0})a} & (m\in L),\\
e^{ik_{+}^{(R)}(m-m_{R})a}|\Phi_{+}^{(R)}\rangle\mathcal{S}^{(RL)}%
e^{ik_{+}^{(L)}(m_{L}-m_{0})a} & (m\in R),
\end{array}
\right.$$ so the GF from the left lead to the right lead is essentially the transmission wave:$$\mathbf{G}_{m\in R,m_{0}\in L}=\frac{a}{iv_{+}^{(L)}}|\Psi_{+}%
^{(L,\mathrm{out})}(m)\rangle\langle\tilde{\Phi}_{-}^{(L)}|,$$ while the GF inside the left lead is essentially the sum of the incident eigenmode $|\Phi_{\pm}^{(L)}(m)\rangle$ and the reflection wave:$$\mathbf{G}_{m\in L,m_{0}\in L}=\mathbf{g}_{m,m_{0}}^{(L)}+\frac{a}%
{iv_{+}^{(L)}}|\Psi_{+}^{(L,\mathrm{out})}(m)\rangle\langle\tilde{\Phi}%
_{-}^{(L)}|.$$
### Green’s function of 2D graphene *p-n* junction: anomalous focusing
Compared with the standard RGF method, the advantages of our GF method lie in its physical transparency and numerical efficiency. Here we demonstrate the first point by using our method to provide a clear physical picture of the anomalous focusing effect [@CheianovScience2007; @ParkNanoLett2008; @MoghaddamPRL2010; @LeeNatPhys2015; @ChenScience2016] across the graphene *p-n* junction described by the tight-binding model. For this purpose, we first obtain the GF of the 2D graphene *p-n* junction from the 1D GFs by a Fourier transform \[Eq. (\[G2D\_FT\])\]. In particular, the GF from the unit cell $(m_{1},n_{1})$ in the *n* region to the unit cell $(m_{2},n_{2})$ in the *p* region, $$\mathbf{G}(\mathbf{R}_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}},E)=\int\frac
{dk_{y}}{2\pi}|\Psi_{+}^{(L,\mathrm{tran})}(\mathbf{R}_{m_{2}n_{2}}%
)\rangle\frac{a}{iv_{+}^{(L)}}\langle\tilde{\Phi}_{-}^{(L)}|, \label{G2D_RL}%$$ is essentially the sum of all transmission wave functions$$|\Psi_{+}^{(L,\mathrm{tran})}(\mathbf{R}_{m_{2}n_{2}})\rangle\equiv
e^{i\mathbf{k}_{+}^{(R)}\cdot(\mathbf{R}_{m_{2}n_{2}}-\mathbf{R}_{m_{R},0}%
)}|\Phi_{+}^{(R)}\rangle\mathcal{S}^{(RL)}e^{i\mathbf{k}_{+}^{(L)}%
\cdot(\mathbf{R}_{m_{L},0}-\mathbf{R}_{m_{1}n_{1}})}, \label{PHI_OUT}%$$ which emanates from the incident eigenmode $|\Phi_{+}^{(L)}(\mathbf{R}%
_{m,n})\rangle=e^{i\mathbf{k}_{+}^{(L)}\cdot(\mathbf{R}_{m,n}-\mathbf{R}%
_{m_{1}n_{1}})}|\Phi_{+}^{(L)}\rangle$ through three steps: propagation to the left interface $\mathbf{R}_{m_{L},0}$ of the junction with wave vector $\mathbf{k}_{+}^{(L)}$, transmission across the interface, and propagation from $\mathbf{R}_{m_{R},0}$ to $\mathbf{R}_{m_{2}n_{2}}$ with wave vector $\mathbf{k}_{+}^{(R)}$. Here $\mathbf{k}_{\pm}^{(L)}$ ($\mathbf{k}_{\pm}%
^{(R)}$) are the wave vectors of the *normal* eigenmodes in the left (right) region, i.e., in the tilted coordinate \[Fig. \[G\_PNJ\](b)\]:$\ \mathbf{k}_{\pm}^{(p)}\cdot\mathbf{e}_{x}=k_{\pm}^{(p)}$ and $\mathbf{k}_{\pm}^{(p)}\cdot\mathbf{e}_{y}\equiv k_{y}$ ($p=L,R$). The GF from $\mathbf{R}_{m_{1}n_{1}}$ to $\mathbf{R}_{m_{2}n_{2}}$ can be measured as the conductance between one STM probe coupled to $\mathbf{R}_{m_{1}n_{1}}$ and another STM probe coupled to $\mathbf{R}_{m_{2}n_{2}}$ through the Landauer-Büttiker formula [@DattaBook1995] ($2e^{2}/h)T(E_{F})$, where the transmission probability $T(E_{F})\propto|\mathbf{G}(\mathbf{R}%
_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}},E_{F})|^{2}$. Therefore, Eq. (\[G2D\_RL\]) directly connects the transmission wave function to the experimentally measurable conductance and hence provides a clear physical picture for observing the anomalous focusing in dual-probe STM measurements, and further reveals some interesting effects.
Let us consider a sharp, symmetric interface at $m_{L}=m_{R}=0$ and $V_{R}=-V_{L}=V_{0}>0$. In this case, the transmission wave simplifies to$$|\Psi_{+}^{(L,\mathrm{tran})}(\mathbf{R}_{m_{2}n_{2}})\rangle\equiv
e^{i(\mathbf{k}_{+}^{(R)}\cdot\mathbf{R}_{m_{2}n_{2}}-\mathbf{k}_{+}%
^{(L)}\cdot\mathbf{R}_{m_{1}n_{1}})}|\Phi_{+}^{(R)}\rangle\mathcal{S}^{(RL)}.
\label{PHI_OUT_2}%$$ Here $\mathbf{k}_{+}^{(R)}$, $\mathbf{k}_{+}^{(L)}$, $|\Phi_{+}^{(R)}\rangle$, and $\mathcal{S}^{(RL)}$ all depend on $k_{y}$ weakly. The strongest dependence on $k_{y}$ comes from the phase factor $e^{i(\mathbf{k}_{+}%
^{(R)}\cdot\mathbf{R}_{m_{2}n_{2}}-\mathbf{k}_{+}^{(L)}\cdot\mathbf{R}%
_{m_{1}n_{1}})}$, which usually oscillates rapidly as a function of $k_{y}$ when $\mathbf{R}_{m_{2}n_{2}}$ and $\mathbf{R}_{m_{1}n_{1}}$ are far away. However, when the energy of the incident electron lies midway in between the Dirac point of the *n* region and the Dirac point of the *p* region (i.e., $E=E_{F}=0$), in the *Cartesian* coordinate system spanned by the orthogonal unit vectors $\mathbf{e}_{X}$ and $\mathbf{e}_{Y}$ \[see Fig. \[G\_PNJ\](b)\], the electron-hole symmetry of graphene dictates that the Fermi wave vector $\mathbf{k}_{+}^{(L)}$ of the right-going eigenmode in the *n* region and the Fermi wave vector $\mathbf{k}_{+}^{(R)}$ of the right-going eigenmode in the *p* region to have the same component along the *p-n* interface (i.e., $\mathbf{k}_{+}^{(L)}\cdot\mathbf{e}_{Y}=\mathbf{k}_{+}^{(R)}\cdot
\mathbf{e}_{Y}$), but opposite components perpendicular to the *p-n* interface (i.e., $\mathbf{k}_{+}^{(L)}\cdot\mathbf{e}_{X}=-\mathbf{k}_{+}^{(R)}%
\cdot\mathbf{e}_{X}$). Therefore, when $\mathbf{R}_{m_{1}n_{1}}$ and $\mathbf{R}_{m_{2}n_{2}}$ are mirror symmetric about the *p-n* interface, the rapidly oscillating phase factor equals unity for all $k_{y}$. In this case, all the transmitted waves have nearly the same phase factor for all $k_{y}$, so they contribute constructively to the GF. This corresponds to electrons flowing out of an electron source at $\mathbf{R}_{m_{1}n_{1}}$ being focused to $\mathbf{R}_{m_{2}n_{2}}$, i.e., the anomalous focusing [@CheianovScience2007]. According to the Landauer-Büttiker formula, the constructive enhancement of the GF could be detected as an enhanced conductance in dual-probe STM measurements.
In addition to locally enhancing the GF, the constructive interference of all the transmission waves also gives rise to two interesting behaviors. First, the phase factor $e^{i(\mathbf{k}_{+}^{(R)}\cdot\mathbf{R}_{m_{2}n_{2}%
}-\mathbf{k}_{+}^{(L)}\cdot\mathbf{R}_{m_{1}n_{1}})}$ and hence the transmission wave and the GF remain invariant when $\mathbf{R}_{m_{2}n_{2}}$ and $\mathbf{R}_{m_{1}n_{1}}$ are moved equally but in opposite directions perpendicular to the *p-n* interface. This would give rise to distance-independent conductance. Second, since each transmission wave contributes constructively to the GF, we have $\mathbf{G}(\mathbf{R}%
_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}},E_{F})\propto$ density of states on the Fermi surface $\propto V_{0}\propto$ carrier concentration. Thus the locally enhanced conductance should increase quadratically with increasing doping level $V_{0}$ (or equivalently the carrier concentration), in contrast to the linear dependence in uniform graphene [@SettnesPRL2014]. These points will be verified in our subsequent numerical simulations of the dual-probe STM measurements, which provide a useful tool, with high spatial resolution, to measure such local transport properties and detect possible zero-energy bound states of the Dirac fermions caused by suitable 2D potential well [@PhysRevLett.102.226803; @PhysRevB.92.165401].
### Comparison of the standard RGF and our GF approach
Compared with the standard RGF that treats the *entire* central region numerically \[see Fig. \[G\_SETUP\](a) for an example\], an important advantage of our approach is that it fully utilizes the translational invariance of all the periodic subregions \[even if they lie inside the central region, such as the middle lead in Fig. \[G\_SETUP\](a)\] to treat these subregions semianalytically, so that only the *truly disordered* subregions need numerical treatment. Therefore, our GF approach is more efficient if the central region contains periodic subregions; otherwise the two approaches are equally efficient. Here we calculate $G(\mathbf{R}_{2},\mathbf{R}_{1},E)$ across a sharp graphene *p-n* junction using these two methods to demonstrate their equivalence and highlight the numerical efficiency of our approach. In the calculation, ${\bf R}_1$ is fixed at a randomly chosen A-sublattice site in the *n* region; ${\bf R}_2$ is swept over all the B-sublattice sites along the $x$ axis from the *n* region to the *p* region. The range of the sweep is from $0.25 \mu$m on the left of ${\bf R}_1$ to $0.75 \mu$m on the right of ${\bf R}_1$. For the RGF method, the central region (infinite along the *p-n* interface) is the smallest region that encloses ${\bf R}_1$, ${\bf R}_2$, and the *p-n* interface. For our GF method, the scatterer region (infinite along the *p-n* interface) consists of one slice at the *p-n* interface. The numerical results from the two approaches always agree with each other up to the machine accuracy. The time cost, however, differs by two orders of magnitude: with 90 Intel cores, the time cost of the standard RGF approach varies from 260 s to 540 s, depending on the position of ${\bf R}_1$ relative to the *p-n* interface, while the time cost of our approach is always less than 3 s. Similar speedup is expected in long multilayer structures with sharp interfaces, such as quantum wells, superlattices, or sharp *p-n-p* junctions.
### Numerical examples
![Transmission and reflection of a right-going eigenmode of energy $E=0$ incident from the *n* region of a sharp (solid lines and dotted lines) or 5nm wide smooth (dashed lines), symmetric (i.e., $-V_{L}=V_{R}=V_{0}$) graphene *p-n* junction. For $V_{0}=0.2$, the range of $k_{y}$ in which the eigenmode is traveling is marked by the double arrow. For $V_{0}=0.02$, the range of $k_{y}$ in which the eigenmode is traveling is much narrower.[]{data-label="G_TRANS"}](fig9Trans.eps){width="\columnwidth"}
In the following (main text and all the figures), we always use the C-C bond length of graphene $a_{\mathrm{C-C}}=0.142$ nm as the unit of length and the nearest-neighbor hopping amplitude $t=3$ eV as the unit of energy. For specificity, we focus on symmetric graphene *p-n* junctions with $V_{R}=-V_{L}=V_{0}>0$. Unless explicitly specified, we always take a typical doping level $V_{0}=0.2$ and set the energy $E=E_{F}=0$, so we denote $G(\mathbf{R}_{2},\mathbf{R}_{1},E)$ by $G(\mathbf{R}_{2},\mathbf{R}_{1})$ for brevity.
As shown in Fig. \[G\_TRANS\], at $V_{0}=0.2$, the tunneling of a right-going traveling eigenmode reproduces the well-known results from the continuum model [@AllainEPJB2011], such as the perfect transmission at normal incidence. For the evanescent eigenmode in a sharp *p-n* junction, however, $|\mathcal{S}%
^{(RL)}|^{2}$ shows a peak, indicating enhanced tunneling of certain evanescent states. When the Fermi level is tuned closer to the Dirac point, i.e., for $V_{0}=0.02$, this enhanced tunneling becomes more pronounced and may be observed by dual-probe STM measurements.
Next we visualize the contributions from different scattering channels to the GF and their interference. For a smooth junction, the spatial map of the GF \[Fig. \[GFtu12\](a)\] shows imperfect focusing [@CheianovScience2007] due to negative refraction across a finite-width *p-n* junction. Let us consider $\mathbf{R}_{m_{2}n_{2}}$ and $\mathbf{R}_{m_{1}n_{1}}$ both in the *n* region and $\mathbf{R}_{m_{2}n_{2}}$ on the right of $\mathbf{R}_{m_{1}n_{1}}$ (i.e., $m_{2}>m_{1}$); the GF $$\begin{aligned}
\mathbf{G}(\mathbf{R}_{m_{2}n_{2}},\mathbf{R}_{m_{1}n_{1}}) & =\int
\frac{dk_{y}}{2\pi}\frac{a}{iv_{+}^{(L)}}\nonumber\\
& \times\lbrack|\Phi_{+}^{(L)}(\mathbf{R}_{m_{2}n_{2}})\rangle+|\Psi
_{+}^{(L,\mathrm{refl})}(\mathbf{R}_{m_{2}n_{2}})\rangle]\langle\tilde{\Phi}_{-}%
^{(L)}|\end{aligned}$$ is essential the sum of all right-going eigenmodes $|\Phi_{+}^{(L)}%
(\mathbf{R}_{m_{2}n_{2}})\rangle$ and all reflection waves $$|\Psi_{+}^{(L,\mathrm{refl})}(\mathbf{R}_{m_{2}n_{2}})\rangle\equiv
e^{i\mathbf{k}_{-}^{(L)}\cdot(\mathbf{R}_{m_{2}n_{2}}-\mathbf{R}_{m_{L},0}%
)}|\Phi_{-}^{(L)}\rangle\mathcal{S}^{(LL)}e^{i\mathbf{k}_{+}^{(L)}%
\cdot(\mathbf{R}_{m_{L,0}}-\mathbf{R}_{m_{1}n_{1}})}.$$ The incident wave contribution coincides with that of pristine graphene \[Fig. \[GFtu12\](b)\]. The reflection wave contribution \[Fig. \[GFtu12\](c)\] tends to vanish perpendicularly to the junction interface, indicative of the chiral tunneling. The interference between the incident and reflection waves \[Fig. \[GFtu12\](d)\] is responsible for the interference pattern in the total GF \[Fig. \[GFtu12\](a)\], which would be directly manifested in dual-probe STM measurements.
![Spatial map of the scaled conductance (i.e., the transmission coefficient $T_{12}$ times $|\mathbf{R}_{2}-\mathbf{R}_{1}|$) in a sharp graphene *p-n* junction as a function of $\mathbf{R}_{2}-\mathbf{R}_{1}$. Here $\mathbf{R}_{1}\equiv(X_{1},Y_{1})$ is fixed at the $A$ sublattice of the unit cell $(0,0)$ and $\mathbf{R}_{2}\equiv(X_{2},Y_{2})$ is swept over all the lattice sites (including both $A$ and $B$ sublattices) of the entire structure.[]{data-label="condmap"}](fig11ConductanceMap.eps){width="\columnwidth"}
Finally we simulate dual-probe STM measurements [@SettnesPRL2014; @SettnesPRB2014] over the graphene *p-n* junction at zero temperature. Following Refs. , we assume that each probe couples to a single carbon site for simplicity, so the Landauer-Büttiker formula [@DattaBook1995] gives the interprobe conductance as $\sigma(\mathbf{R}_{2},\mathbf{R}_{1})=(2e^{2}/h)T_{12}%
(E_{F})=\Gamma_{1}\Gamma_{2}|\mathbf{\bar{G}(R}_{2},\mathbf{R}_{1},E_{F})|^{2}$, where $\mathbf{\bar{G}(R}_{2},\mathbf{R}_{1},E_{F})$ is the GF incorporating the self-energy corrections from the STM probes and $\Gamma_{1,2}$ are coupling constants between the STM probes and the graphene. Usually, $\Gamma_{1,2}$ have a sensitive exponential dependence on the distance between the STM probe and the graphene sample, but their specific values do not affect the shape of the signal. Therefore, following Ref. , we always rescale the maximum of $T_{12}$ to unity. In Fig. \[condmap\], the real-space conductance map shows a pronounced focusing due to negative refraction [@CheianovScience2007]. Other observable electron optics features include the high transparency of the junction near normal incidence, i.e., chiral tunneling [@KatsnelsonNatPhys2006], and the interference pattern between the incident and reflection waves. Recently, negative refraction in graphene *p-n* junctions was observed [@LeeNatPhys2015], but the measurement via macroscopic contacts only gives a spatially averaged result. Here our simulation shows that dual-probe STM measurements can further provide spatially resolved interference pattern; i.e., dual-probe STM could be an ideal experimental technique for studying local transport and quantum interference phenomenon.
![Transmission coefficient $T_{12}$ between two STM probes at mirror symmetric locations about the junction interface. (a) $T_{12}$ vs. interprobe distance. (b) $T_{12}$ vs. $V_{0}$. Here, the width of the smooth junction is 5 nm for (a) and inset of (b), and the maximum of $T_{12}$ is always rescaled to 1 in each panel.[]{data-label="depend"}](fig12Transmission.eps){width="\columnwidth"}
Now we demonstrate numerically some interesting features of the dual-probe STM measurements in graphene *p-n* junctions \[see the physical analysis following Eq. (\[PHI\_OUT\_2\])\]. First, when the two probes are mirror symmetric about the junction interface, $T_{12}$ is nearly independent of the interprobe distance \[Fig. \[depend\](a)\], in contrast to the $1/R$ decay in uniform graphene [@SettnesPRL2014]. For a sharp junction, this behavior can be attributed to the anomalous focusing [@CheianovScience2007]. However, the existence of the same behavior for a smooth junction interface indicates that it has a different physical origin, i.e., the cancellation of the propagation phases due to the matching of the electron Fermi surfaces in the *n* region and the hole Fermi surface in the *p* region [@ZhangPRB2016]. The distance-independent response could change qualitatively the spatial scaling of many physical quantities, such as the Friedel oscillation induced by an impurity and the carrier-mediated RKKY interaction between two localized magnetic moments. Second, the conductance across a sharp junction scales quadratically with the junction potential:$\ T_{12}\propto V_{0}^{2}$ \[red line in Fig. \[depend\](b)\], which differs qualitatively from the linear scaling $T_{12}\propto V_{0}$ of uniform graphene [@SettnesPRL2014]. When the junction becomes smooth \[inset of Fig. \[depend\](b)\], the quadratic dependence still persists for small $V_{0}$ (electron’s Fermi wavelength $\gg$ junction width), but recovers the linear scaling behavior in uniform graphene for large $V_{0}$. This can be attributed to the gradual destruction of the anomalous focusing when the junction width becomes larger than the Fermi wavelength [@CheianovPRB2006].
Conclusions
===========
We have presented a numerically efficient and physically transparent formalism to calculate and understand the Green’s function (GF) of a general layered structure. In contrast to the commonly used recursive GF method that directly calculates the GF through the Dyson equations, our approach converts the calculation of the GF to the generation and subsequent propagation of a scattering wave function emanating from a local excitation. This viewpoint provides analytical expressions of the GF in terms of a generalized scattering matrix. This identifies the contributions of individual scattering channels to the GF and hence allows this information to be extracted quantitatively from dual-probe STM experiments. We further derive an analytical rule to construct the GF of a general layered system, which could significantly reduce the computational time cost and enable quantum transport calculations for large samples. Application of this formalism to simulate the two-dimensional conductance map of a realistic graphene *p-n* junction demonstrates the possibility of observing the spatial interference caused by negative refraction and further reveals a few interesting features, such as the distance-independent conductance and its quadratic dependence on the carrier concentration, as opposed to the linear dependence in uniform graphene. In addition to conventional mesoscopic quantum transport, it would be interesting to apply our GF approach to the investigation other electron interference phenomena, such as the carrier-mediated RKKY interaction between local magnetic moments, the impurity-induced Friedel oscillation, and using dual-probe STM measurements to detect possible zero-energy bound states in graphene caused by suitable 2D potentials [@PhysRevLett.102.226803; @PhysRevB.92.165401].
ACKNOWLEDGMENTS
===============
This work was supported by the MOST of China (Grants No. 2015CB921503 and No. 2014CB848700), the NSFC (Grants No. 11274036, No. 11322542, No. 11434010, and No.11504018), and the NSFC program for “Scientific Research Center” (Grant No. U1530401). We acknowledge the computational support from the Beijing Computational Science Research Center (CSRC).
Calculation of eigenmodes {#APPEND_BULKMODE}
=========================
When the determinant of the $M\times M$ hopping matrix $\mathbf{t}^{\dagger
}$ or $\mathbf{t}$ is nonzero, the $2M$ eigenmodes can be obtained by letting $\lambda\equiv e^{ika}$ and rewriting Eq. (\[MEQ1\]) or equivalently $[-\mathbf{t}^{\dagger}+\lambda(z-\mathbf{h})-\lambda^{2}\mathbf{t}%
]|\Phi\rangle\equiv0\mathbf{\ }$as a generalized $2M\times2M$ eigenvalue problem$$%
\begin{bmatrix}
0 & 1\\
-\mathbf{t}^{\dagger} & z-\mathbf{h}%
\end{bmatrix}%
\begin{bmatrix}
|\Phi\rangle\\
\lambda|\Phi\rangle
\end{bmatrix}
=\lambda%
\begin{bmatrix}
1 & 0\\
0 & \mathbf{t}%
\end{bmatrix}%
\begin{bmatrix}
|\Phi\rangle\\
\lambda|\Phi\rangle
\end{bmatrix}
, \label{GEIG}%$$ where $z=E+i0^{+}$. In the numerical calculation, we remove the infinitesimal imaginary part of $z$, i.e., $z=E$. Then Eq. (\[GEIG\]) is solved for $2M$ solutions $\{\lambda\equiv e^{ika},|\Phi\rangle\}$, where $k$ could be either real (traveling modes) or complex (evanescent modes). Next the $2M$ eigenmodes are classified into $M$ right-going ones and $M$ left-going ones: the former consists of traveling modes (i.e., $\operatorname{Im}k=0$) with a positive group velocity and evanescent modes decaying exponentially to the right (i.e., $\operatorname{Im}k>0$), while the latter consists of traveling modes (i.e., $\operatorname{Im}k=0$) with a negative group velocity and evanescent modes decaying exponentially to the left (i.e., $\operatorname{Im}%
k<0$). There is an alternative but less accurate method to calculate and classify the eigenmodes. In this approach, the infinitesimal imaginary part of $z$ is replaced by a finite but small positive number $\eta$, i.e., $z=E+i\eta$. Then Eq. (\[GEIG\]) is solved for the $2M$ solutions $\{\lambda\equiv e^{ika},|\Phi\rangle\}$. Next, according to the sign of $\operatorname{Im}k$, the $2M$ eigenmodes are classified into $M$ right-going ones with $\operatorname{Im}k>0$ and $M$ left-going ones with $\operatorname{Im}k<0$. In the limit $\eta\rightarrow0^{+}$, the imaginary part $\operatorname{Im}k$ may either vanish (i.e., traveling modes) or remain finite (i.e., evanescent modes). Obviously, the second approach is accurate only in the limit of $\eta\rightarrow0^{+}$, so we always use the first approach in the main text.
When the determinant of the hopping matrix $\mathbf{t}$ vanishes, solving Eq. (\[GEIG\]) would give some trivial evanescent eigenmodes. Suppose that $M_{0}$ out of the $M$ eigenvalues of the hopping matrix $\mathbf{t}^{\dagger
}$ or $\mathbf{t}$ are zero, and the corresponding eigenvectors are $\{|\Phi_{+,\alpha}\rangle\}$ (for $\mathbf{t}^{\dagger}$) and $\{|\Phi
_{-,\alpha}\rangle\}$ (for $\mathbf{t}$), where $\alpha=1,2,\cdots,M_{0}$. Then there would be $2M_{0}$ trivial evanescent eigenmodes, including $M$ right-going ones $\lambda_{+,\alpha}=0,|\Phi_{+,\alpha}\rangle$ and $M_{0}$ left-going ones $\lambda_{-,\alpha}=\infty$, $|\Phi_{-,\alpha}\rangle$. The former correspond to $|\Phi(m-1)\rangle=|\Phi_{+,\alpha}\rangle,$ $|\Phi(m)\rangle=|\Phi(m+1)\rangle=0$ in Eq. (\[SE\]), while the latter correspond to $|\Phi(m+1)\rangle=|\Phi_{-,\alpha}\rangle,$ $|\Phi
(m)\rangle=|\Phi(m-1)\rangle=0$ in Eq. (\[SE\]). As a result, propagation matrices $\mathbf{P}_{-}$ and $\mathbf{P}_{+}^{-1}$ both diverge. However, this does not affect our formulas, which only contain $\mathbf{P}_{+}$ and $\mathbf{P}_{-}^{-1}$ due to causality. The only problem is that for a trivial evanescent eigenmode $(s,\alpha)$, the generalized group velocity $v_{s,\alpha}$ \[Eq. (\[VSA\])\] is not well defined. For example, when $\lambda_{+,\alpha}=0$ and hence $\lambda_{-,\alpha}=\infty$, the generalized velocity $v_{+,\alpha}=v_{-,\alpha}^{\ast}=-ia\lambda_{-,\alpha}^{\ast}%
\langle\Phi_{-,\alpha}|\mathbf{t}^{\dagger}|\Phi_{+,\alpha}\rangle$ involves the product of $\lambda_{-,\alpha}^{\ast}=\infty$ and $\mathbf{t}^{\dagger
}|\Phi_{+,\alpha}\rangle=0$. This singular problem can be avoided by adding sufficiently small numbers $\{\epsilon\}$ to $\mathbf{t}$ and $\mathbf{t}%
^{\dagger}$ to make their determinant nonzero and take the limit $\{\epsilon\rightarrow0\}$ at the end of the calculation.
Taking the graphene junction as an example, the 1D left or right lead is characterized by the hopping matrix $\mathbf{t}$ and unit cell Hamiltonian $V+\mathbf{h}_{0}$, where $V=V_{L}$ (left lead) or $V_{R}$ (right lead) and $\mathbf{h}_{0}$, $\mathbf{t}$ are given by Eqs. (\[T\]) and (\[H0\]). Here the hopping matrix $\mathbf{t}$ has one zero eigenvalue with eigenvector $|\Phi_{-,0}\rangle=[0,1]^{T}$, while $\mathbf{t}^{\dagger}$ has one zero eigenvalue with eigenvector $|\Phi_{+,0}\rangle=[1,0]^{T}$. This gives rise to two trivial evanescent eigenmodes: the right-going one $\lambda_{+,0}%
=0,|\Phi_{+,0}\rangle$ and the left-going one $\lambda_{-,0}=\infty$, $|\Phi_{-,0}\rangle$, for which the generalized group velocities $v_{+,0}=v_{-,0}^{\ast}$ are not well defined. To cure this problem, we add a small number $\epsilon$ to the off-diagonal of the hopping matrix, so that $\mathbf{t}$ and $\mathbf{t}^{\dagger}$ become $$\mathbf{t}(\epsilon)=%
\begin{bmatrix}
0 & \epsilon\\
t & 0
\end{bmatrix}
,\ \mathbf{t}^{\dagger}(\epsilon)=%
\begin{bmatrix}
0 & t\\
\epsilon & 0
\end{bmatrix}
.$$ Then using the first-order perturbation theory, we obtain $$\begin{aligned}
\lambda_{+,0}(\epsilon) & \approx\frac{-\epsilon}{t\left( e^{-iak_{y}%
}+1\right) },\ |\Phi_{+,0}(\epsilon)\rangle\approx%
\begin{bmatrix}
1\\
\frac{-\epsilon E}{t^{2}(e^{-iak_{y}}+1)}%
\end{bmatrix}
,\\
\lambda_{-,0}(\epsilon) & \approx\frac{t\left( e^{iak_{y}}+1\right)
}{-\epsilon},\ \ |\Phi_{-,0}(\epsilon)\rangle\approx%
\begin{bmatrix}
\frac{-\epsilon E}{t^{2}(e^{iak_{y}}+1)}\\
1
\end{bmatrix}
.\end{aligned}$$ Substituting into Eq. (\[VSA\]) and taking the limit $\epsilon\rightarrow0$ gives $$v_{+,0}=v_{-,0}^{\ast}=iat(e^{-iak_{y}}+1).$$
Scattering of a partial wave {#APPEND_EQUIVALENCE}
============================
Let us consider a scatterer connected to a semi-infinite left lead (whose conversion matrix is $\mathbf{g}$) and prove that a right-going incident partial wave $|\Phi_{\mathrm{in}}(m)\rangle$ is equivalent to a local excitation $|\Phi_{\mathrm{loc}}\rangle_{m_{0}}\equiv\mathbf{g}^{-1}%
|\Phi_{\mathrm{in}}(m_{0})\rangle$ at an arbitrary site $m_{0}\leq m_{L}$ ($m_{L}$ is the left surface of the scatterer), in the sense that they produce the same scattering state at $m\geq m_{0}$. For this purpose, we use $|\Psi(m)\rangle$ for the conventional scattering state emanating from the incident wave $|\Phi_{\mathrm{in}}(m)\rangle$ and $|\Phi(m)\rangle$ for the scattering state emanating from the local excitation $|\Phi_{\mathrm{loc}%
}\rangle_{m_{0}}$ at $m_{0}$. We notice that $|\Psi(m)\rangle$ and $|\Phi(m)\rangle$ obey the same Schrödinger equation for $m\geq m_{0}+1$, and the same uniform Schrödinger equation $$-\mathbf{t}^{\dagger}|\Phi(m-1)\rangle+(z-\mathbf{h})|\Phi(m)\rangle
-\mathbf{t}|\Phi(m+1)\rangle=0$$ for $m\leq m_{0}-1$. The difference lies at $m=m_{0}$: $$\begin{aligned}
& -\mathbf{t}^{\dagger}|\Psi(m_{0}-1)\rangle+(z-\mathbf{H}_{m_{0},m_{0}%
})|\Psi(m_{0})\rangle-\mathbf{H}_{m_{0},m_{0}+1}|\Psi(m_{0}+1)\rangle
\nonumber\\
& =0,\\
& -\mathbf{t}^{\dagger}|\Phi(m_{0}-1)\rangle+(z-\mathbf{H}_{m_{0},m_{0}%
})|\Phi(m_{0})\rangle-\mathbf{H}_{m_{0},m_{0}+1}|\Phi(m_{0}+1)\rangle
\nonumber\\
& =|\Phi_{\mathrm{loc}}\rangle_{m_{0}},\end{aligned}$$ and the boundary conditions: $|\Phi(m)\rangle$ should be finite at $m\rightarrow-\infty$, while the right-going part of $|\Psi(m)\rangle$ should equal the incident wave on the left of the scatterer, i.e., $|\Psi
_{+}(m)\rangle=|\Phi_{\mathrm{in}}(m)\rangle$ for $m\leq m_{L}$. The former gives $|\Phi(m_{0}-1)\rangle=\mathbf{P}_{-}^{-1}|\Phi(m_{0})\rangle$, while the latter gives $$\begin{aligned}
|\Psi(m_{0}-1)\rangle & =\mathbf{P}_{+}^{-1}|\Psi_{+}(m_{0})\rangle
+\mathbf{P}_{-}^{-1}|\Psi_{-}(m_{0})\rangle\\
& =\mathbf{P}_{-}^{-1}|\Psi(m_{0})\rangle+(\mathbf{P}_{+}^{-1}-\mathbf{P}%
_{-}^{-1})|\Phi_{\mathrm{in}}(m_{0})\rangle.\end{aligned}$$ Using these relations to eliminate $|\Phi(m_{0}-1)\rangle$ and $|\Psi
(m_{0}-1)\rangle$ from the equations for $|\Phi(m)\rangle$ and $|\Psi
(m)\rangle$ at $m=m_{0}$, and using Eq. (\[EQUALITY\]), we see that they become identical and contains the same source $|\Phi_{\mathrm{loc}}%
\rangle_{m_{0}}$. Therefore, $|\Phi(m)\rangle|_{m\geq m_{0}}$ and $|\Psi(m)\rangle|_{m\geq m_{0}}$ obeys exactly the same set of closed equations and natural boundary conditions (i.e., they should be finite for $m\rightarrow+\infty$); thus they are identical. Applying this equivalence principle to a scatterer connected to a semi-infinite left lead $L$ and a semi-infinite right lead $R$ gives Eqs. (\[PHI\_S1\]) and (\[PHI\_S2\]) of the main text.
Green’s function of one scatterer: simple examples {#APPEND_EXAMPLE}
==================================================
Here we give a few simple examples for the GF of an infinite (or semi-infinite) system containing a single scatterer. As the first example, a semi-infinite lead (with unit cell Hamiltonian $\mathbf{h}$, hopping $\mathbf{t}$, and propagation matrices $\mathbf{P}_{\pm}$) consisting of the unit cells $m\leq0$ can be regarded as a single-unit-cell scatterer at $m=0$ connected to a semi-infinite left lead. Then the conversion matrix of this scatterer is $$\mathbb{G}^{(\mathrm{L})}=(z-\mathbf{h}-\mathbf{t}^{\dagger}\mathbf{P}%
_{-}^{-1})^{-1}=(\mathbf{tP}_{-})^{-1}, \label{GL}%$$ where we have used Eq. (\[EQUALITY\]) in the second step. The GF of the entire system is$$\mathbf{G}_{m,m_{0}}^{(\mathrm{L})}=\mathbf{g}_{m,m_{0}}+\mathbf{P}_{-}%
^{m}(\mathbb{G}^{(\mathrm{L})}\mathbf{g}^{-1}-\mathbf{I})\mathbf{g}_{0,m_{0}%
}\mathbf{.} \label{GL_MM0}%$$ Taking $m=m_{0}=0$ gives $\mathbf{G}_{0,0}^{(\mathrm{L})}=\mathbb{G}%
^{(\mathrm{L})}$. By using Eqs. (\[GL\]) and (\[G00\]), we have $\mathbb{G}^{(\mathrm{L})}\mathbf{g}^{-1}-\mathbf{I}=-\mathbf{P}_{-}%
^{-1}\mathbf{P}_{+}$ and hence recover the results by Sanvito *et al*. [@SanvitoPRB1999]: $\mathbf{G}_{m,m_{0}}^{(\mathrm{L})}=\mathbf{g}%
_{m,m_{0}}-\mathbf{P}_{-}^{m-1}\mathbf{P}_{+}^{1-m_{0}}\mathbf{g}$.
As the second example, a semi-infinite lead consisting of the unit cells $m\geq0$ can be regarded as a single-unit-cell scatterer at $m=0$ connected to a semi-infinite right lead. Then the conversion matrix of this scatterer is$$\mathbb{G}^{(\mathrm{R})}=(z-\mathbf{h}-\mathbf{tP}_{+})^{-1}=(\mathbf{t}%
^{\dagger}\mathbf{P}_{+}^{-1})^{-1}.$$ The GF of the entire system is$$\mathbf{G}_{m,m_{0}}^{(\mathrm{R})}=\mathbf{g}_{m,m_{0}}+\mathbf{P}_{+}%
^{m}(\mathbb{G}^{(\mathrm{R})}\mathbf{g}^{-1}-\mathbf{I})\mathbf{g}_{0,m_{0}},$$ where $\mathbb{G}^{(\mathrm{R})}\mathbf{g}^{-1}-\mathbf{I}=-\mathbf{P}%
_{+}\mathbf{P}_{-}^{-1}$. Taking $m=m_{0}=0$ gives $\mathbf{G}_{0,0}%
^{(\mathrm{R})}=\mathbb{G}^{(\mathrm{R})}$.
The third example is an interface at $m=0$ (with unit cell Hamiltonian $\mathbf{H}_{0,0}$) connected to two semi-infinite leads $L$ and $R$. In this case the conversion matrix of the interface is $$\mathbb{G}^{(\mathrm{I})}=(z-\mathbf{H}_{0,0}-\mathbf{t}_{L}^{\dagger
}(\mathbf{P}_{-}^{(L)})^{-1}-\mathbf{t}_{R}\mathbf{P}_{+}^{(R)})^{-1},$$ where $\mathbf{t}_{L}$ ($\mathbf{t}_{R}$) is the nearest-neighbor hopping in the left (right) lead. The GFs of the entire system are given by$$\begin{aligned}
\mathbf{G}_{m\geq0,m_{0}\leq0} & =(\mathbf{P}_{+}^{(R)})^{m}\mathbb{G}%
^{(\mathrm{I})}(\mathbf{g}^{(L)})^{-1}\mathbf{g}_{0,m_{0}}^{(L)},\\
\mathbf{G}_{m\leq0,m_{0}\geq0} & =(\mathbf{P}_{-}^{(L)})^{m}\mathbb{G}%
^{(\mathrm{I})}(\mathbf{g}^{(R)})^{-1}\mathbf{g}_{0,m_{0}}^{(R)},\\
\mathbf{G}_{m\leq0,m_{0}\leq0} & =\mathbf{g}_{m,m_{0}}^{(L)}+(\mathbf{P}%
_{-}^{(L)})^{m}[\mathbb{G}^{(\mathrm{I})}(\mathbf{g}^{(L)})^{-1}%
-\mathbf{I}]\mathbf{g}_{0,m_{0}}^{(L)},\\
\mathbf{G}_{m\geq0,m_{0}\geq0} & =\mathbf{g}_{m,m_{0}}^{(R)}+(\mathbf{P}%
_{+}^{(R)})^{m}[\mathbb{G}^{(\mathrm{I})}(\mathbf{g}^{(R)})^{-1}%
-\mathbf{I}]\mathbf{g}_{0,m_{0}}^{(R)}.\end{aligned}$$
[109]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
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[^1]: Since $G(E)$ describes the scattering of both traveling and *evanescent* waves, while $S(E)$ only describes the scattering of traveling waves, $G(E)$ and $S(E)$ are equivalent only in the asymptotic regions of the lead, i.e., sufficiently far way from the scatterer such that the evanescent waves vanish completely.
[^2]: Lattice models with hopping between $p$th nearest neighbors can always be reduced to nearest-neighbor hopping by retarding $p$ successive unit cells as one composite unit cell.
|
---
abstract: 'A theoretical study of the surface energy-loss function of freestanding Pb(111) thin films is presented, starting from the single monolayer case. The calculations are carried applying the linear response theory, with inclusion of the electron band structure by means of a first-principles pseudopotential approach using a supercell scheme. Quantum-size effects on the plasmon modes of the thinnest films are found in qualitative agreement with previous work based on the jellium model. For thicker films, results show a dispersionless mode at all thicknesses, in agreeement with electron energy-loss measurements. For sizeable values of the momentum, the raising of the surface plasmon with increasing thickness is retrieved.'
author:
- 'X. Zubizarreta'
- 'E. V. Chulkov'
- 'V. M. Silkin'
title: 'Quantum effects on the loss function of Pb(111) thin films: an *ab initio* study.'
---
INTRODUCTION
============
In thin metallic films, the confinement in the direction perpendicular to the film plane gives rise to the quantization of the electronic wave functions. As a result of the appearance of the so-called quantum-well states (QWS),[@chiassp00] the properties of the metallic slabs might strongly depend on the exact thickness of the film. This dependence is a purely quantum phenomena known as quantum-size effect (QSE), which often appears as an oscillatory dependence of several physical properties on the film thickness.
Thin lead films exhibit important quantum-size oscillations in the layer-by-layer growth,[@hincepl89] first observed by He-atom scattering and attributed to interference with the quantum-well states. The latter modulate the electron density of states (DOS) at the Fermi level $(E_F)$, causing oscillations with varying thickness in the superconducting critical temperature and the upper critical field,[@ozernph06; @sklyprl11] interlayer distances,[@jiaprb06] island height distributions,[@oterprb02] zone-center phonon frequencies,[@ynduprl08; @brauprb09] electronic transport,[@jaloprl96] photoemission properties,[@kircprb07] work functions [@kimpnas10] and quasiparticle lifetimes.[@hobrprb09; @kirenp10] Also, recently superconductivity was discovered in a single lead monolayer (ML) on silicon.[@zhannph10] Thus, lead films have become an important model system for exploring electronic and structural properties of metals on the nanoscale.[@jiajpsj07]
However, to the best of our knowledge, there are few experimental studies on the surface dielectric response of Pb thin slabs,[@jaloprb02; @jalocoma04; @puccprb06] and no theoretical works. Thus, the aim of the present work is performing a computational systematic study of the surface energy-loss function of Pb(111) films with varying thickness, starting from the single monolayer case, up to the 15 ML thick slab.
An approximate description of thin film plasmons is given by the solution of the Maxwell equations for the proper geometry.[@ritcpr57] It leads to the coupling between the classical surface plasmons of the two different surfaces of the film. The resulting coupled modes of the film disperse as [@ritcpr57; @yuanprb06; @pitarpp07] $$\omega_{\pm} = \frac{\omega_{p}}{\sqrt{2}}(1 \pm e^{-qL})^{1/2},\label{classical_dispersion}$$ where $\omega_{p}$ is the bulk plasmon frequency, which is given by $\omega_p = \sqrt{4\pi n/m^*}$ with $n$ being the average electron density and $m^*$ the electron effective mass, which in terms of the density parameter $r_s$ standing for the average inter-electron distance reads as $\omega_p = \sqrt{3/r_s^3\,m^*}$. The energy splitting between the modes depends on the film thickness L and the in-plane momentum transfer $q$. The low-energy mode $\omega_{-}$ corresponds to a symmetric induced charge profile in the direction perpendicular to the film plane, whereas the high-energy mode $\omega_{+}$ corresponds to an antisymmetric one.[@yuanprb06] As L increases, the coupling between the two modes decreases. In the limit L$\gg$1/$q$ the two film modes are decoupled and the two classical surface plasmons of frequency $\omega_{p}/\sqrt{2}$ are retrieved. This model ignores the electronic structure of the film. This is a serious drawback since the ground state electronic structure has been shown to strongly affect the surface response to external perturbations. More detailed classical models showed the dependence of the surface plasmon dispersion on the microscopic details of the surface electronic density profile.[@bennprb70; @schwprb82]
On a more quantitative level, the jellium model [@langprb70] has been used to study the quantum-mechanical electrodynamical response of metal slabs,[@eguiprl83; @dobsprb92; @schaprb94] gaining basic insight into the nature of electronic excitations of metallic films. As an example, Yuan and Gao have shown,[@yuanprb06] using the jellium model with the electron density corresponding to Ag, the disappearance of the antisymmetric mode $\omega_{+}$ for $q \rightarrow 0$ when the film thickness is comparable to the Fermi wavelength. Instead, a few discrete interband peaks were found.[@yuanprb06]
A more precise description of the electron band structure in the direction perpendicular to the film plane,[@chulss97; @chulss99] allowing to describe the surface states which are missing in a jellium model, was recently used to study new collective electronic excitations at metal surfaces [@silkprb05; @pohlepl10; @krasprb10] and thin metal films.[@silkprb11] However, the recipe of the improved one-dimensional potential [@chulss97; @chulss99] can not give a satisfactory description of the electronic structure of Pb(111) films. Thus, in the present work a first-principles approach is used to study the dielectric response of Pb(111) films. Indeed, using an *ab initio* calculation scheme possible anisotropy effects can be studied, which are missing in jellium models or in using the potentials of Refs. and , as they assume in-plane free-electron-like behavior.
In general, the plasmon modes of the Pb(111) films are found to follow qualitatively the classical dispersion relation Eq. (\[classical\_dispersion\]). However, for the thinnest slabs QSE are found. A dispersionless mode is found at all thicknesses, replacing in practice the surface plasmon as the short wavelength limit of the symmetric plasmon mode. The raising of the surface plasmon with increasing thickness is found at short wavelengths. Also, comparison of the surface energy-loss function with the momentum transfer **q** along different high-symmetry directions showed no sizeable anisotropy effects.
The rest of the paper is organized as follows. In Sec. II the details of the *ab initio* calculation of the surface loss function using a supercell scheme are shown. In Sec. III the calculated ground state electronic structure properties are presented, while in Sec. IV the results on the surface loss function are analyzed in detail. Finally, the main conclusions are drawn in Sec. V. Unless otherwise stated, atomic units are used throughout, i.e., $e^2=\hslash=m_e=1$.
CALCULATION METHOD
==================
When a perturbing electric charge is located far from one side of the film the differential cross section for its scattering with energy $\omega$ and in-plane momentum transfer **q** is proportional to the imaginary part of the surface response function *g*$(\textbf{q},\omega$) defined as [@persprb85] $$\textsl{g}(\textbf{q},\omega) = - \dfrac{2\pi}{q}\int dz \int dz' \chi(z,z',\textbf{q},\omega)e^{q(z+z')},\label{g}$$ which depends on the film electronic properties only ($q = |\textbf{q}|$). This quantity is relevant in the description of surface collective excitations measured in electron energy-loss experiments.[@liebphs87; @liebsch97] Here $\chi(z,z',\textbf{q},\omega)$ is the density response function of an interacting electron system that determines, within linear response theory, the electron density $n^{\rm{ind}}(z,\textbf{q},\omega)$ induced in the system by an external potential $V^{\rm{ext}}(z,\textbf{q},\omega)$ according to $$n^{\rm{ind}}(z,\textbf{q},\omega) = \int dz'\chi(z,z',\textbf{q},\omega)V^{\rm{ext}}(z',\textbf{q},\omega).\label{nind}$$ The collective electronic excitations in thin films then can be traced to the peaks in the surface loss function defined as the imaginary part of *g*, Im\[*g*($\textbf{q},\omega$)\].
{width="100.00000%"}
In the framework of time-dependent density functional theory,[@rungeprl84; @petersprl96] $\chi$ is the solution of the integral equation $$\begin{gathered}
\chi(z,z',\textbf{q},\omega) = \chi^{0}(z,z',\textbf{q},\omega) + \int dz_{1} \int dz_{2}\chi^{0}(z,z',\textbf{q},\omega)
\\ \times [v_{c}(z_{1},z_{2},\textbf{q})+ K_{XC}(z_{1},z_{2},\textbf{q},\omega)]\chi(z_{2},z',\textbf{q},\omega),\label{chi_int} \end{gathered}$$ with $\chi^{0}$ being the response function of the noninteracting Kohn-Sham electrons. In Eq. (\[nind\]) $v_{c}(z,z',\textbf{q}) = -\frac{2\pi}{q}e^{q|z-z'|}$ stands for the two-dimensional (2D) Fourier transform of the bare Coulomb potential and $K_{XC}$ accounts for the exchange-correlation (XC) effects. In the present work, we use the random-phase approximation (RPA) where $K_{XC}$ is set to zero, i.e., the dynamical short-range exchange-correlation effects are ignored. Previous studies of collective excitations at the surfaces [@nagao10; @nagaprl01; @tsuess91; @silkprl04] and in the bulk [@aryaprl94; @krasprb99] of many “metallic” systems suggest that XC effects should have little impact on the study of Pb films. For a periodic system, the polarizability can be expressed as a matrix in the basis of the reciprocal space vectors $\left\lbrace\textbf{G}\right\rbrace$. As a consequence, Eq. (\[chi\_int\]) becomes a matrix equation. Then, once the ground state has been obtained, the starting point of the calculation of the surface response function is the evaluation of the matrix elements of the noninteracting polarizability
$$\chi^{0}_{\textbf{G},\textbf{G}'}(\textbf{q},\omega) = \frac{2}{S}\sum^{SBZ}_{\textbf{k}}\sum_{n}^{occ}\sum_{n'}^{unocc}\frac{f_{\textbf{k},n}-f_{\textbf{k}+\textbf{q},n'}}{E_{\textbf{k},n}-E_{\textbf{k}+\textbf{q},n'}+(\omega+i\eta)}\langle\phi_{\textbf{k},n}|e^{-i(\textbf{q}+\textbf{G})\cdot r}|\phi_{\textbf{k}+\textbf{q},n'}\rangle\langle\phi_{\textbf{k}+\textbf{q},n'}|e^{i(\textbf{q}+\textbf{G}')\cdot r}|\phi_{\textbf{k},n}\rangle,$$
\[chi0\] where *n* (*n’*) is an occupied (unoccupied) band index, **k** is in the two-dimensional surface Brillouin zone (SBZ), $f_{\textbf{k},n}$ are Fermi factors and $E_{\textbf{k},n}\ \ (\phi_{\textbf{k},n})$ are Kohn-Sham energies (wave functions). Actually in order to speed up the calculations, following Refs. and , first the spectral function is calculated and from its knowledge the imaginary and real parts of $\chi^{0}_{\textbf{G},\textbf{G}'}$ are obtained. Finally, the expression for the surface response function in the case of a periodically repeated slab reads $$\textsl{g}(\textbf{q},\omega) = - \dfrac{2\pi}{q}\int dz \int dz' \chi_{\textbf{G}=0,\textbf{G}'=0}(z,z',\textbf{q},\omega)e^{q(z+z')},\label{g_per}$$ Even though only the $\textbf{G}=\textbf{G}'=0$ matrix element of $\chi_{\textbf{G},\textbf{G}'}$ enters Eq. (\[g\_per\]), the full three-dimensional (3D) nature of the polarizability is implicity taken into account via the evaluation of Eq. (\[chi\_int\]) as a matrix equation.
In order to save computational time, $\chi^{0}_{\textbf{G},\textbf{G}'}$ has been calculated retaining only $\textbf{G} = (0,0,G_{z})$ reciprocal space vectors. Physically, this means that lateral crystal local field effects [@adlpr62] were neglected. This approach was already found to give indistinguishable results compared with the calculations carried using the 3D **G’**s for metal surfaces.[@silkprl04] All important 3D effects are included in the evaluation of $\chi^{0}_{\textbf{G},\textbf{G}'}$ through the use of the fully 3D Bloch functions and their respective one-electron energies.
In the present work Pb(111) films are represented by freestanding slabs infinite in the *xy* plane and periodically repeated in the *z* direction, separated by a vacuum region whose thickness here is fixed in all cases as 10 interlayer distances of the lead atoms of the film in the *z* direction. Films are not relaxed, representing ideal cuts of the face-centered cubic bulk Pb in the (111) direction with the bulk experimental lattice parameter of $4.95\ \mathring{A}$. Thus, the in-plane lattice parameter is $a=3.50\ \mathring{A}$, the interlayer distance is $c=2.86\ \mathring{A}$ and the vacuum region thickness is $d=28.6\ \mathring{A}$. However, 4 - 6 ML thick films were also allowed to relax in the *z* direction and their band structure showed small changes compared with their unrelaxed counterparts.
For the density functional theory (DFT) ground state calculations, the electron-ion interaction is represented by norm-conserving non-local pseudopotentials,[@bacheleprb82] and the LDA approximation is chosen for the exchange and correlation potential, with the use of the Perdew-Zunger [@pezuprb81] parametrization of the XC energy of Ceperley and Alder.[@cealprl80] Well-converged results have been found with a kinetic energy cut-off of $\sim$220 eV, including from $\sim2200$ (1 ML) to $\sim5300$ (15 ML) plane-waves in the expansion of the Bloch states.
For 1 - 4 ML thick films, the Hamiltonian was also solved including the SO term fully self consistently. As a centrosymmetric supercell was used in the calculations, due to the Kramers degeneracy [@tinkam71] the electron energy bands are doubly degenerate also when the SO coupling is included in the Hamiltonian (see Fig. \[bs\]).
The calculation of $\chi^{0}_{\textbf{G},\textbf{G}'}(\textbf{q},\omega)$ was carried out using a Monkhorst Pack 192$\times$192$\times$1 (96$\times$96$\times$1) grid of [**k**]{} vectors as the hexagonal SBZ sampling with 3169 (817) **k** vectors in the irreducible part of the SBZ for the 1-5 ML (6-15 ML) thick films. Up to 500 bands were included in the evaluation of $\chi^{0}_{\textbf{G},\textbf{G}'}(\textbf{q},\omega)$ for all thicknesses. The width of the Gaussian replacing the delta function in the evaluation of $\chi^{0}_{\textbf{G},\textbf{G}'}(\textbf{q},\omega)$ was set to 0.15 eV, a value which gave smooth results while not hiding any feature on the surface loss function of the films. Well converged results are found including 750 plane waves in the expansion of the wave functions in the calculation of $\chi^{0}_{\textbf{G},\textbf{G}'}(\textbf{q},\omega)$ and expanding the size of the polarizability matrix up to 60 **G** vectors.
ELECTRONIC STRUCTURE RESULTS
============================
The results of the ground state calculations showed bilayer oscillations as a function of the slab thickness on the density of states at the Fermi level and on the work function, with a beating pattern of period 9 ML superimposed (not shown). This is in agreement with previous experimental and theoretical studies (see i.e. Refs. and ).
In Fig. \[bs\] the calculated electronic band structure of Pb(111) films of several thicknesses is shown. For a N ML thick film, each electron state energy level is unfolded in N subbands. The subbands below -6 eV are of *s* character. They are separated by a gap from the 3N subbands of *p* character which form the Fermi surfaces of the slabs. As can be seen from Fig. \[bs\], the width of the gap is already fixed as $\sim$2 eV for the 3 ML thick film.
Around the SBZ center ($\overline{\Gamma}$ point) bands show a parabolic free-electron-like dispersion. The *p* bands around $\overline{\Gamma}$ present a $p_z$ character, while acquiring an increasing $p_{x,y}$ component as they loss their parabolic-like dispersion moving away from $\overline{\Gamma}$.[@kirenp10] The $p_z$ states represent the QWS of the Pb(111) films. The present work found that the inverse of the energy separation of the QWS around $E_F$ is linearly proportional to the film thickness, in good agreement with a previous study.[@weiprb02]
As lead is a heavy element (atomic number 82), SO interaction has sizeable effects on its energy spectrum. As an example, in bulk Pb the SO-induced splitting at the BZ center is $\sim$3 eV, and several degeneracies are lifted throughout the BZ.[@zubiprb11] In Fig. \[bs\] the band structure and DOS for the 1 - 3 ML thick films is shown with (dashed lines) and without (solid lines) SO coupling included in the Hamiltonian. As can be readily seen, SO effects are remarkable only for the single monolayer case, which becomes semimetallic when the SO interaction is switched on, as a result of the avoiding of the band-crossings present for the scalar-relativistic system around the Fermi level. As the slab thickness is increased, also the filling of the phase space by the unfolding of the subbands increases. Because of the fast filling of the phase space, SO effects on the ground state of Pb(111) films become small for slabs as thin as 3 ML (see Fig. \[bs\]), as avoiding of the band-crossings is the only sizeable SO effect on their band structure. As a consequence, SO effects are not expected to affect qualitatively the films surface loss function, except for the somewhat artificial semimetallic single Pb(111) monolayer. Thus, in the present work only scalar-relativistic calculations are reported. Note however that a recent first-principles study has shown the inclusion of the SO coupling as necessary in the calculation of the electron-phonon coupling and the superconducting temperature of Pb(111) films.[@sklyprb13]
SURFACE LOSS FUNCTION RESULTS
=============================
{width="48.00000%"}
.\[ani\]
Isotropy
--------
{width="100.00000%"}
In Fig. \[ani\] the calculated surface loss function for the 3MLs thick Pb(111) film, with $\textbf{q}$ along two different high-symmetry directions, namely $\overline{\Gamma}-\overline{M}$ \[panel (a)\] and $\overline{\Gamma}-\overline{K}$ \[panel (b)\] is shown. It is clear that Im\[$\textsl{g}(\textbf{q},\omega)$\] exhibits a highly isotropic character. In all the carried tests the same isotropic behaviour of the surface loss function was found independently of the film thickness. Thus, from here on only results for $\textbf{q}$ along $\overline{\Gamma}-\overline{M}$ are shown, as the grid used in this high-symmetry direction is finer than the one along $\overline{\Gamma}-\overline{K}$.
General results
---------------
The general results of the present work are shown in Fig. \[main\]. In order to get insight, the dispersion of the classical modes $\omega_{\pm}=\omega_{\pm}(q)$ given by Eq. (\[classical\_dispersion\]) is represented by solid green curves. For each freestanding slab, $\omega_{\pm}(q)$ are plotted for an effective thickness corresponding to a number of interlayer distances equal to the number of MLs forming the slab, as the jellium edge in the first-principles calculations was fixed at half an interlayer distance away from the outermost atomic layers.
The results of the present work as plotted in Fig. \[main\] show several modes of different character. First, the low-energy symmetric mode is detected for the thinnest slabs at small momentum transfer values, closely following the dispersion described by the low-energy $\omega_{-}$ mode of Eq. (\[classical\_dispersion\]) for all thicknesses as represented in Fig. \[main\] by the bottom green line in each panel. However, notice that it disappears upon entering the almost dispersionless peak present around $\omega \simeq 7$ eV for all thicknesses.
Also, the high-energy plasmon mode analogous to the classical thin film $\omega_{+}$ mode is found for thicknesses greater than 2MLs. Note that it is placed at too high energies in comparison with the predictions of Eq. (\[classical\_dispersion\]). Unfortunately, calculations including the 5$d$ semicore electrons are too computationally demanding in the supercell scheme used here. Thus, the influence of the semicore electrons on Im\[$\textsl{g}(\textbf{q},\omega$)\] has not been checked . The inclusion of the polarizable 5$d$ semicore electrons through the use of a model dielectric function $\varepsilon_d$ [@liebsch97] is ambiguous and its use has been discarded in this present study. Note the 5$d$ electrons have been shown to play a crucial role in the dielectric response of bulk Pb, more precisely in the optics and dynamics of the main bulk plasmon.[@zubiprb13b]
{width="90.00000%"}
In Ref. using jellium calculations it was shown that the antisymmetric mode disappears for film thicknesses comparable or smaller than the Fermi wavelength of the metal when $q \rightarrow 0$. Instead, peaks corresponding to discrete interband transitions show up. For Pb, using the value $r_s^{\rm Pb}=2.298$, one finds $\lambda_F^{\rm Pb} = 7.52$ a.u. which is roughly 1.4 times the interlayer distance in Pb(111) films. Thus, the 1 and 2 MLs thick Pb(111) films present electronic effective thicknesses equal to 0.7 and 1.4 times $\lambda_F^{\rm Pb}$, respectively. As seen in Fig. \[main\], our results are in agreement with the work of Ref. as far as the disappereance of the high-energy mode for thin films is concerned. In the surface loss function of the single monolayer shown in Fig. \[main\], a manifold of interband peaks is present for energy transfers $\omega \gtrsim 11$ eV \[see also the black solid curve in Fig. \[thick\] (a)\], where the high-energy mode should be present (see the upper green line in the first panel of Fig. \[main\]). This is a manifestation of strong QSE in the surface-loss function of the single Pb(111) monolayer. The 2 ML Im\[$\textsl{g}(\textbf{q},\omega$)\] results (see Fig. \[main\]) correspond to the transition between the two different thickness regimes, at $L\simeq1.4\lambda_F^{\rm Pb}$.
An important conclusion of the present work is the large difference in spectral weight between the low- and high-energy modes of the film, in sharp contrast with the results reported in Ref. for Ag slabs modeled by the jellium approximation. The low-energy mode analogous to the classical symmetric $\omega_{-}$ plasmon appears as a faint feature in comparison with the rest of the peaks present in Im\[$\textsl{g}(\textbf{q},\omega$)\]. On the contrary, the high-energy mode is the most intense feature in the surface loss function of freestanding Pb(111) films, except for the single monolayer. In the latter case, a slightly upwards dispersing interband peak raises at energies $\omega \simeq 5.5 - 6$ eV. It exhibits the highest intensity \[see Fig. \[thick\] (a) and (b)\] together with a vanishing linewidth at momentum transfer smaller than 0.1 a.u. This long-living mode stems from transitions between the highest occupied and lowest unoccupied QWSs around the SBZ center (see Fig. \[bs\]), representing strong QSE. Once more the 2 MLs results represent the crossover with larger thicknesses for which the quantization of the states is not reflected in the same fashion in the calculated surface loss function. Nevertheless, in the evaluated Im\[$\textsl{g}(\textbf{q},\omega$)\] corresponding to the 2 MLs thick slab, still two interband peaks similar to the long-living mode present in the single monolayer are found, overlapping with each other. However, their intensity is greatly decreased in comparison with the corresponding feature of the response of the 1ML Pb(111) film.
In Fig. \[n\_ind\] the real part of the two-dimensional Fourier transform of the induced density \[see Eq. (\[nind\])\] Re\[$n^{\rm{ind}}(z,\textbf{q},\omega)$\] for the 4 MLs thick slab is shown. The results correspond to a momentum transfer of $q$ = 0.014 a.u. First, note the antisymmetric distribution of the induced density with respect to $z$=0. Second, for spatial positions inside the film, several changes of the sign of Re\[$n^{\rm{ind}}$\] are found. In panel (b) results for $4\leqslant\omega\leqslant8$ eV are zoomed in. Interestingly a sharp change of phase of Re\[$n^{\rm{ind}}$\] can be recognized at $\omega \simeq 7$ eV, signaling about the presence of the dispersionless peak seen in Im\[$\textsl{g}(\textbf{q},\omega$)\] at this energy. This is a general behaviour found at all thicknesses at $\omega \simeq 7$ eV.
Moreover, the fingerprint of the low-energy mode analogous to the classical symmetric $\omega_{-}$ plasmon is found, as marked by the circle in panel (b) of Fig. \[n\_ind\]. As can be seen, $\omega \sim$ 5 eV is the only energy at which there is a noticeable weight of Re\[$n^{\rm{ind}}(z,\textbf{q},\omega)$\] at the center of the slab, $z$ = 0. This notable distortion in the general antisymmetric distribution of the real part of the two-dimensional Fourier transform of the induced density signals about the presence of the symmetric plasmon mode. This faint but appreciable $\omega_{-}-$like fingerprint has been found at all thicknesses in which the symmetric mode could be resolved.
![(Color online) Surface loss function for different values of $\textbf{q}$ along $\overline{\Gamma}-\overline{M}$ and different film thicknesses. Black solid, red dashed, green dashed-dotted and blue dashed-dotted-dotted curves represent results for 1, 3, 5, and 8 MLs thick Pb(111) films, respectively. The thick orange (grey) solid curve stands for the results deduced from bulk calculations without (with) inclusion of the 5$d$ electrons. The vertical dashed line marks the classical Pb surface plasmon energy of 9.57 eV, while the shaded energy interval corresponds to the electron energy loss experimental value of 10.6$\pm$0.2 eV.[@powepps60][]{data-label="thick"}](thick.png){width="45.00000%"}
Thickness dependence
--------------------
In order to relate the results for different thicknesses, several cuts of Im\[$\textsl{g}(\textbf{q}_\parallel,\omega$)\] are plotted in Fig. \[thick\] comparing the surface loss function of Pb(111) films of distinct thicknesses for the same momentum transfer values of $\textbf{q}$ along the $\overline{\Gamma}-\overline{M}$ high-symmetry direction.
In panel (a) of Fig. \[thick\] the black curve at $\omega \gtrsim 11$ eV shows the manifold of interband peaks which replaces a single high-energy antisymmetric mode for the Pb monolayer. An additional important feature in the surface loss function results for the 1 ML slab is the long-living interband peak found at small momentum transfer, seen at $\omega = 5.5$ eV in panels (a) and (b) of Fig. \[thick\].
For the 3, 5 and 8 MLs thick films the surface plasmon is already present at $q$=0.1256 a.u., as seen in panel (c) of Fig. \[thick\]. Note that it presents a remarkably smaller intensity than the high-energy antisymmetric mode. On the other hand, the low-energy symmetric mode can not be seen in the scale of Fig. \[thick\], as it is a faint feature (see Fig. \[main\]).
Surface plasmon
---------------
![(Color online) Scalar-relativistic surface loss function of the 15 MLs thick Pb(111) film, $\textbf{q}$ along $\overline{\Gamma}-\overline{M}$. The green lines stand for the dispersion of the classical thin film modes as given by Eq. (\[classical\_dispersion\]). The pink square represents the experimental data of Ref. of $\omega_{s}^{\rm{exp}}=10.6\pm0.2$ eV. The same colour code as in Fig. \[main\] is used.[]{data-label="15ML"}](15ML.png){width="45.00000%"}
In Fig. \[thick\] the vertical dashed line marks the classical surface plasmon energy $\omega_{s}=\omega_{p}/\sqrt{2}=\sqrt{1.5r_s^{-3}}$, which for the averaged valence electron density of bulk lead $r_{s}^{\rm Pb}=2.298$ gives the value $\omega_{s}^{\rm Pb}$ = 9.57 eV. Also, the results of the experimental electron energy-loss measurements of 10.6$\pm$0.2 eV [@powepps60] are represented by the thin shaded area. As can be seen, the classical expression gives a too low value of the surface plasmon energy by about 1 eV.
On the other hand, in the optical limit ($q \rightarrow 0$) the surface response function can be calculated from the bulk dielectric function as [@liebphs87; @liebsch97] $$\textsl{g}(q\rightarrow0,\omega) = \frac{\varepsilon^{\rm{bulk}}(q\rightarrow0,\omega)-1}{\varepsilon^{\rm{bulk}}(q\rightarrow0,\omega)+1},\label{g_opt}$$ and thus the surface loss function is $$\rm{Im}[\textsl{g}(q\rightarrow0,\omega)] \propto - \rm{Im}\left[\frac{1}{\varepsilon^{bulk}(q\rightarrow0,\omega)+1}\right].\label{im_g_opt}$$ In Fig. \[thick\] the orange (grey) thick solid curve represents $\rm{Im}[\textsl{g}(q\rightarrow0,\omega)]$ calculated using Eq. (\[im\_g\_opt\]) and including the 5$d$ electrons in the core (valence). The energy of this peak at half width at half maximum (HWHM) is of 10.85 (9.3) eV with the semicore electrons excluded from (included in) the valence configuration. When using the bulk dielectric function obtained without including the 5$d$ electrons, the value retrieved is close to the experimental one of 10.6$\pm$0.2 eV. However, the agreement is worsen upon taking the semicore electrons into account in the evaluation of $\varepsilon^{bulk}(q\rightarrow0,\omega)$. Note that the surface loss function obtained from a bulk calculation (without the semicore) through Eq. (\[im\_g\_opt\]) is in qualitative agreeement with the slab surface plasmon for thicknesses greater than 2 MLs at momentum transfer values where the modes $\omega_{\pm}$ are uncoupled, see panel (c) in Fig. \[thick\].
![(Color online) Surface plasmon dispersion for the 7 MLs thick Pb(111) film as a function of $\textbf{q}$ (along $\overline{\Gamma}-\overline{M}$). The circles represent the calculated values of the HWHM position at each *q*. The red solid line is a linear fit of the computational results, while the green dashed one stands for the surface plasmon dispersion in a hydrodynamic approach of the jellium semiinfinite surface (see the text). The shaded orange square marks the experimental interval of $\omega_{s}^{\rm{exp}}=10.6\pm0.2$ eV.[@powepps60][]{data-label="7ML"}](7MLOK.png){width="48.00000%"}
Surprisingly, Im$[\textsl{g}(q\rightarrow0,\omega)]$ calculated from the knowledge of $\varepsilon^{bulk}(q\rightarrow0,\omega)$ shows a faint peak at 7 eV, mimicking the dispersionless feature which plays the role of the short wavelength limit of the symmetric mode $\omega_{-}$ in the thinnest films. This signals about the bulk-like character of the aforementioned dispersionless interband mode.
From Figs. \[main\] and \[15ML\], it seems the surface plasmon disperses roughly linearly with the momentum transfer. In Fig. \[7ML\] the calculated dispersion $\omega_{s} = \omega_{s}(q)$, with $\textbf{q}$ along $\overline{\Gamma}-\overline{M}$, is shown for the 7 ML thick Pb(111) film. Thicker films did not present any remarkable difference in the surface plasmon dispersion. The values of $\omega_{s}$ were evaluated at the position of the HWHM. As can be seen, the surface plasmon presents a fairly linear dispersion as a function of the momentum transfer for $q \gtrsim 0.1$ a.u.
The straight line in Fig. \[7ML\] is the result of fitting $\omega(q) = A + B\cdot q$ for $q > 0.1$ a.u. The obtained values of the fitting parameters are $A$ = 10.58 eV and $B$ = 7.35 eV$\cdot$a.u. It is interesting to compare this findings with a simple model giving a similar behaviour of $\omega_s = \omega_s(q)$.
In a semiinfinite jellium surface, using the so-called on-step hydrodynamic approach,[@pitarpp07; @lundthe83] the following expression for the dispersion of the surface plasmon is found at long wavelengths: $$\omega_{s}(q) = \frac{\omega_{p}}{\sqrt{2}} + \frac{\beta q}{2}, \label{hydro}$$ where $\omega_{p}/\sqrt{2}=\sqrt{1.5r_{s}^{-3}}$ and $\beta=\sqrt{3/5}(v_{F}/2)$ [@pitarpp07; @lundthe83], being $v_{F} = (9\pi/4)^{1/3}r_{s}^{-1}$ the Fermi velocity of a free-electron gas of average valence electron density parameter $r_s$. Using $r_{s}^{\rm Pb}$ one gets $\omega_{p}^{\rm Pb}/\sqrt{2} = 9.57$ eV and $\beta^{\rm Pb}/2$ = 8.801 eV$\cdot$a.u. This dispersion is plotted in Fig. \[7ML\] as a green dashed line, while the orange square shows the energy interval for the experimentally determined value of $\omega_{s}^{\rm exp}=10.6\pm0.2$ eV.[@powepps60] The dispersion derived from the hydrodynamic approach fails in reproducing a correct value for the optical surface plasmon energy (as pointed above). Note that strictly speaking, Eq. (\[hydro\]) is valid for $q\ll2\omega_{p}/\beta$.[@pitarpp07; @lundthe83] In the case of lead, this gives the condition $q \ll 1.087$ a.u.
Finally, note it is difficult to deduce a value of $\omega_s(q\rightarrow0)$ from the present calculations, as the surface plasmon disperses with the momentum transfer in contrast to the classical picture described by Eq. (\[classical\_dispersion\]). In addition, the well-known negative dispersion of the surface plasmon as a function of $\textbf{q}$ in the long wavelength limit is not retrieved in the present work, as even for the thickest film studied (15 MLs) the low- and high- energy modes are splitted for the smallest values of $\textbf{q}$ used.
CONCLUSIONS
===========
In the present work, the surface loss function of thin Pb(111) films has been studied by means of a first-principles pseudopotential approach using a supercell scheme.
For 1 and 2 MLs thick films strong QSE have been found. The high-energy mode is completely absent in the dielectric response of the single monolayer. This is a direct consequence of the quantization of the electronic states, leading instead to the appearance of discrete interband transitions in the high-energy range at small momentum transfer *q* \[see Fig. \[thick\] (a)\]. This is in agreement with a previous work based on the jellium model.[@yuanprb06]
Incorporation of the full 3D *ab initio* band structure also shows a new feature. It does not disperse with the momentum transfer for films thicker than 2 MLs, presenting an energy of $\omega \sim 7$ eV. In practice, this new mode plays the role of the classical surface plasmon as the long-$q$ limit of the low-energy thin film mode, as $\omega_{-}$ disappears upon coupling to the dispersionless peak. To the best of our knowledge, this is the first work predicting the existence of this new mode as the short wavelength limit of the low-energy mode, replacing the role of the classical surface plasmon of energy $\omega_{s}=\omega_{p}/\sqrt{2}$. Indeed, in Ref. a value of 7.2$\pm$0.1 eV was reported as the average energy of a feature below the surface plasmon energy in EELS measurements. We identify this feature as the dispersionless mode found in the present *ab initio* study. Surprisingly, the optical surface loss function evaluated from bulk calculations \[see Eq. (\[g\_opt\])\] also shows a faint peak at $\sim$ 7 eV.
Also, the surface loss function calculated from Eq. (\[g\_opt\]) is in agreeement with the first-principle results \[see Fig. \[thick\] (c) and (d)\]. As regards the surface plasmon dispersion, a linear dependence with $q$ has been found in the present work. Once its dispersion is fitted to a linear function of the momentum transfer, extrapolation of the fitting to $q \rightarrow 0$ gives a value of 10.58 eV, in agreement with the experimental $\omega_{s}^{\rm exp}$ = 10.6$\pm$0.2 eV.[@powepps60]
New electron energy loss spectroscopy measurements on Pb(111) thin films are highly desirable to check the present predictions and gain further insight in the dynamics of collective electronic excitations of nanostructured systems and the consequences of the quantization of the electronic states on them.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We are grateful to Iñigo Aldazabal for technical help in computational optimization. We also acknowledge financial support from the Spanish MICINN (No. FIS2010-19609-C02-01), the Departamento de Educación del Gobierno Vasco, and the University of the Basque Country (No. GIC07-IT-366-07).
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|
---
abstract: 'The relativistic two-particle quantum mixtures are studied from the topological point of view. The mixture field variables can be transformed in such a way that a kinematical decoupling of both particle degrees of freedom takes place with a residual coupling of purely algebraic nature (“exchange coupling”). Both separated sets of particle variables induce a certain map of space-time onto the corresponding “exchange groups”, i.e. $SU(2)$ and $SU(1,1)$, so that for the compact case ($SU(2)$) there arises a pair of winding numbers, either odd or even, which are a topological characteristic of the two-particle Hamiltonian.'
author:
- |
S. Pruß-Hunzinger, S. Rupp, and M. Sorg\
[*II. Institut für Theoretische Physik der Universität Stuttgart,*]{}\
[*Pfaffenwaldring 57, D-70550 Stuttgart, Germany*]{}
title: |
Topological Quantum Numbers\
of\
Relativistic Two-Particle Mixtures
---
Introduction
============
Many of the great successes of modern field theory, classical and quantum, are undoubtedly due to the use of topological methods. Indeed, these methods did not only help classifying and subdividing the set of solutions of the field equations, as e.g. in monopole [@CrGoNa] and instanton [@FeUh; @DoKr] theory, but they also generated unexpected new relationships between apparently unrelated branches in particle theory; e.g. it has been observed within the framework of the string theories that the Chern-Simons theory, as a quantum field theory in three dimensions, is intimately related to the invariants of knots and links [@Wi; @At] which first had been discussed from a purely mathematical point of view [@Jo]. The list of examples of a fruitful cooperation of topology and physics could easily be continued but may be sufficient here in order to demonstrate the significance of topological investigations of physical theories.
The present paper is also concerned with such a topological goal, namely to elaborate the topological invariants which are encoded into the Hamiltonian of the ${\Bbb C}^2$ realization of [*Relativistic Schrödinger Theory (RST)*]{}, a recently established alternative approach to relativistic many-particle quantum mechanics [@So97; @MaRuSo01; @RuSo01] (for a critical discussion of some of the deficiencies of the conventional Bethe-Salpeter equation see ref. [@Gr]). The present new approach works for an arbitrary number of scalar [@MaRuSo01] and spin particles [@MaSo99], but for the present purposes we restrict ourselves to a scalar two-particle system which requires the ${\Bbb C}^2$ realization of RST. This means that the RST wave functions $\Psi(x)$ constitute a section of a complex vector-bundle over space-time as the base space and the two-dimensional complex plane ${\Bbb C}^2$ as the typical fibre. Such a construction clearly displays the main difference to the conventional approach where the Hilbert space of a two-particle system is taken as the tensor product of the one-particle spaces, whereas in RST one takes the Whitney sum of the single-particle bundles. Thus the latter construction suggests a fluid-dynamic interpretation of the formalism in contrast to the conventional approach whose tensor-product construction rather meets with Born’s probabilistic interpretation.
Now it should be evident that such differences in the mathematical formalism necessarily will imply also different descriptions of the physical phenomena. For the present topological context, the phenomenon of [*entanglement*]{} becomes important. In the conventional theory, an entangled $N$-particle system requires the use of either the symmetric or anti-symmetric wave functions $\Psi_{\pm}(x_1,x_2,...x_n)$, see e.g. equation (\[e3:3\]) below. For the relativisitic treatment, this automatically would introduce $N$ time variables $x^0_1...\,x^0_N$ for such a quantum state $\Psi_{\pm}$ which makes its physical interpretation somewhat obscure (however in the non-relativistic limit, there are no such problems and one may interpret the (anti-)symmetrization postulate simply as a modification of the topology of the configuration space [@LeMy]). Naturally, for the Whitney sum construction such a process of (anti-)symmetrization is not feasible because the $N$-particle wave function $\Psi(x)$ remains an object defined at any event $x$ of the underlying space-time. Consequently the conventional matter dichotomy of (anti-)symmetric states must be incorporated into RST in a different way, namely in form of the positive and negative [*mixtures*]{}. It has already been observed that the physical properties of positive RST mixtures resemble the symmetrized states of the conventional theory whereas the negative mixtures appear as the RST counterpart of the anti-symmetrized conventional states [@RuSo01_2]. Thus, in view of such a physical and mathematical dichotomy of the RST solutions, one naturally asks whether perhaps those positive and negative RST mixtures carry different topological features in analogy to the topological modification of the conventional configuration space [@LeMy].
Subsequently these questions are studied in detail and the results are the following: the positive mixtures carry a pair of topological quantum numbers (“winding numbers”) which are either even or odd, whereas the negative mixtures have always trivial winding numbers. This comes about because the positive mixtures induce a map from space-time onto the (compact) group $SU(2)$ whereas the corresponding map of the negative mixtures refers to the (non-compact) group $SU(1,1)$. Clearly the winding number of negative mixtures, as the value of the pullback of the invariant volume form over $SU(1,1)$ upon some three-cycle $C^3$ of space-time, must necessarily produce a trivial result; whereas for the compact case $SU(2)$ one observes the emergence of a pair of even or odd winding numbers.
The procedure to obtain these results is the following: In [**Sect. II**]{}, the general theory is briefly sketched for the sake of sufficient self-containment of the paper. [**Sect. III**]{} then presents the specialization of the general formalism to the two-particle case to be treated during the remainder of the paper. Here, the emphasis lies on the right parametrization of the theory: it is true, the wave function parametrization reveals the fact that the general two-particle mixture may be described in terms of four single-particle Klein-Gordon wave functions $\{ \psi_1$, $\psi_2$; $\phi_1$, $\phi_2\}$ cf. equations (\[e3:70\]). These four wave functions undergo ($2+2$)-pairing such that either pair $\{ \psi_1$, $\phi_1\}$ and $\{ \psi_2$, $\phi_2\}$ builds up a four-current $j_{1 \mu}[\psi_1,\phi_1]$ and $j_{2 \mu}[\psi_2,\phi_2]$ as the sources of some vector potential $A_{2 \mu}$ and $A_{1 \mu}$ which then enters the covariant derivatives for the members of the other pair, (\[e3:66\])-(\[e3:67\]). However such a wave function parametrization is extremely ineffective for the purpose of detecting the topological peculiarities of the theory, and therefore one resorts to the parametrization in terms of [*exchange fields*]{}. The point here is that either of the two particle degrees of freedom can be equipped by its own triplet of exchange fields $\{X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu} \}$ ($a=1,2$) so that the corresponding dynamical equations become decoupled, see the source equations (\[e3:36\]) and the curl equations (\[e3:37\]). The entanglement of both particles arises now as an [*algebraic*]{} constraint upon the two sets of exchange fields, cf. (\[e3:40\])-(\[e3:41\]).
The advantage of such a parametrization in terms of exchange fields is made evident in [**Sect. IV**]{}, where the curl equations for the exchange fields are identified with the Maurer-Cartan structure equations over the “exchange groups”, namely the compact $SO(3)$ for the positive mixtures and the non-compact $SO(1,2)$ for the negative mixtures. Consequently the RST exchange fields can be thought to be generated by some map from space-time to the corresponding exchange group so that their curl equations (\[e3:37\]) are automatically satisfied and it remains to obey the source equations (\[e3:36\]) together with the exchange coupling conditions (\[e3:20\])-(\[e3:21\]). The latter condition can be satisfied by coupling the group elements ${{\mathcal{G}}}_a$ for both particles ($a=1,2$) in the right way. For this purpose one has to look for a convenient parametrization of the exchange groups so that the exchange coupling condition adopts a simple form in terms of the selected group parameters. In this context, the optimal parametrization is achieved in terms of the well-known Euler angles $\{ \gamma_1$, $\gamma_2$, $\gamma_3 \}$ relative to which the exchange coupling condition is expressed in an almost trivial way, cf. equation (\[e4:28\]).
However for the purpose of attaining a concrete geometric picture of the emergence of topological quantum numbers ([**Sect. V**]{}), it is more instructive to resort to the universal covering groups, i.e. $SU(2)$ for the positive mixtures (and $SU(1,1)$ for the negative mixtures). The reason is that the group $SU(2)$ is topologically equivalent to the 3-sphere $S^3$ and therefore also to the 3-cycles $C^3$ of space-time. On the other hand, both sets of exchange fields induce a map ${{\mathcal{G}}}_{a (x)}$ from space-time into the exchange group, i.e. $SU(2) \sim S^3$ for the positive mixtures, and thereby associate two winding numbers $Z_{(a)}$ ($a=1,2$) (\[e5:14\]) to any three-cycle $C^3$ as the number of times this cycle is wound onto the exchange group. Since the exchange fields are constituents of the Hamiltonian ${{\mathcal{H}}}_{\mu}$, the pair of winding numbers $Z_{(a)}$ are a topological characteristic of the Hamiltonian. As expected, the exchange coupling condition between both particle degrees of freedom should establish some correlation between the winding numbers, but this correlation turns out to be so weak that it simply results in the property of both winding numbers being either odd or even. As an example of a non-trivial exchange field configuration, with unit winding numbers, the method of stereographic projection is applied ([**Sect. VI**]{}) which compactifies the Euclidean 3-space $E_3$ (as a time-slice of space-time) to $S^3$ by one-point compactification.
General RST Dynamics
====================
One of the crucial points for the subsequent topological discussions refers to an adequate parametrization of the RST field system. The reason for this is that the topological characteristics of the solutions become manifest only through an adequate choice of the field variables, whereas the general RST dynamics (in operator form) does itself not suggest such a preferred choice. Therefore we briefly list the key features of the general RST dynamics and then concentrate upon a certain parametrization being most helpful for the present topological purposes.
Matter Dynamics
---------------
The basic building blocks of the theory consist of (i) the matter dynamics, (ii) Hamiltonian dynamics, and (iii) gauge field dynamics [@MaRuSo01]. Restricting ourselves here to a system of $N$ scalar particles, one uses an $N$-component wave function $\Psi$ when the particles are “disentangled” but one resorts to an Hermitian $N\times N$ matrix ${{\mathcal{I}}}$ for the “entangled” (i.e. mixture) situation. The motion of matter is then governed by the Relativistic Schrödinger Equation (RSE) for the wave function $\Psi$ $$\label{e2:1}
i\hbar c\, {{\mathcal{D}}}_{\mu}\Psi={{\mathcal{H}}}_{\mu}\cdot \Psi \; ,$$ or by the Relativistic von Neumann Equation (RNE) for the “[*intensity matix*]{}” ${{\mathcal{I}}}$ $$\label{e2:2}
{{\mathcal{D}}}_{\mu}{{\mathcal{I}}}= \frac{i}{\hbar c}\,[{{\mathcal{I}}}\cdot\overline{{{\mathcal{H}}}}_{\mu}-{{\mathcal{H}}}_{\mu}\cdot{{\mathcal{I}}}]\;,$$ which constitutes the “[*matter dynamics*]{}”. The wave functions $\Psi_{(x)}$, to be considered as sections of an appropriate (complex) vector bundle over pseudo-Riemannian space-time as the base space, describe a pure RST state of the N-particle system; whereas the intensity matrix ${{\mathcal{I}}}$, as an operator-valued bundle section, is to be used for a mixture configuration. Clearly, the pure states can also be considered as a special kind of mixture, for which the intensity matrix ${{\mathcal{I}}}$ then degenerates to a simple tensor product $$\label{e2:3}
{{\mathcal{I}}}\Rightarrow \Psi \otimes \bar{\Psi} \; .$$ Obviously, for such a degenerate situation, the intensity matrix ${{\mathcal{I}}}$ must obey the [*Fierz identity*]{} [@MaRuSo01] $$\label{e2:4}
{{\mathcal{I}}}^2-\rho\cdot {{\mathcal{I}}}\equiv 0$$ where the scalar density $\rho$ is given by the trace of ${{\mathcal{I}}}$ $$\label{e2:5}
\rho \doteqdot {\operatorname{tr}}{{\mathcal{I}}}\; .$$
Hamiltonian Dynamics
--------------------
The Hamiltonian ${{\mathcal{H}}}_{\mu}$, emerging in the matter dynamics (\[e2:1\])-(\[e2:2\]), is itself a (non-Hermitian) dynamical object of the theory and therefore must be determined from its own field equations (“[*Hamiltonian dynamics*]{}”), namely from the [*integrability condition*]{} $$\label{e2:6}
{{\mathcal{D}}}_{\mu}{{\mathcal{H}}}_{\nu}-{{\mathcal{D}}}_{\nu}{{\mathcal{H}}}_{\mu}+\frac{i}{\hbar c}[{{\mathcal{H}}}_{\mu},{{\mathcal{H}}}_{\nu}] = i \hbar c \, {{\mathcal{F}}}_{\mu\nu}$$ and the [*conservation equation*]{} $$\label{e2:7}
{{\mathcal{D}}}^{\mu}{{\mathcal{H}}}_{\mu}-\frac{i}{\hbar c}{{\mathcal{H}}}^{\mu}\cdot{{\mathcal{H}}}_{\mu}=-i\hbar c \left(\frac{{{\mathcal{M}}}c}{\hbar}\right)^{2}\;.$$ Here, the first equation (\[e2:6\]) guarantees the (local) existence of solutions for the matter dynamics (\[e2:1\])-(\[e2:2\]) via the bundle identities for $\Psi$ $$\label{e2:8}
[{{\mathcal{D}}}_{\mu}{{\mathcal{D}}}_{\nu}-{{\mathcal{D}}}_{\nu}{{\mathcal{D}}}_{\mu}]\, \Psi={{\mathcal{F}}}_{\mu \nu} \cdot \Psi$$ or for ${{\mathcal{I}}}$, resp. $$\label{e2:9}
[{{\mathcal{D}}}_{\mu}{{\mathcal{D}}}_{\nu}-{{\mathcal{D}}}_{\nu}{{\mathcal{D}}}_{\mu}]\, {{\mathcal{I}}}=[{{\mathcal{F}}}_{\mu \nu},{{\mathcal{I}}}] \; .$$ The meaning of the second equation (\[e2:7\]) refers to the validity of certain conservation laws, such as charge conservation to be expressed in terms of the current operator ${{\mathcal{J}}}_{\mu}$ $$\label{e2:10}
{{\mathcal{D}}}^{\mu} {{\mathcal{J}}}_{\mu} \equiv 0 \; ,$$ or energy-momentum conservation to be written in terms of the energy-momentum operator ${{\mathcal{T}}}_{\mu \nu}$ as $$\label{e2:11}
{{\mathcal{D}}}^{\mu} {{\mathcal{T}}}_{\mu \nu}+\frac{i}{\hbar c}[\,\overline{{{\mathcal{H}}}}^{\mu}\cdot {{\mathcal{T}}}_{\mu \nu}-{{\mathcal{T}}}_{\mu \nu}\cdot {{\mathcal{H}}}^{\mu}]=0 \; .$$
Gauge Field Dynamics
--------------------
The third building block of the RST dynamics refers to the gauge field ${{\mathcal{F}}}_{\mu \nu}$ (“bundle curvature”) which is defined in terms of the gauge potential ${{\mathcal{A}}}_{\mu}$ (“bundle connection”) as usual $$\label{e2:12}
{{\mathcal{F}}}_{\mu \nu} = \nabla_{\mu} {{\mathcal{A}}}_{\nu} - \nabla_{\nu} {{\mathcal{A}}}_{\mu} + [{{\mathcal{A}}}_{\mu},{{\mathcal{A}}}_{\nu}] \; .$$ The gauge potentials are Lie-algebra valued 1-forms over space-time and do also enter the gauge-covariant derivative ${{\mathcal{D}}}$ in the usual way, i.e. for the wave functions $\Psi$ $$\label{e2:13}
{{\mathcal{D}}}_{\mu} \Psi \doteqdot \partial_{\mu}\Psi+ {{\mathcal{A}}}_{\mu} \cdot \Psi \; ,$$ or similarly for the operators such as the intensity matrix ${{\mathcal{I}}}$ $$\label{e2:14}
{{\mathcal{D}}}_{\mu} {{\mathcal{I}}}\doteqdot \partial_{\mu}{{\mathcal{I}}}+ [{{\mathcal{A}}}_{\mu}, {{\mathcal{I}}}]$$ or the field strength ${{\mathcal{F}}}_{\mu \nu}$ $$\label{e2:15}
{{\mathcal{D}}}_{\lambda}{{\mathcal{F}}}_{\mu \nu} = \nabla_{\lambda} {{\mathcal{F}}}_{\mu \nu} + [ {{\mathcal{A}}}_{\lambda},{{\mathcal{F}}}_{\mu \nu}] \; .$$
Now in order to close the RST dynamics for the operators, one has to specify some dynamical equations for the gauge field ${{\mathcal{F}}}_{\mu \nu}$. Here, our nearby choice is the (non-abelian generalization of) Maxwell’s equation $$\label{e2:16}
{{\mathcal{D}}}^{\mu} {{\mathcal{F}}}_{\mu \nu} = 4 \pi \alpha_{\ast}\, {{\mathcal{J}}}_{\nu} \; ,$$
($\alpha_{\ast}=$ coupling constant).
This is surely a consistent choice because the generally valid bundle identity $$\label{e2:17}
{{\mathcal{D}}}^{\mu}{{\mathcal{D}}}^{\nu}{{\mathcal{F}}}_{\mu \nu} \equiv 0$$ immediately implies the charge conservation law (\[e2:10\]) which is independently implied by the RST dynamics itself. In order to see this in some more detail, one first decomposes the Lie-algebra valued objects whith respect to the (anti-Hermitian) generators $\tau^a$ of the gauge group
\[e2:18\] $$\begin{aligned}
{{\mathcal{A}}}_{\mu} &= A_{a \mu} \tau^{a}\\
{{\mathcal{F}}}_{\mu \nu} &= F_{a \mu \nu} \tau^{a}\\
{{\mathcal{J}}}_{\mu} &= j_{a \mu} \tau^{a} \; .\end{aligned}$$
Then one defines the current densities $j_{a \mu}$ in terms of velocity operators $v_{a \mu}$ through $$\label{e2:19}
j_{a \mu}={\operatorname{tr}}({{\mathcal{I}}}\cdot v_{a \mu})$$ where the velocity operators read in terms of the Hamiltonian ${{\mathcal{H}}}_{\mu}$ $$\label{e2:20}
v_{a \mu} = \frac{i \hbar c}{2}(\, \overline{{{\mathcal{H}}}}_{\mu}\cdot ({{\mathcal{M}}}c^2)^{-1}\cdot \tau_{a} + \tau_{a}\cdot ({{\mathcal{M}}}c^2)^{-1}\cdot {{\mathcal{H}}}_{\mu}) \; .$$
By this arrangement, the charge conservation law (\[e2:10\]) reads in component form $$\label{e2:21}
D^{\mu}j_{a \mu} \equiv 0$$ with the gauge covariant derivatives being defined in terms of the structure constants $C^{bc}{}_{a}$ of the gauge group through: $$\label{e2:22}
D_{\mu} j_{a \nu} \doteqdot \nabla_{\mu} j_{a \nu} + C^{bc}{}_{a}\, A_{b
\mu} j_{c \nu}$$
($\, [\tau^a,\tau^b]=C^{ab}{}_c \tau^c$).
However the continuity equations (\[e2:21\]) are found to be guaranteed just by the matter dynamics (\[e2:2\]) together with the RST conservation equation (\[e2:7\]) where the mass operator ${{\mathcal{M}}}$ is assumed to be covariantly constant $$\label{e2:23}
{{\mathcal{D}}}_{\mu}\, {{\mathcal{M}}}\equiv 0 \; .$$
A similar result does also apply to the energy-momentum conservation (\[e2:11\]) which in the presence of gauge interactions (as source of energy-momentum) must be generalized to $$\label{e2:24}
{{\mathcal{D}}}^{\mu}{{\mathcal{T}}}_{\mu \nu}+\frac{i}{\hbar c}[\,\overline{{{\mathcal{H}}}}^{\mu}\cdot {{\mathcal{T}}}_{\mu \nu}-{{\mathcal{T}}}_{\mu \nu}\cdot {{\mathcal{H}}}^{\mu}] = {\mathfrak{f}}_{\nu} \; .$$ Defining here the energy-momentum operator of the scalar RST matter through [@MaRuSo01] $$\label{e2:25}
{{\mathcal{T}}}_{\mu \nu} = \frac{1}{2} \left\{ \overline{{{\mathcal{H}}}}_{\mu} \cdot ({{\mathcal{M}}}c^2)^{-1}\cdot {{\mathcal{H}}}_{\nu} + \overline{{{\mathcal{H}}}}_{\nu} \cdot ({{\mathcal{M}}}c^2)^{-1}\cdot {{\mathcal{H}}}_{\mu} - g_{\mu \nu}[ \overline{{{\mathcal{H}}}}^{\lambda} \cdot ({{\mathcal{M}}}c^2)^{-1}\cdot {{\mathcal{H}}}_{\lambda} - {{\mathcal{M}}}c^2] \right\} \; ,$$ one finds for the “[*force operator*]{}” ${\mathfrak{f}}_{\nu}$ $$\label{e2:26}
{\mathfrak{f}}_{\nu} = \frac{i \hbar c}{2 }\, [ \overline{{{\mathcal{H}}}}^{\mu}\cdot ({{\mathcal{M}}}c^2
)^{-1} \cdot {{\mathcal{F}}}_{\mu \nu} + {{\mathcal{F}}}_{\mu \nu}\cdot ({{\mathcal{M}}}c^2)^{-1}\cdot {{\mathcal{H}}}^{\mu} ] \; .$$ Now one can introduce the “[*energy-momentum density*]{}” $T_{\mu \nu}$ of the RST matter through $$\label{e2:27}
T_{\mu \nu} \doteqdot {\operatorname{tr}}\, ({{\mathcal{I}}}\cdot {{\mathcal{T}}}_{\mu \nu})$$ the source of which is given by the “[*Lorentz force density*]{}” $f_{\nu}$ $$\label{e2:28}
f_{\nu} = {\operatorname{tr}}\ ( {{\mathcal{I}}}\cdot {\mathfrak{f}}_{\nu} )=\hbar c F_{a \mu \nu}j^{a \mu} \; .$$ Clearly if no gauge fields are present ($F_{a \mu \nu} \equiv 0$), the force density $f_{\nu}$ vanishes and the RST matter system is closed $$\label{e2:29}
\nabla^{\mu} T_{\mu \nu}\equiv 0 \; .$$ For non-trivial gauge interactions one can add their energy-momentum density to that of the RST matter and then one also has the closedness relation (\[e2:29\]) for the total system consisting of RST matter and gauge fields [@MaRuSo01].
In this way, a logically consistent dynamical system for elementary matter has been set up which automatically implies the most significant conservation laws (namely for charge and energy-momentum). However, beyond these pleasant features of the theory, the corresponding solutions carry a non-trivial topology which is not less interesting than their physical implications. This will be demonstrated now for the two-particle systems.
Two-Particle Systems
====================
Once the general RST dynamics has been set up in abstract form, its concrete realizations must be adapted to the spin type and number $N$ of particles as well as to the gauge type of interactions among those particles. The most simple realization is the ${\Bbb{C}}^1$-realization with electromagnetic interactions which is equivalent to the conventional one-particle Klein-Gordon theory [@MaSo99_2]. The next complicated case is the ${\Bbb{R}}^2$-realization which may be understood as a kind of embedding theory for the ${\Bbb{C}}^1$-realization so that a point particle becomes equipped with additional structure [@RuSiSo00]. Then comes the ${\Bbb{C}}^2$-realization which is adequate for describing systems of two scalar particles, either acted upon by the weak [@OcSo96] or electromagnetic interactions [@MaRuSo01]. The highest-order realization, which has been considered up to now, is the ${\Bbb{C}}^4$-realization for a Dirac electron subjected to the gravitational interactions [@SiSo97].
Mixtures and Pure States
------------------------
In the present paper we want to study the topological features of scalar two-particle systems with electromagnetic interactions. This means that we have to evoke the ${\Bbb{C}}^2$-realization of RST where the typical vector fibre for the wave functions $\Psi$ is the two-dimensional complex space ${\Bbb{C}}^2$ such that $\Psi$ becomes a two-component wave function $$\label{e3:1}
\Psi(x)= \left(
\begin{array}{c} \psi_1(x) \\ \psi_2(x) \end{array}
\right) = \left(
\begin{array}{c} L_1 e^{-i\alpha_1} \\ L_2 e^{-i\alpha_2} \end{array}\; .
\right)$$ Such a field configuration is considered as the RST counterpart of the simple product states of the conventional quantum theory: $$\label{e3:2}
\Psi_{(1,2)}=\Psi_{(1)}\otimes \Psi_{(2)} \; .$$ The entangled states of the conventional theory, either symmetrized or anti-symmetrized $$\label{e3:3}
\Psi_{\pm (1,2)}=\frac{1}{\sqrt{2}}\big(\Psi_{1(1)}\otimes \Psi_{2(2)}\pm \Psi_{2(1)}\otimes \Psi_{1(2)}\big) \; ,$$ have their RST counterparts as the positive and negative mixtures which must be described by an (Hermitian) intensity matrix ${{\mathcal{I}}}$ such that $\det {{\mathcal{I}}}> 0$ for the positive mixtures and $\det {{\mathcal{I}}}< 0$ for the negative mixtures.
Quite analogously as in the conventional theory a superselection rule forbids the transitions between the symmetric and the anti-symmetric states $\Psi_{\pm}$, the RST dynamics also implies such a superselection rule, namely by preserving the sign of $\det {{\mathcal{I}}}$ [@HuMaSo01]. More concretely, the intensity matrix ${{\mathcal{I}}}$ (as any other operator such as ${{\mathcal{H}}}_{\mu}$, ${{\mathcal{F}}}_{\mu \nu}$, etc.) may be decomposed with respect to a certain operator basis; this then yields the corresponding scalar densities $\rho_a$, $s_a$ ($a=1,2$) as the components of ${{\mathcal{I}}}$. Choosing two orthogonal projectors ${{\mathcal{P}}}_a$ ($a=1,2$) and two permutators $\Pi_a$ ($a=1,2$) as such an “orthogonal” operator basis for the present ${\Bbb{C}}^2$-realization
\[e3:4\] $$\begin{aligned}
{{\mathcal{P}}}_a \cdot {{\mathcal{P}}}_b &= \delta_{ab} \cdot {{\mathcal{P}}}_b \\
tr \, {{\mathcal{P}}}_a &=1 \\
{{\mathcal{P}}}_1 + {{\mathcal{P}}}_2 &= {\bf 1}\\
\big\{\Pi_a,{{\mathcal{P}}}_a \big\} & = \Pi_a \;, \quad \forall a,b \\
\big[ {{\mathcal{P}}}_1,\Pi^a \big]&=-\big[{{\mathcal{P}}}_2,\Pi^a\big] = i \epsilon^a{}_{b}\, \Pi^b\\
\big\{\Pi_a,\Pi_b\big\}& =2 \delta_{ab} \cdot {\bf 1} \\
\big[ \Pi_a,\Pi_b\big]& =2 i \epsilon_{ab} \, ({{\mathcal{P}}}_1-{{\mathcal{P}}}_2) \; ,\end{aligned}$$
the decomposition of the intensity matrix looks as follows $$\label{e3:5}
{{\mathcal{I}}}=\rho_a {{\mathcal{P}}}^a +\frac{1}{2}s_a \, \Pi^a \; .$$
(summation of indices in juxtaposition).
Once the scalar densities for the two-particle systems have been defined now, one could convert the general RNE (\[e2:2\]) into the corresponding field equations for $\rho_a$, $s_a$ (\[e3:5\]). For an analysis of these field equations it is very instructive to parametrize the four densities $\rho_a$, $s_a$ ($a=1,2$) in terms of three [*renormalization factors*]{} $\{Z_T$, $Z_R$, $Z_O\}$ and an amplitude field $L$
\[e3:6\] $$\begin{aligned}
\rho_1+\rho_2\doteqdot \rho &=Z_T \cdot L^2 \\
\rho_1-\rho_2 \doteqdot q &=Z_R \cdot L^2 \\
s=\sqrt{s^as_a} &=Z_O \cdot L^2 \; .\end{aligned}$$
Here the amplitude field $L$ emerges through the following observation: when the Hamiltonian ${{\mathcal{H}}}_{\mu}$ is decomposed into its Hermitian part ${{\mathcal{K}}}_{\mu}$ (“kinetic field”) and anti-Hermitian part ${{\mathcal{L}}}_{\mu}$ (“localization field”) according to $$\label{e3:7}
{{\mathcal{H}}}_{\mu}=\hbar c ({{\mathcal{K}}}_{\mu}+i{{\mathcal{L}}}_{\mu}) \; ,$$ then the integrability condition (\[e2:6\]) says that the trace $L_{\mu}$ of the localization field ${{\mathcal{L}}}_{\mu}$ has vanishing curl $$\label{e3:8}
\nabla_{\mu}L_{\nu}-\nabla_{\nu}L_{\mu}=0 \; .$$
($L_{\mu}\equiv {\operatorname{tr}}{{\mathcal{L}}}_{\mu}$).
Therefore $L_{\mu}$ can be generated by some scalar amplitude field $L_{(x)}$ through $$\label{e3:9}
L_{\mu}=\frac{\partial_{\mu}L^2}{L^2}=2\frac{\partial_{\mu}L}{L}\; ,$$ and it is just this amplitude field $L_{(x)}$ (\[e3:9\]) which has been used for the parametrization (\[e3:6\]) of the densities $\rho_a$, $s_a$.
With this arrangement, the RNE (\[e2:2\]) can be transcribed directly to the corresponding field equations for the renormalization factors $\{Z_T$, $Z_R$, $Z_O\}$ which then admit the following first integral $$\label{e3:10}
Z_T^2-(Z_R^2+Z_O^2)=\sigma_{\ast} \; .$$ Here the “[*mixture index*]{}” $\sigma_{\ast}$ is an integration constant and thus subdivides the density configuration space into two parts: $\sigma_{\ast}=+1$ for the positive mixtures and $\sigma_{\ast}=-1$ for the negative mixtures; the intermediate value $\sigma_{\ast}=0$ applies to the pure states which mark the border between the two types of mixtures, see fig.1. Observe here that these dynamically disconnected two types of RST mixtures ($\sigma_{\ast}=\pm 1$) are implied by the RST dynamics itself, whereas their conventional analogue (i.e. the interdiction of transitions between the symmetric and anti-symmetric states) is an extra postulate which “[*cannot be deduced from the other principles of quantum mechanics*]{}” [@Ba]. In this sense, RST is of less postulative character than the conventional theory.
Reparametrization
-----------------
As we shall readily see, the topological properties of the RST field configurations are rather encoded in the components of the Hamiltonian ${{\mathcal{H}}}_{\mu}$, not in the densities $\rho_a$, $s_a$ as the components of the intensity matrix ${{\mathcal{I}}}$ (\[e3:5\]). Here it is important to remark that the Hamiltonian ${{\mathcal{H}}}_{\mu}$ decomposes into two conceptually different parts: the single-particle fields and the exchange fields. Naturally the single-particle constituent ${}^{(S)}{{\mathcal{H}}}_{\mu}$ of ${{\mathcal{H}}}_{\mu}$ is obtained through projection by the single-particle projectors ${{\mathcal{P}}}_a$ $$\label{e3:11}
{}^{(S)}{{\mathcal{H}}}_{\mu}\doteqdot {{\mathcal{P}}}_a \cdot {{\mathcal{H}}}_{\mu} \cdot {{\mathcal{P}}}^a \; ,$$ whereas the exchange part ${}^{(\Pi)}{{\mathcal{H}}}_{\mu}$ is the remainder $$\label{e3:12}
{}^{(\Pi)}{{\mathcal{H}}}_{\mu}={{\mathcal{H}}}_{\mu} -{}^{(S)}{{\mathcal{H}}}_{\mu} \; .$$ Clearly, the entangling exchange interactions between the two particles are contained in the exchange constituent ${}^{(\Pi)}{{\mathcal{H}}}_{\mu}$ (\[e3:12\]) whereas for a disentangled situation the Hamiltonian ${{\mathcal{H}}}_{\mu}$ consists of the single-particle contribution ${}^{(S)}{{\mathcal{H}}}_{\mu}$ alone. Now, as has been remarked expressly, the right parametrization of the RST field degrees of freedom is crucial for the detection of the topological characteristics. Here it has turned out very helpful to parametrize the exchange part ${}^{(\Pi)}{{\mathcal{H}}}_{\mu}$ of the Hamiltonian (\[e3:12\]) in a redundant way by six exchange fields $\{ X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$ ($a=1,2$) obeying the following curl equations [@RuSo01_2]
\[e3:37\] $$\begin{aligned}
\nabla_{\mu}X_{a\nu}-\nabla_{\nu}X_{a\mu}&=2\sigma_{\ast}[\Lambda_{a\mu}\Gamma_{a\nu}-\Lambda_{a\nu}\Gamma_{a\mu}]\\
\nabla_{\mu}\Gamma_{a\nu}-\nabla_{\nu}\Gamma_{a\mu}&= 2\left[X_{a\mu}\Lambda_{a\nu}-X_{a\nu}\Lambda_{a\mu} \right]\\
\nabla_{\mu}\Lambda_{a\nu}-\nabla_{\nu}\Lambda_{a\mu}&=2\left[\Gamma_{a\mu}X_{a\nu}-\Gamma_{a\nu}X_{a\mu} \right] \;.\end{aligned}$$
Clearly these curl equations are a direct consequence of the general integrability condition (\[e2:6\]). The redundancy of this parametrization is expressed by an algebraic constraint upon the exchange fields which effectively reduces their number from six to four, namely for the positive mixtures ($\sigma_{\ast}=+1$)
\[e3:20\] $$\begin{aligned}
X_{1 \mu}&=\sinh \beta \cdot \Gamma_{1 \mu}+\cosh \beta \cdot \Gamma_{2 \mu} \\
X_{2 \mu}&=\sinh \beta \cdot \Gamma_{2 \mu}+\cosh \beta \cdot \Gamma_{1 \mu} \; ,\end{aligned}$$
and similarly for the negative mixtures ($\sigma_{\ast}=-1$)
\[e3:21\] $$\begin{aligned}
X_{1 \mu}&=\cosh \beta \cdot \Gamma_{1 \mu}+\sinh \beta \cdot \Gamma_{2 \mu} \\
X_{2 \mu}&=\cosh \beta \cdot \Gamma_{2 \mu}+\sinh \beta \cdot \Gamma_{1 \mu} \; .\end{aligned}$$
Here, the mixture variable $\beta$ is merely a transformed version of the former variable $\lambda$ [@RuSo01] $$\label{e3:22}
\lambda \doteqdot \frac{1}{1+\frac{\sigma_{\ast}}{Z_O}}=
\begin{cases}
\frac{1}{2}\Big(1+\tanh \beta\Big),\quad \sigma_{*}=+1\\
\frac{1}{2}\Big(1+\coth \beta\Big),\quad \sigma_{*}=-1\;.
\end{cases}$$
It will readily become evident, in what way the newly introduced exchange triplets $\{X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$ lead us directly to the topological properties of the exchange subsystem.
In order to finally close the RST dynamics, we also have to specify the field equation for the mixture variable $\beta$ (\[e3:22\]), or $\lambda$ resp. In the last end, the desired equation must be traced back to the RNE (\[e2:2\]) since the mixture variables $\lambda$ (or $\beta$) have been defined in terms of the renormalization factor $Z_O$ (\[e3:6\]c) which is part of the overlap density $s$ as a component of the intensity matrix ${{\mathcal{I}}}$ (\[e3:5\]). Thus, if one follows the mixture variable through all its reparametrizations, one finally ends up with the following field equation for $\beta$: $$\label{e3:38}
\partial_{\mu}\beta=
\begin{cases}
2 \cosh{\beta}\left(\Lambda_{1 \mu}+\Lambda_{2 \mu}\right);\quad \sigma_{\ast}=+1\\
2 \sinh{\beta}\left(\Lambda_{1 \mu}+\Lambda_{2 \mu}\right);\quad \sigma_{*}=-1 \; .
\end{cases}$$ Obviously this result identifies the sum of both exchange fields $\Lambda_{1 \mu}+\Lambda_{2 \mu}$ as a gradient field
\[e3:39\] $$\begin{aligned}
\nabla_{\mu}(\Lambda_{1 \nu}+\Lambda_{2 \nu}) -\nabla_{\nu}(\Lambda_{1 \mu}+\Lambda_{2 \mu}) &=0 \\
\Lambda_{1 \mu}+\Lambda_{2 \mu} &=\frac{1}{2}\sigma_{\ast} \partial_{\mu}\epsilon \; .\end{aligned}$$
This gradient property could also have been discovered by eliminating the mixture variable $\beta$ from the link between both exchange fields ${X}_{a \mu}$ and $\Gamma_{a \mu}$ (\[e3:20\])-(\[e3:21\]) which yields two quadratic relations between these fields, namely a symmetric one $$\label{e3:40}
\Gamma_{1\mu}\Gamma_{1\nu}+\sigma_{*}X_{1\mu}X_{1\nu}=\Gamma_{2\mu}\Gamma_{2\nu}+\sigma_{*}X_{2\mu}X_{2\nu}$$ and an anti-symmetric one $$\label{e3:41}
X_{1\mu} \Gamma_{1\nu}-X_{1\nu}\Gamma_{1\mu}=-\Big[X_{2\mu}\Gamma_{2\nu}-X_{2\nu}\Gamma_{2\mu}\Big]\;.$$ However this latter relation just implies the present gradient condition (\[e3:39\]) when the curl relations (\[e3:37\]c) for the exchange fields $\Lambda_{a \mu}$ are observed.
For specifying also the source equations for the exchange triplets $\{ X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$ one has to observe their coupling to the single-particle subsystem which therefore must first be suitably parametrized. Such a parametrization is obtained by using a doublet of “[*kinetic fields*]{}” ${{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}$ and “[*amplitude fields*]{}” ${\Bbb{L}}_a$ which are required to obey the following “[*single particle dynamics*]{}” [@RuSo01_2]
\[e3:22\_kurz\] $$\begin{aligned}
\label{e3:35a}
\nabla_{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{a\nu}-\nabla_{\nu}{{\stackrel{\circ}{\Bbb{K}}}}_{a\mu}&=F_{a\mu\nu} \\
\label{e3:35b}
\nabla^{\mu}({{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}+X_{a \mu})+2{\Bbb{L}}_a ({{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}+X_{a \mu})&=0\\
\label{e3:32}
\square{{\Bbb{L}}}_{a}+{{\Bbb{L}}}_{a}\Big\{ \Big(\frac{M c}{\hbar}\Big)^{2}-({{\stackrel{\circ}{\Bbb{K}}}}_a+X_{a \mu})({{\stackrel{\circ}{\Bbb{K}}}}_a{}^{\mu}+X_a{}^{\mu}) \Big\}&=\sigma_{*}{{\Bbb{L}}}_{a}\big\{\Lambda_{a\mu}{\Lambda_{a}}^{\mu}+\Gamma_{a\mu
}{\Gamma_{a}}^{\mu}\big\}\;.\end{aligned}$$
Clearly these source equations are rigorously deducible again from the general conservation equation (\[e2:7\]) which is the origin also for the two charge conservation laws ($a=1,2$) $$\label{e3:30}
\nabla^{\mu}j_{a \mu}=0 \; ,$$ following from the general source equations (\[e2:21\]) for the present case of an abelian gauge group (i.e. $U(1)\times U(1))$. The RST currents $j_{a \mu}$ (\[e2:19\]) themselves read in terms of the single-particle variables $$\label{e3:28}
j_{a \mu} =\frac{\hbar}{Mc}{\Bbb{L}}_a{}^2( {{\stackrel{\circ}{\Bbb{K}}}}_{a \mu} +X_{a \mu}) \; ,$$ where the squares of the amplitudes ${\Bbb{L}}_a$ can be identified with the single-particle densities $\rho_a$ (\[e3:5\]): ${\Bbb{L}}_a{}^2\equiv \rho_a$.
Once an adequate parametrization of the single-particle subsystems has been achieved now, it becomes a straightforward matter to make explicit their coupling to the exchange fields $\{X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$:
\[e3:36\] $$\begin{aligned}
\nabla^{\mu}X_{a \mu}&=-2{\Bbb{L}}_a{}^{\mu}X_{a \mu}-\nabla^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}-2{\Bbb{L}}_a{}^{\mu}\,{{\stackrel{\circ}{\Bbb{K}}}}_{a \mu} \\
\nabla^{\mu}\Gamma_{a\mu}&=-2{\Gamma_{a}}^{\mu}{{\Bbb{L}}}_{a\mu}-2 {\Lambda_{a}}^{\mu}\,
{{\stackrel{\circ}{\Bbb{K}}}}_{a\mu}\\
\nabla^{\mu}\Lambda_{a\mu}&=-2{\Lambda_{a}}^{\mu}{{\Bbb{L}}}_{a\mu}+2{\Gamma_{a}}^{\mu}\,
{{\stackrel{\circ}{\Bbb{K}}}}_{a\mu}\; .\end{aligned}$$
These source equations are relevant for treating concrete physical problems (e.g. the Helium problem [@RuSo01_2]) but they are not needed for the deduction of the subsequent topological results which are based exclusively upon the curl equations of the exchange fields (\[e3:37\]).
Average Exchange Fields
-----------------------
An alternative way for the redundant description of a system, obeying some constraint in order to remove the redundancy, is to minimalize the number of dynamical variables so that no longer a constraint has to be imposed. Thus, for the present situation one introduces in place of the six exchange fields $\{X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$ only three “[*average*]{}” exchange fields $\{ \tilde{{{\mathbb{X}}}}$, $\tilde{{{\mathbb{\Gamma}}}}$, $\tilde{{{\mathbb{\Lambda}}}}\}$
\[e3:26\_kurz\] $$\begin{aligned}
\tilde{{{\mathbb{X}}}}&=\tilde{X}_{\mu} {{\mathbb{d}}}x^{\mu}\\
\tilde{{{\mathbb{\Gamma}}}}&=\tilde{\Gamma}_{\mu} {{\mathbb{d}}}x^{\mu}\\
\tilde{{{\mathbb{\Lambda}}}}&=\tilde{\Lambda}_{\mu} {{\mathbb{d}}}x^{\mu}\end{aligned}$$
which are required to obey the curl equations
\[e3:59\] $$\begin{aligned}
{{\mathbb{d}}}\tilde{{\mathbb{X}}}&=\sigma_{\ast}\, \tilde{{\mathbb{\Lambda}}}\wedge \tilde {{\mathbb{\Gamma}}}\\
{{\mathbb{d}}}\tilde{{\mathbb{\Gamma}}}&=\tilde{{\mathbb{X}}}\wedge \tilde {{\mathbb{\Lambda}}}\\
\label{e3:61}
{{\mathbb{d}}}\tilde{{\mathbb{\Lambda}}}&=-4 \tilde{{\mathbb{X}}}\wedge \tilde{{\mathbb{\Gamma}}}\; .\end{aligned}$$
Additionally, one introduces an arbitrary scalar field $\epsilon(x)$ (“[*exchange angle*]{}”) and if one builds up the original exchange fields $\{{{\mathbb{X}}}_{a}=X_{a \mu} {{\mathbb{d}}}x^{\mu}$, ${{\mathbb{\Gamma}}}_{a}= \Gamma_{a \mu} {{\mathbb{d}}}x^{\mu}$, ${{\mathbb{\Lambda}}}_{a}= \Lambda_{a \mu} {{\mathbb{d}}}x^{\mu}\}$ by means of these new variables according to ($\sigma_{\ast}=+1$)
\[e3:62\] $$\begin{aligned}
{{\mathbb{X}}}_1&=\cos\frac{\epsilon}{2} \cdot \tilde{{\mathbb{X}}}+\sin \frac{\epsilon}{2} \cdot \tilde{{\mathbb{\Gamma}}}\\
{{\mathbb{\Gamma}}}_1&=\cos\frac{\epsilon}{2} \cdot \tilde{{\mathbb{\Gamma}}}-\sin \frac{\epsilon}{2} \cdot \tilde{{\mathbb{X}}}\\
{{\mathbb{\Lambda}}}_1 &= \frac{1}{2}\, \{ \tilde{{\mathbb{\Lambda}}}+\frac{1}{2}\, {{\mathbb{d}}}\epsilon \}\\
{{\mathbb{X}}}_2&=\cos\frac{\epsilon}{2} \cdot \tilde{{\mathbb{\Gamma}}}+\sin \frac{\epsilon}{2} \cdot \tilde{{\mathbb{X}}}\\
{{\mathbb{\Gamma}}}_2&=\cos\frac{\epsilon}{2} \cdot \tilde{{\mathbb{X}}}-\sin \frac{\epsilon}{2} \cdot \tilde{{\mathbb{\Gamma}}}\\
{{\mathbb{\Lambda}}}_2 &= -\frac{1}{2}\, \{ \tilde{{\mathbb{\Lambda}}}-\frac{1}{2}\, {{\mathbb{d}}}\epsilon \} \; ,\end{aligned}$$
then [*both*]{} the curl equations (\[e3:37\])
\[e3:29\_kurz\] $$\begin{aligned}
{{\mathbb{d}}}{{\mathbb{X}}}_a &=2\sigma_{\ast} {{\mathbb{\Lambda}}}_a \wedge {{\mathbb{\Gamma}}}_a\\
{{\mathbb{d}}}{{\mathbb{\Gamma}}}_a&=2 {{\mathbb{X}}}_a \wedge {{\mathbb{\Lambda}}}_a\\
{{\mathbb{d}}}{{\mathbb{\Lambda}}}_a&=2 {{\mathbb{\Gamma}}}_a\wedge {{\mathbb{X}}}_a\end{aligned}$$
[*and*]{} the exchange constraints (\[e3:40\])-(\[e3:41\]) are satisfied simultaneously! For the negative mixtures ($\sigma_{\ast}=-1$) a similar parametrization in terms of the average fields and exchange angle $\epsilon$ is possible but this is not presented here on account of the topological trivialtity of the negative mixtures.
Wave Function Description
-------------------------
Up to now, the RST field system has been parametrized by the Hamiltonian component fields and the densities as the components of the intensity matrix ${{\mathcal{I}}}$. Such a parametrization is very helpful for deducing the subsequent topological results, however for practical purposes in view of the physical applications of the theory it is more convenient to parametrize the RST system by means of wave functions. Moreover the latter parametrization yields further insight into the relationship of the new theory with the conventional quantum theory which per se is based upon the concept of wave functions.
In place of the densities $\rho_a$, $s_a$ as the components of the intensity matrix ${{\mathcal{I}}}$ (\[e3:5\]), the latter operator can also be parametrized in the general case by two two-particle wave functions $\Psi$ and $\Phi$ of the kind (\[e3:1\]): $$\label{e3:64}
{{\mathcal{I}}}=I_{11} \Psi \otimes \overline{\Psi} +I_{12} \Psi \otimes \overline{\Phi}+I_{21} \Phi \otimes \overline{\Psi}+I_{22} \Phi \otimes \overline{\Phi} \; .$$ Here the constants $I_{ab}$ ($a,b=1,2$) must form a Hermitian matrix ${{\mathcal{I}}}_{\ast}$ (i.e. $\overline{{{\mathcal{I}}}}_{\ast}={{\mathcal{I}}}_{\ast}$) in order that the intensity matrix ${{\mathcal{I}}}$ be also Hermitian. The RNE (\[e2:2\]) for the intensity matrix ${{\mathcal{I}}}$ is obeyed whenever either one of the two wave functions $\Psi$ and $\Phi$ obeys the RSE (\[e2:1\])
\[e3:65\] $$\begin{aligned}
i\hbar c {{\mathcal{D}}}_{\mu}\Psi&={{\mathcal{H}}}_{\mu}\Psi\\
i \hbar c {{\mathcal{D}}}_{\mu}\Phi&={{\mathcal{H}}}_{\mu}\Phi \; .\end{aligned}$$
Here the covariant derivative (\[e2:13\]) of the two-particle wave functions reads in components $$\label{e3:66}
{{\mathcal{D}}}_{\mu}\Psi = \left(
\begin{array}{c}
D_{\mu} \psi_1\\ D_{\mu} \psi_2
\end{array} \right)
= \left(
\begin{array}{c}
\partial_{\mu}\psi_1-iA_{1 \mu}\psi_1 \\
\partial_{\mu}\psi_2-iA_{2 \mu}\psi_2
\end{array}
\right) \; ,$$ resp. for the second wave function $\Phi$ $$\label{e3:67}
{{\mathcal{D}}}_{\mu}\Phi = \left(
\begin{array}{c}
D_{\mu} \phi_1\\ D_{\mu} \phi_2
\end{array} \right)
= \left(
\begin{array}{c}
\partial_{\mu}\phi_1-iA_{1 \mu}\phi_1 \\
\partial_{\mu}\phi_2-iA_{2 \mu}\phi_2
\end{array}
\right) \; ,$$ where the gauge potential ${{\mathcal{A}}}_{\mu}$ (\[e2:18\]a) has been decomposed with respect to the two $U(1) \times U(1)$ generators $\tau_a$ $$\label{e3:68}
\tau_a=-i{{\mathcal{P}}}_a \; .$$ Observe here that the first components $\psi_1$, $\phi_1$ are acted upon by the first gauge potential $A_{1 \mu}$ whereas the second potential $A_{2 \mu}$ acts upon the second components $\psi_2$, $\phi_2$.
Now one can easily show by differentiating once more the RSE’s (\[e3:65\]) and using the conservation equation (\[e2:7\]) for the Hamiltonian ${{\mathcal{H}}}_{\mu}$ that the two-particle wave functions $\Psi$ and $\Phi$ must obey the Klein-Gordon equation (KGE)
\[e3:69\] $$\begin{aligned}
{{\mathcal{D}}}^{\mu}{{\mathcal{D}}}_{\mu}\Psi+\Big(\frac{{{\mathcal{M}}}c}{\hbar}\Big)^2 \Psi&=0\\
{{\mathcal{D}}}^{\mu}{{\mathcal{D}}}_{\mu}\Phi+\Big(\frac{{{\mathcal{M}}}c}{\hbar}\Big)^2 \Phi&=0 \; .\end{aligned}$$
In components, this result says that the general intensity matrix ${{\mathcal{I}}}$ is composed of four single-particle wave functions $\psi_1$, $\psi_2$, $\phi_1$, $\phi_2$ which obey the Klein-Gordon equations
\[e3:70\] $$\begin{aligned}
D^{\mu}D_{\mu}\psi_1+\Big(\frac{M_1 c}{\hbar}\Big)^2 \psi_1&=0\\
D^{\mu}D_{\mu}\phi_1+\Big(\frac{M_1 c}{\hbar}\Big)^2 \phi_1&=0 \\
D^{\mu}D_{\mu}\psi_2+\Big(\frac{M_2 c}{\hbar}\Big)^2 \psi_2&=0\\
D^{\mu}D_{\mu}\phi_2+\Big(\frac{M_2 c}{\hbar}\Big)^2 \phi_2&=0\; .\end{aligned}$$
Observe here that the first two equations (\[e3:70\]a)-(\[e3:70\]b) are governed by the gauge potential $A_{1 \mu}$ whereas the second half (\[e3:70\]c)-(\[e3:70\]d) is referred to the gauge potential $A_{2 \mu}$. Thus the interactions in our two-particle mixture become fixed by specifying the way in which the potentials $A_{a \mu}$ ($a=1,2$) are tight up to the RST currents $j_{a \mu}$, see below.
It has been demonstrated that, in order to avoid unphysical self-interactions [@MaRuSo01], the link between the potentials and currents must be made in such a way that the curvature $F_{1 \mu \nu}$ of the first particle couples to the second current $j_{2 \mu}$ $$\label{e3:71}
\nabla^{\mu}F_{1 \mu \nu}=4 \pi \alpha_{\ast} j_{2 \nu}$$ and vice versa for the second curvature component $$\label{e3:72}
\nabla^{\mu}F_{2 \mu \nu}=4 \pi \alpha_{\ast} j_{1 \nu} \; .$$ Thus it remains to be shown in what way both currents $j_{a \mu}$ ($a=1,2$) of the two-particle mixture are built up by the single-particle wave functions $\psi_a$, $\phi_a$. However this connection between wave functions and currents has already been clarified by the former prescriptions (\[e2:19\])-(\[e2:20\]); one simply has to insert there the present form of the intensity matrix ${{\mathcal{I}}}$ (\[e3:64\]) in order to find $$\begin{aligned}
\label{e3:73}
j_{a \mu}=i\frac{\hbar}{2Mc}\big\{ &I_{11}&[\psi^{\ast}_a(D_{\mu}\psi_a)-(D_{\mu}\psi^{\ast}_a)\psi_a]\nonumber \\
+ &I_{12}&[\phi^{\ast}_a(D_{\mu}\psi_a)-(D_{\mu}\phi^{\ast}_a)\psi_a]\nonumber \\
+ &I_{21}&[\psi^{\ast}_a(D_{\mu}\phi_a)-(D_{\mu}\psi^{\ast}_a)\phi_a]\nonumber \\
+&I_{22}&[\phi^{\ast}_a(D_{\mu}\phi_a)-(D_{\mu}\phi^{\ast}_a)\phi_a]\big\} \; .\end{aligned}$$ By prescribing special values to the constant matrix elements $I_{ab}$ (\[e3:64\]) one can show that the conventional Hartree and Hartree-fock approaches just form the non-relativistic approximations of the present two-particle RST (see a separate paper). Obviously in this wave-function picture, the RST entanglement consists in the fact that the sources $j_{a \mu}$ of the field strengths $F_{a \mu \nu}$ are a mixture of one-particle and interference currents.
It is instructive also to observe the degeneration of the general intensity matrix ${{\mathcal{I}}}$ to its pure-state form (\[e2:3\]) $$\label{e3:74}
{{\mathcal{I}}}\Rightarrow \Psi^{\prime}\otimes \overline{\Psi}^{\prime}$$ which is the tensor product of a two-particle state $\Psi^{\prime}$ and its Hermitian conjugate $\overline{\Psi}^{\prime}$. Such a degeneration occurs when the constant matrix ${{\mathcal{I}}}_{\ast}=\{I_{ab}\}$ (\[e3:64\]) becomes itself degenerate, i.e $$\label{e3:75}
\det {{\mathcal{I}}}_{\ast} \equiv I_{11}I_{22}-I_{12}I_{21}=0 \; .$$ For such a situation one can parametrize the four constant matrix elements $I_{ab}$ by two complex numbers $p$ and $b$ such that $$\label{e3:76}
\begin{array}{lr}
I_{11}=p^{\ast}p \quad&\quad I_{12}=b^{\ast}p \\
I_{21}=p^{\ast}b \quad&\quad I_{22}=b^{\ast}b
\end{array} \; .$$ This arrangement gives rise to recollect the four Klein-Gordon states $\{\psi_1$, $\psi_2$; $\phi_1$, $\phi_2\}$ into only two pure one-particle states $\psi_1^{\prime}$, $\psi_2^{\prime}$ according to
\[e3:77\] $$\begin{aligned}
\psi_1^{\prime}&=p\psi_1+b\phi_1\\
\psi_2^{\prime}&=p\psi_2+b\phi_2 \; .\end{aligned}$$
The meaning of the new states (\[e3:77\]) is immediately evident by introducing the restricted form of the matrix ${{\mathcal{I}}}_{\ast}$ (\[e3:76\]) into the RST currents (\[e3:73\]) which yields
\[e3:78\] $$\begin{aligned}
j_{1 \mu}& \Rightarrow j_{1 \mu}^{\prime}=i\frac{\hbar}{2Mc} \big[\psi_1^{\ast \prime} (D_{\mu} \psi_1^{\prime})-(D_{\mu}\psi_1^{\ast \prime})\psi_1^{\prime} \big]\\
j_{2 \mu}& \Rightarrow j_{2 \mu}^{\prime}=i\frac{\hbar}{2Mc} \big[\psi_2^{\ast \prime} (D_{\mu} \psi_2^{\prime})-(D_{\mu}\psi_2^{\ast \prime})\psi_2^{\prime} \big] \; .\end{aligned}$$
Maurer-Cartan Forms
===================
The curl relations (\[e3:29\_kurz\]) for the single-particle exchange fields $\{{{\mathbb{X}}}_{a}$, ${{\mathbb{\Gamma}}}_{a}$, ${{\mathbb{\Lambda}}}_{a}\}$, as well as the curl equations for the average exchange fields $\{\tilde{{\mathbb{X}}}$, $\tilde{{\mathbb{\Gamma}}}$, $\tilde{{\mathbb{\Lambda}}}\}$ (\[e3:59\]), are of a certain well-known structure in mathematics which is known as [*Maurer-Cartan structure equations*]{} [@GoSc]. The exploitation of this mathematical structure yields a deeper insight into the RST subsystem of exchange fields. The Maurer-Cartan forms (${{\mathcal{T}}}^j$, say; $j=1..$dim$\,{{\mathfrak{g}}}$) are left-invariant 1-forms over a Lie group $G$ with Lie algebra ${{\mathfrak{g}}}$. Let ${{\mathfrak{g}}}$ be spanned by a set $\{ \tau_k \}$ of left-invariant vector fields ($k=1..$dim$\, {{\mathfrak{g}}}$) with structure constants $C^l{}_{jk}$ $$\label{e4:1}
[\tau_j,\tau_k]=C^l{}_{jk}\tau_l \; .$$ The Maurer-Cartan forms ${{\mathcal{T}}}^j$ may then be taken as the dual objects of the Lie algebra generators $\tau_k$, i.e. the values of ${{\mathcal{T}}}^j$ upon $\tau_k$ is then given as usual by $$\label{e4:2}
<{{\mathcal{T}}}^j|\tau_k>=\delta^j{}_k \; .$$
It is possible to equip the Lie algebra ${{\mathfrak{g}}}$ (as a linear vector space) with a metric $g$, the [*Killing-Cartan form*]{}, such that any pair $\tau_j$, $\tau_k$ of generators is mapped into a real (or complex) number $g_{jk}$ $$\label{e4:3}
g(\tau_j,\tau_k)=g_{jk} \; .$$ Such a metric can be realized by means of the adjoint representation of the generators, i.e. one puts $$\label{e4:4}
g_{jk}=-\frac{1}{\mbox{dim}\, {\mathfrak{g}}}\cdot {\operatorname{tr}}\{({\mathfrak{Ad}}\tau_j)\cdot({\mathfrak{Ad}}\tau_k)\}$$ where the adjoint representation is given by $$\label{e4:5}
({\mathfrak{Ad}}\tau_j)^k{}_l=C^k{}_{jl} \; .$$ Alternatively one can look upon the metric $g$ as a bijective map from the Lie algebra $\mathfrak{g}$ to its dual space $\overline{\mathfrak{g}}$, in which the Maurer-Cartan forms are living $$\begin{aligned}
\label{e4:6}
g: \quad \mathfrak{g} \Rightarrow \overline{\mathfrak{g}} \nonumber \\
g^{-1}: \quad \overline{\mathfrak{g}}\Rightarrow\mathfrak{g}\end{aligned}$$ i.e. $$\begin{aligned}
\label{e4:7}
g(\tau_j)=g_{jk}{{\mathcal{T}}}^k \\
g^{-1}({{\mathcal{T}}}^j)=g^{jk}\tau_k \nonumber\\
(g^{jk}g_{kl}=\delta^j{}_l)\; . \nonumber\end{aligned}$$ Consequently, one can realize the duality relations (\[e4:2\]) by means of the Killing-Cartan form (\[e4:4\]), namely by putting $$\label{e4:8}
<{{\mathcal{T}}}^j|\tau_k>=g(g^{-1}({{\mathcal{T}}}^j), \tau_k)=g^{jl}g(\tau_l,\tau_k)=g^{jl}g_{lk}=\delta^j{}_k \; .$$
In the present context, the point with the Maurer-Cartan forms is now that they obey the structure equations [@GoSc] $$\label{e4:9}
{{\mathbb{d}}}{{\mathcal{T}}}^j=-\frac{1}{2}\, C^j{}_{kl}\, {{\mathcal{T}}}^k \wedge {{\mathcal{T}}}^l \; .$$ This looks already very similar to the curl relations for the exchange fields $\tilde{{\mathbb{X}}}_{a}$, $\tilde{{\mathbb{\Gamma}}}_{a}$, $\tilde{{\mathbb{\Lambda}}}_{a}$ (\[e3:59\]) or also for the fields ${{\mathbb{X}}}_{a}$, ${{\mathbb{\Gamma}}}_{a}$, ${{\mathbb{\Lambda}}}_{a}$ (\[e3:29\_kurz\]). Thus one may suppose that one could generate the desired exchange fields via the Maurer-Cartan forms over suitable Lie groups $G$. This method would then consist in establishing a suitable map from space-time onto the group $G$ ($x \Rightarrow {{\mathcal{G}}}_{(x)} \in G$) and considering the corresponding pullback of the Maurer-Cartan forms from $G$ to space-time. More concretely, let the group element ${{\mathcal{G}}}$ corresponding to the space-time event $x$ be ${{\mathcal{G}}}_{(x)}$, the Maurer-Cartan pullback $\tilde{{{\mathcal{C}}}}$ appears then in terms of ${{\mathcal{G}}}_{(x)}$ as $$\label{e4:10}
\tilde{{{\mathcal{C}}}}={{\mathcal{G}}}\cdot {{\mathbb{d}}}{{\mathcal{G}}}^{-1}={{\mathcal{C}}}_{\mu}{{\mathbb{d}}}x^{\mu} \; ,$$ i.e. in components (exploiting the isomorphism of ${{\mathfrak{g}}}$ and its dual $\overline{{{\mathfrak{g}}}}$ via the metric map (\[e4:6\]) $$\label{e4:11}
{{\mathcal{C}}}_{\mu}=E^j{}_{\mu}\tau_j={{\mathcal{G}}}_{(x)}\cdot \partial_{\mu} {{\mathcal{G}}}_{(x)}{}^{-1} \; .$$ On behalf of the structure equations (\[e4:9\]) the Maurer-Cartan pullback $\tilde{{{\mathcal{C}}}}={{\mathbb{E}}}^j\tau_j$ obeys the relation $$\label{e4:12}
{{\mathbb{d}}}\tilde{{{\mathcal{C}}}}+\tilde{{{\mathcal{C}}}}\wedge\tilde{{{\mathcal{C}}}}=0 \; ,$$ or written in the component fields ${{\mathbb{E}}}^j={{\mathbb{E}}}^j{}_{\mu}{{\mathbb{d}}}x^{\mu}$ $$\label{e4:13}
{{\mathbb{d}}}{{\mathbb{E}}}^j=-\frac{1}{2} C^j{}_{kl}{{\mathbb{E}}}^k\wedge{{\mathbb{E}}}^l \; .$$
Thus we are left with the problem of finding the right groups $G$ in order to identify their Maurer-Cartan pullbacks ${{\mathbb{E}}}^j$ with our former exchange fields, with the corresponding curl equations being then satisfied automatically.
Exchange Groups
---------------
It should not come as a surprise that the ordinary rotation group $SO(3)$ is the appropriate group for the positive mixtures and similarly the Lorentz group $SO(1,2)$ in ($1+2$) dimensions is adequate for the negative mixtures. The reason for this is that the structure constants for both groups can be taken as the totally anti-symmetric permutation tensor $\epsilon_{ijk}$ ($\epsilon_{123}=+1$), cf. (\[e4:1\]) $$\label{e4:14}
[\tau_j,\tau_k]=\epsilon^l{}_{jk}\tau_l$$ such that the components of the generators become for their adjoint representation (\[e4:5\]) $$\label{e4:15}
(\tau_j)^k{}_l=\epsilon^k{}_{jl} \; .$$ The corresponding Killing-Cartan metric $g_{jk}$ (\[e4:4\]) turns out as the (pseudo-)Euclidean case $\eta_{jk}$ $$\label{e4:16}
\{ \eta_{jk}\}=
\left(
\begin{array}{ccc} 1&0&0 \\ 0&\sigma_{\ast}&0\\0&0& \sigma_{\ast}
\end{array}
\right)$$
($\epsilon^k{}_{jl}=\eta^{ki}\epsilon_{ijl}$, etc.).
By this arrangement, the comparison of the former curl equations for the exchange fields ${{\mathbb{X}}}_a$, ${{\mathbb{\Gamma}}}_a$, ${{\mathbb{\Lambda}}}_a$ (\[e3:29\_kurz\]) with the present Maurer-Cartan structure equations (\[e4:13\]) admit the following two identifications
\[e4:17\] $$\begin{aligned}
{{\mathbb{E}}}_{(a)}{}^1 &=-2\sigma_{\ast}{{\mathbb{X}}}_a\\
{{\mathbb{E}}}_{(a)}{}^2 &=\mp 2 {{\mathbb{\Gamma}}}_a\\
{{\mathbb{E}}}_{(a)}{}^3 &=\pm 2{{\mathbb{\Lambda}}}_a \; .\end{aligned}$$
Furthermore, one can also compare the structure equations (\[e4:13\]) to the former curl equations for the average exchange fields $\tilde{{{\mathbb{X}}}}$, $\tilde{{\mathbb{\Gamma}}}$, $\tilde{{\mathbb{\Lambda}}}$ (\[e3:59\]) which again allows us to identify these fields with certain Maurer-Cartan forms $\tilde{{\mathbb{E}}}^j$ due to both groups $SO(3)$ and $SO(1,2)$
\[e4:18\] $$\begin{aligned}
\tilde{{\mathbb{E}}}^1 &\equiv -2\sigma_{\ast}\tilde{{\mathbb{X}}}\\
\tilde{{\mathbb{E}}}^2 &\equiv 2 \tilde{{\mathbb{\Gamma}}}\\
\tilde{{\mathbb{E}}}^3 &\equiv 2 \tilde{{\mathbb{\Lambda}}}\; .\end{aligned}$$
Euler Angles
------------
For the subsequent topological discussion a very helpful parametrization of the Maurer-Cartan forms (and thus also of the exchange fields) is obtained by specifying the generating group element ${{\mathcal{G}}}_{(x)}$ (\[e4:10\]) in terms of the “[*Euler angles*]{}” $\gamma_j$ ($j,k,l \ =1,2,3$) [@ChDeDi] $$\label{e4:19}
{{\mathcal{G}}}_{(\gamma_1,\gamma_2,\gamma_3)}={{\mathcal{G}}}_{1(\gamma_1)}\cdot {{\mathcal{G}}}_{2(\gamma_2)}\cdot {{\mathcal{G}}}_{3(\gamma_3)} \; .$$ Here the individual factors ${{\mathcal{G}}}_{j(\gamma_j)}$ are chosen as follows:
\[e4:20\] $$\begin{aligned}
{{\mathcal{G}}}_{1(\gamma_1)}&=\exp[\gamma_1\cdot \tau_3]\\
{{\mathcal{G}}}_{2(\gamma_2)}&=\exp[\gamma_2\cdot \tau_2]\\
{{\mathcal{G}}}_{3(\gamma_3)}&=\exp[\gamma_3\cdot \tau_3] \; ,\end{aligned}$$
and then the components ${{\mathbb{E}}}^j$ (\[e4:13\]) of the Maurer-Cartan form $\tilde{{{\mathcal{C}}}}$ (\[e4:10\]) are immediately obtained in terms of the Euler angles $\gamma_j$, namely for the positive mixtures as follows ($\sigma_{\ast}=+1$)
\[e4:21\] $$\begin{aligned}
{{\mathbb{E}}}^1 &=\sin \gamma_1 \cdot {{\mathbb{d}}}\gamma_2-\cos \gamma_1\cdot \sin \gamma_2\cdot {{\mathbb{d}}}\gamma_3\\
{{\mathbb{E}}}^2 &=-\sin \gamma_1 \cdot \sin \gamma_2\cdot {{\mathbb{d}}}\gamma_3 - \cos \gamma_1\cdot {{\mathbb{d}}}\gamma_2\\
{{\mathbb{E}}}^3 &=-{{\mathbb{d}}}\gamma_1-\cos \gamma_2\cdot {{\mathbb{d}}}\gamma_3\; ,\end{aligned}$$
and similarly for the negative mixtures ($\sigma_{\ast}=-1$)
\[e4:22\] $$\begin{aligned}
{{\mathbb{E}}}^1 &=\sinh \gamma_1 \cdot {{\mathbb{d}}}\gamma_2-\cosh \gamma_1\cdot \sinh \gamma_2\cdot {{\mathbb{d}}}\gamma_3\\
{{\mathbb{E}}}^2 &=\sinh \gamma_1 \cdot \sinh \gamma_2\cdot {{\mathbb{d}}}\gamma_3 - \cosh \gamma_1\cdot {{\mathbb{d}}}\gamma_2\\
{{\mathbb{E}}}^3 &=-{{\mathbb{d}}}\gamma_1-\cosh \gamma_2\cdot {{\mathbb{d}}}\gamma_3\; .\end{aligned}$$
Observe here that for the negative mixtures ($\sigma_{\ast}=-1$) there exists a second inequivalent parametrization by Euler angles in addition to the first parametrization (\[e4:20\]), namely
\[e4:23\] $$\begin{aligned}
{{\mathcal{G}}}_{1(\gamma_1)}&=\exp[\gamma_1\cdot \tau_3]\\
{{\mathcal{G}}}_{2(\gamma_2)}&=\exp[\gamma_2\cdot \tau_1]\\
{{\mathcal{G}}}_{3(\gamma_3)}&=\exp[\gamma_3\cdot \tau_3] \; .\end{aligned}$$
Due to this latter parametrization the Maurer-Cartan forms for the negative mixtures adopt a somewhat different shape, i.e.
\[e4:24\] $$\begin{aligned}
{{\mathbb{E}}}^1 &=-\{\cosh \gamma_1 \cdot {{\mathbb{d}}}\gamma_2-\sinh \gamma_1\cdot \sin \gamma_2\cdot {{\mathbb{d}}}\gamma_3\}\\
{{\mathbb{E}}}^2 &=\sinh \gamma_1\cdot {{\mathbb{d}}}\gamma_2 - \cosh \gamma_1 \cdot \sin \gamma_2\cdot {{\mathbb{d}}}\gamma_3\\
{{\mathbb{E}}}^3 &=-{{\mathbb{d}}}\gamma_1-\cos \gamma_2\cdot {{\mathbb{d}}}\gamma_3\; .\end{aligned}$$
The discussion of the negative mixtures runs quite analogously as for the positive mixtures, but since the negative-mixture topology is trivial we restrict ourselves to the positive case exclusively.
The point with the introduction of the Maurer-Cartan forms is now that our exchange fields ${{\mathbb{X}}}_a$, ${{\mathbb{\Gamma}}}_a$, ${{\mathbb{\Lambda}}}_a$ can be expressed in terms of the Euler angles $\gamma_{j(a)}$ for either particle ($a=1,2$) and thus the exchange coupling conditions (\[e3:20\])-(\[e3:21\]) can be transcribed to the corresponding coupling conditions for the Euler angles $\gamma_{j(1)}$ (first particle) and $\gamma_{j(2)}$ (second particle). This however gives a very simple result as we shall see readily. For the positive mixtures ($\sigma_{\ast}=+1$) the average exchange fields $\tilde{{\mathbb{X}}}$, $\tilde{{\mathbb{\Gamma}}}$, $\tilde{{\mathbb{\Lambda}}}$ adopt the following forms by simply combining the Maurer-Cartan identifications (\[e4:18\]) with the Euler angle parametrization (\[e4:21\])
\[e4:25\] $$\begin{aligned}
\tilde{{\mathbb{X}}}&\equiv-\frac{1}{2}\,\tilde{{\mathbb{E}}}^1 =-\frac{1}{2}\,\{\sin \tilde{\gamma_1} \cdot {{\mathbb{d}}}\tilde{\gamma_2}-\cos \tilde{\gamma_1}\cdot \sin \tilde{\gamma_2}\cdot {{\mathbb{d}}}\tilde{\gamma_3}\}\\
\tilde{{\mathbb{\Gamma}}}&\equiv -\frac{1}{2}\,\tilde{{\mathbb{E}}}^2 = \frac{1}{2}\,\{\sin \tilde{\gamma_1} \cdot \sin \tilde{\gamma_2}\cdot {{\mathbb{d}}}\tilde{\gamma_3} + \cos \tilde{\gamma_1}\cdot {{\mathbb{d}}}\tilde{\gamma_2}\}\\
\tilde{{\mathbb{\Lambda}}}&\equiv\tilde{{\mathbb{E}}}^3 =-{{\mathbb{d}}}\tilde{\gamma_1}-\cos \tilde{\gamma_2}\cdot {{\mathbb{d}}}\tilde{\gamma_3}\; .\end{aligned}$$
By use of this result, the single-particle exchange fields ${{\mathbb{X}}}_a$, ${{\mathbb{\Gamma}}}_a$, ${{\mathbb{\Lambda}}}_a$ (\[e3:62\]) look then as follows
\[e4:26\] $$\begin{aligned}
{{\mathbb{X}}}_1&=-\frac{1}{2}\,\{\sin (\tilde{\gamma_1}-\frac{\epsilon}{2}) \cdot {{\mathbb{d}}}\tilde{\gamma_2}- \sin \tilde{\gamma_2}\cdot \cos( \tilde{\gamma_1}-\frac{\epsilon}{2})\cdot {{\mathbb{d}}}\tilde{\gamma_3}\}\\
{{\mathbb{\Gamma}}}_1&=\frac{1}{2}\,\{\cos (\tilde{\gamma_1}-\frac{\epsilon}{2})\cdot {{\mathbb{d}}}\tilde{\gamma_2}+\sin \tilde{\gamma_2} \cdot \sin (\tilde{\gamma_1}-\frac{\epsilon}{2})\cdot {{\mathbb{d}}}\tilde{\gamma_3}\} \\
{{\mathbb{\Lambda}}}_1&=-\frac{1}{2}\,\{{{\mathbb{d}}}(\tilde{\gamma_1}-\frac{\epsilon}{2})+\cos \tilde{\gamma_2}\cdot {{\mathbb{d}}}\tilde{\gamma_3}\}\\
{{\mathbb{X}}}_2&=\frac{1}{2}\,\{\cos (\tilde{\gamma_1}+\frac{\epsilon}{2})\cdot {{\mathbb{d}}}\tilde{\gamma_2}+\sin \tilde{\gamma_2} \cdot \sin (\tilde{\gamma_1}+\frac{\epsilon}{2})\cdot {{\mathbb{d}}}\tilde{\gamma_3}\} \\
{{\mathbb{\Gamma}}}_2&=\frac{1}{2}\,\{\sin \tilde{\gamma_2}\cdot \cos( \tilde{\gamma_1}+\frac{\epsilon}{2})\cdot {{\mathbb{d}}}\tilde{\gamma}_3- \sin (\tilde{\gamma_1}+\frac{\epsilon}{2}) \cdot {{\mathbb{d}}}\tilde{\gamma_2}\}\\
{{\mathbb{\Lambda}}}_2&=\frac{1}{2}\,\{{{\mathbb{d}}}(\tilde{\gamma_1}+\frac{\epsilon}{2})+\cos \tilde{\gamma_2}\cdot {{\mathbb{d}}}\tilde{\gamma_3}\} \; .\end{aligned}$$
On the other hand, these single-particle exchange fields may also be written in terms of the single-particle angles $\gamma_{j(a)}$ (\[e4:21\]) via the Maurer-Cartan identifications (\[e4:17\]) as ($\sigma_{\ast}=+1$)
\[e4:27\] $$\begin{aligned}
{{\mathbb{X}}}_a & \equiv-\frac{1}{2}\,{{\mathbb{E}}}^1{}_{(a)} =-\frac{1}{2}\,\{\sin \gamma_{1(a)} \cdot {{\mathbb{d}}}\gamma_{2(a)}-\cos \gamma_{1(a)}\cdot \sin \gamma_{2(a)}\cdot {{\mathbb{d}}}\gamma_{3(a)}\}\\
{{\mathbb{\Gamma}}}_a & \equiv-\frac{1}{2}\,{{\mathbb{E}}}^2{}_{(a)} = \frac{1}{2}\,\{\sin \gamma_{1(a)} \cdot \sin \gamma_{2(a)}\cdot {{\mathbb{d}}}\gamma_{3(a)} + \cos \gamma_{1(a)}\cdot {{\mathbb{d}}}\gamma_{2(a)}\}\\
{{\mathbb{\Lambda}}}_a &\equiv\frac{1}{2}\, {{\mathbb{E}}}^3{}_{(a)} =-\frac{1}{2}\,\{{{\mathbb{d}}}\gamma_{1(a)}+\cos \gamma_{1(a)}\cdot {{\mathbb{d}}}\gamma_{3(a)}\}\; .\end{aligned}$$
Thus, comparing both versions (\[e4:26\]) and (\[e4:27\]) immediatly yields the desired exchange coupling conditions upon the Euler angles $\gamma_{j(a)}$ ($\sigma_{\ast}=+1$) in a very simple form:
\[e4:28\] $$\begin{aligned}
\gamma_{1(1)}&=\tilde{\gamma_1}-\frac{\epsilon}{2}\\
\gamma_{1(2)}&=\frac{\pi}{2}-(\tilde{\gamma_1}+\frac{\epsilon}{2})\\
\gamma_{2(1)}&=\tilde{\gamma_2}\\
\gamma_{2(2)}&=\pi-\tilde{\gamma_2}\\
\gamma_{3(1)}&=\gamma_{3(2)}=\tilde{\gamma_3} \; .\end{aligned}$$
Clearly this coupling of the Euler angles $\gamma_{j(a)}$ actually represents a coupling of both group elements ${{\mathcal{G}}}_{a(x)}$ which necessarily must result also in the corresponding coupling of the Maurer-Cartan form ${{\mathbb{E}}}^j{}_{(a)}$
\[e4:28\_2\] $$\begin{aligned}
{{\mathbb{E}}}^1{}_{(2)}&=\cos \epsilon \cdot {{\mathbb{E}}}^2{}_{(1)}+\sin \epsilon \cdot {{\mathbb{E}}}^1{}_{(1)}\\
{{\mathbb{E}}}^2{}_{(2)}&=\cos \epsilon \cdot {{\mathbb{E}}}^1{}_{(1)}-\sin \epsilon \cdot {{\mathbb{E}}}^2{}_{(1)}\\
{{\mathbb{E}}}^3{}_{(2)}&={{\mathbb{E}}}^3{}_{(1)}+{{\mathbb{d}}}\epsilon \; .\end{aligned}$$
Summarizing the present results with the Euler angles one encounters a very pleasant result; namely the parametrization of the exchange fields ${{\mathbb{X}}}_a$, ${{\mathbb{\Gamma}}}_a$, ${{\mathbb{\Lambda}}}_a$ yields a general form of these exchange fields which automatically obeys those relatively complicated exchange coupling conditions (\[e3:20\]); and additionally all the curl relations for the exchange fields are also obeyed automatically! Furthermore, the Euler angle parametrization provides one with a convenient possibility to incorporate certain symmetries into the desired RST solutions. For instance, imagine that we want to look for static solutions where all RST fields become time-independent, e.g. think of the bound solutions in an attractive potential. For such a situation one expects that all scalar fields, such as the Euler angles, cannot depend upon time $t$ but will exclusively depend on the space position ${\vec r}$. Moreover, the time-component of the four-vectors $X_{a \mu}$, $\Gamma_{a\mu}$, $\Lambda_{a\mu}$ must also be time-independent, e.g. we put
\[e4:34\] $$\begin{aligned}
X_{a \mu}&=X_a(\vec{r})\cdot \hat{t}_{\mu}\\
\Gamma_{a \mu}&=\Gamma_a(\vec{r})\cdot \hat{t}_{\mu} \; .\end{aligned}$$
All these conditions can easily be satisfied by putting for the Euler angles
\[e4:35\] $$\begin{aligned}
\epsilon&=\epsilon(\vec{r})\\
\tilde{\gamma_1}&=\tilde{\gamma_1}(\vec{r})\\
\tilde{\gamma_2}&=K_{\ast}\cdot t \quad \mbox{($K_{\ast}=$const.)}\\
\tilde{\gamma_3}&=\mbox{const.}\end{aligned}$$
This then yields the static form of the exchange fields:
\[e4:36\] $$\begin{aligned}
X_{1 \mu}&=-\frac{1}{2}\; K_{\ast} \cosh (\tilde{\gamma_1}+\frac{\epsilon}{2}) \cdot \hat{t}_{\mu}\\
\Gamma_{1 \mu}&=-\frac{1}{2}\;K_{\ast} \sinh(\tilde{\gamma_1}+\frac{\epsilon}{2})\cdot \hat{t}_{\mu}\\
\Lambda_{1 \mu}&=-\frac{1}{2}\; \partial_{\mu} (\tilde{\gamma_1}+\frac{\epsilon}{2})\\
X_{2 \mu}&=-\frac{1}{2}\; K_{\ast} \cosh (\tilde{\gamma_1}-\frac{\epsilon}{2}) \cdot \hat{t}_{\mu}\\
\Gamma_{2 \mu}&=\frac{1}{2}\;K_{\ast} \sinh(\tilde{\gamma_1}-\frac{\epsilon}{2})\cdot \hat{t}_{\mu}\\
\Lambda_{2 \mu}&=\frac{1}{2}\; \partial_{\mu} (\tilde{\gamma_1}-\frac{\epsilon}{2})\; .\end{aligned}$$
It was exactly this form of the exchange fields which has been applied for a treatment of the bound two-particle states in an attractive Coulomb force field [@RuSo01_2].
Three-Vector Parametrization
----------------------------
For the subsequent computation of winding numbers it is very helpful to consider also an alternative parametrization of the group elements ${{\mathcal{G}}}_{(x)}$ (\[e4:20\]), namely the parametrization by a “three-vector” $\{\xi^j;j=1,2,3\}$ $$\label{e4:37}
{{\mathcal{G}}}_{(x)}=\exp[\xi^j{}_{(x)}\tau_j] \; .$$ It is true, expressing the exchange coupling condition in terms of this three-vector $\vec{\xi}$ is somewhat cumbersome; but on the other hand the parametrization (\[e4:37\]) enables one to easily recognize the way how the topological quantum numbers come about in RST. Therefore it is instructive to consider the three-vector parametrization in some detail.
The adjoint transformation $G \rightarrow G$ $$\label{e4:38}
{{\mathcal{G}}}_{(x)}\rightarrow {{\mathcal{G}}}^{\prime}{}_{(x)}={{\mathcal{S}}}\cdot {{\mathcal{G}}}_{(x)}\cdot {{\mathcal{S}}}^{-1}$$
(${{\mathcal{S}}}\in G$)
acts over the groups $G=SO(3)$ and $G=SO(1,2)$ in a somewhat different way. Since the three-vector $\vec{\xi}=\{\xi^j\}$ changes under the transformation (\[e4:38\]) according to $$\label{e4:39}
\xi^j \rightarrow \xi^{\prime j}=S^j{}_k \xi^k \; ,$$ where the matrix $\{ S^j{}_k\}$ is identical to ${{\mathcal{S}}}$, the three-vector $\vec{\xi}$ can be rotated into any direction in group space for $SO(3)$ but not for $SO(1,2)$. The reason is that the Killing metric $g_{jk}$ (\[e4:4\]) is invariant with respect to these transformations i.e. $$\label{e4:40}
\eta_{jk}S^j{}_lS^k{}_m=\eta_{lm}$$ and therefore the “length” of the three-vector is preserved $$\label{e4:41}
\xi^{\prime j}\xi^{\prime}{}_j=\xi^j\xi_j \equiv \eta_{ij}\xi^i\xi^j \; .$$ This condition does not restrict the possible end configurations of $\vec{\xi}$ for $SO(3)$ but it does for $SO(1,2)$. More concretely, since the Killing metric $\eta_{jk}$ (\[e4:16\]) is indefinite for the negative mixtures ($\sigma_{\ast}=-1$), the three-vector $\vec{\xi}$ can never cross the “light-cone” $\xi^j\xi_j=0$ and thus the adjoint transformation (\[e4:38\]) leaves invariant two three-dimensional subspaces in $SO(1,2)$, namely those which have $\xi^j\xi_j>0$ and $\xi^j\xi_j<0$. This is the reason why one has to resort to two different parametrizations of the exchange fields for the negative mixtures.
In detail, when $\vec{\xi}$ is “time-like” (i.e. $\xi^j\xi_j>0$) the Maurer-Cartan components ${\Bbb E}^j_{(a)}$ (\[e4:11\]) are found to be of the following form for the positive and negative mixtures ($\sigma_{\ast}=+1$) $$\label{e4:42}
{{\mathbb{E}}}^j=-u^j{{\mathbb{d}}}\xi-\sin \xi \cdot {{\mathbb{d}}}u^j+(\cos \xi -1)\epsilon^j{}_{kl}u^k{{\mathbb{d}}}u^l \; .$$ Here, a nearby decomposition of the three-vector $\vec{\xi}$ into a unit vector $\vec{u}$ ($u^j u_j=1$) and the “length” $\xi$ has been used, i.e. $$\label{e4:43}
\xi^j=\xi \cdot u^j \; .$$
Change of Parametrization
-------------------------
When different parametrizations are found to be advantageous for different purposes, it becomes necessary to specify the transformations which connect those various forms of parametrizations. The three-vector $\vec{\xi}$ parametrizing the group element ${{\mathcal{G}}}$ (\[e4:37\]) can be expressed in terms of ${{\mathcal{G}}}$ itself in the following way: first extract the “length” $\xi$ of the three-vector through ($\sigma_{\ast}=+1$) $$\label{e4:45}
\cos \xi =-1+\frac{1}{2} {\operatorname{tr}}{{\mathcal{G}}}$$ and then find the unit vector $\vec{u}=\{u^j\}$ (\[e4:43\]) through $$\label{e4:46}
u^k\eta_{kj}=u_j=-\frac{1}{2\sin \xi} {\operatorname{tr}}({{\mathcal{G}}}\cdot \tau_j) \; .$$ Thus, referring to the Euler parametrization of ${{\mathcal{G}}}$ (\[e4:19\])-(\[e4:20\]), yields the length of the three-vector $\vec{\xi}$ in terms of the Euler angles as ($\sigma_{\ast}=+1$) $$\label{e4:47}
\cos \xi=\frac{1}{2} \Big\{ \cos \gamma_2-1+(1+\cos \gamma_2)\cdot \cos(\gamma_1+\gamma_3) \Big\}$$ and then the unit vector $\vec{u}=\{u^j\}$ is found from (\[e4:46\]) as ($\sigma_{\ast}=+1$)
\[e4:48\] $$\begin{aligned}
u^1&=\frac{\sin \gamma_2}{2 \sin \xi} \Big( \sin \gamma_3 - \sin \gamma_1 \Big)\\
u^2&=\frac{\sin \gamma_2}{2 \sin \xi} \Big( \cos \gamma_1 + \cos \gamma_3 \Big)\\
u^3&=\frac{1+\cos \gamma_2}{2 \sin \xi} \sin (\gamma_1 +\gamma_3) \; .\end{aligned}$$
The advantage of working with both parametrizations comes now into play when we transcribe the exchange coupling condition between both particles from the Euler parametrizations (\[e4:28\]), where this condition looks very simple, to the three-vector parametrizations for which that coupling condition would have never been found directly. Thus, for the positive mixtures we obtain the three-vector $\vec{\xi}$ in terms of the average Euler angles $\tilde{\gamma_j}$ and exchange angle $\epsilon$ by combining the present result (\[e4:47\])-(\[e4:48\]) with the former coupling conditions (\[e4:28\]) in order to find for the first particle ($\sigma_{\ast}=+1$)
\[e4:52\] $$\begin{aligned}
\cos \xi_{(1)}&=\frac{1}{2} \Big\{ \cos \tilde{\gamma_2} -1+(1+\cos \tilde{\gamma_2}) \cdot \cos (\tilde{\gamma_1}+\tilde{\gamma_3} -\frac{\epsilon}{2})\Big\} \\
u^1{}_{(1)}&=\frac{\sin \tilde{\gamma_2}}{2 \sin \xi_{(1)}} \Big\{ \sin \tilde{\gamma_3} - \sin (\tilde{\gamma_1} -\frac{\epsilon}{2})\Big\} \\
u^2{}_{(1)}&=\frac{\sin \tilde{\gamma_2}}{2 \sin \xi_{(1)}} \Big\{ \cos (\tilde{\gamma_1}-\frac{\epsilon}{2})+\cos \tilde{\gamma_3} \Big\} \\
u^3{}_{(1)}&=\frac{1+\cos \tilde{\gamma_2}}{2 \sin \xi_{(1)}} \sin(\tilde{\gamma_1}+\tilde{\gamma_3}-\frac{\epsilon}{2})\end{aligned}$$
and similarly for the second particle
\[e4:53\] $$\begin{aligned}
\cos \xi_{(2)}&=\frac{1}{2} \Big\{ -(1+\cos \tilde{\gamma_2})+(1-\cos \tilde{\gamma_2}) \cdot \sin(\tilde{\gamma_3}-\tilde{\gamma_1}-\frac{\epsilon}{2}) \Big\}\\
u^1{}_{(2)}&=\frac{\sin \tilde{\gamma_2}}{2 \sin \xi_{(2)}} \Big\{ \sin \tilde{\gamma_3} -\cos(\tilde{\gamma_1} +\frac{\epsilon}{2}) \Big\}\\
u^2{}_{(2)}&=\frac{\sin \tilde{\gamma_2}}{2 \sin \xi_{(2)}} \Big\{ \cos \tilde{\gamma_3} + \sin (\tilde{\gamma_1}+\frac{\epsilon}{2}) \Big\}\\
u^3{}_{(2)}&=\frac{1-\cos \tilde{\gamma_2}}{2 \sin \xi_{(2)}} \cos (\tilde{\gamma_3} -\tilde{\gamma_1} -\frac{\epsilon}{2}) \; .\end{aligned}$$
Winding Numbers ($\sigma_{\ast}=+1$)
====================================
These results become now relevant for revealing the correlation of the topological quantum numbers of both particles where this correlation is a consequence of the exchange coupling. First recall that either particle requires its own map ${{\mathcal{G}}}_{a(x)}$ ($a=1,2$) from space-time to the exchange group $G$ (i.e. $SO(3)$ or $SO(1,2)$). However, on behalf of the exchange coupling conditions, both group elements ${{\mathcal{G}}}_{a(x)}$ become coupled as shown, e.g., by the present three-vector parametrizations (\[e4:52\])-(\[e4:53\]). This coupling of both group elements ${{\mathcal{G}}}_a$ can be viewed in such a way that an “average” group element $\tilde{{{\mathcal{G}}}}$ is introduced for both particles. This average $\tilde{{{\mathcal{G}}}}$ may be parametrized, e.g. by the average three-vector $\tilde{\xi}^j=\tilde{\xi}\cdot \tilde{u}^j$ and then the individual group elements ${{\mathcal{G}}}_a={{\mathcal{G}}}_a(\xi^j{}_{(a)})$ are constructed from the average group element $\tilde{G} (\tilde{\xi}^j)$ by shifting the average $\tilde{\xi}^j$ to the individual $\xi^j{}_{(a)}$ for either particle as shown by the present results, e.g. (\[e4:52\])-(\[e4:53\]) for the positive mixtures. Clearly this latter map $\tilde{{{\mathcal{G}}}} \Rightarrow {{\mathcal{G}}}_a$ ($a=1,2$) is described by the left action of certain elements ${{\mathcal{S}}}_{(a)}$ upon the average group element $\tilde{{{\mathcal{G}}}}$. Thus, any three-cycle $C^3$ of space-time is first mapped into the exchange group $SO(3)$ via $\tilde{{{\mathcal{G}}}}(x)$ and thereby generates the “average” winding number $\tilde{Z}[C^3]$ as the number of times how often the exchange group is covered when the original $C^3$ is swept out once. Subsequently, the maps $\tilde{{{\mathcal{G}}}} \rightarrow {{\mathcal{G}}}_{(a)}$ mediated by the elements ${{\mathcal{S}}}_{(a)}$ generate the corresponding winding numbers $\tilde{l}_{(a)}$ in an analogous way, such that ultimately the exchange group becomes covered $\tilde{Z}\cdot \tilde{l}_{(a)}$ times, yielding the total winding numbers $Z_{(a)}$ ($a=1,2$) as $$\label{e5:1}
Z_{(a)}=\tilde{Z} \cdot \tilde{l}_{(a)} \; .$$
On the other hand, one may consider the map from space-time to the exchange group $G=SO(3)$ with the first Euler angle $\tilde{\gamma_1}$ being kept constant, however the exchange angle $\epsilon$ being assumed as the given non-trivial space-time function. This map induces identical winding numbers ($Z_{\epsilon}$, say) for both particles, as may be seen from the exchange coupling (\[e4:28\]). Since the average winding number $\tilde{Z}$ applies for the case when the exchange angle $\epsilon$ is kept constant, the general situation (\[e5:1\]) is associated with the sum/difference of both winding numbers $\tilde{Z}$ and $Z_{\epsilon}$, i.e.
\[e5:2\] $$\begin{aligned}
Z_{(1)}&=\tilde{Z}\cdot \tilde{l}_{(1)} = \tilde{Z}-Z_{\epsilon}\\
Z_{(2)}&=\tilde{Z}\cdot \tilde{l}_{(2)} = \tilde{Z}+Z_{\epsilon} \; .\end{aligned}$$
This however says for the “interior” winding numbers $\tilde{l}_{(a)}$
\[e5:3\] $$\begin{aligned}
\tilde{l}_{(1)} &=1-\frac{Z_{\epsilon}}{\tilde{Z}} \\
\tilde{l}_{(2)}&=1+\frac{Z_{\epsilon}}{\tilde{Z}} \; ,\end{aligned}$$
i.e. the ratio of winding numbers $\frac{Z_{\epsilon}}{\tilde{Z}}$ must also be an integer ($\tilde{n}$, say) $$\label{e5:4}
\frac{Z_{\epsilon}}{\tilde{Z}}\doteqdot \tilde{n} \; .$$ Consequently the winding numbers $Z_{(a)}$ of both particles ($a=1,2$) are correlated in the following manner ($\tilde{n}=0,\pm 1,\pm 2,...$)
\[e5:6\] $$\begin{aligned}
Z_{(1)}&=(1-\tilde{n})\tilde{Z}\\
Z_{(2)}&=(1+\tilde{n})\tilde{Z} \; .\end{aligned}$$
This result says that [*both winding numbers $Z_{(a)}$ are either odd or even but are otherwise completely unrestricted*]{}.
Subsequently we give a more rigorous treatment of these rather heuristic arguments.
Invariant Volume
----------------
It should be immediately obvious from the very definition of the Maurer-Cartan form $\tilde{{{\mathcal{C}}}}$ (\[e4:10\]) that it transforms under the adjoint map when its generating group element ${{\mathcal{G}}}$ is acted upon by some (constant) element ${{\mathcal{S}}}$ of the exchange group $G$
\[e5:7\] $$\begin{aligned}
{{\mathcal{G}}}&\Rightarrow {{\mathcal{G}}}^{\prime}= {{\mathcal{S}}}\cdot {{\mathcal{G}}}\\
\tilde{{{\mathcal{C}}}}&\Rightarrow \tilde{{{\mathcal{C}}}}^{\prime}={{\mathcal{S}}}\cdot\tilde{{{\mathcal{C}}}} \cdot {{\mathcal{S}}}^{-1} \; .\end{aligned}$$
Therefore, referring to the adjoint representation of the exchange group $G$, it is possible to introduce a volume 3-form ${{\mathbb{V}}}$ over $G$ $$\label{e5:8}
{{\mathbb{V}}}=-g_{\ast} {\operatorname{tr}}({{\mathcal{C}}}\wedge {{\mathcal{C}}}\wedge {{\mathcal{C}}})$$ with some real normalization constant $g_{\ast}$. Obviously, this volume form is invariant under the left action of ${{\mathcal{S}}}$ (\[e5:7\]) and thus attributes an invariant volume $V$ to the exchange group $G$ if the latter is compact (i.e. for $SO(3)$ but not for $SO(1,2)$); $$\label{e5:9}
V=\int_G {{\mathbb{V}}}$$ (Haar measure [@ChDeDi]).Decomposing here the pullback $\tilde{{{\mathcal{C}}}}$ of the Maurer-Cartan form ${{\mathcal{C}}}$ into its components ${{\mathbb{E}}}^j$ (\[e4:11\]) yields for the corresponding pullback $\tilde{{{\mathbb{V}}}}$ of the volume form ${{\mathbb{V}}}$ $$\label{e5:10}
\tilde{{{\mathbb{V}}}}=-g_{\ast}{{\mathbb{E}}}^j \wedge {{\mathbb{E}}}^k \wedge {{\mathbb{E}}}^l \cdot {\operatorname{tr}}(\tau_j\cdot \tau_k \cdot \tau_l)=g_{\ast}\epsilon_{ijk}{{\mathbb{E}}}^i\wedge {{\mathbb{E}}}^j \wedge {{\mathbb{E}}}^k$$ which applies to both kinds of mixtures ($\sigma_{\ast}=\pm 1$).
However, a crucial difference between both types of mixtures becomes now evident when expressing the volume form ${{\mathbb{V}}}$ in terms of the Euler variables $\gamma_j$: namely, for the positive mixtures one finds by referring to their Maurer-Cartan forms (\[e4:21\]) $$\label{e5:11}
{{\mathbb{V}}}_+=3!\, g_{\ast} \sin \gamma_2 {{\mathbb{d}}}\gamma_1 \wedge {{\mathbb{d}}}\gamma_2 \wedge {{\mathbb{d}}}\gamma_3$$
($\sigma_{\ast}=+1$)
and similarly for the negative mixtures by use of their Maurer-Cartan forms (\[e4:22\]) $$\label{e5:12}
{{\mathbb{V}}}_-=3!\, g_{\ast} \sinh \gamma_2 {{\mathbb{d}}}\gamma_1 \wedge {{\mathbb{d}}}\gamma_2 \wedge {{\mathbb{d}}}\gamma_3$$
($\sigma_{\ast}=-1$).
Obviously the exchange group $SO(3)$ is covered once when the Euler angles sweep out their possible range of values $0<(\gamma_1$, $\gamma_3) \le 2 \pi$, $0 \le \gamma_2 \le \pi$, so that the invariant volume $V$ (\[e5:9\]) becomes ($\sigma_{\ast}=1$) $$\label{e5:13}
V=6g_{\ast}\int_0^\pi \sin \gamma_2 d\gamma_2 \int_0^{2\pi}d\gamma_1 \int_0^{2 \pi} d \gamma_3=48 g_{\ast} \pi^2 \; ,$$ whereas such a definite volume $V$ cannot be attributed to the negative mixtures.
Concerning the fixing of the normalization constant $g_{\ast}$, it is better to refer this to the universal covering group $SU(2)$ of $SO(3)$ because this has the same topology as the 3-cycle $C^3$ in space-time over which the pullback of the volume form ${{\mathbb{V}}}$ has to be integrated over for any particle ($a=1,2$) in order to define its winding number $Z_{(a)}$ $$\label{e5:14}
Z_{(a)} = <\tilde{{{\mathbb{V}}}}|C^3>\doteqdot \oint_{C^3}\tilde{{{\mathbb{V}}}}$$ as the number of times to cover $SO(3)$ when $C^3$ is swept out once. Since for a diffeomorphic map the universal covering group $SU(2)$ is swept out once when its homomorphic image $SO(3)$ is covered twice we put $g_{\ast}=(96\pi^2)^{-1}$ and thus find for the volume form ${{\mathbb{V}}}$ (\[e5:11\]) of the positive mixtures ($\sigma_{\ast}=+1$) $$\label{e5:15}
{{\mathbb{V}}}_{(a)}=\frac{1}{(4\pi)^2} \sin \gamma_{2(a)} {{\mathbb{d}}}\gamma_{1(a)} \wedge {{\mathbb{d}}}\gamma_{2(a)} \wedge {{\mathbb{d}}}\gamma_{3(a)} \; .$$
Once the volume forms ${{\mathbb{V}}}$ of both particles have thus been obtained, it becomes easy to verify the preceding guess for the winding numbers $Z_{(a)}$ (\[e5:2\])-(\[e5:3\]). Indeed, one merely has to substitute the exchange coupling conditions (\[e4:28\]) into those volume forms ${{\mathbb{V}}}_{(a)}$ (\[e5:15\]) and then to compute the individual winding numbers $Z_{(a)}$ (\[e5:14\]). This yields immediately
\[e5:16\] $$\begin{aligned}
\tilde{Z}&=\frac{1}{(4\pi)^2} \oint_{C^3} \sin \tilde{\gamma_2} {{\mathbb{d}}}\tilde{\gamma_1}\wedge {{\mathbb{d}}}\tilde{\gamma_2} \wedge {{\mathbb{d}}}\tilde{\gamma_3}\\
Z_{\epsilon}&=\frac{1}{(4\pi)^2}\oint_{C^3}\sin \tilde{\gamma_2} ({{\mathbb{d}}}\frac{\epsilon}{2}) \wedge {{\mathbb{d}}}\tilde{\gamma_2} \wedge {{\mathbb{d}}}\tilde{\gamma_3} \end{aligned}$$
and thus the presumed result (\[e5:6\]) is verified. Here the interior winding numbers $\tilde{l}_{(a)}$ (\[e5:1\]) are given by $$\label{e5:17}
\tilde{l}_{(a)}=\oint_{SO(3)} \frac{\partial (\gamma_1,\gamma_2,\gamma_3)}{\partial (\tilde{\gamma_1},\tilde{\gamma_2},\tilde{\gamma_3})} \cdot \sin \tilde{\gamma_2} {{\mathbb{d}}}\tilde{\gamma_1}\wedge {{\mathbb{d}}}\tilde{\gamma_2} \wedge {{\mathbb{d}}}\tilde{\gamma_3}\; ,$$ where the Jacobian of the interior transformation $\tilde{{{\mathcal{G}}}} \rightarrow {{\mathcal{G}}}_a$ multiplies the invariant volume element of the exchange group $SO(3)$.
Universal Covering group
------------------------
Properly speaking, the RST exchange fields do not refer to the exchange group itself but rather to its Lie algebra via the corresponding Maurer-Cartan form $\tilde{{{\mathcal{C}}}}$ (\[e4:10\]). Therefore one can take as the exchange groups also the universal covering groups of $SO(3)$ and $SO(1,2)$, namely $SU(2)$ and $SU(1,1)$, resp. For the compact case, one constructs a unit four-vector $U^{\alpha}$ ($U^{\alpha}U_{\alpha}=\eta_{\alpha \beta}U^{\alpha}U^{\beta}=1$, $\eta_{\alpha \beta}=\mbox{diag} [1,1,1,1]$) from the three-vector $\vec{\xi}$ (\[e4:43\]) in the following way:
\[e5:18\] $$\begin{aligned}
U^0&=\cos \frac{\xi}{2}\\
U^j&=\sin\frac{\xi}{2}\cdot u^j \;.\end{aligned}$$
This yields the $SU(2)$ group element ${{\mathcal{G}}}(x)$ as $$\label{e5:19}
{{\mathcal{G}}}=U^0\cdot {\bf 1}-iU^j\sigma_j$$ where $\sigma_j$ are the Pauli matrices as ususal. The associated Mauerer-Cartan form $\tilde{{{\mathcal{C}}}}$ (\[e4:10\])-(\[e4:11\]) reads now $$\label{e5:20}
{{\mathcal{C}}}_{\mu}=-\frac{i}{2} E^j{}_{\mu} \sigma_j$$ with the Maurer-Cartan forms $E^j{}_{\mu}$ given by (\[e4:42\]). Clearly, for the negative mixtures one would introduce the unit four-vector $U^{\alpha}$ through
\[e5:21\] $$\begin{aligned}
U^0&=\cosh \frac{\xi}{2}\\
U^j&=\sinh \frac{\xi}{2}\cdot u^j \end{aligned}$$
and would replace the $SU(2)$ generators $\tau_j(=-\frac{i}{2}\sigma_j)$ by the corresponding ${\mathfrak{su}}(1,1)$ generators in order to find the curresponding Maurer-Cartan forms.
The pleasant effect with the choice of the universal covering group $SU(2)$ for the positive mixtures is now that its nomalized volume form $\tilde{{{\mathbb{V}}}}$ (\[e5:10\]) $$\label{e5:22}
\tilde{{{\mathbb{V}}}}=-\frac{1}{24 \pi^2} {\operatorname{tr}}(\tilde{{{\mathcal{C}}}} \wedge \tilde{{{\mathcal{C}}}} \wedge \tilde{{{\mathcal{C}}}}) =-\frac{1}{96\pi^2}\epsilon_{jkl}{{\mathbb{E}}}^j \wedge {{\mathbb{E}}}^k \wedge {{\mathbb{E}}}^l$$ reads in the three-vector parametrization $$\label{e5:23}
\tilde{{{\mathbb{V}}}}=\frac{1}{(2\pi)^2} \sin^2 \frac{\xi}{2} {{\mathbb{d}}}\xi \wedge \Big\{\frac{1}{2} \epsilon_{jkl} u^j({{\mathbb{d}}}u^k)\wedge ({{\mathbb{d}}}u^l) \Big\} \; ,$$ or even more concisely in the four-vector parametrization
$$\label{e5:24}
\tilde{{{\mathbb{V}}}}=\frac{3!}{2\pi^2} \epsilon_{\alpha \beta \gamma \delta} U^{\alpha}({{\mathbb{d}}}U^{\beta}) \wedge ({{\mathbb{d}}}U^{\gamma}) \wedge ({{\mathbb{d}}}U^{\delta}) \; .$$
Obviously this is just the volume form over the 3-sphere $S^3$, parametrized by the unit four-vector $U^{\alpha}$ (\[e5:18\]) and divided by the invariant volume $V=2 \pi^2$ (\[e5:9\]) of the compact exchange group $SU(2)$. As a consequence we actually arrived at our original goal of obtaining integer winding numbers $Z_{(a)}$ (\[e5:14\]) for any particle $a=1,2$, namely by specifying two group elements ${{\mathcal{G}}}_{(a)}$ of the $SU(2)$ form (\[e5:19\]) parametrized by two four-vectors ($U_{(a)}{}^{\alpha}$) which however must be subjected to the exchange coupling condition. Thus we do now have at hand not only a group theoretical method of generating consistent pairs of exchange fields $\{ X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$ for our two-particle system but simultaneously we gained a method of classifying the corresponding two-particle solutions by an (even or odd) pair of topological quantum numbers.
Kinematics of Exchange Coupling
-------------------------------
The transition to the universal covering groups does not only yield some further insight into the topology of the mixture configurations but it helps also to get a concrete geometric picture of the exchange coupling condition. This comes about by looking for the exchange constraints between both four-vectors $U^{\alpha}{}_{(1)}$ and $U^{\alpha}{}_{(2)}$, as the group parameters for the exchange group elements ${{\mathcal{G}}}_{(1)}$ and ${{\mathcal{G}}}_{(2)}$ for either particle, cf. (\[e5:19\]). In other words, we have to transcribe the former exchange coupling condition between the Euler angles (\[e4:28\]) to the present four-vectors $U^{\alpha}{}_{(a)}$ (\[e5:18\]), $a=1,2$. Clearly, this goal can be achieved by eliminating the average Euler angles $\tilde{\gamma_1}$, $\tilde{\gamma_2}$, $\tilde{\gamma_3}$ from the three-vector parametrizations (\[e4:52\])-(\[e4:53\]).
The point of departure is here again a convenient reparametrization of the four-vector components $U^{\alpha}{}_{(a)}$ so that the exchange coupling condition adopts a very simple shape when expressed in these new parameters. The key point is here a ($2+2$)-splitting of the group space, i.e. we put for either particle ($a=1,2$)
\[e5:25\] $$\begin{aligned}
(U_{||(a)})^2&=(U^1{}_{(a)})^2+(U^2{}_{(a)})^2\\
(U_{\perp (a)})^2&=(U^0{}_{(a)})^2+(U^3{}_{(a)})^2\end{aligned}$$
and thus consider the projections of the four-vectors $U^{\alpha}{}_{(a)}$ into the ($1,2$)-plane and the ($0,3$)-plane. Now for these projections one easily finds from the three-vector parametrization (\[e4:52\])-(\[e4:53\]) the following cross relations
\[e5:26\] $$\begin{aligned}
(U_{\perp (1)})^2&=(U_{||(2)})^2=\cos^2 \Big(\frac{\tilde{\gamma_2}}{2} \Big)\\ (U_{\perp (2)})^2&=(U_{||(1)})^2=\sin^2 \Big(\frac{\tilde{\gamma_2}}{2} \Big) \; .\end{aligned}$$
This means that when the projection of $U^{\alpha}{}_{(1)}$ into the ($0,3$)-plane is maximal, the corresponding projection of $U^{\alpha}{}_{(2)}$ is minimal and vice versa. Such an effect suggests to parametrize those projections in the following way (first particle, $a=1$)
\[e5:27\] $$\begin{aligned}
U^0{}_{(1)}&=\cos \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \cos \zeta_{(1)}\\
U^3{}_{(1)}&=\cos \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \sin \zeta_{(1)}\\
U^1{}_{(1)}&=\sin \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \cos \eta_{(1)}\\
U^2{}_{(1)}&=\sin \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \sin \eta_{(1)}\; ,\end{aligned}$$
and similarly for the second particle ($a=2$)
\[e5:28\] $$\begin{aligned}
U^0{}_{(2)}&=\sin \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \cos \zeta_{(2)}\\
U^3{}_{(2)}&=\sin \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \sin \zeta_{(2)}\\
U^1{}_{(2)}&=\cos \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \cos \eta_{(2)}\\
U^2{}_{(2)}&=\cos \Big(\frac{\tilde{\gamma_2}}{2} \Big)\cdot \sin \eta_{(2)}\; .\end{aligned}$$
Thus the remaining task is to reveal the exchange coupling condition upon the angles $\zeta_{(a)}$ and $\eta_{(a)}$ in the corresponding two-planes.
In the next step, the remaining two Euler angles $\tilde{\gamma_1}$ and $\tilde{\gamma_3}$ are expressed by the four-vector components $U^{\alpha}{}_{(a)}$ in the following way
\[e5:29\] $$\begin{aligned}
\cos \Big(\tilde{\gamma_1}+\tilde{\gamma_3}-\frac{\epsilon}{2}\Big)&=\frac{(U^0{}_{(1)})^2-(U^3{}_{(1)})^2}{(U^0{}_{(1)})^2+(U^3{}_{(1)})^2}=\cos (2 \zeta_{(1)} )\\
\sin \Big(\tilde{\gamma_1}+\tilde{\gamma_3}-\frac{\epsilon}{2}\Big)&=2\, \frac{U^0{}_{(1)}\cdot U^3{}_{(1)}}{(U_{\perp}{}_{(1)})^2} =\sin (2 \zeta_{(1)})\\
\sin \Big(\tilde{\gamma_3}-\tilde{\gamma_1}-\frac{\epsilon}{2}\Big)&=-\frac{(U^0{}_{(2)})^2-(U^3{}_{(2)})^2}{(U^0{}_{(2)})^2+(U^3{}_{(2)})^2}=-\cos (2 \zeta_{(2)} )\\
\cos \Big(\tilde{\gamma_3}-\tilde{\gamma_1}-\frac{\epsilon}{2}\Big)&=2\, \frac{U^0{}_{(2)}\cdot U^3{}_{(2)}}{(U_{\perp}{}_{(2)})^2} =\sin (2 \zeta_{(2)})\; .\end{aligned}$$
From here one recognizes immediately that the angles $\zeta_{(a)}$ in the ($0,3$)-plane are linked to the Euler angles $\tilde{\gamma_1}$, $\tilde{\gamma_3}$ as follows
\[e5:30\] $$\begin{aligned}
2 \zeta_{(1)}&=\tilde{\gamma_1}+\tilde{\gamma_3}-\frac{\epsilon}{2}\\
2 \zeta_{(2)}&=\frac{\pi}{2}-\tilde{\gamma_1}+\tilde{\gamma_3}-\frac{\epsilon}{2} \; . \end{aligned}$$
Finally, the last step consists in eliminating one of the Euler angles $\tilde{\gamma_1}$ or $\tilde{\gamma_3}$ from the first and second components $U^1{}_{(a)}$, $U^2{}_{(a)}$ of both particles. Thus, eliminating $\tilde{\gamma_1}$ yields by use of (\[e5:30\]) the following two equations
\[e5:31\] $$\begin{aligned}
\zeta_{(1)}&=\eta_{(1)}+\tilde{\gamma_3}-\frac{\pi}{2}\\
\zeta_{(2)}&=-\frac{\pi}{2}+\eta_{(2)}+\tilde{\gamma_3} \; , \end{aligned}$$
and, similarly, eliminating $\tilde{\gamma_3}$ yields
\[e5:32\] $$\begin{aligned}
\zeta_{(1)}&=\tilde{\gamma_1}-\eta_{(1)}-\frac{\epsilon}{2}+\frac{\pi}{2} \\
\zeta_{(2)}&=-\tilde{\gamma_1}-\eta_{(2)}-\frac{\epsilon}{2}-\pi \; . \end{aligned}$$
Clearly, both sets of equations (\[e5:31\]) and (\[e5:32\]) must be consistent when the former relationships are to be respected, which provides us with the final form of exchange coupling:
\[e5:33\] $$\begin{aligned}
\zeta_{(2)}&=\frac{3\pi}{4}-\eta_{(1)}-\frac{\epsilon}{2}\\
\eta_{(2)}&=-\zeta_{(1)}-\frac{\epsilon}{2}-\frac{\pi}{4} \; . \end{aligned}$$
This pleasant result allows us now to explicitly construct pairs of exchange fields which obey [*both*]{} the required curl relations [*and*]{} the exchange coupling conditions! Obviously we simply can [*choose*]{} one of both four-vectors $U^{\alpha}{}_{(x)}$ over space time, say $U^{\alpha}{}_{(1)}$, and then the second four-vector $U^{\alpha}{}_{(2)}$ is immediately given by equations (\[e5:28\]) together with (\[e5:33\]). Clearly the passage from the chosen $U^{\alpha}{}_{(1)}$ to the second vector $U^{\alpha}{}_{(2)}$ is still parametrized by the exchange angle $\epsilon$ which however can also be chosen arbitrarily! This may be seen more explicitly by recasting the transition from the first particle (\[e5:27\]) to the second particle (\[e5:28\]) into the form of an $SO(4)$ rotation $$\label{e5:34}
U^{\alpha}{}_{(2)}=S^{\alpha}{}_{\beta}\,U^{\beta}{}_{(1)} \; .$$ Here the rotation matrix $S^{\alpha}{}_{\beta}(\epsilon)$ is given by two $O(2)$ elements ${{\mathcal{R}}}$ through $$\label{e5:35}
S^{\alpha}{}_{\beta}=\left( \begin{array}{cc} 0 &{{\mathcal{R}}}\big(\frac{\epsilon}{2}+\frac{\pi}{4}\big) \\ {{\mathcal{R}}}\big(\frac{\epsilon}{2}-\frac{3\pi}{4}\big) &0 \end{array} \right)$$ with the $O(2)$ element ${{\mathcal{R}}}$ being given by $$\label{e5:36}
{{\mathcal{R}}}_{(\alpha)}=\left( \begin{array}{cc} \cos \alpha & -\sin \alpha \\ -\sin \alpha & -\cos \alpha \end{array} \right) \; .$$
Example: Stereographic Projection
=================================
As the simplest demonstration of the present results, one may put to zero the exchange angle $\epsilon$ which then turns the $SO(4)$ matrix ${{\mathcal{S}}}$ (\[e5:34\])-(\[e5:36\]) into a constant matrix ${\overset{\;\circ}{{{\mathcal{S}}}}{}}$ $$\label{e6:1}
{\overset{\;\circ}{{{\mathcal{S}}}}{}}=\left( \begin{array}{cc} 0 & {{\mathcal{R}}}_{\left(\frac{\pi}{4}\right)} \\ {{\mathcal{R}}}_{\left(-\frac{3 \pi}{4}\right)} & 0 \end{array} \right) \; .$$ For such a situation, both winding numbers $Z_{(a)}$ (\[e5:14\]) must necessarily coincide with the average winding number $\tilde{Z}$ because the winding number $Z_{\epsilon}$ (\[e5:16\]b) is trivially zero. The coincidence of $Z_{(a)}$ and $\tilde{Z}$ may also be seen from the four-vector version (\[e5:24\]) of the volume three-forms ${{\mathbb{V}}}_{(a)}$ since the constant $SO(4)$ element ${\overset{\;\circ}{{{\mathcal{S}}}}{}}$ escapes the exterior differentiation and just reproduces the $SO(4)$ invariant permutation tensor $$\label{e6:2}
\epsilon_{\mu \nu \lambda \sigma}= (\det {\overset{\;\circ}{{{\mathcal{S}}}}{}}) \cdot {\overset{\;\circ}{{{\mathcal{S}}}}{}}^{\alpha}{}_{\mu}\cdot {\overset{\;\circ}{{{\mathcal{S}}}}{}}^{\beta}{}_{\nu} \cdot {\overset{\;\circ}{{{\mathcal{S}}}}{}}^{\gamma}{}_{\lambda}\cdot {\overset{\;\circ}{{{\mathcal{S}}}}{}}^{\delta}{}_{\sigma} \cdot \epsilon_{\alpha \beta \gamma \delta} \; .$$ Clearly $\det {\overset{\;\circ}{{{\mathcal{S}}}}{}}=+1$ because ${\overset{\;\circ}{{{\mathcal{S}}}}{}}$ is an element of the proper rotation group.
Stereographic Projection
------------------------
Thus we are left with the problem of determining some differentiable four-vector field $U^{\alpha}{}_{(1)}$ over space-time from which the second four-vector $U^{\alpha}{}_{(2)}$ can then be simply constructed by means of the $SO(4)$ rotation ${\overset{\;\circ}{{{\mathcal{S}}}}{}}$ (\[e6:1\]), cf. (\[e5:34\]). Here, it is well-known that Euclidean 3-space (as a time slice of space-time $x^0=\mbox{const.}$) can be compactified to the 3-sphere $S^3$ by stereographic projection [@GoSc]. This procedure yields the following differentiable four-vector field $U^{\alpha}{}_{(x)}$ over $E_3$:
\[e6:3\] $$\begin{aligned}
U^0{}_{(1)}&=\frac{r^2-a^2}{r^2+a^2}\\
U^j{}_{(1)}&=2a^2\frac{x^j}{a^2+r^2} \; .\end{aligned}$$
This immediately yields for the average Euler angle $\tilde{\gamma_2}$ (\[e5:27\]) $$\label{e6:4}
\sin \big( \frac{\tilde{\gamma_2}}{2}\big)=\frac{2 a r \sin \theta}{a^2+r^2}$$ where spherical polar coordinates ($r$, $\theta$, $\phi$) have been used in place of the Cartesian parameters $\{x^j\}$. Hence for spatial infinity ($r=\infty$) the four-vector $U^{\alpha}{}_{(1)}$ points to the north pole of $S^3$ ($\tilde{\gamma_2}=0$), and for the origin ($r=0$) of $E_3$ to the south pole ($\tilde{\gamma_2}=\pi$), see fig. 2.
Once the first four-vector $U^{\alpha}{}_{(1)}$ has been fixed now, the corresponding second four-vector $U^{\alpha}{}_{(2)}$ can be constructed in a straight-forward manner by means of the $SO(4)$ rotation process
\[e6:5\] $$\begin{aligned}
U^0{}_{(2)}&=\frac{2ar \sin \theta}{a^2+r^2} \cos \big(\phi-\frac{3 \pi}{4} \big)\\
U^3{}_{(2)}&=-\frac{2ar \sin \theta}{a^2+r^2} \sin \big(\phi-\frac{3 \pi}{4} \big)\\
U^1{}_{(2)}&=\frac{1}{\sqrt{2}}\cdot \frac{r^2-a^2-2ar \cos \theta}{a^2+r^2}\\
U^2{}_{(2)}&=-\frac{1}{\sqrt{2}}\cdot \frac{r^2-a^2+2ar \cos \theta}{a^2+r^2}\; .\end{aligned}$$
Observe here that the second four-vector $U^{\alpha}{}_{(2)}$ (\[e6:5\]) cannot be made identical to the first one $U^{\alpha}{}_{(1)}$ (\[e6:3\]), because the exchange coupling condition always requires a [*non-trivial*]{} $SO(4)$ rotation ${{\mathcal{S}}}$ (\[e5:35\])!
Winding Numbers
---------------
Since for the present exchange configuration both winding numbers coincide with the average winding number $\tilde{Z}$($=Z_{(1)}=Z_{(2))}$), it may be sufficient here to consider only the first number $Z_{(1)}$. The corresponding volume three-form $\tilde{{{\mathbb{V}}}}_{(1)}$ (\[e5:23\]) reads $$\label{e6:6}
\tilde{{{\mathbb{V}}}}_{(1)}=\frac{1}{(2\pi)^2} \sin^2 \frac{\xi_{(1)}}{2}\, {{\mathbb{d}}}\xi_{(1)} \wedge \big\{ \frac{1}{2} \epsilon_{j k l}u^j{}_{(1)}\, {{\mathbb{d}}}u^k{}_{(1)} \wedge {{\mathbb{d}}}u^l{}_{(1)} \big\}$$ where the three-vector length $\xi_{(1)}$ is obtained from the four-vector $U^{\alpha}{}_{(1)}$ (\[e5:18\]) through $$\label{e6:7}
\sin^2 \frac{\xi_{(1)}}{2}=U^j{}_{(1)}U_{j(1)}=1-(U^0{}_{(1)})^2 \; .$$ Thus, when the four-vector $U^{\alpha}{}_{(1)}$ is generated by stereographic projection as shown by equations (\[e6:3\]), one finds for $\xi_{(1)}$
\[e6:8\] $$\begin{aligned}
\sin \frac{\xi_{(1)}}{2}=\frac{2ar}{a^2+r^2} \\
\cos \frac{\xi_{(1)}}{2}=\frac{a^2-r^2}{a^2+r^2}\; , \end{aligned}$$
i.e. we put $\xi_{(1)}=2 \pi$ for $r=0$ and $\xi_{(1)}=0$ for $r=\infty$. Furthermore, the unit three-vector $u^j{}_{(1)}$ simply becomes the normalized position vector of the Euclidean three-plane ($x^0=\mbox{const.}$) $$\label{e6:9}
u^j{}_{(1)}=\frac{x^j}{r} \doteqdot \hat{x}^j \; .$$
Consequently, the general volume form: $\tilde{{{\mathbb{V}}}}_{(1)}$ (\[e6:6\]) becomes specialized here to $$\label{e6:10}
\tilde{{{\mathbb{V}}}}_{(1)}=-\Big(\frac{2}{\pi}\Big)^2 \frac{a^3}{(a^2+r^2)^3}\,r^2{{\mathbb{d}}}r \wedge (\sin \theta\, {{\mathbb{d}}}\theta\wedge {{\mathbb{d}}}\phi)$$ and obviously contains as its angular part the volume two-form over the sphere $S^2$. The integrations can therefore be done in a straightforward way and yield the expected result for the winding number $Z_{(1)}$ ($=Z_{(2)}$) $$\label{e6:11}
Z_{(1)}=-\Big(\frac{2}{\pi}\Big)^2 a^3 \int_{r=o}^{\infty}\frac{r^2 dr}{(a^2+r^2)^3} \int_{\theta=0}^{\pi} d\theta \sin \theta \int_{\phi=0}^{2 \pi} d\phi=-1 \; .$$ Clearly the Euclidean 3-space $E_3$ is wound up to the 3-sphere $S^3$ just once by the stereographic projection, whereby spatial infinity ($r=\infty$) is mapped to the north pole (which reverses the conventional orientation and gives a minus sign for the winding number).
Exchange Fields
---------------
Once the generating group elements ${{\mathcal{G}}}_a$ of the $SU(2)$ form (\[e5:19\]) have been determined with the corresponding four-vector parametrizations $U^{\alpha}{}_{(a)}$ being displayed through equations (\[e6:3\]) and (\[e6:5\]), it is an easy matter to compute the associated Maurer-Cartan forms ${{\mathbb{E}}}^j{}_{(a)}$. Here it is convenient to resort to the three-vector parametrization (\[e4:42\]) of the Maurer-Cartan forms where the length $\xi_{(1)}$ of the first three-vector $\vec{\xi}_{(1)}$ (\[e4:43\]) has been specified through equations (\[e6:8\]) and the unit three-vector $u^j{}_{(1)}$ is given by (\[e6:9\]). With these prerequisites the Maurer-Cartan forms for the first particle are found to be of the following form $$\label{e6:12}
{{\mathbb{E}}}^j{}_{(1)}=\frac{4a}{a^2+r^2} \hat{x}^j {{\mathbb{d}}}r-4ar\frac{a^2-r^2}{(a^2+r^2)^2}{{\mathbb{d}}}\hat{x}^j-8\frac{a^2r^2}{(a^2+r^2)^2}\epsilon^j{}_{kl}\hat{x}^k{{\mathbb{d}}}\hat{x}^l \; .$$
Concerning the Maurer-Cartan forms of the second particle ${{\mathbb{E}}}^j{}_{(2)}$, one can refer to their link (\[e4:28\_2\]) to the first particle’s expressions ${{\mathbb{E}}}^j{}_{(1)}$ with the present specialization of putting the exchange angle to zero. This immediately yields for the second particle
\[e6:13\] $$\begin{aligned}
{{\mathbb{E}}}^1{}_{(2)}&={{\mathbb{E}}}^2{}_{(1)}\\
{{\mathbb{E}}}^2{}_{(2)}&={{\mathbb{E}}}^1{}_{(1)}\\
{{\mathbb{E}}}^3{}_{(2)}&=-{{\mathbb{E}}}^3{}_{(1)} \; ,\end{aligned}$$
or explicitly
\[e6:14\] $$\begin{aligned}
{{\mathbb{E}}}^1{}_{(2)}&\equiv -2 {{\mathbb{X}}}_2 = \frac{4a}{a^2+r^2}\hat{x}^2{{\mathbb{d}}}r-4ar\frac{a^2-r^2}{(a^2+r^2)^2}{{\mathbb{d}}}\hat{x}^2-8\frac{a^2r^2}{(a^2+r^2)^2}[\hat{x}^3{{\mathbb{d}}}\hat{x}^1-\hat{x}^1{{\mathbb{d}}}\hat{x}^3]\\
{{\mathbb{E}}}^2{}_{(2)}&\equiv -2 {{\mathbb{\Gamma}}}_2 = \frac{4a}{a^2+r^2}\hat{x}^1{{\mathbb{d}}}r-4ar\frac{a^2-r^2}{(a^2+r^2)^2}{{\mathbb{d}}}\hat{x}^1-8\frac{a^2r^2}{(a^2+r^2)^2}[\hat{x}^2{{\mathbb{d}}}\hat{x}^3-\hat{x}^3{{\mathbb{d}}}\hat{x}^2]\\
{{\mathbb{E}}}^3{}_{(2)}&\equiv 2 {{\mathbb{\Lambda}}}_2 = -\frac{4a}{a^2+r^2}\hat{x}^3{{\mathbb{d}}}r+4ar\frac{a^2-r^2}{(a^2+r^2)^2}{{\mathbb{d}}}\hat{x}^3+8\frac{a^2r^2}{(a^2+r^2)^2}[\hat{x}^1{{\mathbb{d}}}\hat{x}^2-\hat{x}^2{{\mathbb{d}}}\hat{x}^1] \; .\end{aligned}$$
Observe here that for the third form ${{\mathbb{E}}}^3{}_{(a)}\equiv 2{{\mathbb{\Lambda}}}_{(a)}$ (\[e6:13\]c) we have on account of the constancy of the exchange angle $\epsilon$ the relationship $$\label{e6:15}
{{\mathbb{\Lambda}}}_1=-{{\mathbb{\Lambda}}}_2$$ which of course just meets with the former derivative of $\epsilon$ (\[e3:39\]b) for vanishing $\epsilon$. Furthermore one can show that for vanishing $\epsilon$ the mixture variable $\beta$ must also vanish and therefore the exchange coupling (\[e3:20\]) says:
\[e6:16\] $$\begin{aligned}
X_{1 \mu}&\equiv \Gamma_{2 \mu}\\
X_{2 \mu}&\equiv \Gamma_{1 \mu}\end{aligned}$$
which is nothing else than the above result (\[e6:13\]).
Single-Particle Fields
----------------------
In Sect. III it has been made clear that the total set of RST variables can be subdivided into the subset of single-particle fields $\{{\Bbb L}_a$, ${{\stackrel{\circ}{\Bbb{K}}}}_{a \mu} \}$ and exchange fields $\{X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}\}$ such that both sets become coupled through the specific nature of the RST dynamics. Now that the exchange fields have been determined by hand in order to clearly display their topological features, one can face the problem of determining the dynamically associated single-particle fields $\{ {\Bbb L}_a$, ${{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}\}$ by solving the single-particle dynamics on the background of the given exchange fields. However, since the RST field equations constitute a very complicated dynamical system, one cannot hope to attain here an exact solution where the single-particle fields would be of a comparably simple form as the present exchange fields (\[e6:12\]) or (\[e6:14\]). Rather, we will be satisfied with convincing ourselves of the consistency of the RST dynamics so that one can be sure that the desired single-particle solution, to be associated to our construction of the exchange field system, does exist in principle.
First, consider the dynamical equations for the kinetic fields ${{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}$ which must consist of a source equation and of a curl equation in order to fix these vector fields, apart from certain boundary conditions. The source equations consist of both equations (\[e3:36\]a), where however the coupling conditions (\[e6:15\])-(\[e6:16\]) are to be taken into account:
\[e6:17\] $$\begin{aligned}
\nabla^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{1 \mu}+2 {\Bbb L}_1{}^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{1 \mu}&=-2 X_1{}^{\mu}({\Bbb L}_{1 \mu}-{\Bbb L}_{2 \mu})-2 \Lambda_1{}^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{2 \mu} \\
\nabla^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{2 \mu}+2 {\Bbb L}_2{}^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{2 \mu}&=2\Gamma_1{}^{\mu}({\Bbb L}_{1 \mu}-{\Bbb L}_{2 \mu})+2 \Lambda_1{}^{\mu}{{\stackrel{\circ}{\Bbb{K}}}}_{1 \mu} \; .\end{aligned}$$
Furthermore, the curl equations for the kinetic fields have already been specified by equations (\[e3:35a\]) where the gauge fields $F_{a \mu \nu}$ emerging there couple back to the RST currents $j_{a \mu}$ (\[e3:28\]) via the Maxwell equations (\[e2:16\]), i.e. for the present two-particle situation $$\label{e6:18}
\nabla^{\mu}F_{a \mu \nu}=4 \pi \alpha_{\ast}j_{a \nu} \; .$$ Observe here that the RST currents $j_{a \mu}$ themselves are to be constructed by means of the single-particle and exchange fields according to $$\label{e6:19}
j_{a \mu}=\frac{\hbar}{M c}{\Bbb L}_a{}^2 \{ {{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}+X_{a \mu}\} \; .$$
Evidently even if the exchange fields $X_{a \mu}$, $\Gamma_{a \mu}$, $\Lambda_{a \mu}$ are known (e.g. by our procedure of stereographic projection (\[e6:12\])-(\[e6:14\])), the kinetic fields still couple through their sources (\[e6:17\]) to the unknown amplitude fields ${\Bbb L}_a$. Therefore the single-particle dynamics must be closed by specifying the field equations for the amplitudes ${\Bbb L}_a$ which are of course given by the wave equations (\[e3:32\]) where the known exchange fields have to be inserted: $$\label{e6:20}
\Box {\Bbb L}_a +{\Bbb L}_a \Big\{ \left(\frac{Mc}{\hbar}\right)^2 -{{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}{{\stackrel{\circ}{\Bbb{K}}}}_a{}^{\mu} - 2 {{\stackrel{\circ}{\Bbb{K}}}}_{a \mu}X_a{}^{\mu}\Big\}=\sigma_{\ast}{\Bbb L}_a \{ \Gamma_{a \mu}\Gamma_a{}^{\mu}+\Lambda_{a \mu}\Lambda_a{}^{\mu}+\sigma_{\ast} {X}_{a \mu}X_a{}^{\mu}\} \; .$$ The right-hand side may be interpreted as the effect of an “exchange potential” $Y_a$ $$\label{e6:21}
Y_a \doteqdot \Gamma_{a \mu}\Gamma_a{}^{\mu}+\Lambda_{a \mu}\Lambda_a{}^{\mu}+\sigma_{\ast} {X}_{a \mu}X_a{}^{\mu} \; .$$ where however both potentials are identical ($Y_1\equiv Y_2$) because of the general quadratic exchange coupling (\[e3:40\]). There is no doubt that the present single-particle dynamics will admit non-trivial solutions which are however too difficult to be constructed analytically (for constructing [*approximate*]{} static solutions see ref. [@RuSo01_2]).
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abstract: 'We present a relation which connects the propagator in the radial (Fock-Schwinger) gauge with a gauge invariant Wilson loop. It is closely related to the well-known field strength formula and can be used to calculate the radial gauge propagator. The result is shown to diverge in four-dimensional space even for free fields, its singular nature is however naturally explained using the renormalization properties of Wilson loops with cusps and self-intersections. Using this observation we provide a consistent regularization scheme to facilitate loop calculations. Finally we compare our results with previous approaches to derive a propagator in Fock-Schwinger gauge.'
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March 27, 1996 TPR–95–31, hep-th/9604015
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[**RADIAL PROPAGATORS AND WILSON LOOPS**]{}
Stefan Leupold[^1]
Institut für Theoretische Physik, Universität Regensburg,
D-93040 Regensburg, Germany
Heribert Weigert[^2]
School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA
0.3cm
Introduction
============
While perturbation theory for gauge fields formulated in covariant gauges is very well established [@pascual] many aspects of non-covariant gauges are still under discussion. In principle one expects physical quantities to be independent of the chosen gauge. However this might lead to the naive conclusion that a quantum theory in an arbitrary gauge is simply obtained by inserting the respective gauge fixing term and the appropriate Faddeev-Popov ghosts in the path integral representation and reading off the Feynman rules. Unfortunately it is not so easy to obtain the correct Feynman rules, i.e. a set of rules yielding the same results for observable quantities as calculations in covariant gauges. Prominent examples are formulations in temporal and axial gauges. Such gauge choices are considered since one expects the Faddeev-Popov ghosts to decouple. However problems even start with the determination of the appropriate free gauge propagators. Temporal and axial gauge choices yield propagators plagued by gauge poles in their momentum space representation. These are caused by the fact that such gauge conditions are insufficient to [*completely*]{} remove the gauge degrees of freedom. The correct treatment of such poles can cause ghost fields to reappear [@cheng], can break translational invariance [@cara] or both [@leroy]. While these problems seem to be “restricted” to the evaluation of the correct gauge propagators and ghost fields, the necessity of introducing even new multi-gluon vertices appears in the Coulomb gauge [@christ]. These additional vertices are due to operator ordering problems which are difficult to handle in the familiar path integral approach. They give rise to anomalous interaction terms at the two-loop level [@doust] and cause still unsolved problems with renormalization at the three-loop level [@taylor].
In this article we are interested in the radial (Fock-Schwinger) gauge condition $$x_\mu A^\mu(x) = 0 \,. \label{eq:fsgaugecond}$$ It found widespread use in the context of QCD sum-rules (e.g. [@shif]). There it is used as being more or less synonymous to the important field strength formula $$\label{eq:fsformula}
A_\mu^{\mbox{\scriptsize rad}}(x) = \int\limits^1_0 \!\! ds \, s x^\nu
F_{\nu\mu}(sx)$$ which enormously simplifies the task of organizing the operator product expansion of QCD n-point functions in terms of gauge invariant quantities by expressing the gauge potential via the gauge covariant field strength tensor. It was introduced long ago [@fock], [@schwing] and rediscovered several times (e.g. [@cron]).
Only a few efforts have been made to establish perturbation theory for radial gauge. The main reason for this is that the gauge condition breaks translational invariance since the origin (in general an arbitrary but fixed point $z$, c.f. (\[eq:arbz\])) is singled out by the gauge condition. Thus perturbation theory cannot be formulated in momentum space as usual but must be set up in coordinate space.
The first attempt to evaluate the free radial propagator was performed in [@kumm]. Later however the function $\Gamma_{\mu\nu}(x,y)$ presented there was shown to be not symmetric [@moda]. Moreover it could not be symmetrized by adding $\Gamma_{\nu\mu}(y,x)$ since the latter is not a solution of the free Dyson equation. It was even suspected in [@moda] that it might be impossible to find a symmetric solution of this equation in four-dimensional space, due to the appearance of divergences even on the level of the [*free*]{} propagator when one uses the field strength formula to derive a free propagator. Indeed we agree with this statement in principle, but we will present an explanation for this problem and a way to bypass it. Other approaches to define a radial gauge propagator try to circumvent the problem (e.g. [@menot]) by sacrificing the field strength formula as given in (\[eq:fsformula\]) which was one of the main reasons the gauge became popular in non-perturbative QCD sum rule calculations [@shif] in the first place. If we are not prepared to do so we are forced to understand the origin of the divergences that plague most of the attempts to define even free propagators in radial gauges and see whether they can be dealt with in a satisfying manner.
In Section \[sec:radgaugecond\] we will make the first and decisive step in this direction by exploring the completeness of the gauge condition (\[eq:fsgaugecond\]) and its relation to the field strength formula and developing a new representation of the gauge potentials via link operators.
In Section \[sec:radprop\] we use this information to relate the divergences encountered in some of the attempts to define radial propagators to the renormalization properties of link operators. We find that even free propagators in radial gauge may feel remnants of the renormalization properties of closed, gauge invariant Wilson loops. Surprising as this seems to be superficially it is not impossible however if we recall that the inhomogeneous term in the gauge transformation has an explicit $1/g$ factor in it. As a result we are able to define a regularized radial propagator using the field strength formula and established regularization procedures for link operators.
Section \[sec:radcalc\] will be devoted to demonstrate the consistency of our approach by calculating a closed Wilson loop using our propagator and relating the steps to the equivalent calculation in Feynman gauge.
In Section \[sec:radren\] we obtain an explicitly finite version of our propagator by completing the renormalization program developed for link operators before we summarize and compare our results to other approaches in the literature in \[sec:radsum\] and shortly discuss the next steps in the program of establishing a new perturbative framework in radial gauges which – although the steps to be performed are quite straightforward – we will postpone for a future publication.
In the following we work in a $D$-dimensional Euclidean space. The vector potentials are given by $$A_\mu(x) \equiv A^a_\mu(x) \,t_a$$ where $t_a$ denotes the generators of an $SU(N)$ group in the fundamental representation obeying $$[t_a,t_b] = i f_{abc} \,t^c$$ and $$\mbox{tr}(t_a t_b) = {1\over 2} \,\delta_{ab} \,.$$ In general the radial gauge condition with respect to $z$ reads $$(x-z)_\mu A^\mu(x) =0 \,. \label{eq:arbz}$$ For simplicity we take $z=0$ after Section \[sec:radgaugecond\]. The results nevertheless can be easily generalized to arbitrary values of $z$.
The gauge condition revisited {#sec:radgaugecond}
=============================
Before we can go ahead and tackle the problem of divergences in the radial gauge propagator we have to establish a clearer picture of the uniqueness of the gauge condition we are about to implement. After all, if we do not succeed to fix the gauge completely we might be naturally confronted with divergences – if not at the free level then later in perturbative calculations. They would be a simple consequence of the incompleteness of the gauge fixing and the zero modes of the propagator which would then necessarily be present. This point has caused a still continuing discussion for the case of axial gauges (e.g. [@leroy]) but is only briefly mentioned in the context of radial gauges (e.g. [@azam]).
Readers who are not interested in the discussion of (in)completeness of radial gauge conditions might skip the following considerations without getting lost and start reading again after eq. (\[eq:gaugetrans\]).
To clarify the question whether the gauge condition (\[eq:arbz\]) is sufficient to completely fix the gauge degrees of freedom we have to catalogue the gauge transformations $U[B](x)$ which transform an arbitrary vector potential $B$ into the field $A$ satisfying (\[eq:arbz\]). A gauge condition is complete if $U[B](x)$ is uniquely determined up to a global gauge transformation. In other words, we want to find all solutions of $$\label{eq:transformations}
(x-z)_\mu\ U[B](x)\left[B^\mu(x) -{1\over i g}
\partial^\mu\right]U[B]^{-1}(x) = 0 \,.$$ It is easily checked that we have an infinite family of such solutions which can all be cast in the form of a product of two gauge transformations of the form $$\label{eq:gaugesol}
U[B](x) = V(z(x)) U[B](z(x),x) \,.$$ Here $$\label{eq:gaugetranssol}
U[B](z(x),x) = {\cal P} \exp i g \int^{z(x)}_x \!d\omega_\mu
B^\mu(\omega)$$ is a link operator whose geometric ingredients are parameterized via its endpoints $x$ and $z(x)$ and the straight line path $\omega$ between them, ${\cal P}$ denotes path ordering.
In particular $z(x)$ is the point where a straight line from $z$ through $x$ and a given closed hyper-surface around $z$ intersect. Since there is a unique relation between these points and the hyper-surface we will also refer to the hyper-surface itself by $z(x)$. This geometry is illustrated in Fig. \[fig:sphericalgeom\].
=
Both the detailed form of $z(x)$ and the local gauge transformation $V(x)$ are completely unconstrained as long as $(x-z).\partial^x z(x)
= 0$. In short, they parameterize the residual gauge freedom not eliminated by (\[eq:fsgaugecond\]). Note that while $V(x)$ is completely arbitrary the solutions (\[eq:gaugesol\]) ask only for its behavior at the given hyper-surface $z(x)$. The simplest and most intuitive choice for $z(x)$ is a spherical hyper-surface around $z$. Introducing the appropriate spherical coordinates it becomes obvious that $V(z(x))$ parameterizes gauge transformations which purely depend on the angles. Clearly the radial gauge condition (\[eq:arbz\]) cannot fix the angular dependence of any gauge transformation in (\[eq:transformations\]).
To eliminate the residual gauge freedom one has to impose a condition which is stronger than (\[eq:arbz\]) and suffices to pin down $V(x)$ up to a global transformation. A possible choice for such a gauge fixing would be the condition $$\begin{aligned}
\label{eq:compgaugefix}
\Box \left(\int_{z(x)}^x \!d\omega.A(\omega) + \int\!d^4\!y \ {1\over
\Box}(z(x),y)\ \partial^y.A(y)\right) \equiv 0\end{aligned}$$ which in addition to the vanishing of the radial component of the gauge potential also implements a covariant gauge on the hyper-surface $z(x)$. Such a gauge for arbitrary $z(x)$ would immediately force us to introduce ghosts into the path integral. Moreover the field strength formula would also be lost as we will illustrate below.
There is one exception to these unwanted modifications however, which may be implemented by contracting the closed surface $z(x)$ to the point $z$. Then the influence of $V(x)$ becomes degenerate with a global transformation and the gauge is completely fixed. Incidentally this is also the only case which entails the field strength formula. To see this we use $$\begin{aligned}
\lefteqn{ \delta U(x,z) = i g \;\Bigg\{ A_\mu(x)U(x,z)dx^\mu
- U(x,z)A_\mu(z)dz^\mu + } \nonumber \\
& & - \int^1_0\!\!ds\, [U(x,w_x)F_{\mu\nu}(w_x)U(w_x,z)]
{dw_x^\mu\over ds}
\left ( {dw_x^\nu\over dx^\alpha}dx^\alpha
+ {dw_x^\nu\over dz^\alpha}dz^\alpha
\right )
\Bigg\} \quad , \nonumber \\
\label{eq:udiff}\end{aligned}$$ (see e.g. [@bralic], [@ElGyuVa86]) to differentiate the link operators in the gauge transformation (\[eq:gaugesol\]) in order to find an expression for the radial gauge field: $$\begin{aligned}
\label{eq:gaugetrans}
A^{\mbox{\scriptsize rad}}_\mu(x) & = & U[A](z,x)\left[A_\mu(x) -{1\over i g}
\partial^x_\mu\right]U[A](x,z)
\nonumber \\ & = &
\int\limits^1_0 \!\! ds \, s {d\omega^\nu\over d s}
\,U[A](z,\omega)\,F_{\nu\mu}(\omega) \,U[A](\omega,z)
\nonumber \\ & = &
\int\limits^1_0 \!\! ds \, s {d\omega^\nu\over d s}
\,F_{\nu\mu}^{\mbox{\scriptsize rad}}(\omega) \,.\end{aligned}$$ This is nothing but (\[eq:fsformula\]) for arbitrary $z$ (note that in this case $\omega = \omega(s)$ is simply given by $\omega(s) = z +
(x-z) s$.) This simple result is only true since $\partial_\mu z(x)
\equiv \partial_\mu z = 0$. For general $z(x)$ there would be an additional term in the above formula reflecting the residual gauge freedom encoded in $V(z(x))$.
This sets the stage for a further exploration of the radial gauge in a context where we can be sure of having completely fixed the gauge in such a way that the field strength formula is guaranteed to be valid. Before we go on to studying the consequences the above has for the implementation of propagators we will introduce yet another representation of the gauge field in this particular complete radial gauge – this time solely in terms of link operators.
From now on we will assume the reference point $z$ to be the origin, but it will always be straightforward to recover the general case without any ambiguities. We will also suppress the explicit functional dependence of link operators on the gauge potential $A$ for brevity.
Let us start with a link operator along a straight line path $$U(x,x') = {\cal P}\exp\left[
i g\int\limits^1_0 \!\!d\omega_\mu A^\mu(\omega)
\right]$$ where now $ \omega(s):= x'+s(x-x')\, $. According to (\[eq:udiff\]) we have $$\partial_\mu^x \, U(x,x') = i g \left[ A_\mu(x) - \!\!\int\limits^1_0
\!\! ds \, s \, {d\omega^\nu\over d s}\ U(x,\omega)\,F_{\nu\mu}(\omega)
\,U(\omega,x) \right] U(x,x')
\label{eq:difstr}$$ which can be used to express the vector potential in terms of the link operator $$\lim_{x'\to x} \partial_\mu^x \, U(x,x') = i g \, A_\mu(x) \,.$$ In the case at hand the fact that $U(0,x) = 1$ in any of the $x.A(x) = 0$ gauges allows us to introduce a new gauge covariant representation $$A^{\mbox{\scriptsize rad}}_\mu(x) =
{1\over i g} \lim_{x'\to x} \partial_\mu^x
\left[ U(0,x)\,U(x,x')\,U(x',0) \right] \label{eq:gaucov}$$ for the Fock-Schwinger gauge field. It is easy to see that this is indeed equivalent to the field strength formula as given in (\[eq:fsformula\]) and consequently satisfies the same complete gauge fixing condition (i.e. (\[eq:compgaugefix\]) for $z(x) \to
z$): $$\begin{aligned}
A^{\mbox{\scriptsize rad}}_\mu(x)
&=&
{1\over i g} \lim_{x'\to x} \partial_\mu^x
\left[ U(0,x)\,U(x,x')\,U(x',0) \right]
\nonumber\\ &=&
{1\over i g} \lim_{x'\to x}
\left[ \partial_\mu^x \,U(0,x) \,U(x,x') \,U(x',0) \right.
\nonumber \\ && \hskip 2cm \left.
+ U(0,x) \,\partial_\mu^x \,U(x,x') \,U(x',0)\right]
\nonumber\\ &=&
{1\over i g} \,\partial_\mu^x \,U(0,x) \,U(x,0)
+ U(0,x) \,A_\mu(x)\,U(x,0)
\nonumber\\ &=&
\int\limits^1_0 \!\! ds \, s x^\nu F^{\mbox{\scriptsize rad}}_{\nu\mu}(sx)
\label{eq:fistr}\end{aligned}$$ where the last step uses (\[eq:difstr\]), mirroring the relations in (\[eq:gaugetrans\]) for $z=0$.
The Radial Gauge Propagator {#sec:radprop}
===========================
Having established the complete gauge fixing we are interested in, it is now straightforward to devise expressions for the propagator as a two-point function. According the above we know that $$\begin{aligned}
\label{eq:propdef}
\lefteqn{ \langle A_\mu(x) \otimes A_\nu(y)\rangle_{\mbox{\scriptsize rad}} }
\nonumber \\& = & \lim_{x'\to x \atop y'\to y}\partial_\mu^x
\partial_\nu^y \, \langle U(0,x)\,U(x,x')\,U(x',0)\otimes
U(0,y)\,U(y,y')\,U(y',0) \rangle \nonumber \\ & = & \int\limits^1_0
\!\! ds \int\limits^1_0 \!\! dt\, sx^\alpha \, ty^\beta \, \langle
U(0,sx)\,F_{\alpha\mu}(sx) \,U(sx,0)\otimes
U(0,ty)\,F_{\beta\nu}(ty) \,U(ty,0) \rangle \ \ .
\nonumber \\ &&\end{aligned}$$
Since we are in a fixed gauge it makes sense to perform a multiplet decomposition and for instance extract the singlet part of this propagator. The latter reduces to the free propagator in the limit $g
\to 0$.
We define $$\mbox{tr}\,\langle A_\mu(x) A_\nu(y) \rangle
=\mbox{tr}(t_a t_b)
\underbrace{\langle A^a_\mu(x) A^b_\nu(y) \rangle^{\mbox{\scriptsize
singlet}}}_{{\textstyle
=:\delta^{ab} D_{\mu\nu}(x,y)}} = {N^2 -1\over 2} \,
D_{\mu\nu}(x,y)$$ to extract $$\begin{aligned}
\lefteqn{\langle A^a_\mu(x) A^b_\nu(y) \rangle^{\mbox{\scriptsize
singlet}}_{\mbox{\scriptsize rad}}
= \delta^{ab} {2\over N^2 -1}
\mbox{tr}\, \langle A_\mu(x) A_\nu(y) \rangle_{\mbox{\scriptsize
rad}} = \delta^{ab} {2\over N^2 -1} }
\label{propa} \\
&& \times {1\over (ig)^2} \lim_{x'\to x \atop
y'\to y}\partial_\mu^x \partial_\nu^y \, \mbox{tr}\,\langle
U(0,x)\,U(x,x')\,U(x',0)\,U(0,y)\,U(y,y')\,U(y',0) \rangle
\,.\nonumber\end{aligned}$$ Obviously $$\begin{aligned}
W_1(x,x',y,y') := {1\over N} \, \mbox{tr}\,\langle
U(0,x)\,U(x,x')\,U(x',0)\,U(0,y)\,U(y,y')\,U(y',0) \rangle
\label{w1def}\end{aligned}$$ is a gauge invariant Wilson loop. Its geometry is illustrated in Fig. \[fig:faech\].
On the other hand, using the second expression in (\[eq:propdef\]) we have an equivalent representation for the singlet part of radial gauge propagator via the field strength formula: $$\begin{aligned}
\lefteqn{\langle A^a_\mu(x) A^b_\nu(y) \rangle_{\mbox{\scriptsize
rad}}^{\mbox{\scriptsize singlet}}
= \delta^{ab} {2\over N^2 -1} }
\label{profi}\\ && \times
\int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\, sx^\alpha \,
ty^\beta \, \mbox{tr}\,\langle U(0,sx)\,F_{\alpha\mu}(sx) \,U(sx,0)
\,U(0,ty)\,F_{\beta\nu}(ty) \,U(ty,0) \rangle \,. \nonumber\end{aligned}$$ Modanese [@moda] tried to calculate the free radial gauge propagator from (\[profi\]) in a $D$ dimensional space-time.[^3] Unfortunately one gets a result which diverges in the limit $D\to 4$.
Since we have performed a complete gauge fixing (at least on the classical level), this comes as a surprise since we certainly do not expect zero mode problems to come into the game as a possible explanation and consequently a way out. Does this mean we are trapped at a dead end or is there another explanation for this seemingly devastating discovery?
Before we try to answer this question let us briefly recapitulate how this divergence makes its appearance: Since the right hand side of (\[profi\]) is gauge invariant we can choose an arbitrary gauge to calculate it. For simplicity we take the Feynman gauge with its free propagator $$\langle A_\mu^a(x) A_\nu^b(y) \rangle_{\mbox{\scriptsize Feyn}}
= \delta^{ab}\, D_{\mu\nu}^{\mbox{\scriptsize Feyn}}(x,y)
= -{\Gamma(D/2-1) \over 4\pi^{D/2}}
\,g_{\mu\nu} \,\delta^{ab}\, [(x-y)^2]^{1-D/2} \,.\label{feynman}$$ Using the free field relations $ U(a,b) = 1 $ and $ F_{\mu\nu}=
\partial_\mu A_\nu - \partial_\nu A_\mu $ we get (for more details see Appendix \[appprop\]) $$\begin{aligned}
\lefteqn{\langle A^a_\mu(x) A^b_\nu(y) \rangle^0_{\mbox{\scriptsize
rad}}=} \nonumber\\&& = -{\Gamma(D/2-1) \over 4\pi^{D/2}}
\,\delta^{ab} \int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\,
sx^\alpha \, ty^\beta \nonumber\\&& \phantom{=}\times \left(
g_{\mu\nu} \partial_\alpha^{sx}\partial_\beta^{ty} +
g_{\alpha\beta}\partial_\mu^{sx}\partial_\nu^{ty} -
g_{\alpha\nu}\partial_\mu^{sx}\partial_\beta^{ty} -
g_{\mu\beta}\partial_\alpha^{sx}\partial_\nu^{ty} \right)
[(sx-ty)^2]^{1-D/2} \nonumber\\ && = -{\Gamma(D/2-1) \over
4\pi^{D/2}} \,\delta^{ab} \, \bigg( g_{\mu\nu} [(x-y)^2]^{1-D/2}
\nonumber\\ && \phantom{mmm} -\partial_\mu^x \int\limits^1_0 \!\! ds
\, x_\nu\,[(sx-y)^2]^{1-D/2} -\partial_\nu^y \int\limits^1_0 \!\! dt
\, y_\mu \,[(x-ty)^2]^{1-D/2} \nonumber\\ && \phantom{mmm}
+\partial_\mu^x \partial_\nu^y \underbrace{\int\limits^1_0 \!\! ds
\int\limits^1_0 \!\! dt\, x\cdot y \,
[(sx-ty)^2]^{1-D/2}}_{\textstyle \sim {\textstyle 1\over
\textstyle 4-D}} \bigg) \,.
\label{divprop}\end{aligned}$$ Thus the radial gauge propagator is singular for arbitrary arguments $x$ and $y$ — with one remarkable exception: It is easy to see that it vanishes for $x=0$ or $y=0$. This is simply a consequence of our task to preserve the field strength formula (\[eq:fsformula\]) which forces the vector field to vanish at the origin (in general at the reference point $z$).
The observation that the radial gauge propagator as calculated here diverges in four-dimensional space raises the question, whether it is perhaps impossible to formulate a quantum theory in radial gauge. This would suggest that the radial gauge condition – in the form that facilitates the field strength formula – is inherently inconsistent (“unphysical”) in contrast to the general belief that it is “very physical” since it allows to express gauge variant quantities like the vector potential in terms of gauge invariant ones. To answer this question we have to understand where this divergence comes from. In the following we will see that for this purpose the complicated looking Wilson loop representation (\[propa\]) is much more useful than the field strength formula (\[profi\]). (Note, however, that the result for the free propagator (\[divprop\]) of course will be the same.)
It is well-known that Wilson loops need renormalization to make them well-defined (see e.g. [@korrad] and references therein). The expansion of an arbitrary Wilson loop $$W(C) = {1\over N} \, \mbox{tr}\,\left\langle {\cal P}\exp\left[
ig\oint_C dx^\mu A_\mu(x) \right] \right\rangle$$ in powers of the coupling constant is given by $$\begin{aligned}
\lefteqn{
W(C) = 1 + {1\over N} \sum\limits_{n=2}^\infty (ig)^n
\oint_C \! dx_1^{\mu_1} \ldots \oint_C \! dx_n^{\mu_n}}
\nonumber \\ && \times
\Theta_C(x_1 > \cdots > x_n) \,\mbox{tr}\,
G_{\mu_1 \ldots \mu_n}(x_1,\ldots,x_n) \label{ptexp}\end{aligned}$$ where $\Theta_C(x_1 > \cdots > x_n)$ orders the points $x_1,\ldots,x_n$ along the contour $C$ and $$G_{\mu_1 \ldots \mu_n}(x_1,\ldots,x_n) :=
\left\langle A_{\mu_1}(x_1) \cdots A_{\mu_n}(x_n) \right\rangle$$ are the Green functions.
In general Wilson loops show ultraviolet singularities in any order of the coupling constant. If the contour $C$ is smooth (i.e. differentiable) and simple (i.e. without self-intersections) the conventional charge and wave-function renormalization — denoted by ${\cal R}$ in the following — is sufficient to make $W(C)$ finite. We refer to [@regul] for more details about renormalization of regular (smooth and simple) loops.
In our example we must apply the renormalization operation ${\cal R}$ to $W_1$ as given in (\[w1def\]). This yields $$\tilde W_1(x,x',y,y';g_R,\mu,D) = {\cal R} W_1(x,x',y,y';g,D)
\label{roper}$$ where $W_1(x,x',y,y';g,D)$ is a regularized expression calculated in $D$ dimensions and $\mu$ is a subtraction point introduced by the renormalization procedure ${\cal R}$. For the purpose of the present work the only important relation is $$g_R = \mu^{(D-4)/2} g + o(g^3) \,. \label{grug}$$
While the operation ${\cal R}$ is sufficient to make regular loops well-defined, new divergences appear if the contour $C$ has cusps or self-intersections. The renormalization properties of such loops are discussed in [@brandt] and [@brandt2]. While the singularities of regular loops appear at the two-loop level (order $g^4$ in (\[ptexp\])) cusps and cross points cause divergences even in leading (non-trivial) order $g^2$.
Since $W_1$ is indeed plagued by cusps and self-intersections a second renormalization operation must be carried out to get a renormalized expression $W_1^R$ from the bare one $W_1$: According to [@brandt] each cusp is multiplicatively renormalizable with a renormalization factor $Z$ depending on the cusp angle. In our case we have four cusps with angles $$\begin{aligned}
\alpha &:=& \angle (x-x',-x) \,, \\
\alpha' &:=& \angle (x',x-x') \,, \\
\beta &:=& \angle (y-y',-y) \,, \\
\beta' &:=& \angle (y',y-y') \,.\end{aligned}$$ The cross point at the origin introduces a mixing between $W_1$ and $$\begin{aligned}
\lefteqn{W_2(x,x',y,y') :=}\nonumber\\
&&\left\langle
{1\over N} \,\mbox{tr}\left[ U(0,x)\,U(x,x')\,U(x',0)\right] \,
{1\over N} \,\mbox{tr}\left[ U(0,y)\,U(y,y')\,U(y',0)\right]
\right\rangle \,.\end{aligned}$$ Again the divergences appearing here are functions of the angles $$\left.
\begin{array}{lcl}
\gamma_{xx'} &:=& \angle (-x,x') \\
\gamma_{yy'} &:=& \angle (-y,y') \\
\gamma_{xy} &:=& \angle (-x,-y) \\
\gamma_{x'y'} &:=& \angle (x',y') \\
\gamma_{x'y} &:=& \angle (x',-y) \\
\gamma_{xy'} &:=& \angle (-x,y')
\end{array}
\right\} \vec\gamma \,.$$ The renormalized Wilson loop $W_1^R$ is given by $$\begin{aligned}
\lefteqn{W_1^R(x,x',y,y';g_R,\mu,
\bar C_\alpha,\bar C_{\alpha'},\bar C_\beta,\bar C_{\beta'},
\bar C_{\vec\gamma})}
\nonumber\\ &&
= \lim_{D\to 4}
\, Z(\bar C_\alpha,g_R,\mu;D) \, Z(\bar C_{\alpha'},g_R,\mu;D)
\, Z(\bar C_\beta,g_R,\mu;D) \, Z(\bar C_{\beta'},g_R,\mu;D)
\nonumber\\ && \phantom{=\lim}
\times \left[ Z_{11}(\bar C_{\vec\gamma},g_R,\mu;D) \,
\tilde W_1(x,x',y,y';g_R,\mu,D)
\right. \nonumber\\ && \phantom{=\lim \times [} \left.
{}+Z_{12}(\bar C_{\vec\gamma},g_R,\mu;D) \,
\tilde W_2(x,x',y,y';g_R,\mu,D)
\right]
\nonumber\\ &&
=: \lim_{D\to 4} \bar W_1(x,x',y,y';g_R,\mu,
\bar C_\alpha,\bar C_{\alpha'},\bar C_\beta,\bar C_{\beta'},
\bar C_{\vec\gamma};D) \label{barw}\end{aligned}$$ where the second renormalization procedure introduces new subtraction points $\bar C_\sigma$ (c.f. [@korrad] and [@brandt] for more details). Of course different renormalization procedures are possible and so the $Z$ factors are not unique. We will return to this point in Section \[sec:radren\] where we specify a renormalization operation which is appropriate for our purposes.
The observation that Wilson loops with cusps and/or cross points show additional divergences has an important consequence for our radial gauge propagator as given in (\[propa\]): Even the free propagator needs renormalization! This provides a natural explanation for the fact that a naive calculation of this object yields an ultraviolet divergent result [@moda]. Note that the usual divergences of Wilson loops which are removed by ${\cal R}$, like e.g. vertex divergences, appear at $o(g^4)$ and thus do not contribute to the free part of the radial gauge propagator, while the cusp singularities indeed contribute since they appear at $o(g^2)$ and affect the free field case due to the factor $1/g^2$ in (\[propa\]).
Now we are able to answer the question whether the radial gauge is “unphysical” or “very physical”. It is just its intimate relation to physical, i.e. gauge invariant, quantities which makes the gauge propagator — even the free one — divergent. One might cast the answer in the following form: [*The propagator diverges because of — and not contrary to — the fact that the radial gauge is “very physical”*]{}.
Consequently the next questions are:
- Is there any use for a divergent expression for the free propagator? Especially: Can we use it to perform (dimensionally regularized) loop calculations?
- Can one find a renormalization program which yields a finite radial gauge propagator?
In the next Section we will perform a one loop calculation of a Wilson loop using the radial gauge propagator (\[divprop\]) and compare the dimensionally regularized result with a calculation in Feynman gauge.
In Section \[sec:radren\] we will explicitly demonstrate that the renormalization program for link operators carries over and allows to derive a finite result for the radial propagator and contrast its properties and use to the regularized version.
Calculating a Wilson Loop in Radial Gauge {#sec:radcalc}
=========================================
We choose the path $$\ell : z(\sigma) = \left\{
\begin{array}{lclcl}
\sigma x &,& \sigma\in [0,1] &,& x\in{\mathbb R}^D \\
w(\sigma -1) &,& \sigma\in [1,2] &,& w(0)=x,\, w(1)=y \\
(3-\sigma)\,y &,& \sigma\in [2,3] &,& y\in{\mathbb R}^D
\end{array}
\right. \label{looppath}$$ It is shown in Fig. \[fig:drop\]. The line $w(\sigma-1)$ is supposed to be an arbitrary curve connecting $x$ and $y$.
First we will perform the calculation of this Wilson loop in Feynman gauge. Using (\[feynman\]) we get in leading order of the coupling constant $$\begin{aligned}
W(\ell) &=&
{1\over N} \, \mbox{tr}\,\left\langle {\cal P}\exp\left[
ig\oint_\ell dz^\mu A_\mu(z) \right] \right\rangle
\nonumber \\ &\approx&
1 + (ig)^2 {N^2 -1\over 2N}
\int\limits^3_0 \!\! d\sigma \!\int\limits^3_0 \!\! d\tau \,
\Theta(\sigma-\tau)\,\dot z^\mu(\sigma) \, \dot z^\nu(\tau) \,
D_{\mu\nu}^{\mbox{\scriptsize Feyn}}(z(\sigma),z(\tau))
\nonumber \\ &=&
1 + (ig)^2 {N^2 -1\over 2N} {1\over 2}
\underbrace{\int\limits^3_0 \!\! d\sigma \!\int\limits^3_0 \!\! d\tau \,
\dot z^\mu(\sigma) \, \dot z^\nu(\tau) \,
D_{\mu\nu}^{\mbox{\scriptsize Feyn}}(z(\sigma),z(\tau))}
_{\textstyle =: I_f} \,. \label{feynres}\end{aligned}$$ To get rid of the $\Theta$-function we have exploited the symmetry property of two-point Green functions $$D_{\mu\nu}^{\mbox{\scriptsize Feyn}}(x,y)
= D_{\nu\mu}^{\mbox{\scriptsize Feyn}}(y,x) \,.$$ Decomposing the contour $\ell$ according to (\[looppath\]) we find that the Feynman propagator in (\[feynres\])) connects each segment of $\ell$ with itself and with all the other segments. Thus $I_f$ is given by $$I_f = \sum\limits_{A = 1}^3 \sum\limits_{B = 1}^3\,(A,B)$$ where $(A,B)$ denotes the contribution with propagators connecting loop segments $A$ and $B$ (c.f. Fig. \[fig:drop\]), e.g. $$\begin{aligned}
(1,2) &=&
\int\limits^1_0 \!\! d\sigma \!\int\limits^1_0 \!\! d\tau \,
x^\mu \, \dot w^\nu(\tau) \,
D_{\mu\nu}^{\mbox{\scriptsize Feyn}}(\sigma x,w(\tau))
\nonumber\\ &=&
-{\Gamma(D/2-1) \over 4\pi^{D/2}}
\int\limits^1_0 \!\! d\sigma \!\int\limits^1_0 \!\! d\tau \,
x^\mu \, \dot w_\mu(\tau) \, [(\sigma x-w(\tau))^2]^{1-D/2} \,.\end{aligned}$$
Next we will evaluate the same Wilson loop in radial gauge. Clearly the first and the third part of the path do not contribute if the radial gauge condition $x_\mu A^\mu(x) = 0$ holds. We insert the free propagator $$\begin{aligned}
\langle A^a_\mu(x) A^b_\nu(y) \rangle^0_{\mbox{\scriptsize rad}}
=: \delta^{ab} D^0_{\mu\nu}(x,y) \end{aligned}$$ from (\[divprop\]) into $$\begin{aligned}
W(\ell) &=&
{1\over N}\, \mbox{tr}\,\left\langle {\cal P}\exp\left[
ig\int\limits_0^1 \!\! d\sigma \,\dot w_\mu(\sigma) A^\mu(w(\sigma))
\right]\right\rangle
\nonumber\\ &\approx&
1 + (i g)^2 {N^2 -1\over 2N}
{1\over 2}
\underbrace{\int\limits^1_0 \!\! d\sigma \!
\int\limits^1_0 \!\! d\tau \,
\dot w^\mu(\sigma)\, \dot w^\nu(\tau) \,
D_{\mu\nu}^0(w(\sigma),w(\tau))}
_{\textstyle =: I_r}
\label{wilir}\end{aligned}$$ and observe that $$\dot w_\mu(\sigma) \,\partial^\mu_{w(\sigma)} = {d\over d\sigma} \,.$$ Thus the integral in (\[wilir\]) reduces to $$\begin{aligned}
\lefteqn{I_r =
-{\Gamma(D/2-1) \over 4\pi^{D/2}}
\Bigg[
\int\limits^1_0 \!\! d\sigma \int\limits^1_0 \!\! d\tau \,
\dot w_\mu(\sigma)\, \dot w^\mu(\tau) \,
[(w(\sigma)-w(\tau))^2]^{1-D/2}}
\nonumber\\&& \phantom{-{\Gamma(D/2-1) \over 4\pi^{D/2}}}
{}+ \int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\,
\Big(
w_\mu(1)\,w^\mu(1)\,[(sw(1)-tw(1))^2]^{1-D/2}
\nonumber\\&&
\phantom{-MM(D/2-1)\int\limits^1_0\!\! ds\int\limits^1_0\!\!dt\,}
{}+ w_\mu(0)\,w^\mu(0)\,[(sw(0)-tw(0))^2]^{1-D/2}
\nonumber\\&&
\phantom{-MM(D/2-1)\int\limits^1_0\!\! ds\int\limits^1_0\!\!dt\,}
{}-w_\mu(1)\,w^\mu(0)\,[(sw(1)-tw(0))^2]^{1-D/2}
\nonumber\\&&
\phantom{-MM(D/2-1)\int\limits^1_0\!\! ds\int\limits^1_0\!\!dt\,}
{}-w_\mu(0)\,w^\mu(1)\,[(sw(0)-tw(1))^2]^{1-D/2}
\Big)
\nonumber\\&& \phantom{-{\Gamma(D/2-1) \over 4\pi^{D/2}}}
{}- \int\limits^1_0 \!\! ds \int\limits^1_0 \!\! d\tau \,
\dot w_\mu(\tau)
\Big(
w^\mu(1) [(sw(1)-w(\tau))^2]^{1-D/2}
\nonumber\\&&
\phantom{-MMMM(D/2-1)\int\limits^1_0\!\! ds\int\limits^1_0\!\!dt\,}
{}- w^\mu(0) [(sw(0)-w(\tau))^2]^{1-D/2}
\Big)
\nonumber\\&& \phantom{-{\Gamma(D/2-1) \over 4\pi^{D/2}}}
{}- \int\limits^1_0 \!\! dt \int\limits^1_0 \!\! d\sigma \,
\dot w_\mu(\sigma)
\Big(
w^\mu(1) [(w(\sigma)-tw(1))^2]^{1-D/2}
\nonumber\\&&
\phantom{-MMMM(D/2-1)\int\limits^1_0\!\! ds\int\limits^1_0\!\!dt\,}
{}- w^\mu(0) [(w(\sigma)-tw(0))^2]^{1-D/2}
\Big)
\Bigg]
\label{lengthy}\\&&
= (2,2) + (3,3) + (1,1) + (3,1) + (1,3) + (3,2) + (1,2) + (2,3) +
(2,1) \,. \nonumber\end{aligned}$$ A careful analysis of (\[lengthy\]) shows that it exactly coincides with the Feynman gauge calculation. This is expressed in the last line where we have denoted which parts of the loop are connected by the Feynman gauge propagator to reproduce (\[lengthy\]) term by term. Thus using the radial gauge propagator as given in (\[divprop\]) yields the same result as the calculation in Feynman gauge. Finally this regularized expression has to be renormalized. This can be performed without any problems according to [@brandt]. Since we are not interested in the Wilson loop itself but in the comparison of the results obtained in radial and Feynman gauge, we will not calculate the renormalized expression for $W(\ell)$.
However a qualitative discussion of the renormalization properties of $W(\ell)$ is illuminating. By construction $W(\ell)$ has at least a cusp at the origin. (Other cusps are possible at $x$ or $y$ or along the line parameterized by $w$, but are not important for our considerations.) To give the right behavior of the Wilson loop the calculation of $W(\ell)$ in an arbitrary gauge must reproduce the cusp singularity. Usually the parameter integrals in the vicinity of the cusp do the job. For gauge choices where the propagator do not vanish in the vicinity of the origin this is automatically achieved. Let us assume for a moment that it is possible to construct a [*finite*]{} radial gauge propagator obeying the field strength formula (\[eq:fsformula\]) and therefore have trivial gauge factors along radial lines. Of course this is nothing but saying that there are no contributions form parts 1 and 3 of the loop, i.e. in the vicinity of the origin. Since the propagator is assumed to be finite, there are no singular integrals corresponding to the cusp at the origin. Thus a finite radial gauge propagator cannot reproduce the correct behavior of the Wilson loop. In turn we conclude that [*a singular radial gauge propagator is mandatory*]{} to get the right renormalization properties of Wilson loops.
However, as we will demonstrate in the next Section, the renormalization procedure for Wilson loops can be used to devise a consistent renormalization program for the radial gauges considered here. We will apply it to write down a finite version of the free radial propagator. The generalization to higher orders in perturbation theory is straightforward. According to our considerations given above we shall show that the renormalized, thus finite version of the free radial propagator is not suitable as an input to perturbative calculations.
The Renormalized Free Propagator {#sec:radren}
================================
We define the renormalized radial gauge propagator by $$\langle A^a_\mu(x) A^b_\nu(y)
\rangle_R^{\mbox{\scriptsize singlet}} :=
\lim_{D\to 4} \delta^{ab} {2N\over N^2 -1}
{1\over (ig_R)^2} \mu^{D-4}
\lim_{x'\to x \atop y'\to y}\partial_\mu^x \partial_\nu^y \,
\bar W_1(x,x',y,y';D) \label{propren}$$ where we have suppressed most of the other variables on which $\bar
W_1$ depends (see (\[barw\])).
From now on we will concentrate on the calculation of the renormalized free propagator $\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0$. The details of the renormalization program are presented in Appendix \[app:renprog\]. Of course the procedure is closely connected to the renormalization of cusp singularities of Wilson loops. The result is $$\begin{aligned}
\lefteqn{\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0 =}
\\ &&
=\lim_{D\to 4} \left(
\delta^{ab} \mu^{D-4} \partial_\mu^x \partial_\nu^y
\left(
{1\over 4\pi^2} {1\over 4-D} (\pi-\gamma_{xy})\cot\gamma_{xy}
\right)
+\langle A^a_\mu(x) A^b_\nu(y) \rangle^0_{\mbox{\scriptsize rad}}
\right)
\nonumber\end{aligned}$$
Before discussing some properties of the renormalized free propagator we shall show that the counter term $$C_{\mu\nu}^{ab}(x,y) :=
\delta^{ab} \mu^{D-4} \partial_\mu^x \partial_\nu^y
\left(
{1\over 4\pi^2} {1\over 4-D} (\pi-\gamma_{xy})\cot\gamma_{xy}
\right)$$ exactly cancels the divergence of the propagator (\[divprop\]), i.e. that $\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0$ really is finite. To this end we use some technical results derived in Appendix \[siint\]. The divergent part of the propagator (\[divprop\]) is given by $$U_{\mu\nu}^{ab}(x,y):=-{\Gamma(D/2-1) \over 4\pi^{D/2}} \,\delta^{ab} \,
\partial_\mu^x \partial_\nu^y
\int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\, x\cdot y \,
[(sx-ty)^2]^{1-D/2} \,.$$ Using (\[appi2\]) and (\[i2neq\]) we find $$\begin{aligned}
U_{\mu\nu}^{ab}(x,y) &=&
{\Gamma(D/2-1) \over 4\pi^{D/2}} \,\delta^{ab} \,
\partial_\mu^x \partial_\nu^y \, I_2(x,-y)
\nonumber\\ &=&
{\Gamma(D/2-1) \over 4\pi^{D/2}} \,\delta^{ab} \,
\partial_\mu^x \partial_\nu^y
\left({1\over 4-D} \,(\pi-\gamma_{xy}) \cot(\pi-\gamma_{xy}) \;
+\mbox{finite} \right)
\nonumber\\ &=&
-{1 \over 4\pi^2} \,\delta^{ab} \,
\partial_\mu^x \partial_\nu^y
\left({1\over 4-D} \,(\pi-\gamma_{xy}) \cot\gamma_{xy} \right) \;
+\mbox{finite}\end{aligned}$$ and thus $$U_{\mu\nu}^{ab}(x,y) + C_{\mu\nu}^{ab}(x,y) = \mbox{finite} \,.$$
Note that if one tries to guess a finite expression like $\langle
A^a_\mu(x) A^b_\nu(y) \rangle_R^0$ one would have to introduce a scale $\mu$ by hand without interpretation. In our derivation this scale appears naturally as the typical renormalization scale of the ${\cal
R}$ operation.
The counter term $C_{\mu\nu}^{ab}(x,y)$ has some interesting properties. It is symmetric with respect to an exchange of all variables and it obeys the gauge condition $$x^\mu C_{\mu\nu}^{ab}(x,y) = 0 = C_{\mu\nu}^{ab}(x,y) \,y^\nu \,.
\label{courad}$$ Thus $\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0$ is finite in the limit $D\to 4$ but still can be interpreted as a gluonic two-point function which fulfills the radial gauge condition $$x^\mu \,\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0 =0 \,.$$ However the counter term $C_{\mu\nu}^{ab}(x,y)$ and thus also $\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0$ is ill-defined at the origin and hence conflicts with the field strength formula (\[eq:fsformula\]). Note that the regularized propagator in contrast to the renormalized propagator is well defined and vanishes if one of its arguments approaches zero, as pointed out after eq. (\[divprop\]). We therefore conclude that we may use the regularized propagator in perturbative calculations and can be ensured to preserve relations like the field strength formula or eqs. (\[eq:fistr\]) or (\[eq:propdef\]) throughout the calculation. Although the counterterms are not well defined at the reference point itself – a property the radial gauge propagator simply inherits from renormalizing the cusp singularity of the underlying Wilson line – physical (gauge invariant) quantities are not affected, they are rendered finite and unambiguous.
Summary and Outlook {#sec:radsum}
===================
In this article we have shown how to calculate the radial gauge propagator in a $D$-dimensional space using Wilson loops. As discovered in [@moda] the free propagator diverges in four-dimensional space. We were able to explain this singular behavior by studying the properties of associated Wilson loops. Furthermore we have shown that the free propagator, in spite of being divergent in four dimensions, can be used for perturbative calculations in a (dimensionally) regularized framework and that the result for a gauge invariant quantity agrees with the calculation in Feynman gauge. Finally we have presented a renormalization procedure for the radial gauge propagator and calculated the explicit form of the renormalized free propagator. We have pointed out that any version of the radial propagator which is finite in four-dimensional space at least cannot reproduce the correct renormalization properties of Wilson loops with cusps at the reference point $z$.
It is instructive to compare the radial gauge propagators as presented here with other approaches: As discussed in Section \[sec:radgaugecond\] the radial gauge condition (\[eq:fsgaugecond\]) does not completely fix the gauge degrees of freedom. Thus the field strength formula $$A_\mu(x) = \int\limits^1_0 \!\! ds \, s x^\nu F_{\nu\mu}(sx) \label{fistrag}$$ is not the only solution of the system of equations[^4] $$\left\{\begin{array}{l}
x_\mu A^\mu(x) =0 \,, \\
F_{\mu\nu}(x) =
\partial^x_\mu A_\nu(x) - \partial^x_\nu A_\mu(x) \,.
\end{array} \right.$$ One might add a function [@moda] $$A_\mu^0(x) = \partial^x_\mu f(x)$$ to (\[fistrag\]) where $f$ is an arbitrary homogeneous function of degree 0. However any $A_\mu^0(x)$ added in in order to modify (\[fistrag\]) is necessarily singular at the origin. Hence regularity at the origin may be used as a uniqueness condition [@cron]. If we relax this boundary condition other solutions are possible, e.g. $$\bar A_\mu(x) = -\int\limits^\infty_1 \!\! ds \, s x^\nu F_{\nu\mu}(sx)
\label{fistrinf}$$ where we must assume that the field strength vanishes at infinity. While (\[fistrag\]) is the only solution which is regular at the origin, (\[fistrinf\]) is regular at infinity. Ignoring boundary conditions for the moment one can construct a radial gauge propagator by [@menot] $${1\over 2} \left( G_{\mu\nu}(x,y) + G_{\nu\mu}(y,x) \right)
\label{menprop}$$ with $$G_{\mu\nu}(x,y) :=
-\int\limits^1_0 \!\! ds \, s x^\alpha
\int\limits^\infty_1 \!\! dt \, t y^\beta
\left\langle F_{\alpha\mu}(sx) F_{\beta\nu}(ty) \right\rangle \,.$$ It turns out that this propagator is finite in four dimensions. However the price one has to pay is that boundary conditions are ignored and thus the object “lives” in the restricted space ${\mathbb R}^4 \setminus\{0\}$ and not in ${\mathbb R}^4$ anymore. In our approach we insist on the field strength formula (\[fistrag\]) widely used in operator product expansions [@shif] and on the regular behavior of vector potentials at the origin [@cron]. One might use the propagator (\[menprop\]) to calculate the $g^2$-contribution to the Wilson loop on the contour (\[looppath\]). It is easy to check that the result differs from the one obtained in (\[wilir\]), (\[lengthy\]). Clearly this is due to the fact that (\[menprop\]) is ill-defined at the origin.
In the above, all calculations were performed in Euclidean space. In Minkowski space Wilson loops show additional divergences if part of the contour coincides with the light cone [@korkor]. Thus we expect the appearance of new singularities also for the radial propagator, at least if one or both of its arguments are light-like. Further investigation is required to work out the properties of the radial gauge propagator in Minkowski space.
To formulate perturbation theory in a specific gauge the knowledge of the correct free propagator is only the first step. In addition one has to check the decoupling of Faddeev-Popov ghosts in radial gauge which is suggested by the algebraic nature of the gauge condition. However the still continuing discussion about temporal and axial gauges might serve as a warning that the decoupling of ghosts for algebraic gauge conditions is far from being trivial (c.f. [@cheng], [@leroy] and references therein). To prove (or disprove) the decoupling of ghosts in radial gauge we expect that our Wilson loop representation of the propagator is of great advantage since it yields the possibility to calculate higher loop contributions in two distinct ways: On the one hand one might use the Wilson loop representation to calculate the full radial propagator up to an arbitrary order in the coupling constant. The appropriate Wilson loop can be calculated in any gauge, e.g. in a covariant gauge. On the other hand the radial propagator might be calculated according to Feynman rules. Since the results should coincide this might serve as a check for the validity and completeness of a set of radial gauge Feynman rules.
[**Acknowledgments:** ]{}
HW wants to thank Alex Kovner for his invaluable patience in his role as a testing ground of new ideas. SL thanks Professor Ulrich Heinz for valuable discussions and support. During this research SL was supported in part by Deutsche Forschungsgemeinschaft and Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie. HW was supported by the U.S. Department of Energy under grants No. DOE Nuclear DE–FG02–87ER–40328 and by the Alexander von Humboldt Foundation through their Feodor Lynen program.
Derivation of the Free Radial Propagator {#appprop}
========================================
The free radial propagator derived form the field strength formula shows a divergence in $D=4$, as already indicated in section \[sec:radprop\], eq. (\[divprop\]). Here we give the details of the algebra leading to this conclusion.
The following relations summarize the steps carried out in the calculation below: $$\begin{aligned}
&& \hskip 2cm
x_\mu \partial_x^\mu = \vert x\vert \,\partial_{\vert x\vert} \,,
\\
T_{\mu\nu}(x,y) &:=&
x^\alpha y^\beta \left(
g_{\mu\nu} \partial_\alpha^{x}\partial_\beta^{y}
+ g_{\alpha\beta}\partial_\mu^{x}\partial_\nu^{y}
- g_{\alpha\nu}\partial_\mu^{x}\partial_\beta^{y}
- g_{\mu\beta}\partial_\alpha^{x}\partial_\nu^{y}
\right)
\\ &=&
g_{\mu\nu} \partial_{\vert x\vert} \partial_{\vert y\vert}
\vert x\vert \,\vert y\vert
-\partial_\mu^x \,x_\nu \,\partial_{\vert y\vert} \vert y\vert
-\partial_\nu^y \,y_\mu \,\partial_{\vert x\vert} \vert x\vert
+\partial_\mu^x \partial_\nu^y \,x\cdot y \,. \nonumber $$ Introducing $\hat x :=x/\vert x\vert$ and $u=s\vert x\vert$, we have for arbitrary $f$: $$\begin{aligned}
&& \hskip 2cm sx_\alpha \,\partial_\beta^{sx} f(sx) = x_\alpha \,\partial_\beta^x
f(sx) \,,
\\ &&
\partial_{\vert x\vert}\int\limits_0^1 \!\! ds \, \vert x\vert f(sx)
= \partial_{\vert x\vert} \int\limits_0^1 \!\! ds \, \vert x\vert
f(s\vert x\vert \hat x) = \partial_{\vert x\vert}
\int\limits_0^{\vert x\vert} \!\! du \, f(u \hat x) = f(\vert x\vert
\hat x) = f(x) \,. \nonumber \\ &&\end{aligned}$$
We get $$\begin{aligned}
\lefteqn{{}-{4\pi^{D/2} \over \Gamma(D/2-1)} \,
\langle A^a_\mu(x) A^b_\nu(y) \rangle^0_{\mbox{\scriptsize rad}}=}
\nonumber\\&&
= \delta^{ab}
\int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\,
T_{\mu\nu}(sx,ty) \,[(sx-ty)^2]^{1-D/2}
\nonumber\\ &&
= \delta^{ab} \,T_{\mu\nu}(x,y)
\int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\,
[(sx-ty)^2]^{1-D/2}
\nonumber\\ &&
= \delta^{ab}
\left(
g_{\mu\nu} \partial_{\vert x\vert} \partial_{\vert y\vert}
\vert x\vert \,\vert y\vert
-\partial_\mu^x \,x_\nu \,\partial_{\vert y\vert} \vert y\vert
-\partial_\nu^y \,y_\mu \,\partial_{\vert x\vert} \vert x\vert
+\partial_\mu^x \partial_\nu^y \,x\cdot y
\right)
\nonumber\\&& \phantom{=}\times
\int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt\,
[(sx-ty)^2]^{1-D/2}
\nonumber\\ &&
= \delta^{ab}
\bigg(
g_{\mu\nu} [(x-y)^2]^{1-D/2}
\nonumber\\ && \phantom{mmm}
-\partial_\mu^x \int\limits^1_0 \!\! ds \,x_\nu \, [(sx-y)^2]^{1-D/2}
-\partial_\nu^y \int\limits^1_0 \!\! dt \,y_\mu \, [(x-ty)^2]^{1-D/2}
\nonumber\\ && \phantom{mmm}
+\partial_\mu^x \partial_\nu^y
\underbrace{\int\limits^1_0 \!\! ds \int\limits^1_0 \!\! dt \, x\cdot y \,
[(sx-ty)^2]^{1-D/2}}_{\textstyle \sim {\textstyle 1\over \textstyle 4-D}}
\bigg) \,.\end{aligned}$$ The divergent part of the double integral in the last line can be found in Appendix \[siint\]. At the moment however the exact form of the divergence is not important.
Renormalization Program for the Free Propagator {#app:renprog}
===============================================
In Section \[sec:radren\] we discussed the effect of renormalization on the free radial propagator. Here derive in detail the appropriate renormalization procedure starting form the renormalization properties of Wilson lines. Only a few of the many possible renormalization constants will contribute to the final result.
Since in the relation between the propagator and the appropriate Wilson loop (\[propa\]) a factor $1/g^2$ is involved all quantities especially all the $Z$’s and $W$’s of eq. (\[barw\]) have to be calculated up to $o(g_R^2)$. We have $$\begin{aligned}
\tilde W_i &=& 1 + (ig_R)^2 \delta\tilde W_i + o(g_R^4) \qquad
(i=1,2)\,, \\ Z &=& 1 + (ig_R)^2 \delta Z + o(g_R^4) \,,\\ Z_{11}
&=& 1 + (ig_R)^2 \delta Z_{11} + o(g_R^4) \,,\\ Z_{12} &=& 0 +
(ig_R)^2 \delta Z_{12} + o(g_R^4)\end{aligned}$$
yielding $$\begin{aligned}
\lefteqn{\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0 =
\lim_{D\to 4} \delta^{ab} {2N\over N^2 -1} \mu^{D-4}}
\label{freepr} \\ && \times
\lim_{x'\to x \atop y'\to y}\partial_\mu^x \partial_\nu^y \,
\left[
\begin{array}{c}
\delta Z(\bar C_\alpha) +\delta Z(\bar C_{\alpha'})
+\delta Z(\bar C_\beta) +\delta Z(\bar C_{\beta'}) \\
\quad
+\delta Z_{11} +\delta Z_{12} +\delta\tilde W_1
\end{array}
\right] \,.
\nonumber \end{aligned}$$ Using the fact that up to $o(g_R^2)$ the two quantities $W_1$ and $\tilde W_1$ are essentially the same[^5] we find $$\begin{aligned}
\lefteqn{\lim_{D\to 4} \delta^{ab} {2N\over N^2 -1} \mu^{D-4}
\lim_{x'\to x \atop y'\to y}\partial_\mu^x \partial_\nu^y \,
\delta\tilde W_1}
\nonumber\\ &&
= \lim_{D\to 4} \delta^{ab} {2N\over N^2 -1}
{1\over (ig)^2}\lim_{x'\to x \atop y'\to y}\partial_\mu^x
\partial_\nu^y \,
\left(1+(ig)^2 \delta W_1 \right)
\nonumber\\ &&
= \lim_{D\to 4} \delta^{ab} {2N\over N^2 -1}
{1\over (ig)^2}\lim_{x'\to x \atop y'\to y}\partial_\mu^x
\partial_\nu^y \,
W_1 \,\Big\vert_{g=0}
\nonumber\\ &&
=\lim_{D\to 4}
\langle A^a_\mu(x) A^b_\nu(y) \rangle^0_{\mbox{\scriptsize rad}} \,.
\label{naipa}\end{aligned}$$ To get the $\delta Z$’s we must calculate $\delta\tilde W_1$ which is straightforward using (\[w1def\]) and (\[roper\]). We only need the Feynman propagator (\[feynman\]) to get $$\begin{aligned}
\lefteqn{\delta\tilde W_1 =
-\mu^{4-D} {N^2 -1\over 2N} {\Gamma(D/2-1) \over 4\pi^{D/2}}}
\nonumber\\ &&
\left[
\left(
\vert x' \vert^{4-D} +\vert x-x' \vert^{4-D} + \vert x \vert^{4-D} +
\vert y' \vert^{4-D} +\vert y-y' \vert^{4-D} + \vert y \vert^{4-D}
\right) I_1
\right.\nonumber\\ &&
{}+ I_2(x',x-x') + I_2(x-x',-x) + I_2(-x,x')
\nonumber \\ &&
{}+ I_2(y',y-y') + I_2(y-y',-y) + I_2(-y,y')
\nonumber \\ &&
{}- I_2(x',-y') + I_2(x',-y) + I_2(y',-x) - I_2(x,-y)
\nonumber \\ &&
{}- I_3(y',-x',y-y') + I_3(y',-x,y-y') - I_3(x'-y',x-x',y'-y)
\nonumber \\ &&
{}- I_3(x',-y',x-x') + I_3(x',-y,x-x')
\Big] \label{dw1l}\end{aligned}$$ with $$I_1:= \int\limits_0^1 \!\! ds \int\limits_0^1 \!\! dt \,\Theta(s-t) \,
{1 \over [(s-t)^2]^{D/2-1}} \,,$$ $$I_2(p,q):= \int\limits_0^1 \!\! ds \int\limits_0^1 \!\! dt \,
{p\cdot q \over [(sp+tq)^2]^{D/2-1}} \,,$$ and $$I_3(m,p,q):= \int\limits_0^1 \!\! ds \int\limits_0^1 \!\! dt \,
{p\cdot q \over [(m+sp+tq)^2]^{D/2-1}} \,.$$ In the following we are interested only in the divergent parts of these integrals. The integrals $I_1$ and $I_2$ are calculated in Appendix \[siint\]. The results are $$I_1 = -{1\over 4-D} + \mbox{finite} \label{inti1}$$ and $$I_2(p,q) = {1\over 4-D} \, \gamma \cot\gamma + \mbox{finite}
\label{inti2}$$ where $\gamma$ is the angle between $p$ and $q$. The integral $I_3$ is finite as long as $m\neq 0$.
To specify the renormalization factors $Z$ we choose the minimal subtraction scheme $K^{\mbox{\scriptsize MS}}_\gamma$ as described in [@korrad]. In dimensional regularization all the divergences are given by sums of pole terms. We define every $Z$ factor to be given just by the respective sum. The important property of this renormalization scheme is that the $Z$ factors depend on the angles only and not on the length of the loop or of any part of the loop. Using (\[inti1\] and (\[inti2\])) the $Z$ factors can be read off from (\[dw1l\]) (c.f. [@brandt2]): $$\begin{aligned}
\delta Z(\bar C_\alpha)
&=&
{N^2 -1\over 2N}{1\over 4\pi^2} {1\over 4-D} \,
(\alpha \cot\alpha -1) \,,
\\
\delta Z(\bar C_{\alpha'})
&=&
{N^2 -1\over 2N}{1\over 4\pi^2} {1\over 4-D} \,
(\alpha' \cot\alpha' -1) \,,
\\
\delta Z(\bar C_\beta)
&=&
{N^2 -1\over 2N}{1\over 4\pi^2} {1\over 4-D} \,
(\beta \cot\beta -1) \,,
\\
\delta Z(\bar C_{\beta'})
&=&
{N^2 -1\over 2N}{1\over 4\pi^2} {1\over 4-D} \,
(\beta' \cot\beta' -1) \,,
\\
\delta Z_{11}
&=&
{N^2 -1\over 2N}{1\over 4\pi^2} {1\over 4-D}
\left[
(\gamma_{x'y} \cot\gamma_{x'y} -1) +
(\gamma_{xy'} \cot\gamma_{xy'} -1)
\right] \,, \nonumber \\ &&
\\
\delta Z_{12}
&=&
{N^2 -1\over 2N}{1\over 4\pi^2} {1\over 4-D}
\left[
\gamma_{xx'} \cot\gamma_{xx'} + \gamma_{yy'} \cot\gamma_{yy'}
\right.\nonumber\\ && \left.
{}- (\pi-\gamma_{x'y'})\cot(\pi-\gamma_{x'y'})
- (\pi-\gamma_{xy})\cot(\pi-\gamma_{xy})
\right] \,.\end{aligned}$$ Now we exploit the fact that only one of the angles, namely $\gamma_{xy}$, depends on $x$ [*and*]{} $y$. All the other ones depend only on $x$ or $y$ separately, or on none of them. This simplifies (\[freepr\]) drastically: $$\begin{aligned}
\lefteqn{\langle A^a_\mu(x) A^b_\nu(y) \rangle_R^0 =
\lim_{D\to 4} \delta^{ab} {2N\over N^2 -1} \mu^{D-4}
\lim_{x'\to x \atop y'\to y}\partial_\mu^x \partial_\nu^y \,
\left( \delta Z_{12} +\delta\tilde W_1 \right)}
\\ &&
=\lim_{D\to 4} \left(
\delta^{ab} \mu^{D-4} \partial_\mu^x \partial_\nu^y
\left(
{1\over 4\pi^2} {1\over 4-D} (\pi-\gamma_{xy})\cot\gamma_{xy}
\right)
+\langle A^a_\mu(x) A^b_\nu(y) \rangle^0_{\mbox{\scriptsize rad}}
\right)
\nonumber\end{aligned}$$ where we have used (\[naipa\]) to get the last expression.
Some Important Integrals {#siint}
========================
The integrals $I_1$ and $I_2$ played an important part in the renormalization procedure of appendix \[app:renprog\] and determine the divergences of the naive free radial propagator introduced in section \[sec:radprop\]. They are discussed in detail below.
To calculate $$I_1:= \int\limits_0^1 \!\! ds \int\limits_0^1 \!\! dt \,\Theta(s-t) \,
{1 \over [(s-t)^2]^{D/2-1}}$$ we introduce the substitution $$g=s-t \quad,\quad h=s+t$$ to get $$\begin{aligned}
I_1 &=&
{1\over 2} \int\limits_0^1 \!\! dg \int\limits_g^{2-g} \!\! dh \, g^{2-D}
= \int\limits_0^1 \!\! dg \, (1-g) \, g^{2-D} \nonumber\\
&=& {\Gamma(2) \Gamma(3-D) \over \Gamma(5-D)} = {1\over (4-D)(3-D)} \,.
\label{i1erg}\end{aligned}$$
For the calculation of $$I_2(p,q):= \int\limits_0^1 \!\! ds \int\limits_0^1 \!\! dt \,
{p\cdot q \over [(sp+tq)^2]^{D/2-1}} \,. \label{appi2}$$ we have to distinguish the two cases $p \neq \alpha q$ where the only divergence that appears is for $s=t=0$ and $p = \alpha q$ with an additional divergence at $s=t\alpha$. Here we will only need the former.
As a first step it is useful to separate off the divergence at the origin by the substitution $$\lambda = s+t \quad,\quad x=s/\lambda \,.$$ This yields $$\begin{aligned}
I_2(p,q) &=& \left( \int\limits_0^{1/2} \!\! dx
\int\limits_0^{1/(1-x)} \!\!\!\! d\lambda \, + \int\limits_{1/2}^1
\!\! dx \int\limits_0^{1/x} \!\! d\lambda \right) \lambda^{3-D}
{p\cdot q \over [(xp+(1-x)q)^2]^{D/2-1}} \nonumber\\ &=&
\int\limits_0^{1/2} \!\! dx \,{(1-x)^{D-4} \over 4-D} \, {p\cdot q
\over [(xp+(1-x)q)^2]^{D/2-1}} \nonumber\\ && +\int\limits_{1/2}^1
\!\! dx \,{x^{D-4} \over 4-D} \, {p\cdot q \over
[(xp+(1-x)q)^2]^{D/2-1}} \label{xint}\,.\end{aligned}$$ As long as $p \neq -q$ holds there are no divergences in the $x$-integration since $$u(x):=xp+(1-x)q$$ never vanishes. We introduce the angle between $p$ and $q$ $$\cos\gamma := {p\cdot q \over \vert p \vert \, \vert q\vert }$$ and the substitution [@korrad] $$e^{2i\psi} = {x\vert p\vert +(1-x)\vert q\vert e^{i\gamma} \over
x\vert p\vert +(1-x)\vert q\vert e^{-i\gamma}} \label{psisub}\,.$$ Note that $\psi$ is nothing but the angle between $p$ and $u(x)$. To perform this substitution in (\[xint\]) we need $$x = \vert q\vert \sin(\gamma-\psi) /N(\psi) \quad,\quad
1-x = \vert p \vert \sin\psi / N(\psi) \,,$$ $$[u(x)]^2 = p^2 q^2 \sin^2\gamma /[N(\psi)]^2
\quad\mbox{and} \quad
{d\psi \over dx} =
- {[N(\psi)]^2 \over \vert p\vert\,\vert q\vert \sin\gamma }$$ with $$N(\psi) := \vert p \vert \sin\psi + \vert q\vert \sin(\gamma-\psi)
\,.$$ In addition it is useful to introduce $$\psi' :=\psi(x=1/2)$$ which is the angle between $p$ and $p+q$ (cf. Fig. \[fig:pqangl\]).
Using all that we end up with $$\begin{aligned}
\lefteqn{
I_2(p,q) = \int\limits_\gamma^{\psi'}\!\! d\psi \,
{-\vert p\vert\,\vert q\vert \sin\gamma \over N^2}
\left({\vert p \vert \sin\psi \over N}\right)^{D-4}
\left({N^2 \over p^2 q^2 \sin^2\gamma} \right)^{D/2-1} {p\cdot q \over 4-D}}
&&\nonumber\\
&& +\int\limits_{\psi'}^0\!\! d\psi \,
{-\vert p\vert\,\vert q\vert \sin\gamma \over N^2}
\left({ \vert q\vert \sin(\gamma-\psi) \over N}\right)^{D-4}
\left({N^2 \over p^2 q^2 \sin^2\gamma} \right)^{D/2-1} {p\cdot q \over 4-D}
\nonumber \\
&=& {- \cos\gamma \sin^{3-D}\gamma\over 4-D} \left( \vert
q\vert^{4-D} \int\limits_\gamma^{\psi'}\!\! d\psi \,
\sin^{D-4}\psi + \vert p\vert^{4-D} \int\limits_{\psi'}^0\!\!
d\psi \, \sin^{D-4}(\gamma-\psi) \right) \nonumber\\
&=& {- \cos\gamma \sin^{3-D}\gamma \over
4-D} \left( \vert q\vert^{4-D}
\int\limits_\gamma^{\psi'}\!\! d\psi \, \sin^{D-4}\psi + \vert
p\vert^{4-D} \int\limits_\gamma^{\gamma-\psi'}\!\! d\psi \,
\sin^{D-4}\psi \right) \nonumber\\ &=& {1\over 4-D} \,\gamma
\cot\gamma \; +\mbox{finite}
\label{i2neq}\,. \end{aligned}$$
P. Pascual and R. Tarrach, “QCD: Renormalization for the Practitioner”, Lecture Notes in Physics, Vol. 194 (Springer, Berlin, 1984). H. Cheng and E.-C. Tsai, Phys. Rev. Lett. [**57**]{} (1986) 511. S. Caracciolo, G. Curci, and P. Menotti, Phys. Lett. [**B113**]{} (1982) 311. J.-P. Leroy, J. Micheli, and G.-C. Rossi, Z. Phys. [**C36**]{} (1987) 305. N.H. Christ and T.D. Lee, Phys. Rev. [**D22**]{} (1980) 939. P. Doust, Ann. Phys. [**177**]{} (1987) 169. P.J. Doust and J.C. Taylor, Phys. Lett. [**B197**]{} (1987) 232. M.A. Shifman, Nucl. Phys. [**B173**]{} (1980) 13. V.A. Fock, Sov. Phys. [**12**]{} (1937) 404. J. Schwinger, Phys. Rev. [**82**]{} (1952) 684. C. Cronström, Phys. Lett. [**B90**]{} (1980) 267. W. Kummer and J. Weiser, Z. Phys. [**C31**]{} (1986) 105. G. Modanese, J. Math. Phys. [**33**]{} (1992) 1523. P. Menotti, G. Modanese, and D. Seminara, Ann. Phys. [**224**]{} (1993) 110. M. Azam, Phys. Lett. [**B101**]{} (1981) 401. N.E. Bralic, Phys. Rev. [**D22**]{} (1980) 3090. H.-Th. Elze, M. Gyulassy, and D. Vasak, Nucl. Phys. [**B276**]{} (1986) 706. G.P. Korchemsky and A.V. Radyushkin, Nucl. Phys. [**B283**]{} (1987) 342. V.S. Dotsenko and S.N. Vergeles, Nucl. Phys. [**B169**]{} (1980) 527. R.A. Brandt, F. Neri, and M.-A. Sato, Phys. Rev. [**D24**]{} (1981) 879. R.A. Brandt, A. Gocksch, M.-A. Sato, and F. Neri, Phys. Rev. [**D26**]{} (1982) 3611. I.A. Korchemskaya and G.P. Korchemsky, Phys. Lett. [**B287**]{} (1992) 169.
[^1]: stefan.leupold@physik.uni-regensburg.de
[^2]: weigert@mnhepw.hep.umn.edu
[^3]: In fact he discussed the Abelian case but this makes no difference for free fields.
[^4]: For simplicity we discuss the QED case here. Aiming at an expression for the free gauge propagator this is no restriction of generality. For non-Abelian gauge groups c.f. [@azam].
[^5]: Only a factor $\mu^{D-4}$ comes in since $g_R$ as given in (\[grug\]) is dimensionless in contrast to $g$.
|
---
abstract: 'Our goal is to build systems which write code automatically from the kinds of specifications humans can most easily provide, such as examples and natural language instruction. The key idea of this work is that a flexible combination of pattern recognition and explicit reasoning can be used to solve these complex programming problems. We propose a method for dynamically integrating these types of information. Our novel intermediate representation and training algorithm allow a program synthesis system to learn, without direct supervision, when to rely on pattern recognition and when to perform symbolic search. Our model matches the memorization and generalization performance of neural synthesis and symbolic search, respectively, and achieves state-of-the-art performance on a dataset of simple English description-to-code programming problems.'
bibliography:
- 'main.bib'
---
Introduction
============
An open challenge in AI is to automatically write code from the kinds of specifications humans can easily provide, such as examples or natural language instruction. Such a system must determine both what the task is and how to write the correct code. Consider writing a simple function which maps inputs to outputs: $$\begin{aligned}
&[2, 3, 4, 5, 6] \to [2, 4, 6]\\
&[5, 8, 3, 2, 2, 1, 12] \to [8, 2, 2, 12]\end{aligned}$$ A novice programmer would not recognize from experience any of the program, and would have to *reason* about the entire program structure from first principles. This reasoning would be done by considering the definitions and syntax of the primitives in the programming language, and finding a way to combine these language constructs to construct an expression with the desired behavior.
A moderately experienced programmer might immediately *recognize*, from learned experience, that because the output list is always a subset of the input list, a [[`filter`]{}]{} function is appropriate: $${{\texttt{filter(input, <HOLE>)}}}$$ where [[`<HOLE>`]{}]{} is a lambda function which filters elements in the list. The programmer would then have to reason about the correct code for [[`<HOLE>`]{}]{}.
Finally, a programmer very familiar with this domain might immediately recognize both the need for a [[`filter`]{}]{} function, as well as the correct semantics for the lambda function, allowing the entire program to be written in one shot: $${{\texttt{{}}}filter(input, lambda x: x\%2==0)}$$ Depending on the familiarity of the domain and the complexity of the problem, humans use a flexible combination of recognition of learned patterns and explicit reasoning to solve programming problems [@lake2017building]. Familiar patterns are used, when they exist, and for unfamiliar code elements, explicit reasoning is employed.
This flexibility is not unique to input-output examples. For example, a natural language specification could be used to further specify the desired program, i.e., “Find the even values in a list." In this case, the process of writing code is analogous. For example, a programmer might learn that “find X in a list" means [[`filter`]{}]{}, and “even" corresponds to the code [[``]{}x%2==0]{}. For a less familiar task, such as “Find values in the list which are powers of two," a programmer might recognize the need for [[`filter`]{}]{}, but would need to reason about how to write a lambda function which classifies powers of two.
We propose a system which mimics the human ability to dynamically incorporate pattern recognition and reasoning to solve programming problems from examples or natural language specification. We show that without direct supervision, our model learns to find good intermediates between pattern recognition and symbolic reasoning components, and outperforms existing models on several programming tasks.
Recent work [@murali2017neural; @dong2018coarse] has attempted to combine learned pattern recognition and explicit reasoning using *program sketches*—schematic outlines of full programs [@solar2008program]. In @murali2017neural, a neural network is trained to output program sketches when conditioned on a spec, and candidate sketches are converted into full programs using symbolic synthesis techniques, which approximate explicit reasoning from first principles.
However, previous systems use static, hand-designed intermediate sketch grammars, which do not allow the system to learn how much to rely on pattern recognition and how much to rely on symbolic search. The neural network is trained to map from spec to a pre-specified sketch, and cannot learn to output a more detailed sketch, if the pattern matching task is easy, or learn to output a more general sketch, if the task is too difficult.
Ideally, a neuro-symbolic synthesis system should dynamically take advantage of the relative strengths of its components. When given an easy or familiar programming task, for example, it should rely on its learned pattern recognition, and output a fuller program with a neural network, so that less time is required for synthesis. In contrast, when given a hard task, the system should learn to output a less complete sketch and spend more time filling in the sketch with search techniques. We believe that this flexible integration of neural and symbolic computation, inspired by humans, is necessary for powerful, domain-general intelligence, and for solving difficult programming tasks.
The key idea in this work is to allow a system to *learn* a suitable intermediate sketch representation between a learned neural proposer and a symbolic search mechanism. Inspired by @murali2017neural, our technique comprises a learned [**neural sketch generator**]{} and a enumerative symbolic [**program [synthesizer]{}**]{}. In contrast to previous work, however, we use a flexible and domain-general sketch grammar, and a novel self-supervised training objective, which allows the network to learn how much to rely on each component of the system. The result is a flexible, domain-general program synthesis system, which has the ability to learn sophisticated patterns from data, comparably to @devlin2017robustfill, as well as utilize explicit symbolic search for difficult or out-of-sample problems, as in @balog2016deepcoder.
Without explicit supervision, our model learns good intermediates between neural network and synthesis components. This allows our model to increase data efficiency and generalize better to out-of-sample test tasks compared to RNN-based models. Concretely, our contributions are as follows:
- We develop a novel neuro-symbolic program synthesis system, which writes programs from input-output examples and natural language specification by learning a suitable intermediate sketch representation between a neural network sketch generator and a symbolic [synthesizer]{}.
- We introduce a novel training objective, which we used to train our system to find suitable sketch representations without explicit supervision.
- We validate our system by demonstrating our results in two programming-by-example domains, list processing problems and string transformation problems, and achieve state-of-the-art performance on the AlgoLisp English-to-code test dataset.
Problem Formulation
===================
Assume that we have a DSL which defines a space of programs, $\mathcal G$. In addition, we have a set of program specifications, or *spec*s, which we wish to ‘solve’. We assume each spec $\mathcal{X}_i$ is satisfied by some true unknown program $F_i$.
If our specification contains a set of IO examples $\mathcal{X}_i = \{(x_{ij}, y_{ij})\}_{j=1..n}$, then we can say that we have solved a task $\mathcal{X}_i$ if we find the true program $F_i$, which must satisfy all of the examples: $$\forall j: F_i(x_{ij}) = y_{ij}$$ Our goal is to build a system which, given $\mathcal{X}_i$, can quickly recover $F_i$. For our purposes, *quickly* is taken to mean that such a solution is found within some threshold time, $\textrm{Time}(\mathcal{X}_i \rightarrow F_i) < t$. Formally, then, we wish to maximize the probability that our system solves each problem within this threshold time: $$\max \log \mathbb{P}\Big[\textrm{Time}(\mathcal{X}_i \rightarrow F_i) < t\Big] \label{eq:formulation}$$ Additionally, for some domains our spec $\mathcal{X}$ may contain additional informal information, such as natural language instruction. In this case, we can apply same formalism, maximizing the probability that the true program $F_i$ is found given the spec $\mathcal X_i$, within the threshold time.
Our Approach: Learning to Infer Sketches
========================================
System Overview:
----------------
Our approach, inspired by work such as @murali2017neural, is to represent the relationship between program specification and program using an intermediate representation called a program sketch. However in contrast to previous work, where the division of labor between generating sketches and filling them in is fixed, our approach allows this division of labor to be learned, without additional supervision. We define a *sketch* simply as a valid program tree in the DSL, where any number of subtrees has been replaced by a special token: [**$<$HOLE$>$** ]{}. Intuitively, this token designates locations in the program tree for which pattern-based recognition is difficult, and more explicit search methods are necessary.
Our system consists of two main components: 1) a [**sketch generator**]{}, and 2) a [**program [synthesizer]{}**]{}.
The [**sketch generator**]{} is a distribution over program sketches given the spec: $q_\phi(sketch|\mathcal{X})$. The generator is parametrized by a recurrent neural network, and is trained to assign high probability to sketches which are likely to quickly yield programs satisfying the spec when given to the [synthesizer]{}. Details about the learning scheme and architecture will be discussed below.
The [**program [synthesizer]{}**]{} takes a sketch as a starting point, and performs an explicit symbolic search to “fill in the holes" in order to find a program which satisfies the spec.
Given a set of test problems, in the form of a set of specs, the system searches for correct programs as follows: The sketch generator, conditioned on the program spec, outputs a distribution over sketches. A fixed number of candidate sketches $\{s_i\}$ are extracted from the generator. This set $\{s_i\}$ is then given to the program [synthesizer]{}, which searches for full programs maximizing the likelihood of the spec. For each candidate sketch, the [synthesizer]{} uses symbolic enumeration techniques to search for full candidate programs which are formed by filling in the holes in the sketch.
Using our approach, our system is able to flexibly learn the optimal amount of detail needed in the sketches, essentially learning how much to rely on each component of the system. Furthermore, due to our domain-general sketch grammar, we are easily able to implement our system in multiple different domains with very little overhead.
\[model\] {width="100.00000%"}
Learning to Infer Sketches via Self-supervision
-----------------------------------------------
By using sketches as an intermediate representation, we reframe our program synthesis problem (Eq. \[eq:formulation\]) as follows: learn a sketch generator $q_\phi(s|\mathcal{X})$ which, given a spec $\mathcal{X}_i$, produces a ‘good’ sketch $s$ from which the [synthesizer]{} can quickly find the solution $F_i$. We may thus wish to maximize the probability that our sketch generator produces a ‘good’ sketch: $$\max_\phi \log \mathbb{P}_{s \sim q_\phi(-|\mathcal{X}_i)}\Big[\textrm{Time}(s \rightarrow F_i) < t\Big] \label{eq:formulation2}$$ where $\textrm{Time}(s \rightarrow F_i)$ is the time taken for the [synthesizer]{} to discover the solution to $\mathcal{X}_i$ by filling the holes in sketch $s$, and $t$ is the [synthesizer]{}’s evaluation budget.
In order to learn a system which is most robust, we make one final modification to Equation (\[eq:formulation2\]): at train time we do not necessarily know what the timeout will be during evaluation, so we would like to train a system which is agnostic to the amount of time it would have. Ideally, if a program can be found entirely (or almost entirely) using familiar patterns, then the sketch generator should assign high probability to very complete sketches. However, if a program is unfamiliar or difficult, the sketches it favors should be very general, so that the [synthesizer]{} must do most of the computation. To do this, we can train the generator to output sketches which would be suitable for a wide distribution of evaluation budgets. This can be achieved by allowing the budget $t$ to be a random variable, sampled from some distribution $\mathcal{D}_t$. Adding this uncertainty to Equation (\[eq:formulation2\]) yields: $$\begin{aligned}
&\max_\phi
\underset{\begin{subarray}{c}t \sim \mathcal{D}_t \\s \sim q_\phi(-|\mathcal{X}_i) \end{subarray}}{\log \mathbb{P}}
\Big[\textrm{Time}(s \rightarrow F_i) < t\Big]
\label{eq:formulation3}\end{aligned}$$ In practice, we can achieve this maximization by self-supervised training. That is, given a dataset of program-spec pairs, for each spec we optimize the *generator* to produce only the sketches from which we can quickly recover its underlying program. Thus, given training data as $(F,\mathcal{X})$ pairs, our training objective may be written: $$\begin{aligned}
&obj = \underset{\begin{subarray}{c}t \sim \mathcal{D}_t \\(F,\mathcal{X}) \sim \mathcal G \end{subarray}}{\mathbb{E}} \log \sum_{s:\textrm{Time}(s\rightarrow F)<t} q_\phi(s|\mathcal{X})
\label{objective}\end{aligned}$$ During each step of training, $t$ is sampled from $\mathcal{D}_t$, and the network is trained to assign high probability to those sketches which can be synthesized into a full program within the budget $t$. Using this training scheme, the network learns to output a distribution of sketches, some of which are very specific and can be synthesized quickly, while others are more general but require more time to synthesize into full programs. This allows the system to perform well with various enumeration budgets and levels of problem difficulty: the system quickly solves “easy" problems with very “concrete" sketches, but also samples more general sketches, which can be used to solve difficult problems for which the system’s learned inductive biases are less appropriate.
Our Implementation
==================
In this section, we discuss our implementation of the above ideas which we use to solve the list processing and string editing tasks discussed above, in a system we call [<span style="font-variant:small-caps;">SketchAdapt</span>]{}.
Seq-to-Seq Neural Sketch Generator
----------------------------------
For our sketch generator, we use a sequence-to-sequence recurrent neural network with attention, inspired by @devlin2017robustfill and @bunel2018leveraging. Our model is inspired by the “Att-A" model in @devlin2017robustfill: the model encodes the spec via LSTM encoders, and then decodes a program token-by-token while attending to the spec. To facilitate the learning of the output grammar, our model also has an additional learned LSTM language model as in @bunel2018leveraging, which reweights the program token probabilities from the seq-to-seq model. This LSTM language model is simply trained to assign high probability to likely sequences of tokens via supervised learning.
Synthesis via Enumeration
-------------------------
Our symbolic sketch [synthesizer]{} is based on @ellis2018learning and @balog2016deepcoder and has two components: a breadth-first probabilistic enumerator, which enumerates candidate programs from most to least likely, and a neural recognizer, which uses the spec to guide this probabilistic enumeration.
The enumerator, based on @ellis2018learning uses a strongly typed DSL, and assumes that all programs are expressions in $\lambda$-calculus. Each primitive has an associated production probability. These production probabilities constitute the parameters, $\theta$, for a probabilistic context free grammar, thus inducing a distribution over programs $p(F|s,\theta)$. Synthesis proceeds by enumerating candidate programs which satisfy a sketch in decreasing probability under this distribution. Enumeration is done in parallel for all the candidate sketches, until a full program is found which satisfies the input-output examples in the spec, or until the enumeration budget is exceeded.
The learned recognizer is inspired by @menon2013machine and the “Deepcoder" system in @balog2016deepcoder. For a given task, an RNN encodes each spec into a latent vector. The latent vectors are averaged, and the result is passed into a feed-forward MLP, which terminates in a softmax layer. The resulting vector is used as the set of production probabilities $\theta$ which the enumerator uses to guide search.
[<span style="font-variant:small-caps;">SketchAdapt</span>]{} succeeds by exploiting the fundamental difference in search capabilities between its neural and symbolic components. Pure-synthesis approaches can enumerate and check candidate programs extremely quickly—we enumerate roughly $3\times10^3$ prog/sec, and the fastest enumerator for list processing exceeds $10^6$ prog/sec. However, generating expressions larger than a few nodes requires searching an exponentially large space, making enumeration impractical for large programs. Conversely, seq2seq networks (and tree RNNs) require fewer samples to find a solution but take much more time per sample (many milliseconds per candidate in a beam search) so are restricted to exploring only hundreds of candidates. They therefore succeed when the solution is highly predictable (even if it is long), but fail if even a small portion of the program is too difficult to infer. By flexibly combining these two approaches, our system searches the space of programs more effectively than either approach alone; [<span style="font-variant:small-caps;">SketchAdapt</span>]{} uses learned patterns to guide a beam search when possible, and fast enumeration for portions of the program which are difficult to recognize. This contrasts with @murali2017neural, where the division of labor is fixed and cannot be learned.
Training
--------
The training objective above (Eq. \[objective\]) requires that for each training program $F$, we know the set of sketches which can be synthesized into $F$ in less than time $t$ (where the synthesis time is given by $\textrm{Time}(s\rightarrow F)$.) A simple way to determine this set would be to simulate synthesis for each candidate sketch, to determine synthesis can succeed in less time than $t$. In practice, we do not run synthesis during training of the sketch generator to determine $\textrm{Time}(s\rightarrow F)$. One benefit of the probabilistic enumeration is that it provides an estimate of the enumeration time of a sketch. It is easy to calculate the likelihood $p(F|s,\theta)$ of any full program $F$, given sketch $s$ and production probabilities $\theta$ given by the recognizer $\theta = r(\mathcal{X})$. Because we enumerate programs in decreasing order of likelihood, we know that search time (expressed as number of evaluations) can be upper bounded using the likelihood: $\textrm{Time}(s\rightarrow F) \leq [p(F|s, \theta)]^{-1}$. Thus, we can use the inverse likelihood to lower bound Equation (\[objective\]) by: $$\begin{aligned}
obj & \geq \underset{\begin{subarray}{c}t \sim \mathcal{D}_t \\(F,\mathcal{X}) \sim \mathcal G \end{subarray}}{\mathbb{E}} \log \sum_{s: p^{-1}(F|s, \theta)<t} q_\phi(s|\mathcal{X})
\label{obj2}\end{aligned}$$ While it is often tractable to evaluate this sum exactly, we may further reduce computational cost if we can identify a smaller set sketches which dominate the log sum. Fortunately we observe that the generator and [synthesizer]{} are likely to be highly correlated, as each program token must be explained by either one or the other. That is, sketches which maximize $q_\phi(s|\mathcal{X})$ will typically minimize $p(F|s, \theta)$. Therefore, we might hope to find a close bound on Equation (\[obj2\]) by summing only the few sketches that minimize $p(F|s,\theta)$. In this work we have found it sufficient to use only a *single minimal sketch*, yielding the objective $obj^*$: $$\begin{aligned}
obj^* & = \underset{\begin{subarray}{c}t \sim \mathcal{D}_t \\(F,\mathcal{X}) \sim \mathcal G \end{subarray}}{\mathbb{E}} \log q_\phi(s^*|\mathcal{X}) \leq obj, \nonumber \\
\textrm{where }s^* & =\argmin_{s:p^{-1}(F|s, \theta)<t} p(F|s, \theta)\end{aligned}$$ This allows us to perform a much simpler and more practical training procedure, maximizing a lower bound of our desired objective. For each full program sampled from the DSL, we sample a timeout $t \sim \mathcal{D}_t$, and determine the sketch with maximum likelihood, for which $p^{-1}(F|s, \theta)<t$. We then train the neural network to maximize the probability of that sketch. Intuitively, we are sampling a random timeout, and training the network to output the easiest sketch which still solves the task within that timeout.
For each full program $F$, we assume that the set of sketches which can be synthesized into $F$ will have enumeration times distributed roughly exponentially. Therefore, in order for our training procedure to utilize a range of sketch sizes, we use an exponential distribution for the timeout: $t \sim \text{Exp}(\alpha)$, which works well in practice.
Our training methodology is described in Algorithm \[trainingalgorithm\], and the evaluation approach is described in Algorithm \[evaluationalgorithm\].
Sketch Generator $q_\phi(sketch| \mathcal{X})$; Recognizer $r_\psi(\mathcal{X}, sketch)$; Enumerator dist. $p(F|\theta, sketch)$, Base Parameters $\theta_{base}$
------------------------------------------------------------------------
*Train Recognizer, $r_\psi$:* Sample $t \sim \mathcal{D}_t$ $sketches, probs$ $\gets$ list all possible sketches of $F$, with probs given by $p(F|s,\theta_{base})$ $sketch$ $\gets$ sketch with largest prob s.t. prob $< t$. $\theta \gets r_\psi(\mathcal{X}, sketch)$ grad. step on $\psi$ to maximize $\log p(F|\theta, sketch)$
------------------------------------------------------------------------
*Train Sketch Generator, $q_\phi$:* Sample $t \sim \mathcal{D}_t$ $\theta \gets r_\psi(\mathcal{X})$ $sketches, probs$ $\gets$ list all possible sketches of $F$, with probs given by $p(F|s,\theta)$ $sketch$ $\gets$ sketch with largest prob s.t. prob $< t$. grad. step on $\phi$ to maximize $\log q_\phi(sketch|\mathcal{X})$
Sketch Generator $q_\phi(sketch| \mathcal{X})$; Recognizer $r_\psi(\mathcal{X}, sketch)$; Enumerator dist. $p(F|\theta, sketch)$
------------------------------------------------------------------------
**function** synthesizeProgram$(\mathcal{X})$ $sketches \gets$ beam search $q_\phi(\cdot|\mathcal{X})$ $\theta_{sketch} \gets r_\psi(\mathcal{X},sketch)$ **while** timeout not exceeded **do** $F \gets$ next full prog. from enumerate($sketch,\theta_{sketch}$) **return** $F$ **end while**
[lll]{} & &\
\
1, \[-101, 63, 64, 79, 119, 91, -56, 47, -74, -33\] &39 &\
4, \[-6, -96, -45, 17, 26, -38, 17, -18, -112, -48\] &8\
2, \[-9, 5, -8, -9, 9, -3, 7, -5, -10, 1\] &\[100, 16\] &\
3, \[-5, -8, 0, 10, 2, -7, -3, -5, 6, 2\] & \[36, 81, 1\]\
{width="48.00000%"} {width="48.00000%"}
Experiments
===========
We provide the results of evaluating [<span style="font-variant:small-caps;">SketchAdapt</span>]{} in three test domains. For all test domains, we compare against two alternate models, which can be regarded as lesioned versions of our model, as well as existing models in the literature:
The [**“[Synthesizer]{} only"**]{} alternate model is equivalent to our program [synthesizer]{} module, using a learned recognition model and enumerator. Instead of enumerating from holes in partially filled-in sketches, the “[Synthesizer]{} only" model enumerates all programs from scratch, starting from a single [**$<$HOLE$>$** ]{} token. This model is comparable to the “Deepcoder" system in @balog2016deepcoder, which was developed to solve the list transformation tasks we examine in subsection \[list\].
The [**“Generator only"**]{} alternate model is a fully seq-to-seq RNN, equivalent in architecture to our sketch generator, trained simply to predict the entire program. This model is comparable to the “RobustFill" model in @devlin2017robustfill, which was developed to solve the string transformation tasks we examine in subsection \[string\]. This model is also comparable to the sequence-to-sequence models in @polosukhin2018neural.
![Performance on string editing problems. Although RobustFill was developed for string editing problems, [[<span style="font-variant:small-caps;">SketchAdapt</span>]{}]{} achieves higher accuracy on these tasks.[]{data-label="rbgraph1"}](rb_new_combined_percentage_camera_ready-eps-converted-to.pdf){width="49.00000%"}
\[sample-table\]
--------------- ------- --
‘Madelaine’ ‘M-’
‘Olague’ ‘O-’
‘118-980-214’ ‘214’
‘938-242-504’ ‘504’
--------------- ------- --
List Processing {#list}
---------------
In our first, small scale experiment, we examine problems that require an agent to synthesize programs which transform lists of integers. We use the list processing DSL from @balog2016deepcoder, which consists of 34 unique primitives. The primitives consist of first order list functions, such as [[`head`]{}]{}, [[`last`]{}]{} and [[`reverse`]{}]{}, higher order list functions, such as [[`map`]{}]{}, [[`filter`]{}]{} and [[`zipwith`]{}]{}, and lambda functions such as [[`min`]{}]{}, [[`max`]{}]{} and [[`negate`]{}]{}. Our programs are semantically equivalent, but differ from those in @balog2016deepcoder in that we use no bound variables (apart from input variables), and instead synthesize a single s-expression. As in @balog2016deepcoder, the spec for each task is a small number of input-output examples. See Table \[listtable\] for sample programs and examples.
Our goal was to determine how well our system could perform in two regimes: *within-sample*, where the test data is similar to the training data, and *out-of-sample*, where the test data distribution is different from the training distribution. We trained our model on programs of length 3, and tested its performance two datasets, one consisting of 100 programs of length 3, and the other with 100 length 4 programs. With these experiments, we could determine how well our system synthesizes easy and familiar programs (length 3), and difficult programs which require generalization (length 4).
During both training and evaluation, the models were conditioned on 5 example input-output pairs, which contain integers with magnitudes up to 128. In Figure \[listresultsT3\], we plot the proportion of tasks solved as a function of the number of candidate programs enumerated per task.
Although a “Generator only" RNN model is able to synthesize many length 3 programs, it performs very poorly on the out-of-sample length 4 programs. We also observe that, while the “[Synthesizer]{} only” model can take advantage of a large enumeration budget and solve a higher proportion of out-of-sample tasks than the “Generator only" RNN, it does not take advantage of learned patterns to synthesize the length 3 programs quickly, due to poor inductive biases. Only our model is able to perform well on both within-sample and out-of-sample tasks.
String Transformations {#string}
----------------------
In our second test domain, we explored programs which perform string transformations, as in @gulwani2011automating. These problems involve finding a program which maps an input string to an output string. Typically, these programs are used to manipulate the syntactic form of the underlying information, with minimal changes to the underlying semantics. Examples include converting a list of [[`FirstName LastName`]{}]{} to [[`LastInitial, Firstname`]{}]{}. These problems have been studied by @gulwani2011automating [@polozov2015flashmeta; @devlin2017robustfill] and others. We show that our system is able to accurately recover these programs.
As our test corpus, we used string editing problems from the SyGuS [@alur2016sygus] program synthesis competition, and string editing tasks used in @ellis2018learning. We excluded tasks requiring multiple input strings or a pre-specified string constant, leaving 48 SyGuS programs and 79 programs from @ellis2018learning. Because we had a limited corpus of problems, we trained our system on synthetic data only, sampling all training programs from the DSL.
Because our system has no access to the test distribution, this domain allows us to see how well our method is able to solve real-world problems when trained only on a synthetic distribution.
Furthermore, the string editing DSL is much larger than the list transformation DSL. This means that enumerative search is both slower and less effective than for the list transformation programs, where a fast enumerator could brute force the entire search space [@balog2016deepcoder]. Because of this, the “[Synthesizer]{} only” model is not able to consistently enumerate sufficiently complex programs from scratch.
We trained our model using self-supervision, sampling training programs randomly from the DSL and conditioning the models on 4 examples of input-output pairs, and evaluated on our test corpus. We plot our results in Figure \[rbgraph1\].
Overall, [[<span style="font-variant:small-caps;">SketchAdapt</span>]{}]{} outperforms the “[Synthesizer]{} only" model, and matches or exceeds the performance of the “Generator only" RNN model, which is noteworthy given that it is equivalent to RobustFill, which was designed to synthesize string editing programs. We also note that the beam size used in evaluation of the “Generator only" RNN model has a large impact on performance. However, the performance of our [[<span style="font-variant:small-caps;">SketchAdapt</span>]{}]{} system is less dependent on beam size, suggesting that the system is able to effectively supplement a smaller beam size with enumeration.
Algolisp: Description to Programs
---------------------------------
![AlgoLisp: varying training data size. We trained our model and baselines on various dataset sizes, and evaluated performance on a held-out test dataset. Our [<span style="font-variant:small-caps;">SketchAdapt</span>]{} system considerably outperforms baselines in the low-data regime.[]{data-label="mainalgolispresults"}](algolisp_init_final_camera_ready.pdf){width="48.00000%"}
------------------------------------------------------- --------------------------------------------------
Consider an array of numbers, [[`( filter a ( lambda1 ( == ( `]{}]{}
find elements in the given array not divisible by two [[``]{} % arg1 2 ) 1)))]{}
You are given an array of numbers, [[`(reduce(reverse(digits(deref (sort a) `]{}]{}
your task is to compute median [[`(/ (len a) 2)))) 0`]{}]{}
in the given array with its digits reversed [[`(lambda2 (+(* arg1 10) arg2)))`]{}]{}
------------------------------------------------------- --------------------------------------------------
\[algolisptable\]
------------------------------------------------------------------------ -------- -------------- -------- --------------
[<span style="font-variant:small-caps;">SketchAdapt</span>]{} (Ours) (88.8) [**90.0**]{} (95.0) [**95.8**]{}
[Synthesizer]{} only (5.2) 7.3 (5.6) 8.0
Generator only (91.4) 88.6 (98.4) 95.6
[<span style="font-variant:small-caps;">SketchAdapt</span>]{}, IO only (4.9) 8.3 (5.6) 8.8
Seq2Tree+Search (86.1) 85.8 - -
SAPS (83.0) 85.2 (93.2) 92.0
------------------------------------------------------------------------ -------- -------------- -------- --------------
: Algolisp results on full dataset
[lll]{} &&\
[<span style="font-variant:small-caps;">SketchAdapt</span>]{} (Ours) &[**34.4**]{}&[**29.8**]{}\
[Synthesizer]{} only &23.7& 0.0\
Generator only &4.5& 1.1\
Our final evaluation domain is the AlgoLisp DSL and dataset, introduced in @polosukhin2018neural. The AlgoLisp dataset consists of programs which manipulate lists of integers and lists of strings. In addition to input-output examples, each specification also contains a natural language description of the desired program. We use this dataset to examine how well our system can take advantage of highly unstructured specification data such as natural language, in addition to input-output examples.
The AlgoLisp problems are very difficult to solve using only examples, due to the very large search space and program complexity (see Table \[algolisptable\]: [<span style="font-variant:small-caps;">SketchAdapt</span>]{}, IO only). However, the natural language description makes it possible, with enough data, to learn a highly accurate semantic parsing scheme and solve many of the test tasks. In addition, because this domain uses real data, and not data generated from self-supervision, we wish to determine how data-efficient our algorithm is. Therefore, we train our model on subsets of the data of various sizes to test generalization.
Figure \[mainalgolispresults\] and Table \[algolisptable\] depict our main results for this domain, testing all systems with a maximum timeout of 300 seconds per task.[^1] When using a beam size of 10 on the full dataset, [<span style="font-variant:small-caps;">SketchAdapt</span>]{} and the “Generator only" RNN baseline far exceed previously reported state of art performance and achieve near-perfect accuracy, whereas the “[Synthesizer]{}s only" model is unable to achieve high performance. However, when a smaller number of training programs is used, [<span style="font-variant:small-caps;">SketchAdapt</span>]{} significantly outperforms the “Generator only" RNN baseline. These results indicates that the symbolic search allows [<span style="font-variant:small-caps;">SketchAdapt</span>]{} to perform stronger generalization than pure neural search methods.
[**Strong generalization to unseen subexpressions:**]{} As a final test of generalization, we trained [<span style="font-variant:small-caps;">SketchAdapt</span>]{} and our baseline models on a random sample of 8000 training programs, excluding all those which contain the function ‘odd’ as expressed by the AlgoLisp subexpression [[``]{}(lambda1(== (% arg1 2) 1))]{} (in python, [[``]{}lambda x: x%2==1]{}). We then evaluate on all 635 test programs containing ‘odd’, as well as the 638 containing ‘even’ ([[``]{}lambda x: x%2==0]{}). As shown in Table \[algolispodd\], the “Generator only" RNN baseline exhibits only weak generalization, solving novel tasks which require the ‘even’ subexpression but not those which require the previously unseen ‘odd’ subexpression. By contrast, [<span style="font-variant:small-caps;">SketchAdapt</span>]{} exhibits strong generalization to both ‘even’ and ‘odd’ programs.
Related Work
============
Our work takes inspiration from the neural program synthesis work of @balog2016deepcoder, @devlin2017robustfill and @murali2017neural. Much recent work has focused on learning programs using deep learning, as in @kalyan2018neural, @bunel2018leveraging, @shin2018improving, or combining symbolic and learned components, such as @parisotto2016neuro, @kalyan2018neural, @chen2017towards, @zohar2018automatic, and @zhang2018neural. Sketches have also been explored for semantic parsing [@dong2018coarse] and differentiable programming [@bovsnjak2017programming]. We also take inspiration from the programming languages literature, particularly Sketch [@solar2008program] and angelic nondeterminism [@bodik2010programming]. Other work exploring symbolic synthesis methods includes $\lambda^2$ [@feser2015synthesizing] and @schkufza2016stochastic. Learning programs has also been studied from a Bayesian perspective, as in EC [@dechter2013bootstrap], Bayesian Program Learning [@lake2015human], and inference compilation [@le2016inference].
Discussion
==========
We developed a novel neuro-symbolic scheme for synthesizing programs from examples and natural language. Our system, [<span style="font-variant:small-caps;">SketchAdapt</span>]{}, combines neural networks and symbolic synthesis by learning an intermediate ‘sketch’ representation, which dynamically adapts its specificity for each task. Empirical results show that [<span style="font-variant:small-caps;">SketchAdapt</span>]{} recognizes common motifs as effectively as pure RNN approaches, while matching or exceeding the generalization of symbolic synthesis methods. We believe that difficult program synthesis tasks cannot be solved without flexible integration of pattern recognition and explicit reasoning, and this work provides an important step towards this goal.
We also hypothesize that learned integration of different forms of computation is necessary not only for writing code, but also for other complex AI tasks, such as high-level planning, rapid language learning, and sophisticated question answering. In future work, we plan to explore the ideas presented here for other difficult AI domains.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Kevin Ellis and Lucas Morales for very useful feedback, as well as assistance using the [EC codebase](https://github.com/ellisk42/ec). M. N. is supported by an NSF Graduate Fellowship and an MIT BCS Hilibrand Graduate Fellowship. L. H. is supported by the MIT-IBM Watson AI Lab.
[^1]: As in @bednarek2018ain, we filter the test and dev datasets for only those tasks for which reference programs satisfy the given specs. The “filtered" version is also used for Figure \[mainalgolispresults\].
|
---
abstract: 'Classification of very high resolution (VHR) satellite images has three major challenges: 1) inherent low intra-class and high inter-class spectral similarities, 2) mismatching resolution of available bands, and 3) the need to regularize noisy classification maps. Conventional methods have addressed these challenges by adopting separate stages of image fusion, feature extraction, and post-classification map regularization. These processing stages, however, are not jointly optimizing the classification task at hand. In this study, we propose a single-stage framework embedding the processing stages in a recurrent multiresolution convolutional network trained in an *end-to-end* manner. The feedforward version of the network, called *FuseNet*, aims to match the resolution of the panchromatic and multispectral bands in a VHR image using convolutional layers with corresponding downsampling and upsampling operations. Contextual label information is incorporated into *FuseNet* by means of a recurrent version called *ReuseNet*. We compared *FuseNet* and *ReuseNet* against the use of separate processing steps for both image fusion, e.g. pansharpening and resampling through interpolation, and map regularization such as *conditional random fields*. We carried out our experiments on a land cover classification task using a Worldview-03 image of Quezon City, Philippines and the ISPRS 2D semantic labeling benchmark dataset of Vaihingen, Germany. *FuseNet* and *ReuseNet* surpass the baseline approaches in both quantitative and qualitative results.'
author:
- 'John Ray Bergado, Claudio Persello, and Alfred Stein[^1] [^2]'
bibliography:
- 'References.bib'
title: Recurrent Multiresolution Convolutional Networks for VHR Image Classification
---
[Bergado : Recurrent fully-convolutional networks for VHR image fusion, classification, and map regularization]{}
Convolutional networks, recurrent networks, land cover classification, VHR image, deep learning.
Introduction
============
of very high resolution (VHR) remotely sensed images allows us to automatically produce maps at a level of detail comparable to conventional in-situ mapping methods. Due to the high spatial resolution of such images, automated classification comes with a set of challenges. One challenge is the inherent low intra-class and high inter-class spectral similarities, inhibiting discrimination of the classes of interest. Conventional methods address this challenge by extracting spatial-contextual features from the image such as texture-describing measures, e.g. gray level co-occurrence matrix (GLCM) [@Haralick1973] and local binary patterns (LBP) [@Ojala2002], or products of morphological operators [@DallaMura2010; @Fauvel2013]. This step is crucial for obtaining discriminative features and accurate classification. However, such feature extraction methods are often disjoint from the supervised classifier, and, hence, not optimized for the task at hand. Deep learning offers a framework to build end-to-end classifiers—directly learning the predictions from the inputs with minimal or no pre-classification and post-classification steps. Features automatically extracted by deep learning based classifiers such as convolutional neural networks (CNN) [@LeCun1998] perform better than intermediate handcrafted features [@Bergado2016; @Mboga2017]. These networks automatically learn spatial-contextual features directly from the input VHR image—effectively integrating the feature extraction step into the training of the classifier as shown in Figure \[fig:ReuseNet\_pipeline\] (b). The design of network architecture, inspired by the model of the visual cortex [@Hubel1962], makes CNN suitable for image analysis and land cover classification.
![Illustration comparing a standard (a), state-of-the-art (b), and proposed (c) piplelines for classifying multiresolution VHR images.[]{data-label="fig:ReuseNet_pipeline"}](./img/ReuseNet_pipeline_v5.png){width="50.00000%"}
Another challenge in the classification of VHR images is the multiresolution nature of the images acquired by space-borne sensors. Most VHR satellite images (e.g. Quickbird, Worldview, IKONOS, and Pleiades) capture panchromatic (PAN) images in a spatial resolution four times of the multispectral (MS) bands. Such mismatch in spatial resolution of the images requires an additional step to fuse these images before performing the semantic analysis. Pansharpening and interpolation-based resampling are common techniques for fusing a multiresolution image [@Ha2013]. Similar to conventional feature extraction methods, the operations to fuse multiresolution bands of a VHR image add another separate pre-classification step that is disjoint from the training of the supervised classifier, and, hence, may not be optimal for the task at hand. CNN extracts a hierarchy of spatial features at different resolutions. We can exploit the multiresolution nature of the VHR data to design a CNN that performs fusion and feature extraction at the same time.
Literature shows that classification accuracy can be improved by using post-classification spatial regularization [@Chen2016; @Paisitkriangkrai2016; @Wang2017]. Methods employing graphical models, such as conditional random fields (CRF) and Markov random fields (MRF), provide a way to perform this spatial regularization step. Similar to the two pre-classification steps described above, a post-classification map regularization technique adds another step independent of the training of the classifier itself—further including a separate objective function to be optimized. For classifying a multiresolution VHR image, a typical classification pipeline would be composed of three main stages: a pre-classification step performing image fusion and feature extraction, a supervised learning algorithm performing the classification, and a post-classification step regularizing the maps obtained from the supervised classification algorithm. This conventional approach is shown in Figure \[fig:ReuseNet\_pipeline\] (a).
Convolutional networks have been recently applied to classify remotely sensed images with very high resolution [@Paisitkriangkrai2016; @Sherrah2016; @Maggiori2017; @Mboga2017; @Persello2017; @Volpi2017; @Zhao2017]. But, aside from [@Xu2017] which used a combination of patch-based CNN and stacked autoencoders to fuse PAN and MS images, the majority of the works did not address the problem of multiresolution VHR images. A patch-based CNN [@Mboga2017] and a fully convolutional network (FCN) [@Persello2017] were used to detect informal settlements from a pansharpened VHR image. Fully convolutional networks were also used to classify urban objects in VHR images both acquired in aerial and space-borne sensors [@Paisitkriangkrai2016; @Sherrah2016; @Maggiori2017; @Volpi2017]. Moreover, [@Paisitkriangkrai2016; @Sherrah2016; @Zhao2017] also utilized a separate post-classification step for map regularization. In this paper, we design a novel single-stage network performing image fusion, classification, and map regularization of a multiresolution VHR image in an end-to-end manner.
Contributions
-------------
We propose a multiresolution convolutional network, called FuseNet, and its recurrent version, called ReuseNet, to perform image fusion, classification, and map regularization of a multiresolution VHR image in an end-to-end fashion. We summarize the main contributions of this paper in: image fusion, map regularization, and sensitivity analysis of network parameters.
### Image fusion
We propose a convolutional network learning how to fuse a multiresolution VHR image, extract spatial features, and classify the latter into classes of interest all at the same time. We call this network FuseNet. It uses convolutional layers with corresponding downsampling and upsampling operations to learn to match and fuse the multiresolution channels of a multispectral VHR image.
### Contextual label dependency through network recurrence
We incorporate recurrence in the FuseNet architecture to model contextual *label-to-label* dependencies and effectively regularize classification maps. We call this improved version ReuseNet. Contextual label dependencies are incorporated in ReuseNet by feeding classification scores of a previous FuseNet instance to a succeeding one. Moreover, we introduce and compare a novel method to initialize the parameters and initial score maps of a ReuseNet.
### Sensitivity analysis
We analyze the sensitivity of the network to some of its chosen hyperparameters. We investigate the effect of varying a number of hyperparameters of our network to the classification performance. The considered hyperparameters are: the bottleneck feature map dimensions, the number of convolutional layers, the input patch sizes, the upsampling operations, and the number of FuseNet instances within a ReuseNet.
Convolutional Networks
======================
Background
----------
Convolutional neural networks are a variant of artificial neural networks connected in a sequential feedforward fashion employing convolutional and pooling (aggregational) operations. Convolutional operations greatly reduce the number of learnable parameters and allows the network to use the same filter to detect the same spatial pattern over different parts of an image. Pooling with downsampling enables the network to learn some degree of translational invariance.
Recurrent neural networks are artificial neural networks employing feedback connection, i.e. connections form a directed cycle. For example, the Jordan network [@Jordan1997] has connections from the output units back to the hidden units. Two key concepts namely, parameter sharing and graph unfolding [@Goodfellow2016 pp. 369–372], allow these networks to accept input sequences of variable lengths while maintaining model complexity—making recurrent networks widely applied to sequential data. However, parameter sharing and graph unfolding can also be used to design networks for different purposes, e.g. application to non-sequential data, while still taking advantage of the benefits, such as model compactness, from the two concepts [@Pinheiro2014].
Deep Networks as Data-flow Graphs
---------------------------------
We can generalize any variant of deep networks by seeing them as data-flow graphs—a graph representing how a set of input data are processed along a possibly branching chain of functions, in the end producing a final set of outputs. Using such a model, we define the networks by three elements: the sets of data they take as an input, the operations they perform in each function block, and the intermediate and final set of outputs they produce. Aside from these three key elements of data-flow graphs, details of a unique configuration and instance of a convolutional network are defined by its *hyperparameters* and *parameters* respectively. Hyperparameters are associated with the configuration of a network architecture and are set to fixed values before training the network. Parameters are values associated to a specific network instance and are learned during network training.
### Input
A convolutional network receives as an input either the whole image itself to be classified or a subset of it, called an input patch. The dimension of this patch is defined by the patch size hyperparameter and the number of bands . A convolutional network accepts an array of pixel values as an input (in the case of the image patches having equal height and width), being the number of patches processed by the network in parallel. Aside from the input image patch, the corresponding reference image can also be considered as an input in terms of data-flow graphs since no operation precedes it.
### Operations {#subsub:ops}
*Convolutions* are the main operations used by convolutional networks. A convolution applies a linear operation on an input image/feature map using a set of learnable kernels. Applying a kernel , composed of a array of learnable parameters, on a input feature map , where is the kernel size, is the number of kernels in each set of kernels, and and are the height and width of the feature map, produces a output feature map . The output at the $i^{\prime}$ row and $j^{\prime}$ column of the $k^{\prime}$ feature map is given by:
\[eq:convolutions\] $$\begin{aligned}
\text{\outfeaturemap}_{k^{\prime}i^{\prime}j^{\prime}} &= \sum\limits_{k=1}^{\text{\nkernels}}\sum\limits_{p=1}^{\text{\kernelsize}}\sum\limits_{q=1}^{\text{\kernelsize}}\text{\featuremap}_{kij} \cdot \text{\kernel}_{kk^{\prime}pq} + \text{\biasvalue}_{k^{\prime}}\label{eq:convolution}\\
i &= i^{\prime}+p-\ceil{\frac{\text{\kernelsize}}{2}}\\
j &= j^{\prime}+q-\ceil{\frac{\text{\kernelsize}}{2}}
\end{aligned}$$
where $_{k^{\prime}}$ is the learnable bias parameter associated with the $k^{\prime}$ feature map. The width and height of the output feature map are given by:
\[eq:outputdim\] $$\begin{aligned}
\text{\outnrows} &= \floor{\frac{\text{\nrows}-\text{\kernelsize}+2\text{\zeropadding}}{\text{\kernelstride}}+1} \label{eq:outnrows}\\
\text{\outncols} &= \floor{\frac{\text{\ncols}-\text{\kernelsize}+2\text{\zeropadding}}{\text{\kernelstride}}+1} \label{eq:outncols}
\end{aligned}$$
where the *zero-padding* is the number of rows and columns of zeros added to the border of the input feature map and the convolutional *stride* is the number of units separating contiguous receptive fields of the kernel on the input feature map.
*Nonlinearity* is applied after the linear operation of a convolution. Since applying a series of linear operations can be reduced to a single linear operation, an elementwise nonlinear function applied between each convolution allows the network to learn more complex input to output mapping. A common choice is the rectifier function
$$\text{\outfeaturemap}_{i^{\prime}j^{\prime}k^{\prime}} = \max(0, \text{\featuremap}_{ijk})\label{eq:rectifier}$$
or a variation of it [@Maas2013; @He2015b; @Clevert2016].
*Pooling* takes an aggregate of values over local regions of the input. A common choice of a pooling function is the average or maximum function. In contrast to convolution, a basic pooling does not have any learnable parameters. Originally, pooling was used to give the network a small degree of translation invariance by summarizing values of the input on non-overlapping windows ($\text{\kernelstride}=\text{\kernelsize}$)—also downsampling the input by a factor of , with proper zero-padding.
*Upsampling* operations are applied to increase the spatial dimensions of input feature maps. Upsampling is important specifically if the network needs to produce output predictions of the same size as the input, i.e. we want to produce a label for each pixel in the input patch. One way to upsample is by employing resampling techniques such as nearest neighbor or bilinear interpolation [@Chen2016]. The original fully convolutional network (FCN) [@Long2015] learns the upsampling operation using *backwards convolution* (or more technically fitting called *transposed convolution*).
*Merging* combines two or more sets of feature maps in a network either by addition or by concatenation. Addition is an elementwise operation performed between feature maps—adding each unit with corresponding indices—hence, all the three dimensions (, , ) must be the same for all inputs [@Long2015]. Concatenation stacks the input feature maps depth-wise—hence, only the spatial dimensions (, ) must be the same.
### Outputs
In data-flow graph terms, the outputs of a convolutional network consist of all the intermediate feature maps, the final class score maps, and the corresponding loss and accuracy calculated using the class score maps and the reference labels. Final class score maps correspond to the units in the last layer of a neural network and its dimension depends on how the task is defined. Authors in [@Volpi2017] categorize the approaches to this task into three variants: 1) patch classification, 2) subpatch labeling, and 3) full patch labeling. In patch classification, we assign a single label to the patch, i.e. the label corresponds to the class of the central pixel of the patch [@Bergado2016; @Volpi2017; @Mboga2017]. In subpatch labeling, we assign labels on a smaller part of the patch corresponding to the area near the center of the patch [@Volpi2017]. Finally, in full patch labeling, we assign labels to all the pixels in the patch [@Long2015; @Sherrah2016; @Badrinarayanan2017; @Volpi2017; @Persello2017]. The last method, aside from being more efficient, also decouples the limit of the input patch size to the number of downsampling operations in the network.
Training Deep Networks {#sub:learning}
----------------------
We train the network by minimizing an objective function in terms of the parameters of the network. For classification involving classes, a cross-entropy loss function is often used given by:
$$\label{eq:crossentropy}
\text{\lossvalue}_{\text{\batchsize}}(\text{\kernel}) = -\sum\limits_{n=1}^{\text{\batchsize}} \text{\targetvector}_{n} \cdot \log(\text{\predictionvector}_{n})$$
where is the loss function value evaluated over samples, $\text{\targetvector}_{n}$ is a binary vector encoding the the target class labels (with the index corrresponding to a class having a value of 1 and 0 otherwise), $\cdot$ denotes the dot product, and $\text{\predictionvector}_{n}$ is the class score maps of a sample $n$ calculated using a *softmax* activation function:
$$\label{eq:softmax}
\text{\predictionvector}_{kij} = \frac{\exp(\text{\featuremap}_{kij})}{\sum\limits_{c=1}^{\text{\nclasses}}\exp(\text{\featuremap}_{cij})}.$$
In this equation, is the softmax score and is the last set of feature maps containing unnormalized class scores at location $ij$.
A stochastic version of the backpropagation with gradient descent algorithm is often used to minimize the objective function [@Bottou2012]. The training is finished after a fixed number of epoch or when a certain convergence criterion is met. We can infer predictions from the final trained network instance by truncating the loss evaluation in the computational graph and taking the index of the maximum class score map value along the class score dimension by
$$\label{eq:inference}
\text{\predictionvalue}_{ij} = \operatorname*{\arg\!\max}_{c} \text{\predictionvector}_{cij}$$
where $\text{\predictionvector}$ and $\text{\predictionvalue}$ are the class score and prediction for location $ij$ respectively.
Regularizing Deep Networks {#sub:regularization}
--------------------------
Deep networks are often prone to overfit the training set. Overfitting occurs when a model reports high accuracy during training but performs poorly on unseen test data. Regularization approaches address the overfitting problem using three common methods: *data augmentation*, *weight decay*, and *early-stopping*. Data augmentation technique increases the number of training samples by permuting them with applicable rotational and/or translational transformations. Data augmentation helps the network to learn relevant invariances that may be present in the input. Weight decay modifies the loss function by
$$\label{eq:weightdecay}
\text{\regularizedloss}(\text{\kernel}) = \text{\lossvalue}(\text{\kernel}) + \text{\weightdecay}\norm{\text{\kernel}}^{2}_{2}$$
adding a penalty proportional to the square of the $\mathit{l_2}$-norm of the weight vector . The weight decay hyperparameter controls the contribution of this penalty to the loss function. Early stopping prematurely stops the training when a criterion measured from a validation set is met.
Proposed Approach
=================
In this paper, we propose a multiresolution convolutional network, called FuseNet, and its recurrent version, called ReuseNet, to perform an end-to-end fusion, classification, and map regularization of a multi-resolution VHR image. ReuseNet is built on top of a fully convolutional network architecture learning to: 1) fuse PAN and MS bands of a VHR image, 2) perform land cover classification on the fused images, and 3) spatially regularize the resulting classification.
{width="90.00000%"}
FuseNet {#sub:fusenet}
-------
The architecture of FuseNet is inspired by several encoder-decoder like convolutional network structures [@Noh2015; @Badrinarayanan2017; @Volpi2017] where the first set of layers learn deep features by a series of convolution, nonlinearity, and maximum pooling with downsampling operations, followed by a second set of layers using upsampling and nonlinearity operations to restore the resolution of the original input image. The main difference of FuseNet with these encoder-decoder architectures is the two initial separate streams of the downsampling part of the network that learns how to fuse two images of different resolution. FuseNet is specifically designed for VHR satellite images with PAN band and MS bands of ground sampling distance ratio of four (e.g. Quickbird, Worldview 2/3, Pleiades, Ikonos). But the architecture can be further generalized to fuse any number of images with different spatial resolutions.
FuseNet accepts two sets of input: an image patch of dimensions 144 taken from PAN image and another patch of dimensions 4 taken from the corresponding location in the MS image. It performs two series of convolution, nonlinearity, and maximum pooling with downsampling to such that the spatial dimensions of the intermediate feature maps match the spatial dimensions of . The nonlinearity operations use an exponential linear activation function [@Clevert2016]. The second input is linearly projected in $k$ dimensions using 11 convolutions such that $k$ matches the number of intermediate feature maps extracted from the first set of input—ensuring a balanced contribution from the two streams of feature. FuseNet then merges the linear projection of with intermediate feature maps extracted from via a concatenation operation.
Additional series of convolution, nonlinearity, and maximum pooling with downsampling operations are applied to the merged feature maps producing the set of feature maps with smallest spatial dimensions—called *bottleneck* [@Volpi2017]. FuseNet then upsamples the bottleneck back to the resolution of using transposed convolutions. The resulting set of feature maps is linearly projected again using 11 convolutions such that the number of feature maps matches . The final class score maps are obtained by applying a softmax activation function. This series of operations can be formulated as a function composition given by:
$$\label{eq:fusenet_ops_long}
\text{\outscore} = s(l_{1}(u(d_{1}(d_{0}(\text{\inputpan}) \oplus l_{0}(\text{\inputms})))))$$
where $d_{i}$ is a series of convolution, nonlinearity, and maximum pooling with downsampling operations, $u$ is the series of upsampling operations via transposed convolution, $l_{i}$ are the linear projections via 11 convolutions, $s$ is the softmax function, and $\oplus$ denotes merging via concatenation. Details of each operation, including the hyperparameter values and dimensions of some chosen intermediate feature maps, are provided in Table \[tab:fusenet\_ops\]. A cross-entropy function following Equation \[eq:crossentropy\] is used to compute the loss in each iteration. Unlabeled pixels are assigned a loss function value of zero.
We described above the default configuration of FuseNet, called FuseNet$_{low}$, performing the fusion at the lower (MS image) resolution. We also tested a network, called FuseNet$_{skip}$, adding skip connections to some lower-level feature maps of FuseNet$_{low}$ [@Long2015]. Figure \[fig:fusenet\] shows a diagram illustrating the general architecture of FuseNet$_{skip}$. Additionally, we experimented with a FuseNet performing the fusion at the resolution of the PAN image, called FuseNet$_{high}$ which is more similar to pansharpening—upsampling first before fusing them with . Tables \[tab:fusenet\_ops\] and \[tab:fusenet\_skip\_ops\] show details of the operations, including dimensions of intermediate output feature maps, used by the FuseNet variants. The format is adapted from [@Simonyan2014]. and denote input patches from the PAN and MS images respectively. IFM and BFM correspond to intermediate and bottleneck feature maps. Convolutions are denoted as “conv$\langle$kernel size $\rangle$-$\langle$number of kernels $\rangle$”. Maximum pooling operations (maxpool) are fixed to have pooling size $_{p}$ and stride $_{p}$ equal to two. Upsampling operations are denoted as “ups$\langle$number of kernels $\rangle$-$\langle$upsampling factor$\rangle$”. Merging operations are denoted as concat for concatenation and add for addition. Fixed upsampling can either be via pansharpening or bilinear interpolation. Consecutive Batch Normalization [@Ioffe2015] and exponential linear activation [@Clevert2016] operations between convolutions and pooling are omitted for brevity. Finally, operations shared by separate streams of feature align with the columns of these streams.
[|c|c|c|c|c|c|]{} & &\
(144) & (4) & (144) & (4) & (144) & (4)\
conv13-16 & conv1-32 & & ups2-16 & &\
maxpool & & & ups2-8 & & fixed\
conv7-32 & & & conv1-4 & & upsampling\
maxpool & & & IFM1 (544) & &\
& & &\
& & &\
& &\
IFM1 (32) & IFM2 (32) &\
&\
&\
\
\
\
\
\
\
\
\
\
\
\
\
\[tab:fusenet\_ops\]
[|>p[4cm]{}|>p[4cm]{}|c|c|]{}\
(144) & (4) &\
conv13-16 & conv1-32 &\
maxpool & &\
conv7-32 & &\
maxpool & &\
IFM1 (32) & IFM2 (32) &\
&\
&\
&\
&\
&\
&\
&\
&\
&\
&\
& IFM1 & IFM5\
& ups6-4 & ups6-8\
& IFM7 (644) & IFM8 (644)\
\
\
\
\[tab:fusenet\_skip\_ops\]
FuseNet implements a full patch labeling approach since it produces labeled image patches of the same dimensions as the input PAN image patch. Inference of final classification map is given by Equation \[eq:inference\] and can be applied to an input image of variable spatial dimension. Application to input of variable size is made possible by the fully-convolutional nature of the network [@Long2015]—allowing it to be applied as an image filter [@Sherrah2016] to any input with spatial dimensions of at least equal to the FCN’s effective receptive field.
ReuseNet {#sub:reusenet}
--------
{width="90.00000%"}
ReuseNet builds on top of the architecture of FuseNet by incorporating recurrent connections. Incorporation of this recurrent architecture in a full patch labeling approach enables the network to learn contextual label-to-label dependencies by feeding output score maps of a FuseNet instance to another instance of itself as an input. Such dependencies are similar to what graphical model (e.g. CRF/MRF) based methods learn in a post-classification regularization inference. For instance, a fully-connected CRF [@Krahenbuhl2011] solves an energy function that penalizes label configurations based on a unary term, often taken as the negative logarithm of the class scores [@Chen2016], and a pairwise term, adding a penalization for pixels with different labels based on image-space and feature-space distances. For ease of notation, let the series of operations performed by FuseNet (Equation \[eq:fusenet\_ops\_long\]) be given by the function $f$: $$\label{eq:fusenet_ops}
\text{\predictionvector} = f(\text{\inputpan}, \text{\inputms})$$ where the ’s are the input of FuseNet, and is the class score map resulting from this input. The operations performed by ReuseNet are given by a recurrent variant $g$:
\[eq:reusenet\_ops\] $$\begin{aligned}
\text{\predictionvector}_{1} &= g(\text{\inputpan}\oplus \text{\predictionvector}_{0},\text{\inputms})\\
\text{\predictionvector}_{\text{\instanceidx}} &= g(\text{\inputpan}\oplus \text{\predictionvector}_{\text{\instanceidx}-1},\text{\inputms})
\end{aligned}$$
where the score map is obtained by applying the same function to a combination of the previous $\text{\instanceidx}-1$ score map and the original FuseNet input as a new input. The recurrent variant $g$ (denoted as FuseNet+ in Figure \[fig:reusenet\]) applies exactly the same operations as $f$ except for the first operation that instead of only taking as an input, this operation takes the concatenation of and a class score map $\text{\predictionvector}_{r}$ associated to the network instance $r$. Figure \[fig:reusenet\] shows a diagram illustrating the general architecture of ReuseNet.
We tested ReuseNet with several number of FuseNet instances (2, 3, and 4), calling each ReuseNet- where is the number of FuseNet instances within the ReuseNet. We also investigated various methods for initializing weights and initial score maps $\text{\predictionvector}_{0}$ of ReuseNet. Plain ReuseNet initializes the score maps with zeros, while ReuseNet$_{map-init}$ initializes the score maps using scores from a pre-trained FuseNet showing the best results in the fusion comparison experiments. We further extend ReuseNet$_{map-init}$ by initializing the weights of the FuseNet instance in the ReuseNet with the same FuseNet that provides the initial score maps. We call this extension ReuseNet$_{map-weights-init}$.
Perspective on Learning the Fusion Approach and Incorporating Recurrence {#sub:reusenet_perpsective}
------------------------------------------------------------------------
Conventional approaches to classify multiresolution images require a separate step to match the resolution of the images. One way is to spatially sharpen the MS images using the PAN image (also called pansharpening) [@Hester2008]. Another possible way is to resample images to match a specific resolution using nearest neighbor or bilinear interpolation. However, these standard fusion techniques are performed independently from the classification problem and are suboptimal. FuseNet provides a streamlined approach including the fusion of the multiresolution images within the learning of the classifier. We expect that coupling and learning the fusion method within a supervised classifier will outperform an approach based on a separate fusion method.
The parameter sharing across FuseNet instances in a ReuseNet is consistent with the definition of a recurrent network, i.e. a recurrent network is a feedforward network that keeps on reusing the same set of weights to cycle through a sequence. The authors in [@Pinheiro2014] view such incorporation of recurrence as a way to increase the contextual window size, equivalent to the patch size in a patch classification approach, of their patch classification based approach while controlling the capacity of the network via inter-instance weight sharing. While both increase in contextual window size and capacity control of a CNN-based image patch classifier helps to improve the latter’s performance, the first benefit is lost in a full patch labeling approach. In a fully convolutional network implementing full patch labeling, the contextual window size does not change as recurrent operations are added to the network since the contextual window size is equivalent to the effective size of the receptive field of the network. The effective size of the receptive field of the network depends on kernel sizes and strides of the network’s convolutional and pooling operations, which are fixed and the same across instances.
In the proposed ReuseNet, recurrence integrates *contextual label information* to our model by considering class score maps as inputs to each FuseNet instance. This allows the model to learn label-to-label dependencies in addition to the spatial contextual information learned from the pixel values, *pixel-to-label* dependencies. This is a form of structured output prediction [@Bakir2007] where interdependencies between outputs may be expressed in terms of constraints restricting permissible output combinations or a more flexible form such as spatial dependencies across different output variables. Graphical models such as conditional random fields [@Lafferty2001] are commonly used for such structured prediction tasks. ReuseNet uses operations in a deep convolutional network to learn features from both the input image and class scores—integrating the learning of label-to-label dependencies from the data instead of explicit image-space and feature-space distances as represented in a pairwise potential of CRF. This allows ReuseNet to be trained end-to-end as opposed to a two-stage approach applying a post-classification MRF/CRF as done in [@Giorgi2014] and [@Sherrah2016].
Data and Experimental Setup
===========================
![Figure showing the true color VHR image together with the locations of the labeled tiles (in blue squares) and the study area: Quezon City, Philippines.[]{data-label="fig:study_area"}](./img/study_area.png){width="50.00000%"}
Dataset Description
-------------------
Tile Number of labeled pixels Set
------ -------------------------- ------------
100 2178768 Training
105 2173602 Training
45 2063971 Validation
78 1977336 Test
82 1961955 Test
: Number of labeled pixels in each tile
\[tab:tile\_samples\]
### Worldview-03 Quezon City dataset
we evaluated the proposed networks in the land cover classification of a dataset covering Quezon City, Philippines. The dataset is composed of a Worldview-03 satellite image of the city acquired on 17$^{th}$ April 2016 and corresponding manually prepared reference images for five chosen tiles (subsets) of the satellite image. The satellite image has a PAN band of 0.3 m resolution and four MS bands (near-infrared, red, green, and blue) of 1.2 m resolution. Reference images were prepared via photointerpretation and set to have the same spatial resolution as the PAN image. The whole satellite image was first divided into regularly-sized image tiles. PAN image tiles have a dimension of 3200 pixels $\times$ 3200 pixels, while MS image tiles have a dimension of 800 pixels $\times$ 800 pixels. Five non-adjacent tiles were sparsely labeled—annotating a pixel with a label belonging to one of the following six classes: impervious surface, building, low vegetation, tree, car, and clutter. Two of the five labeled tiles were used for training (100 and 105), one for validation (45), and the remaining two for testing (78 and 82). Training samples are composed of pairs of image patches with dimensions (taken from the MS image) and 44 (taken from the PAN image tile). Figure \[fig:study\_area\] shows the VHR image and the corresponding locations of labeled tiles in the study area while Table \[tab:tile\_samples\] shows the number of labeled pixels in each image tile. Training samples were normalized to have a value between zero and one. The reference image patches have been converted into a “one-hot” encoding—a vector having zero values except for the index corresponding to the code of the class.
### ISPRS Vaihingen dataset
for the ReuseNet experiments, we utilized the ISPRS 2D semantic labeling benchmark dataset of Vaihingen as an additional dataset [@Cramer2010]. We adopted the experimental setup used in [@Sherrah2016; @Volpi2017], employing the same training and validation tiles, to provide comparable results. We followed the sampling done in [@Sherrah2016], except that data augmentation was not applied—resulting in less training samples. The method discussed in [@Mousa2017] was employed to extract the normalized DSM.
Comparison of methods {#sub:methods_comparison}
---------------------
For the image fusion part, we compared FuseNet against two other baseline approaches: one using pansherpening and another using bilinear interpolation to match the resolution of to the resolution of . We call these two baseline approaches Net$_{pansharp}$ and Net$_{bilinear}$. Net$_{pansharp}$ applies Gram-Schmidt pansharpening technique [@Laben2000]. Only the pansharpened image is fed as an input to Net$_{pansharp}$. Net$_{bilinear}$ upsamples the resolution of the MS image to match the resolution of the PAN image using bilinear interpolation. The upsampled MS images are then merged to the PAN image using concatenation. The architecture of the network after the fusion is kept the same to have a fair comparison among the different approaches (see details of the FuseNet variants in Table \[tab:fusenet\_ops\]). Additionally, we compared a SegNet [@Badrinarayanan2017] trained on the first three principal components of the pansharpened image, since SegNet only accepts three inputs. We found that discarding one band (NIR) considerably degrades the results.
We compared ReuseNet against FuseNet using fully-connected CRF [@Krahenbuhl2011] (FuseNet+CRF) to assess the capability of our classifiers to spatially regularize the classification results. The FuseNet+CRF baseline is similar to the approach adopted in [@Chen2016; @Sherrah2016] but applied to PAN and MS images with different spatial resolutions. Spatial and feature space distances in the pairwise potentials of the fully-connected CRF are computed from the PAN image. We performed a grid-search of the CRF parameters, i.e. the weights and standard deviations of the appearance and smoothness kernels, and used the set of the parameters with the highest accuracy on the validation tile. We fixed the number of iterations to 10 for the mean field approximation algorithm used to perform inference in a fully-connected CRF.
We also performed a sensitivity analysis of a few chosen hyperparameters of FuseNet$_{low}$. We varied the bottleneck feature map dimensions, number of convolutional layers (in the downsampling part of the network), input patch sizes, and upsampling methods—performing the experiments in this order. We took the hyperparameter value that maximizes the overall accuracy on the validation tile and fix it for the succeeding sets of experiments. We experimented using bottleneck feature map dimensions: $16\times16$, $8\times8$, $4\times4$, $2\times2$, and $1\times1$. After fixing the bottleneck feature map dimension, we increased the number of convolutional layers preceding the last downsampling operation—effectively increasing the number of convolutional layers from 8 to 14 in steps of two. We investigated varying patch sizes of $(4\text{\patchsize}, \text{\patchsize})$: $(32, 8)$, $(64, 16)$, $(96, 24)$, $(128, 32)$. For the upsampling operations, we explored two additional methods using nearest neighbor and bilinear interpolation to upsample the feature map and then performing $3\times3$ convolutions after each upsampling operation.
We trained all the networks using a set of 17409 image patches taken from the training tiles and used 8255 image patches taken from the validation tile for early-stopping. We performed a random sampling with the constraint that the pixel near the center of the image patch is labeled. This may produce overlapping patches unlike the systematic gridwise sampling approach used in [@Sherrah2016]. Gridwise sampling reduces the number of training patches since the reference images is sparsely labeled, only around five percent of the pixels are labeled. The total loss value computed over a mini-batch is the total loss of all pixels divided by the number of labeled pixels within the mini-batch.
The FuseNets are trained using backpropagation with stochastic gradient descent setting the initial learning rate $\text{\scheduler}=0.01$, momentum $\text{\momentum}=0.9$, mini-batch size $\text{\batchsize}=32$, and maximum number of epochs $\text{\nepoch}=240$. We decrease the learning rate in a stepwise manner as done in [@He2016]—multiplying it by a factor of 0.1 after 60 and 180 epochs. The weights were initialized as in [@Glorot2010]. We did not find dropout to be helpful; hence, we only used an $\mathit{l_2}$-weight decay penalty—setting $\text{\weightdecay}=0.001$—and a variant of early-stopping to regularize FuseNet. For early stopping, the classification accuracy on the validation set is calculated every epoch and the last model with the best validation accuracy is fixed to be the final instance of the model.
The FuseNet instances within a ReuseNet are identical, sharing the same network configuration and parameters. Each instance also couples a cross-entropy loss function with each of their score map. The total objective loss of a ReuseNet is the average of the cross-entropy loss values from all the FuseNet instances. We also used the same backpropagation with stochastic gradient descent setting as training a FuseNet with the initial learning rate $\text{\scheduler}=0.01$, momentum $\text{\momentum}=0.9$, mini-batch size $\text{\batchsize}=32$, and maximum number of epochs $\text{\nepoch}=240$. Likewise, we decreased the learning rate in a stepwise manner—multiplying it by a factor of 0.1 after 60 and 180 epochs. For regularization, we only used an $\mathit{l_2}$-weight decay penalty—setting $\text{\weightdecay}=0.001$. We can infer classification map from a ReuseNet in the same manner of inference as a FuseNet, with one additional option: to extract different predictions from each FuseNet instance.
For applying ReuseNet on the ISPRS Vaihingen dataset, we employed a feedforward network similar to the No-downsampling FCN proposed by [@Sherrah2016] truncating the last two layers (fc5 and fc6) before softmax activation and entirely removing all maximum pooling without downsampling operations. With only convolutional layers (without pooling), we call this network AllConvNet. The network was trained on 12717 training patches as opposed to the 123330 training patches in [@Sherrah2016]. Although having less parameters and having trained with a smaller number of training samples, AllConvNet provided comparable results with the original No-downsampling FCN while requiring less operations. We trained AllConvNet for 150000 iterations as reported in [@Sherrah2016]. ReuseNet versions of AllConvNet were applied to the ISPRS Vaihingen dataset and were compared to the best results of both [@Sherrah2016] and [@Volpi2017]. All the networks in this additional set of experiments were trained using a variant of SGD proposed in [@Zeiler2012].
Accuracy Assessment
-------------------
We compared the results of the different approaches using global measures: 1) overall classification accuracy (OA), 2) the Kappa coefficient ($\kappa$), 3) average class accuracy (AA), 4) and average class-F1 scores (F1). OA is given by:
$$\label{eq:OA}
OA = \frac{\sum\limits_{i=1}^{C}n_{ii}}{n}$$
where $n_{ii}$ is the number of samples classified as class $i$ in both the the predictions and reference images, $n$ is the total number of labeled samples in the reference images, and $C$ is the number of classes, whereas $\kappa$ is given by:
$$\label{eq:Kappa}
\kappa = \frac{
n\sum\limits_{i=1}^{C}n_{ii} - \sum\limits_{i=1}^{C}n_{i+}n_{+i}
}
{
n^{2} - \sum\limits_{i=1}^{C}n_{i+}n_{+i}
}$$
where $n_{i+}$ and $n_{+i}$ are the number of samples classified as class $i$ in the predictions and reference images respectively. Both OA and $\kappa$ provides the rate of correctly classified pixels with the latter compensating for random agreement in classification. These global measures, however, are biased toward frequently occurring classes—meaning, classes with less frequencies have relatively little impact to the two measures. Unlike OA and $\kappa$, AA and F1 provides average of measures independent of class distribution. AA is given by:
$$\label{eq:AA}
AA = \frac{1}{C}\sum\limits_{i=1}^{C}
\frac{n_{ii}}
{n_{i+}}$$
while F1 is given by:
$$\label{eq:F1}
F1 = \frac{1}{C}
\sum\limits_{i=1}^{C}
\frac{
2\frac{n_{ii}}
{n_{i+}}
\frac{n_{ii}}
{n_{+i}}
}{\frac{n_{ii}}
{n_{i+}}
+
\frac{n_{ii}}
{n_{+i}}}$$
AA computes the average within-class rate of correctly classified pixels, while F1 calculates the harmonic mean of the precision (user’s accuracy) and recall (producer’s accuracy). We also observe and comment on the quality of the resulting classified maps.
Results and Discussion
======================
FuseNet {#fusenet}
-------
Network OA (%) $\kappa$ (%) AA (%) F1 (%)
------------------------------ ----------- -------------- ----------- -----------
Net$_{bilinear}$ 84.76 78.70 81.99 77.48
Net$_{pansharp}$ 86.87 81.53 82.76 77.86
SegNet [@Badrinarayanan2017] 88.11 83.17 83.96 77.01
FuseNet$_{high}$ 88.03 83.18 89.79 79.06
FuseNet$_{low}$ 91.63 88.03 92.91 **82.90**
FuseNet$_{skip}$ **91.90** **88.43** **93.46** 81.74
: Comparison of fusion approaches
\[tab:fusion\_comparison\]
![PAN, MS, and reference images in the tiles used for testing. Corresponding legend is shown.[]{data-label="fig:test_tiles"}](./img/test_tiles_v3.png){width="50.00000%"}
{width=".8\textwidth"}
Table \[tab:fusion\_comparison\] shows the results of accuracies comparing different fusion approaches. The numerical results are evaluated using all the labeled pixels in the two test tiles (see Table \[tab:tile\_samples\] for the total number test samples). FuseNet$_{skip}$ scores the highest in all the four numerical metrics, except for F1 where FuseNet$_{low}$ scores the highest. FuseNet$_{low}$ outperforms both the variants using fixed upsampling (Net$_{pansharp}$, SegNet, and Net$_{bilinear}$) and the variants learning the upsampling but fusing at the scale of the image with higher resolution (FuseNet$_{high}$). Observing each metric: FuseNet$_{low}$ gains about 3–6% in OA, 4–9% in $\kappa$, 3–10% in AA, and 1–5% in F1 against the other baselines (with the exemption of FuseNet$_{skip}$). FuseNet$_{skip}$ further increases the numerical results of FuseNet$_{low}$ in the first three metrics by about 0.2–0.5% but degrades the F1 by about 1.2%.
We have two relevant observations: 1) learning the fusion can improve the classification of PAN and MS VHR images with different resolutions; 2) fusing at the scale of the image with lower resolution results in better classification than performing the fusion at the scale of the image with higher resolution. The first point demonstrates our expected effectiveness of coupling and learning the fusion operation within a supervised classifier. One explanation for the second point could be the placement of upsampling layers. Introducing upsampling layers early in the network—as done in FuseNet$_{high}$—may produce artifacts that can degrade its performance.
Figure \[fig:test\_tiles\] shows the PAN, MS, and reference images of the tiles used for testing. Figure \[fig:fusenet\_plots\] shows the classification results of two FuseNet variants (FuseNet$_{high}$ and FuseNet$_{skip}$) and one baseline method (Net$_{pansharp}$) on two selected areas of the test tiles. The most noticeable misclassifications are found in large and high-rise buildings, in both test tiles, and an overpassing road in tile 78. The facades and rooftops of the buildings are often mistaken to be impervious surfaces by the classifiers; while the overpassing road is mistaken to be a building. These regions can appear to have similar spectral characteristics and can only be distinguished by presence of other cues such as appearing to be elevated. However, with the absence of elevation information, such cues are not directly incorporated in the input data. Manually distinguishing arguably vaguely-defined classes such as low-vegetation and impervious surface can also be problematic, especially in the PAN image, with the lack of ancillary information such as elevation. Adding a digital elevation model or a digital surface model can help address the misclassification of these regions. The cars are also generally misclassified by all the classifiers which is, aside from being underrepresented in terms of the number of labeled pixels, due to the lack of spatial resolution of the MS bands and the cars’ spectral similarity with other classes (such as impervious surface and buildings) in the PAN band. Overall, FuseNet$_{skip}$ generally has less errors in the facade of large buildings, lessen the artifacts noticeably present in the other techniques, and has better delineation of classes with irregular boundaries such as trees and low-vegetation—providing the best classification results among all the FuseNet variants. We, therefore, apply recurrence to FuseNet$_{skip}$ architecure to build the ReuseNet instances.
ReuseNet {#sub:reusenet_results}
--------
### Worldview-03 Quezon City dataset
Network OA (%) $\kappa$ (%) AA (%) F1 (%)
------------- ----------- -------------- ----------- -----------
FuseNet 91.90 88.43 93.46 81.74
FuseNet+CRF 93.07 90.08 **94.71** 81.72
ReuseNet-2 92.82 89.69 94.09 82.64
ReuseNet-3 92.98 89.88 94.54 85.42
ReuseNet-4 **93.49** **90.58** 94.53 86.67
ReuseNet-5 92.74 89.53 92.78 **87.29**
: Comparison of map regularization approaches on Worldview-03 Quezon City dataset
\[tab:regularization\_comparison\]
{width="80.00000%"}
Table \[tab:regularization\_comparison\] shows the accuracies obtained by comparing different classification techniques on the Worldview-03 Quezon City dataset. We found that both the ReuseNet instances and the baseline method FuseNet+CRF improves the numerical results of the plain FuseNet$_{skip}$ gaining around: 0.9–1.5% in OA, 1.2–2.1% in $\kappa$, and 0.6–1.2% in AA. For the F1, however, FuseNet+CRF method performs worse than the plain FuseNet losing 0.02%; while all the other ReuseNet instances improves the F1 by around 0.9–5.5%. ReuseNet-4 outperforms all the other classifiers in all the metrics except for AA and F1—where both ReuseNet-3 and FuseNet+CRF outperform it by some margin in AA (0.01% and 0.18% respectively) and ReuseNet-5 considerably outperforms it in F1 by 0.62%. In particular, all the ReuseNets consistently show better F1 compared to both FuseNet and FuseNet+CRF—gaining almost 6%. These expected relatively smaller gains in numerical accuracy is consistent with what the author in [@Sherrah2016] found—applying a post-classification CRF to an FCN to classify extremely high resolution aerial imagery increases the overall classification accuracy by around 0.1–1.0%. More noticeable changes are expected in the resulting improved regularity of the classified maps.
The numerical results above supports our assertion that introducing contextual label information through recurrence in an FCN applying a full-patch labeling approach can improve the classification of a VHR image. Such incorporation of label information allows our classifier to learn both pixel-to-label and label-to-label contextual dependencies. We can develop an intuition of these two dependencies by using an analogy to photointerpretation. We can easily imagine that it is easier to label a pixel when viewed with its neighboring pixels. This setup is analogous to the improvements a spatial-contextual classifier, like a CNN applying a patch classification, approach bring over a simple pixel-based classifier. But we can also see that it is easier to label a pixel when, aside from viewing its neighboring pixels, its surrounding pixels’ labels are given. With contextual label information, the classifier can learn and leverage class spatial co-occurrences. Additionally, we observe that adding more FuseNet instances to the ReuseNet until $\text{\ninstances}=4$ increases the score of all metrics, except for the average class accuracy where ReuseNet-3 marginally outperforms ReuseNet-4. Adding one more instance only improves the F1 score and degrades the other three metrics. We can interpret this addition of FuseNet instances as a way to increase ReuseNet’s capacity to refine contextual label information fed to it as latter FuseNet instances receive more refined labels.
Figure \[fig:reusenet\_plots\] shows classification results of the best performing ReuseNet, the baseline method FuseNet+CRF, and the plain FuseNet. Both FuseNet+CRF and ReuseNet instances improves the quality of the resulting classified map by producing more regularized classification. We also observe that locations of the errors are carried over from the results of the FuseNet classifier from which both FuseNet+CRF and ReuseNet are based from. However, the occurrences of the errors are diminished especially on the facades of the large buildings. Detection of isolated cars in roads were also improved. Overall, results of ReuseNet-4 show better-quality classified maps by reducing noise in the classification (such as island of impossibly small buildings), further improving delineation of classes with irregular boundaries, and reducing misclassification in regions with ambiguous spectral characteristics such as facades and rooftops of high rise buildings.
### ISPRS Vaihingen dataset
Network OA (%) $\kappa$ (%) AA (%) F1 (%)
------------------------------------ ----------- -------------- ----------- -----------
No-downsampling FCN [@Sherrah2016] 87.17 –.– –.– –.–
CNN-FPL [@Volpi2017] 87.83 83.83 81.35 83.58
AllConvNet 86.98 82.71 87.17 85.46
FCN in [@Sherrah2016] with CRF 87.90 –.– –.– –.–
ReuseNet-2 87.11 82.89 85.09 85.38
ReuseNet-3 **88.08** **84.18** **87.29** **87.24**
ReuseNet-4 87.64 83.59 87.18 86.81
: Comparison of map regularization approaches on ISPRS Vaihingen dataset
Unreported values in the reference are denoted by “–.–”
\[tab:regularization\_comparison\_additional\]
{width="80.00000%"}
Table \[tab:regularization\_comparison\_additional\] shows the accuracies obtained by comparing different classification techniques on the ISPRS dataset. These results are in agreement with the results from the previous dataset. All the ReuseNet versions of AllConvNet improve the resuslts on all the four metrics except for AA and F1 of ReuseNet-2 (2.08% in AA and 0.06% in F1 respectively). ReuseNet-3, the best performing network, considerably improves all the numerical results of the plain AllConvNet by 1.1% in OA and is comparable and even greater than the 0.73% gain after a post-classification CRF in [@Sherrah2016], 1.47% in $\kappa$, 0.12% in AA, and 1.78% in F1. ReuseNet-3 also outperforms best results reported in both [@Sherrah2016] and [@Volpi2017].
These results reconfirm that introducing contextual label information through recurrence in an FCN applying a full-patch labeling approach can improve the classification of a VHR image. Similarly, qualitative improvements—such as holes in building being filled, better delineation of all classes in general, lesser artifacts—in the resulting classified maps are observed when ReuseNet is applied as shown in Figure \[fig:reusenet\_plots\_additional\].
### Different initializations
![Plots showing results of quantitive metrics comparing different ReuseNet initializations; plain, map-init, and map-weights-init correspond to intializing the ReuseNet with zero-score maps, scores from a previously-trained FuseNet, and scores and weights from a previously-trained FuseNet respectively.[]{data-label="fig:reusenet_inits"}](./img/ReuseNet_inits_v2.png){width="50.00000%"}
Figure \[fig:reusenet\_inits\] shows results of quantitive metrics on the three different ReuseNet initializations. There is low variation in the OA and $\kappa$. The trend of the two global scores is also inconsistent across the ReuseNet instances. For ReuseNet-2 and ReuseNet-3, the scores increases marginally (around 0.5% for OA and 0.8% for $\kappa$) when initialized with both the scores and weights from a previously-trained FuseNet. But for ReuseNet-4, there is a minor drop in both the scores (around 0.2% for both scores) when the two intialization methods are introduced. This could mean that increasing the FuseNet instances to a certain amount already provides enough room to a ReuseNet for “label refinement” such that gains from the initialization methods are compensated.
Introducing both initialization methods to a ReuseNet degrades the AA by around 0.9–5.2%. Applying only the initialization using scores from a FuseNet instance (map-init) degrades the F1 by around 0.9–12.2%. Interestingly, the F1 improve by around 0.8–6.1% when both initialization methods are introduced (map-weights-init). Decrease in AA can only imply an increase in false positive predictions in most of the classes; while increase in F1 could either mean decrease in false positive predictions or decrease in false negative predictions or both in most of the classes. The results therefore show that the initialization methods promote higher recall rate (decrease in false negatives) in underrepresented classes such as cars.
Sensitivity Analysis
--------------------
![Plots showing the results of sensitivity analysis. Patch sizes are written as “$\langle$(4, )$\rangle$”. N-neighbor denotes nearest neighbor interpolation.[]{data-label="fig:sensitivity"}](./img/Sensitivity_analysis_v4.png){width="45.00000%"}
Fig. \[fig:sensitivity\] shows the results of the sensitivity analysis performed on four chosen hyperparameters of FuseNet: 1) bottleneck feature map dimensions, 2) number of convolutional layers (in the downsampling part of the network), 3) input patch sizes, and 4) upsampling methods. We got the highest validation accuracy of 90.35% using a bottleneck feature map dimension of 44 pixels. Decreasing the dimension more than the optimal we found severely degrades the classification resulting to large uniform areas producing stamp-like patterns (especially for 11). Increasing the dimensions produces much noisier classification. Fixing the bottleneck size dimension to 44 and further increasing the number of convolutional layers (without downsampling) did not produce any improvements in the validation accuracy. Increasing the number of these convolutional layers within the bottleneck feature maps effectively increases the receptive field (footprint size in the input layer containing the PAN image patch) of the succeeding units by at least half of the size of kernels used in the convolutional layers. Hence, the results show that: with only eight convolutional layers (with downsampling), we can learn enough contextual information for accurate classification.
We found the optimal patch sizes of 6464 for the and 1616 for . Further increasing the patch sizes results in overclassification of a single class (impervious surface). Increasing the patch size also increases the proportion of frequently occurring classes in the training sample, possibly resulting into overclassification. Whereas, decreasing the patch size limits the contextual information incorporated in the input, and, hence, can degrade the classification results. Lastly, we find using transposed convolution for learned upsampling to perform better than using interpolation for fixed upsampling (bilinear and nearest neighbor). This result supports the expected flexibility of empirically learning the upsampling operation directly from data.
Conclusion and Future Works
===========================
In this paper, we presented a recurrent multiresolution convolutional network named ReuseNet to classfiy VHR satellite images. The operations for fusing the bands with different resolutions are learned within convolutional layers with corresponding downsampling and upsampling operations to match the resolution of the images. Regularization of the resulting classified maps is achieved by incorporating contextual label information through the recurrent architecture of ReuseNet. Additionally, we investigated various ways to initialize ReuseNet. The effect of varying a set of chosen network hyperparameters to the classification accuracy of the network was explored. Both numerical and qualitative results show the advantages of incorporating image resolution matching and contextual label learning within the training of the classifier. To this end, we provided a single-stage classification pipeline incorporating image fusion, feature extraction, and map regularization, all combined in a convolutional network trained in an end-to-end manner.
We designed the presented network architecture such that it can easily be adapted to other multiresolution image datasets. Inclusion and leverage of contextual label information is also separate from the design of the fusion network in the sense that it can be implemented on network classifying single-resolution images. For future work, we plan to fuse images from different sensors (e.g. Sentinel-2) and classify classes of higher abstraction such as land use instead of land cover.
[John Ray Bergado]{} Biography text here.
[Claudio Persello]{} Biography text here.
[Alfred Stein]{} Biography text here.
[^1]: Acknowledgement here.
[^2]: Manuscript received Month DD, 2017; revised Month DD, 2017.
|
---
abstract: 'Massive stars influence their surroundings through radiation, winds, and supernova explosions far out of proportion to their small numbers. However, the physical processes that initiate and govern the birth of massive stars remain poorly understood. Two widely discussed models are monolithic collapse of molecular cloud cores and competitive accretion. To learn more about massive star formation, we perform and analyze simulations of the collapse of rotating, massive, cloud cores including radiative heating by both non-ionizing and ionizing radiation using the FLASH adaptive mesh refinement code. These simulations show fragmentation from gravitational instability in the enormously dense accretion flows required to build up massive stars. Secondary stars form rapidly in these flows and accrete mass that would have otherwise been consumed by the massive star in the center, in a process that we term fragmentation-induced starvation. This explains why massive stars are usually found as members of high-order stellar systems that themselves belong to large clusters containing stars of all masses. The radiative heating does not prevent fragmentation, but does lead to a higher Jeans mass, resulting in fewer and more massive stars than would form without the heating. This mechanism reproduces the observed relation between the total stellar mass in the cluster and the mass of the largest star. It predicts strong clumping and filamentary structure in the center of collapsing cores, as has recently been observed. We speculate that a similar mechanism will act during primordial star formation.'
author:
- Thomas Peters
- 'Ralf S. Klessen, Mordecai-Mark Mac Low, Robi Banerjee'
title: 'Limiting Accretion onto Massive Stars by Fragmentation-Induced Starvation'
---
Introduction {#sec:intro}
============
Understanding the formation of massive stars is of pivotal importance in modern astrophysics. Massive stars are rare and short lived. However, they are also very bright and allow us to reach out to the far ends of the universe. For example, the most distant galaxies in the Hubble Ultra Deep Field [@beckwithetal06] are all characterized by vigorous high-mass star formation. Understanding the origin of massive stars, at present and at early times, therefore is a prerequisite to understanding cosmic history. In our own Milky Way, high-mass stars contribute the bulk of the UV radiation field. This radiation can ionize hydrogen and dissolve molecules. Expanding [H [ii]{}]{} regions, bubbles of ionized gas surrounding massive stars, have been identified as an important source of interstellar turbulence. The complex interplay between gas dynamics and radiation thus is a key element of the evolution of the interstellar medium (ISM) and the Galactic matter cycle. Furthermore, stars are the primary source of chemical elements heavier than the hydrogen, helium, and lithium that were produced in the Big Bang. Again, massive stars have a disproportionally large share in the enrichment history of the Galaxy and the universe as a whole. It is the violent supernova explosions associated with the death of high-mass stars that contribute most of the metals. Because they insert large amounts of energy and momentum input, supernovae at the same time strongly stir up the ISM and provide for the effective mixing of the newly bread elements.
Despite their importance, the physical processes that initiate and control the build-up of massive stars are still not well understood and subject of intense debate [@maclow04; @zinnyork07; @mckee07]. Because their formation time is short, of order of $10^5\,$yr, and because they form deeply embedded in massive cloud cores, very little is known about the initial and environmental conditions of high-mass stellar birth. In general high-mass star forming regions are characterized by more extreme physical conditions than their low-mass counterparts, containing cores of size, mass, and velocity dispersion roughly an order of magnitude larger than those of cores in low-mass regions [e.g. @jijmeyad99; @garliz99; @kurtzetal00; @beutheretal07; @motteetal08]. Typical sizes of cluster-forming clumps are $\sim 1\,$pc, they have mean densities of $n \sim 10^5$ cm$^{-3}$, masses of $\sim 10^3\,$M$_\odot$ and above, and velocity dispersions ranging between $1.5$ and $4 \,$km$\,$s$^{-1}$. Whenever observed with high resolution, these clumps break up in even denser cores, that are believed to be the immediate precursors of single or gravitationally bound multiple massive protostars.
Massive stars usually form as members of multiple stellar systems [@hohaschik81; @lada06; @zinnyork07] which themselves are parts of larger clusters [@ladalada03; @dewitetal04; @testietal97]. This fact adds additional challenges to the interpretation of observational data from high-mass star forming regions as it is difficult to disentangle mutual dynamical interactions from the influence of individual stars [e.g. @gotoetal06; @linzetal05]. Furthermore, high-mass stars reach the main sequence while still accreting. Their Kelvin-Helmholtz pre-main sequence contraction time is considerably shorter than their accretion time. Once a star has reached a mass of about $10\,$M$_\odot$ its spectrum becomes UV dominated and it begins to ionize its environment. This means that accretion as well as ionizing and non-ionizing radiation needs to be considered in concert [@keto02b; @keto03; @keto07; @petersetal10a; @petersetal10b]. It has been realized decades ago that in simple 1-dimensional collapse models the outward radiation force on the accreting material should be significantly stronger than the inward pull of gravity [@larsstarr71; @kahn74; @wolfcas87] in particular when taking dust opacities into account. Since stars as massive as $100 - 150\,$M$_\odot$ have been observed [@bonanosetal04; @figer05; @rauwetal05] a simple spherically symmetric approach to high-mass star formation is doomed to fail.
Consequently, two different models for massive star formation have been proposed. The first one takes advantage of the fact that high-mass stars always form as members of stellar clusters. If the central density in the cluster is high enough, there is a chance that low-mass protostars collide and so successively build up more massive objects [@bonbatezin98]. As the radii of protostars usually are considerably larger than the radii of main sequence stars in the same mass range this could be a viable option. However, the stellar densities required to produce very massive stars are still extremely high and seem inconsistent with the observed values of Galactic star clusters [e.g. @pozwetal10 and references therein]. An alternative approach is to argue that high-mass stars form just like low-mass stars by accretion of ambient gas that goes through a rotationally supported disk caused by angular momentum conservation [@mckeetan03]. Indeed such disk structures are observed around a number of high-mass protostars [@chini04; @chini06; @jiangetal08; @daviesetal10]. Their presence breaks any spherical symmetry that might have been present in the initial cloud and thus solves the opacity problem. Radiation tends to break out along the polar axis, while matter is transported inwards through parts of the equatorial plane shielded by the disk. Hydrodynamic simulations in two [@yorke02] and three (@krumholzetal09, @kuiperetal10, in prep.) dimensions using a flux-limited diffusion approach to the transport of non-ionizing radiation strongly support this picture. @krumholzetal09 find non-axisymmetric Rayleigh-Taylor instabilities using gray radiation transfer, whereas @kuiperetal10, using frequency-dependent radiation transfer, do not find such instabilities, but nevertheless find strong disk accretion. Strong accretion continues when ionizing radiation is included using ray-tracing methods in three dimensions [@daleetal05; @daleetal07b; @petersetal10a; @petersetal10b]. In a clustered environment or in individual collapsing cores where the disk becomes gravitationally unstable, the three-dimensional models show that material flows along dense, opaque filaments whereas the radiation escapes through optically thin channels of low-density material. It has been demonstrated that even ionized material can be accreted, if the accretion flow is strong enough. [H [ii]{}]{} regions are gravitationally trapped at that stage, but soon begin to rapidly fluctuate between trapped and extended states, in agreement with observations [@petersetal10a; @galvmadetal10inprep]. Over time, the same ultracompact [H [ii]{}]{} region can expand anisotropically, contract again, and take on any of the observed morphological classes [@woodchurch89; @kurtzetal94; @petersetal10a; @petersetal10b]. In their extended phases, expanding [H [ii]{}]{} regions drive bipolar neutral outflows characteristic of high-mass star formation [@petersetal10a].
Another key fact that any theory of massive star formation must account for is the apparent presence of an upper mass limit. No star more massive than $100 - 150\,$M$_\odot$ has been observed [@massey03]. This holds for the Galactic field, however, it is also true for young star clusters that are massive enough so that purely random sampling of the initial mass function (IMF) [@kroupa02; @chabrier03] without upper mass limit should have yielded stars above $150\,$M$_\odot$ (@weidkroup04 [@figer05; @oeycla05; @weidetal10], see however, @selmel08). This immediately raises the question of what is the physical origin of this apparent mass limit. It has been speculated before that radiative stellar feedback might be responsible for this limit [see, e.g., @zinnyork07] or alternatively that the internal stability limit of stars with non-zero metallicity lies in this mass regime [@appen70a; @appen70b; @appen87; @baraetal01]. However, fragmentation could also limit protostellar mass growth. Indeed, this is what we see in the simulations discussed here. The likelihood of fragmentation to occur and the number of fragments to form depends sensitively on the physical conditions in the star-forming cloud and its initial and environmental parameters [see, e.g., @krumetal10; @girietal10]. Understanding the build-up of massive stars, therefore, requires detailed knowledge about the physical processes that initiate and regulate the formation and dynamical evolution of the molecular clouds these stars form in [@vazsemetal09].
We argue that ionizing radiation, just like its non-ionizing, lower-energy counterpart, cannot shut off the accretion flow onto massive stars. Instead it is the dynamical processes in the gravitationally unstable accretion flow that inevitably occurs during the collapse of high-mass cloud cores that control the mass growth of individual protostars. Accretion onto the central star is shut off by the fragmentation of the disk and the formation of lower-mass companions that intercept inward moving material. We call this process fragmentation-induced starvation and argue that it occurs unavoidably in regions of high-mass star formation where the mass flow onto the disk exceeds the inward transport of matter due to viscosity only and thus renders the disk unstable to fragmentation. We speculate that fragmentation-induced starvation is important not only for present-day star formation but also in the primordial universe during the formation of metal-free Population III stars. Consequently, we expect these stars to be in binary or small number multiple systems and to be of lower mass than usually inferred [@abeletal02; @brommetal09]. Indeed, current numerical simulations provide the first hints that this might be the case [@clarketal08; @turketal09; @stacyetal10].
In the current study, we analyze the simulations by @petersetal10a with special focus on the mass growth history of the individual stars forming and the physical processes that influence their accretion rate. We briefly review the numerical method we use and the assumptions and approximations behind it in Section \[sec:method\]. Then we describe our findings in Section \[sec:results\] and discuss them in the context of present-day and primordial star formation in Section \[sec:discussion\]. We summarize and conclude in Section \[sec:summary\].
Method and Assumptions {#sec:method}
======================
We present three-dimensional, radiation-hydrodynamical simulations of massive star formation that include heating by ionizing and non-ionizing radiation using the adaptive-mesh code FLASH [@fryxell00]. We use our improved version of the hybrid-characteristics raytracing method [@rijk06; @petersetal10a] to propagate the radiation on the grid and couple sink particles [@federrathetal10], which we use as models of protostars, to the radiation module via a prestellar model [@petersetal10a].
The simulations start with a $1000\,$M$_\odot$ molecular cloud. The cloud has a constant density core of $\rho = 1.27 \times 10^{-20}\,$ gcm$^{-3}$ within a radius of $r = 0.5\,$pc and then falls off as $r^{-3 / 2}$ until $r = 1.6\,$pc. The initial temperature of the cloud is $T = 30\,$K. The whole cloud is set up in solid body rotation with an angular velocity $\omega = 1.5 \times 10^{-14}\,$s$^{-1}$ corresponding to a ratio of rotational to gravitational energy $\beta = 0.05$ and a mean specific angular momentum of $j = 1.27 \times 10^{23}\,$cm$^2$s$^{-1}$.
We follow the gravitational collapse of the molecular cloud with the adaptive mesh until we reach a cell size of $98\,$AU. We create sink particles at a cut-off density of $\rho_\mathrm{crit} = 7 \times 10^{-16}\,$gcm$^{-3}$. All gas within the accretion radius of $r_\mathrm{sink} = 590\,$AU above $\rho_\mathrm{crit}$ is accreted to the sink particle if it is gravitationally bound to it. The Jeans mass on the highest refinement level is $M_\mathrm{jeans} = 0.13$M$_\odot$.
The adaptive mesh technique allows us to resolve the gravitational collapse of the gas from the parsec scale down to a few hundred AU. At higher spatial resolution of only several ten AU, the gas becomes optically thick to non-ionizing radiation, and scattering effects must be taken into account. Since our cut-off density is more than two orders of magnitude smaller than the onset of the optically thick regime at $\sim 10^{-13}$gcm$^{-3}$ [e.g. @larson69] and we are focussing in our analysis on large-scale effects on the stellar cluster scale, we expect the raytracing approximation to be valid. Feedback by radiation pressure, which plays a role on the very small scales and is neglected in our model, is dynamically unimportant on these large scales [@krummatzner09].
We discuss three simulations (see Table \[tab:colsim\]). In the first simulation (run A), we only allow for the formation of a single sink particle and suppress the formation of secondary sink particles artificially by introducing the dynamical temperature floor $${T_\mathrm{min}}= \frac{G \mu}{\pi {k_\mathrm{B}}} \rho (n \Delta x)^2$$ with Newton’s constant $G$, mean molecular weight $\mu$, Boltzmann’s constant ${k_\mathrm{B}}$, local gas density $\rho$, and cell size $\Delta x$. The temperature floor ensures the sufficient resolution of the Jeans length with $n \geq 4$ cells, which avoids artificial fragmentation [@truelove97]. The suppression of secondary sink formation guarantees that the accretion flow around the massive star is not weakened by the fragmentation of the disk, which would otherwise inevitably lead to the formation of companion stars that limit the growth of the massive stars in the cluster (see Section \[sec:results\]). In the second simulation (run B), this dynamical temperature floor is not applied, and a whole stellar cluster forms during the course of the simulation. The third simulation (run D) is a control run that allows us to study the influence of radiation feedback on the stellar cluster evolution. As in run B, a small stellar cluster forms, but the stars emit neither ionizing nor non-ionizing radiation.
To demonstrate convergence of our study, we have also run an additional simulation at twice the numerical resolution. This simulation, run D+, which we ran for 0.66 Myr, is identical to run D, except that the cell size at highest grid refinement is $49\,$AU, the sink particle radius is $r_\mathrm{sink} = 312\,$AU and the sink particle cut-off density is $\rho_\mathrm{crit} = 2.5 \times 10^{-15}\,$gcm$^{-3}$. Since we ran it for only a fraction of the time covered by the other simulations, we will not discuss it at length, but the model ran for long enough to show that the relation between total stellar cluster mass and the mass of the largest star is the same is in the lower resolution simulations (see Section \[subsec:compaccr\] for further discussion).
The numerical method along with its inherent physical limitations is discussed in detail in @petersetal10a.
Results {#sec:results}
=======
[ccccccc]{} Name & Resolution & Radiative Feedback & Multiple Sinks & ${M_\mathrm{sinks}}$(M$_\odot$) & ${N_\mathrm{sinks}}$ & ${M_\mathrm{max}}$(M$_\odot$)\
Run A & 98 AU & yes & no & 72.13 & 1 & 72.13\
Run B & 98 AU & yes & yes & 125.56 & 25 & 23.39\
Run D & 98 AU & no & yes & 151.43 & 37 & 14.64\
![ [*top*]{} Total accretion history of all sink particles combined forming in runs A, B, and D. While the heating by non-ionizing radiation does not affect the total star formation rate, the ionizing radiation appreciably reduces the total rate at which gas converts into stars once the most massive object has stopped accreting and its [H [ii]{}]{} region can freely expand. This effect is not compensated by triggered star formation at the ionization front, which is never observed during the simulation. The slope of the total accretion history in run B goes down because the massive stars (dashed line) accrete at a decreased rate, while the low-mass stars (dotted line) keep accreting at the same rate. [*bottom*]{} Instantaneous accretion rate as function of time of the first sink particle to form in the three runs. These are generally the most massive sink particles during most of the simulation. While the accretion onto the star in run A never stops, the massive stars in run B and D are finally starved of material. Since the radiative heating suppresses fragmentation in run B, the final mass of the star is almost twice as high as in run D.[]{data-label="fig:accretion"}](rad_nonrad_detail.pdf){width="8cm"}
![ [*top*]{} Total accretion history of all sink particles combined forming in runs A, B, and D. While the heating by non-ionizing radiation does not affect the total star formation rate, the ionizing radiation appreciably reduces the total rate at which gas converts into stars once the most massive object has stopped accreting and its [H [ii]{}]{} region can freely expand. This effect is not compensated by triggered star formation at the ionization front, which is never observed during the simulation. The slope of the total accretion history in run B goes down because the massive stars (dashed line) accrete at a decreased rate, while the low-mass stars (dotted line) keep accreting at the same rate. [*bottom*]{} Instantaneous accretion rate as function of time of the first sink particle to form in the three runs. These are generally the most massive sink particles during most of the simulation. While the accretion onto the star in run A never stops, the massive stars in run B and D are finally starved of material. Since the radiative heating suppresses fragmentation in run B, the final mass of the star is almost twice as high as in run D.[]{data-label="fig:accretion"}](accretionrate.pdf){width="8cm"}
Accretion History
-----------------
In this section we compare the protostellar mass growth rates from our three runs, with a single sink particle (run A), multiple sinks and radiative heating (run B), and multiple sinks with no radiative heating (run D). As already discussed by @petersetal10a, when only the central sink particle is allowed to form (run A), nothing stops the accretion flow to the center. Figure \[fig:accretion\] shows that the central protostar grows at a rate $\dot{M} \approx 5.9 \times 10^{-4}\,$M$_{\odot}\,$yr$^{-1}$ until we stop the calculation when the star has reached $72\,$M$_{\odot}$. The growing star ionizes the surrounding gas, raising it to high pressure. However this hot bubble soon breaks out above and below the disk plane, without affecting the gas flow in the disk[^1] midplane much. In particular, it cannot halt the accretion onto the central star. Similar findings have also been reported from simulations focussing on the effects of non-ionizing radiation acting on somewhat smaller scales [@yorke02; @krumkleinmckee07; @krumholzetal09; @sigalottietal09]. Radiation pressure is not able to stop accretion onto massive stars and is dynamically unimportant, except maybe in the centers of dense star clusters near the Galactic center [@krummatzner09].
The situation is different when the disk can fragment and form multiple sink particles. Initially the mass growth of the central protostar in runs B and D is comparable to the one in run A. As soon as further protostars form in the gravitationally unstable disk, they begin to compete with the central object for accretion of disk material. However, unlike in the classical competitive accretion picture [@bonnell01a; @bonetal04], it is not the most massive object that dominates and grows disproportionately fast. Figure \[fig:accretion\] shows that, although the accretion rates of the most massive stars ($M \geq
10\,$M$_\odot$) steadily decrease, the low-mass stars ($M <
10\,$M$_\odot$), keep accreting at the same rate. Although the detailed mass distribution of the low-mass stars depends on numerical resolution [@federrathetal10], the net effect of their accretion should not, so long as they can form at all. The successive formation of low-mass objects in the disk at increasing radii limits subsequent growth of the more massive objects in the inner disk. Material that moves inwards through the disk accretes preferentially onto the sinks at larger radii.
The key to the process, as already discussed by @bate00, is that fragments that form later at larger radii will preferentially accrete larger-angular momentum material. This will happen on a much shorter time than the timescales required to redistribute this angular momentum to other parts of the disk at larger radii by viscous or gravitational torques. Similar effects are discussed in the disk fragmentation studies by @kratteretal10.
This behavior is found in models of low-mass protobinary disks, where again the secondary accretes at a higher rate than the primary. Its orbit around the common center of gravity scans larger radii and hence it encounters material that moves inwards through the disk before the primary star. This drives the system towards equal masses and circular orbits [@bateetal97; @bate00]. In our simulations, after a certain transition period hardly any gas makes it all the way to the center and the accretion rate of the first sink particle drops to almost zero. This is the essence of the fragmentation-induced starvation process. In run B, it prevents any star from reaching a mass larger than $25\,$M$_\odot$. The Jeans mass in run D is smaller than in run B because of the lack of accretion heating, and consequently the highest mass star in run D grows to less than $15\,$M$_\odot$. In comparison, @krumkleinmckee07 described simulations starting with a ten times less massive core than ours, using both an isothermal equation of state and including accretion heating. In both cases, objects with roughly half the mass of our most massive object formed. The isothermal case formed an only slightly less massive object than the one including accretion heating, just as in our models. The smaller final masses of their objects compared to ours seems mainly just to be a result of their smaller initial core masses.
Inspection of Figure \[fig:accretion\] reveals additional aspects of the process. We see that the total mass of the sink particle system increases at a faster rate in the multiple sink simulations, run B and D, than in the single sink case, run A. This is understandable, because as more and more gas falls onto the disk it becomes more and more unstable to fragmentation, so as time goes by additional sink particles form at larger and larger radii. Star formation occurs in a larger volume of the disk, and mass growth is not limited by the disk’s ability to transport matter to its center by gravitational or viscous torques (compare Section \[sec:diskinst\]). As a result the overall star-formation rate is larger than in run A.
Since the accretion heating raises the Jeans mass and length in run B, the total number of sink particles is higher in run D than in run B, and the stars in run D generally reach a lower mass than in run B (compare Section \[subsec:compaccr\]). These two effects cancel out to lead to the same overall star formation rate for some time. At one point in the evolution, however, also the total accretion rate of run B drops below that of run D. At time $t \approx 0.68\,$Myr the accretion flow around the most massive star has attenuated below the value required to trap the [H [ii]{}]{} region. It is able to break out and affect a significant fraction of the disk area. A comparison with the mass growth of run D clearly shows that there is still enough gas available to continue constant cluster growth for another $50\,$kyr or longer, but the gas can no longer collapse in run B. Instead, it is swept up in a shell surrounding the expanding [H [ii]{}]{} region.
It is also notable that the expanding ionization front around the most massive stars does not trigger any secondary star formation, which suggests that triggered star formation [@elmelad77] may not be as efficient as expected, at least on the scales considered here.
{width="8cm"} {width="8cm"}
Figure \[fig:history\] shows the individual accretion histories of each of the sink particles in run B and run D. Radiative heating cannot prevent disk fragmentation but raises the local Jeans mass. Hence, as discussed above, much fewer stars form in run B than in run D, and the mass of the most massive stars in run B is higher. It is also evident from the figure that star formation is much more intermittent in the case with radiation feedback (run B). The reason for this behavior is that the star formation process is controlled by the local Jeans mass, which depends to a large degree on how the filaments in the disk shield the radiation (compare Section \[sec:diskinst\]), as previously noted by @daleetal05 [@daleetal07b] and @krumholzetal09. Shielding can lower the Jeans mass temporarily and thereby allow gravitational collapse that would not have occurred otherwise.
The accretion histories also reveal large differences between sink particles within the same simulation. For example, in run D a sink particle that forms around $t \approx 0.688\,$Myr accretes very rapidly and becomes one of the most massive stars in the cluster while other stars only accrete sporadically, interrupted with long intervals of no accretion at all, and some others even accrete only shortly after their formation. There are several reasons for these differences. First, accretion can stop when stars are ejected from the cluster by $N$-body interactions. This happens mostly to low-mass stars with masses $M \lesssim 2$M$_\odot$. Second, intermediate-mass stars can be starved of material by the formation of companions. This is the essential element of the fragmentation-induced starvation scenario. A star that can grow to intermediate masses is embedded in an accretion flow from a larger gas reservoir. If this reservoir is gravitationally unstable, it can collapse to form companions. This may explain why the intermediate-mass stars in run B seem to have shorter time intervals of no accretion. Since the Jeans mass is higher in this case, the formation of companions is suppressed, and this allows the star to accrete for longer times. Third, the stars are not fixed at a position in the disk plane but move quickly within the disk by gravitational interaction with their surrounding dense gas and with nearby stars. Hence, they are not tied to the filament they form in but can move away from it. If the star moves into a low-density void in the disk, accretion will stop until the star is embedded in higher density gas again. The sink particle in run D mentioned above accretes so vigorously because it moves along with its parental filament over a long time and no companions form around it. Fourth, accretion can stop when the star resides within a low-density [H [ii]{}]{} region. This applies to the first two massive stars in run B. Their accretion flow had become so weak that the ionizing radiation was able to isolate the stars from the high-density gas in the disk and stopped accretion.
Disk Growth and Disk Instability {#sec:diskinst}
--------------------------------
To quantify the accretion of mass onto the disk and the subsequent instability, we calculate the amount of mass in a control volume that encloses the disk at all times. This control volume is the same for all runs and has dimensions $0.24\,$pc$\times 0.24\,$pc$\times 0.015\,$pc. We show in Figure \[fig:massinvolume\] the mass ${M_\mathrm{disk}}$ of non-accreted gas contained in the control volume, the mass ${M_\mathrm{sinks}}$ contained in all sink particles as well as the total mass ${M_\mathrm{tot}}= {M_\mathrm{disk}}+ {M_\mathrm{sinks}}$ for runs A, B and D. The disk mass in run A is much larger than in the multiple sink runs B and D since the mass cannot be absorbed by secondary sink particles. Instead, the mass accumulates in the disk plane. Consequently, the star in run A is embedded in a strong accretion flow at all times, which facilitates accretion at an approximately constant rate despite radiation feedback. The evolution in runs B and D differs from the mass growth in run A as soon as secondary sink particles form. The continuous formation of new sink particles in an expanding region around the central star in these runs keeps ${M_\mathrm{disk}}$ almost constant. Figure \[fig:massinvolume\] shows that at $t \approx 0.64\,$Myr, outflow of material from the central [H [ii]{}]{} region [@petersetal10a] reduces ${M_\mathrm{disk}}$ even further in run B, so that it falls slightly below the case without feedback (run D) at late times. However, accretion onto secondary sink particles had already cleared the central region, so that this is a relatively minor effect. The accretion onto sink particles is not affected by the central ionization and stays at a constant rate until the [H [ii]{}]{} region can break free (compare Figure \[fig:accretion\]).
![Disk growth and sink particle formation. The plot shows the mass ${M_\mathrm{disk}}$ of non-accreted gas contained in a control volume around the disk, the mass ${M_\mathrm{sinks}}$ of all sink particles and the total mass ${M_\mathrm{tot}}= {M_\mathrm{disk}}+ {M_\mathrm{sinks}}$ for run A (green), run B (blue) and run D (red). The disk mass in runs B and D is kept almost constant by subsequent sink particle formation, while the disk in run A continuously grows. The deviation between the disk masses in runs B and D at late times is caused by ionization-driven outflows, but these do not affect the total star formation rate.[]{data-label="fig:massinvolume"}](massinvolume_new.pdf){width="8cm"}
We investigate the initial instability of the disk with an analysis of the Toomre $Q$-parameter $$Q = \left| \frac{2 c_\mathrm{s} \Omega}{\pi \Sigma G} \right|$$ with sound speed $c_\mathrm{s}$, angular velocity $\Omega$, surface density $\Sigma$, and Newton’s constant $G$. The disk is linearly stable for $Q > 1$ and linearly unstable for $Q < 1$ [@toomre64; @goldlynd65]. The initial phases of disk instability in run B are displayed in Figure \[fig:toomreq\]. The figure shows slices of density, temperature and $Q$ in the disk plane for four different times. One can clearly see that the most unstable parts of the disk are the filamentary structures that form as the disk grows in mass. The heating by stellar radiation can suppress instability locally, but shielding by the dense filaments prevents the whole disk from becoming stable and restricts the heating to small regions near the center of the disk that are surrounded by filaments. This shielding makes it possible for star formation to progress radially outwards despite accretion heating by the stars. The disk remains sufficiently cool at the inner edge for gravitational instability to set in and star formation proceeds inside-out in the disk plane. Hence, the effect of the filamentary structures in the disk is twofold: they are so dense that they render the disk unstable locally; and, because of their high density, they can effectively shield the radiation from the stellar cluster near the center of the disk, so that radiative heating does not stabilize the outer parts of the disk. Indeed, high-resolution observations of high-mass accretion disk candidates [@beutetal09] provide some evidence for fragmentation and the presence of substructure on $\sim 1000\,$AU scales (as also proposed by @krumetal07).
{width="450pt"}
Since the filaments that shield the outer parts of the disk from radiation are optically thick, with an optical depth for non-ionizing radiation of several tens, it is important to estimate the degree to which our simulations, based on a raytracing technique to propagate the radiation on the grid, are affected by the lack of heating by diffuse radiation. To test this, we use the adaptive-mesh radiative transfer program RADMC-3D[^2]. RADMC-3D is based on the standard Monte Carlo method of @bjorkmanwood01 in combination with Lucy’s method of treating optically thin regions [@lucy99]. It is the successor of the RADMC code [@dullemdom04] and has been used previously to generate maps of dust emission from these simulations [@petersetal10b]. We have calculated self-consistent dust temperatures of the simulation snapshot shown in the last row of Figure \[fig:toomreq\]. Assuming that dust and gas temperatures in the disk plane are equal, we can compare the Monte Carlo dust temperature with the simulation gas temperature, as shown in Figure \[fig:diffuseplot\]. The comparison demonstrates that direct heating dominates over diffuse radiation in regions more than $\sim500$ AU away from the stars. The diffusely heated regions lie completely within the region heated in any case by direct radiation, so our raytracing method accurately describes the shielding by the filaments of the cold disk region where secondary fragmentation proceeds.
![Comparison of heating by direct and diffuse radiation. The slices show the gas temperature from the simulation snapshot in the last row of Figure \[fig:toomreq\] ([ *top*]{}) and the Monte Carlo dust temperature generated by RADMC-3D for the same snapshot ([*bottom*]{}). The dust temperature from diffuse radiation lies below the gas temperature almost everywhere outside the region heated by direct radiation ([*contour*]{}). The black dots indicate the positions of sink particles.[]{data-label="fig:diffuseplot"}](diffuseplot_new.pdf){width="167pt"}
To illustrate the tendency of star formation to occur at increasingly larger disk radii, we show the disk radius at which new sink particles form as a function of time for runs B and D in Figure \[fig:radtim\]. Because the accretion heating sets in already with the very first stars that form in run B, sink formation is suppressed at small disk radii initially. The massive stars slowly spiral outwards with time, so that at $t \approx 0.67\,$Myr their radiation can be shielded by filaments in the disk. Within these filaments, the gas then cools until local collapse sets in and two sink particles form near the center of the disk. Once the filament has dissolved, the gas heats up again and no further sink particles form in the inner disk region.
![Radius of sink formation as a function of time for run B (*top*) and run D (*bottom*). Because of the absence of radiation heating, both the Jeans length and the Jeans mass are smaller in run D than in run B, giving rise to the formation of more numerous, but generally lower-mass stars. In both simulations, sink formation gradually occurs at larger disk radii. The accretion heating by the first stars to form suppresses sink formation at small disk radii in run B until relatively late in the cluster evolution.[]{data-label="fig:radtim"}](radiivstime_runb.pdf){width="8cm"}
![Radius of sink formation as a function of time for run B (*top*) and run D (*bottom*). Because of the absence of radiation heating, both the Jeans length and the Jeans mass are smaller in run D than in run B, giving rise to the formation of more numerous, but generally lower-mass stars. In both simulations, sink formation gradually occurs at larger disk radii. The accretion heating by the first stars to form suppresses sink formation at small disk radii in run B until relatively late in the cluster evolution.[]{data-label="fig:radtim"}](radiivstime_rund.pdf){width="8cm"}
Discussion {#sec:discussion}
==========
Initial Conditions
------------------
### Density Profile
As we argue above, the fragmentation behavior of the disk forming around the massive central star depends sensitively on the initial and boundary conditions, i. e. on the physical properties of the high-mass cloud core. One of the key parameters that determines whether fragmentation becomes widespread during the collapse of a massive cloud core is its initial density profile. Numerical simulations indicate that density profiles with flat inner core are more susceptible to fragmentation, while centrally concentrated cores (for example such as singular isothermal spheres with $\rho \propto r^{-2}$) usually form only one or at most a few objects [@girietal10].
The density structure of prestellar cores is typically estimated through the analysis of dust emission or absorption using near-IR extinction mapping of background starlight, mapping of millimeter/submillimeter dust continuum emission, and mapping of dust absorption against the bright mid-IR background emission [@bertaf07]. A main characteristic of the density profiles derived with the above techniques is that they require a central flattening. The density of low-mass cores is almost constant within radii smaller than $2500 - 5000\,$AU with typical central densities of 10$^{5}$ – 10$^{6}$ cm$^{-3}$ [@motetal98; @wamoan99]. A popular approach is to describe these cores as truncated isothermal (Bonnor-Ebert) sphere [@ebert55; @bonnor56], that often provides a good fit to the data [@bacmanetal01; @alvesetal01; @kandorietal05]. Bonnor-Ebert spheres are equilibrium solutions of self-gravitating gas bounded by external pressure. However, such density structure is not unique. Numerical calculations of the dynamical evolution of supersonically turbulent clouds show that transient cores forming at the stagnation points of convergent flows exhibit similar morphology [@baklva03; @klessenetal05; @banerjeeetal09]. The situation is less clear when it comes to high-mass cores [@beutheretal07], because most high-mass cores studied to date show at least some sign of star formation or turn out to consist of several sub-condensations when observed with high enough resolution [e.g. @beutetal05; @beuthen09]. However, large-scale surveys, e.g. as conducted in the Cygnus-X region [@motteetal07], indicate that high-mass cores are in many aspects similar to scaled-up versions of low-mass cores. We followed these lines of reasoning and chose an initial density profile with flat inner core and $r^{-1.5}$ density profile outside a radius of $r=0.5\,$pc. Although we begin with smooth rather than turbulent initial conditions, by the time ionizing radiation begins to be emitted, gravitational fragmentation has already produced substantial density perturbations.
### Rotation
The second important parameter is the initial rotation of this core, which defines the total amount of angular momentum in the system and hence the radial extent of the protostellar accretion disk. To assess the relevance of fragmentation-induced starvation for high-mass star formation we therefore must compare our choice of $\beta = 0.05$ for the ratio between rotational and potential energy in our $1000\,M_\odot$ core both with observational data of molecular cloud cores and with numerical simulations of core formation. Our choice of values correspond to a specific angular momentum of $j = 1.27 \times 10^{23}\,$cm$^2$s$^{-1}$.
The high-mass cloud cores observed by @pirogovetal03 with masses up to a few thousand solar masses show specific angular momenta $j$ from a few $\times 10^{22}\,$cm$^2$s$^{-1}$ up to $10^{23}\,$cm$^2$s$^{-1}$, corresponding to $\beta$-values up to $0.07$. We took this sample as motivation for our choice of parameters. Interesting in this context is also the large-scale rotational motion of molecular gas around a cluster of hypercompact [H [ii]{}]{} regions in G20.08-0.14 N seen by @gavmadetal09. This flow has a radius of about $0.5\,$pc, roughly consistent with our simulation. Similar observations were made by @keto90 for G10.6-0.4 and @welchetal87 for W94A.
It needs to be noted, however, that although the specific angular momenta of observed cores exhibit considerable scatter there is a clear trend of decreasing $j$ with decreasing core mass $M$. For example, the mean values in the sample discussed by @goodmanetal93 are $j \approx 6 \times 10^{21}\,$cm$^2$s$^{-1}$ with masses up to several hundred solar masses, while @casellietal02 studied low-mass cores with typically only a few solar masses and find $j \approx 7 \times 10^{20}\,$cm$^2$s$^{-1}$. In all cases the typical values for the ratio between rotational energy and potential energy is $\beta \lesssim 0.05$. This trend continues down to stellar scales. A binary G star with a orbital period of 3 days has $j \approx 10^{19}\,$cm$^2$s$^{-1}$, while the spin of a typical T Tauri star is a few $\times 10^{17}\,$cm$^2$s$^{-1}$. Our own Sun rotates only with $j\approx 10^{15}\,$cm$^2$s$^{-1}$. That means, during the process of star formation most of the initial angular momentum is removed from the collapsed object [@boden95].
If we look for guidance from numerical simulations of molecular cloud formation and fragmentation [@maclow04; @klessenetal09] we see a very similar trend. @gammieetal03 find a typical mean specific angular momentum of $j = 4 \times 10^{22}\,$cm$^2$s$^{-1}$, while @jappkles04 find values in the range $10^{20}\,$cm$^2$s$^{-1} \lesssim j \lesssim10 ^{21}\,$cm$^2$s$^{-1}$ for low-mass cores in the simulations by @klessetal98 and @klesburk01 [@klesburk00] depending on the evolutionary state of protostellar collapse. See also @tillpudr07 and @offetal08. Altogether, we find that our initial conditions are consistent with simulations of core formation.
Relation to Other Theoretical Models
------------------------------------
### Monolithic Collapse Models {#subsec:moncol}
The monolithic collapse model rests on the similarity between the shape of the observed core mass distribution [@motetal98; @johnstone00; @johnstone06; @ladaetal08] and the stellar initial mass function, IMF [@kroupa02; @chabrier03]. It assumes a one-to-one relation between the distributions with only a constant efficiency factor separating the two functions. Each molecular cloud core collapses to form a single star or at most a close binary system, with protostellar feedback processes determining the efficiency of the process [@matzner00]. This model reduces the problem of the origin of the IMF to the problem of determining the mass spectrum of bound cores, although strictly speaking the idea that the IMF is set by the mass spectrum of cores is independent of any particular model for the origin of that mass spectrum. Arguments to explain the core mass distribution generally rely on the statistical properties of turbulence [@klessen01; @padnor02; @hencha08; @hencha09], which generate structures with a pure powerlaw mass spectrum. The thermal Jeans mass in the cloud then imposes the flattening and turn-down in the observed mass spectrum.
However, there are a number of caveats. Many of the prestellar cores found in observational surveys appear to be stable entities and thus are unlikely to be in a state of active star formation. In addition, the simple interpretation that one core forms on average one star, and that all cores contain the same number of thermal Jeans masses, leads to a timescale problem [@clark07] that requires differences in the core mass function and the IMF. Other concern about this model comes from hydrodynamic simulations, which seem to indicate that massive cores should fragment into many stars rather than collapsing monolithically [@dobbsetal05; @clabon06; @bonbat06; @federrathetal10]. This objection is, however, weakened by the fact that magnetic fields are able to reduce the level of fragmentation on scales of molecular clouds as a whole [@heitschetal01] as well as of collapsing cloud cores [@henfro08; @hentey08]. In addition, radiative feedback also is able to reduce the number of fragments that form during the collapse of high-mass cores [@krumkleinmckee07] as well as low-mass cores [@offetal09]. But again, all current simulations indicate that neither accretion heating nor ionizing radiation can prevent the fragmentation of the massive dense disk that builds up during protostellar collapse, nor can they stop accretion onto the massive star [@yorke02; @krumholzetal09; @petersetal10a]. Instead, the heating merely increases the average mass of the fragments. This is also supported by analytic estimates comparing typical accretion rates onto the disk with its ability to transport matter inwards through viscous and gravitational torques [@krattmatz06; @kratteretal10]. Our current results agree with these studies.
### Competitive Accretion {#subsec:compaccr}
A second model for the origin of the IMF, called competitive accretion, focuses on the interaction between protostars, and between a protostellar population and the gas cloud around it [@bonnell01a; @bonnell01b; @bonbat02; @batbon05]. In the competitive accretion picture the origin of the peak in the IMF is similar to the monolithic collapse model, it is set by the Jeans mass in the prestellar gas cloud. However, rather than fragmentation in the gas phase producing a spectrum of core masses, each of which collapses down to a single star or star system, in the competitive accretion model all gas fragments down to roughly the Jeans mass. Prompt fragmentation therefore creates a mass function that lacks the powerlaw tail at high masses that we observe in the stellar mass function. This part of the distribution forms via a second phase in which Jeans mass-protostars compete for gas in the center of a dense cluster. The cluster potential channels mass towards the center, so stars that remain in the center grow to large masses, while those that are ejected from the cluster center by $N$-body interactions remain low mass [@klesburk00; @bonetal04]. In this model, the apparent similarity between the core and stellar mass functions is an illusion, because the observed cores do not correspond to gravitationally bound structures that will collapse to stars [@clabon06; @smithetal08].
The competitive accretion picture has been challenged, on the grounds that the kinematic structure observed in star-forming regions is inconsistent with the idea that protostars have time to interact with one another strongly before they completely accrete their parent cores [@krumetal05; @andreetal07].
Taken at face value, competitive accretion models show a correlation between the mass of the most massive star ${M_\mathrm{max}}$ and the total cluster mass ${M_\mathrm{sinks}}$ during the whole cluster evolution that is roughly ${M_\mathrm{max}}\propto {M_\mathrm{sinks}}^{2 / 3}$ [@bonetal04]. This correlation has been argued to represent a way to observationally confirm competitive accretion [@krumbon07] and is in fact in good agreement with observations [@weidkroup06; @weidetal10]. However, we find that our simulations also reproduce the observed relation between ${M_\mathrm{max}}$ and ${M_\mathrm{sinks}}$.
Figure \[fig:compaccr\] shows ${M_\mathrm{sinks}}$ as function of ${M_\mathrm{max}}$ for run A, run B and run D, the higher resolution convergence study run D+ and the relation ${M_\mathrm{max}}= 0.39 {M_\mathrm{sinks}}^{2/3}$, which was found by @bonetal04 as a fit to their simulation data. Over the whole cluster evolution, the curve for run D lies above this fit, while the curve for run B always lies below it. The fit agrees with our simulation data as well as it does to that of @bonetal04. The higher resolution simulation run D+ was only carried out for a fraction of the total time of the other simulations, but its data also agrees very well with the fit, demonstrating convergence of the results from the lower resolution simulations. This indicates that the scaling is not unique to competitive accretion, but can also be found with the fragmentation-induced starvation scenario and hence cannot be used as an observational confirmation of competitive accretion models.
The agreement of both models to this prediction is surprising because the accretion behavior of the most massive stars is totally different. Whereas the massive stars in competitive accretion simulations automatically accrete large fractions of the available gas because they reside in the center of the gravitational potential during the whole cluster evolution, the massive stars in fragmentation-induced starvation models have continuously decreasing accretion rates since they are starved of material by other cluster members and, in the final phase, by feedback from ionizing radiation. It seems that the observed relation between ${M_\mathrm{sinks}}$ and ${M_\mathrm{max}}$ is a very general result of protostellar interaction in a common cluster environment, and not an unambiguous sign of competitive accretion at work.
We can gain further insight from Figure \[fig:compaccr\]. It shows that the accretion heating suppresses low-mass star formation and that for all times the simulation with feedback contains a more massive star relative to the whole cluster mass than the simulation without feedback. When the most massive star in run D reaches $10\,$M$_\odot$, much more gas is used up to create additional low-mass stars than is funneled towards the most massive one. Thus, the growth of the most massive star in run D is much more ineffective than in run B. Remarkably, all four simulations in @bonetal04 show the turn-off towards accretion onto low-mass stars around $10\,$M$_\odot$, and only one simulation has formed a star more massive than $10\,$M$_\odot$. A very similar turn-off (indicated by an arrow in Figure \[fig:compaccr\]) at the same mass scale is found in run D, but the radiative heating in run B shifts the turn-off towards higher masses, so that the scaling is followed closely over a larger mass range. In fact, since the turn-off occurs only at the end of the simulation and some of the massive stars are still accreting at this point, it is unknown up to what mass the scaling will continue. However, these results clearly indicate that radiative feedback is necessary to reproduce the scaling for masses beyond $10\,$M$_\odot$ observed by @weidetal10.
![The total mass in sink particles ${M_\mathrm{sinks}}$ as a function of the most massive star in the cluster ${M_\mathrm{max}}$. We plot the curves for run A, run B and run D, the convergence study run D+ as well as the fit from the competitive accretion simulations @bonetal04. All simulations follow the competitive accretion prediction with good accuracy. The maximum mass ${M_\mathrm{max}}$ in run B is always larger than in run D for a fixed cluster mass ${M_\mathrm{sinks}}$. The turn-off away from the scaling relation (indicated by arrows) is shifted towards higher masses by radiative feedback. The higher resolution study run D+ was not followed long enough to show the turn-off.[]{data-label="fig:compaccr"}](competitive_accretion_highres.pdf){width="7.5cm"}
Relevance for Primordial Star Formation
---------------------------------------
We note that the initial conditions adopted here may also be appropriate for the formation of metal-free Population III stars. The very first generation of stars in the universe is thought to form at a redshift of $z \sim 15$ in relatively unperturbed and quiescent halos with masses of $\sim 10^6\,$M$_\odot$ in dark matter and about $10^5\,$M$_\odot$ in baryons [@bromlars04; @brommetal09]. This gas has primordial composition, i.e. basically consists of hydrogen and helium, and cools mostly via H$_2$ line emission [e.g. @annietal97; @glover05; @glovjapp07]. These halos have a well-defined density structure and rotational profile [@abeletal02] which leads to the build up of an extended accretion disk around the central object (see, e.g., @yoshidaetal08 for numerical simulations, or @tanmckee04 for an analytical model). Just as discussed above, if the mass flow onto the disk exceeds its capacity for transporting material inward by viscous torques, the disk becomes susceptible to gravitational instability and fragments. There are first indications that this indeed happened during Pop III star formation from high-resolution numerical simulations that follow gravitational collapse beyond the formation of the central hydrostatic protostar [@clarketal08; @turketal09; @stacyetal10]. We propose that fragmentation-induced starvation may be the physical process that determines the masses of Pop III stars.
The situation is very different in atomic cooling halos [@wiseabel07; @greifetal09]. Once the virial mass of a halo exceeds a value of $\sim 10^8\,$M$_\odot$, the infalling gas gets partially ionized in the virialization shock. The enhanced abundance of free electrons triggers rapid H$_2$ formation and some fraction of the gas cools down very rapidly. As a result there are streams of cold gas that can travel all the way to the center [see also @dekeletal09] and the overall velocity structure of the gas becomes highly chaotic and turbulent. We expect that these environmental conditions prevent the build-up of a large-scale accretion disk in the center of the halo and our model of fragmentation-induced starvation should therefore not be applicable. However, a more thorough analysis awaits the execution of detailed, high-resolution numerical simulations.
Summary and Conclusion {#sec:summary}
======================
We have presented a detailed analysis of the fragmentation-induced starvation scenario, which was first introduced by @petersetal10a. We have studied the accretion history and disk instability in collapse simulations of massive star formation. We have compared the results from a full simulation with radiation feedback and multiple sink particles with two control runs, one with multiple sink particles but without radiation feedback and one with radiation feedback but only a single sink particle. The combined analysis of all three simulations allows us to establish fragmentation-induced starvation as a workable model of massive star formation. We have also compared the new model to the monolithic collapse and competitive accretion models and speculated on the relevance of fragmentation-induced starvation for primordial star formation.
The basic principle of fragmentation-induced starvation is star formation in a rotating gravitationally unstable flow. Gravitational instability in such a flow leads to fragmentation of the accretion flow and the formation of companions around the central massive star that starve the central star of infalling material. In our simulations, star formation occurs within a rotationally flattened, disk-like structure. As more and more material falls onto the disk, star formation proceeds radially outwards, keeping the total disk mass roughly constant. The accretion history of individual stars is tightly connected with the position of these stars relative to the dense filaments and low-density voids in the disk. The fact that most if not all high-mass stars are observed in higher-order multiple stellar systems that themselves belong to more massive star clusters provides strong evidence for the widespread occurrence of fragmentation that forms the basis of our discussion.
We find that accretion heating does not prevent fragmentation of the disk, but leads to a higher local Jeans mass. As a result, fewer stars form with than without radiation feedback. The accretion heating shifts the masses of the most massive stars up, but leaves the average stellar mass almost unaffected. The accretion heating does not change the overall star formation rate of the whole stellar cluster. Feedback by ionizing radiation is unable to stop protostellar growth if the accretion flow is strong enough. However, if the accretion flow weakens due to starvation, an [H [ii]{}]{} region can expand and terminate the accretion process. The growing [H [ii]{}]{} regions reduce the total star formation rate and do not trigger star formation at the ionization front.
Our model is able to explain the observed morphologies [@petersetal10b] and time variability [@galvmadetal10inprep] of ultracompact [H [ii]{}]{} regions. We find that we can consistently reproduce the observed relation between the total mass of the star cluster ${M_\mathrm{sinks}}$ and the maximum stellar mass in the cluster ${M_\mathrm{max}}$, ${M_\mathrm{max}}\propto {M_\mathrm{sinks}}^{2 / 3}$. This relation seems to be the general outcome of protostellar interaction in a common cluster environment rather than being a signpost of competitive accretion only, as previously claimed. In fact, the dynamical processes discussed here exhibit exactly the opposite behavior of competitive accretion, rather than run-away accretion onto the most massive star together with the suppression of the growth of lower-mass objects, we see that angular momentum conservation and the presence of lower-mass objects limit the mass growth of massive stars.
Our simulations provide evidence for the rejection of proposals that the observed maximum stellar mass of $\sim 100\,$M$_\odot$ is set by radiative feedback. When disk fragmentation is artificially suppressed (run A) the central protostar accretes material at very high rate unimpeded by the intense UV radiation it emits without any indications of an upper limit [see also @petersetal10a]. When we permit disk fragmentation to occur, it is the process of fragmentation-induced starvation that prevents the stellar mass to become larger than $\sim 25\,$M$_\odot$ with our choice of initial conditions. We expect more massive, more centrally-condensed, and/or slower rotating cloud cores to lead to more massive protostars. Indeed, extensive parameter studies [@girietal10] show that the initial density profile dominates the accretion behavior, explaining the formation of a $40\,$M$_\odot$ star from a $100\,$M$_\odot$ core in @krumholzetal09. The alternative view is to attribute the apparent stellar mass limit to internal stability constraints.
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[^1]: We will hereafter refer to the flattened, dense, accretion flow that forms in the midplane of our rotating core as a disk. However, it is not necessarily a true Keplerian, viscous, accretion disk, which probably only forms within the central few hundred astronomical units, unresolved by our models.
[^2]: http://www.mpia.de/homes/dullemon/radtrans/radmc-3d/index.html
|
---
author:
- 'M. Jurkovic,[^1] L. Szabados, J. Vinkó,'
- 'B. Csák'
title: Pulsation and Orbit of AU Pegasi
---
Introduction
============
AU Pegasi has been classified as a Type II Cepheid. It is a remarkable object among Cepheids because of its highly unstable pulsation period (Szabados, 1977; Harris et al., 1979) and membership in a spectroscopic binary system with the shortest known orbital period among binaries involving a Cepheid component (53.3 days, Harris et al., 1984).
The temporal behaviour of the pulsation period of AU Pegasi was thoroughly discussed by Vinkó et al. (1993), who followed the period changes for the interval J.D. 2,433,100–2,448,600. While the pulsation period was practically constant before J.D.2,438,000, it was subjected to a strong and almost linear increase between 1964 ($P_{\rm puls} = 2.391$ days) and 1986 ($P_{\rm puls} = 2.412$ days), which corresponds to a yearly increase of about 0.1 per cent.
The analysis of the more recent photometric data, however, indicated that the pulsation period became stable at the value of about 2.411 days at the beginning of the 1990s. The light variations folded on this period are, however, not repetitive, which called for a deeper study of AU Peg.
Observational data and their analysis
=====================================
Photometric and radial velocity data of AU Pegasi have been analysed. Fourier analysis of photometric data from All Sky Automated Survey (2003–2006) and from observationsmade at Piszkéstető Mountain Station of the Konkoly Observatory (1994–2005) was carried out with Period04 (Lenz & Breger, 2005). Radial velocity data obtained by theMoscow CORAVEL group were taken from the papers by Gorynya et al. (1995, 1998). In the calculation of the orbital elements a circular orbit was assumed at first. The orbital parameters were obtained by iteration, in order to separate the orbital and pulsational velocities.
Results
=======
Photometry
----------
In Fig. \[jurkovic\_fig1\], we have plotted the photometric phase curve of AU Peg based on the data obtained during the ASAS project (Pojmanski, 2002). Fig. \[jurkovic\_fig2\] shows a part of its Fourier power spectrum where the spectral peak at $f_0=0.4147$ c/d belongs to the main pulsation frequency and the second highest peak is the alias of $1-f_0$. Fig. \[jurkovic\_fig3\] is the spectral window of the AU Peg ASAS data.
![Folded light curve of AU Peg from ASAS data[]{data-label="jurkovic_fig1"}](jurkovic_fig1.eps){width="83mm" height="62mm"}
![A part of the Fourier spectrum of AU Peg obtained from the ASAS light curve[]{data-label="jurkovic_fig2"}](jurkovic_fig2.eps){width="83mm" height="62mm"}
![Spectral window of ASAS data[]{data-label="jurkovic_fig3"}](jurkovic_fig3.eps){width="83mm" height="62mm"}
In Fig. \[jurkovic\_fig4\] and Fig. \[jurkovic\_fig5\] the folded light curve of AU Peg constructed from the data taken at Piszkéstető Mountain Station and the relevant part of the Fourier power spectrum of this light curve are shown. Fig. \[jurkovic\_fig6\] shows the spectral window. Again the main pulsation frequency appears at $f_0=0.4147$ accompanied by its alias, $1-f_0$. This pulsation frequency seems to be stable throughout the observation interval and corresponds to the period of 2.41138 days.
![Phased $V$ light curve of AU Peg from the Piszkéstető data[]{data-label="jurkovic_fig4"}](jurkovic_fig4.eps){width="83mm" height="62mm"}
![A part of the Fourier spectrum of the Piszkéstető $V$ light curve[]{data-label="jurkovic_fig5"}](jurkovic_fig5.eps){width="83mm" height="62mm"}
![Spectral window of Piszkéstető data[]{data-label="jurkovic_fig6"}](jurkovic_fig6.eps){width="83mm" height="62mm"}
Because the scatter in the phase curve exceeds the value expected for a Type II Cepheid, both data sets were analysed further. Secondary periodicity was detected on the residual power spectra with the frequency of $f_1=0.5870$ for the ASAS light curve (Fig. \[jurkovic\_fig7\]) and $f_1=0.5873$ for the Piszkéstető Mountain Station data (from both $V$ and $B$ photometric bands), respectively. Note that the peak appearing at $f=0.41566$ c/d is a 1-day alias of the combination frequency $2f_0+f_1$.
![A part of the residual Fourier power spectrum of ASAS light curve after prewhitening with $f_0$ and its harmonics[]{data-label="jurkovic_fig7"}](jurkovic_fig7.eps){width="83mm" height="62mm"}
The ratio of the frequencies is $f_0/f_1=0.706$ which is typical of double mode classical Cepheids. Moreover, the frequencies of the linear combinations, $f_0+f_1$, $f_1-f_0$, and $2f_0+f_1$ also appear in the respective Fourier spectra. From this we conclude that AU Peg is very likely to be a classical Cepheid rather than a Type II Cepheid.
The error of the frequencies was estimated from the half width of the spectral peaks. In the case of $f_0$ and $f_1$ the error of the ASAS data is $\sigma=0.0005$, while for the Piszkéstető data the error is $\sigma=0.0001$.
Other photometric data sets were also analysed in order to reveal a second pulsation mode. Fourier analysis of photometric data from the Hipparcos measurements(J.D. 2,447,889–J.D. 2,448,972, ESA 1997) showed the main frequency at $f_0=0.4147$, but the data are very noisy so the secondary periodicity could not be detected. The double mode nature of AU Peg can be verified by analysing previous data published by Harris (1980), Harris et al. (1979) and from a subsample of the observational data given bySzabados (1977). These data were obtained when AU Pegasi showed a significant monotonous period increase. The individual data sets are, however, short enough to use an appropriate ‘instantaneous’ pulsation period when searching for secondary periodicity. The Fourier analysis indicates presence of double mode pulsation in the case of all three photometric series. From the data obtained by Harris et al. (1979) frequencies of $f_0=0.4162$ and $f_1=0.5911$ c/d can be deduced, Harris’ (1980) data set can be well described by $f_0=0.4160$ and $f_1=0.5891$ c/d, while Szabados’ (1977) data indicate double-mode pulsation with the frequencies $f_0=0.41652$ and $f_1=0.5898$ c/d (and their harmonics and linear combinations). From these frequencies it follows that the period of the first overtone varied simultaneously with the period of the fundamental mode, and the $f_0/f_1$ frequency ratio does not differ significantly from the present value.
Spectroscopy
------------
Spectroscopic data used in the analysis were taken from Gorynya et al. (1995, 1998) who published the Moscow CORAVEL radial velocities. Their radial velocity data were decomposed into the pulsational radial velocity curve (Fig. \[jurkovic\_fig8\]) and the orbital radial velocity curve (Fig. \[jurkovic\_fig9\]) – proving the presence of the companion, which was earlier detected by Harris et al. (1979, 1984) and Vinkó et al. (1993).
![Pulsational radial velocity curve of AU Pegasi from the Moscow CORAVEL data[]{data-label="jurkovic_fig8"}](jurkovic_fig8.eps){width="83mm" height="62mm"}
![Orbital radial velocity curve of AU Pegasi assuming a circular orbit[]{data-label="jurkovic_fig9"}](jurkovic_fig9.eps){width="83mm" height="62mm"}
Assuming a circular orbit we computed the following orbital elements of the binary system:\
$P_{\mathrm{orb}}=53.26\pm0.3 ~{\mathrm{days}}$\
$T_0=2447739.496$\
$\phi=-41.68\pm0.01$\
$v_0=-1.96\pm0.42 ~{\mathrm{km\,s^{-1}}}$\
$K=-44.86\pm0.57 ~{\mathrm{km\,s^{-1}}}$\
$a_1\sin i=0.2196\pm0.003 ~{\mathrm{AU}}$\
$f(m_2)=0.49\pm0.02 ~{\mathrm{M_\odot}}$.\
We also tried to fit an elliptical orbit to the radial velocities plotted in Fig. \[jurkovic\_fig9\]. The best solution was found at $e=0.02\pm0.01$ excentricity. Therefore, the circular orbit computed above describes this binary system in a satisfactory manner.
The period – radius relation for Cepheids given byGieren et al. (1998) $$\log R=0.751(\pm0.026)\log P+1.070(\pm0.008)$$ can be used for deriving the radius of AU Pegasi. Substituting the period of the fundamental mode (in days) into the above formula, we get that the radius of AU Pegasi is $R=22.75\pm0.67~R_\odot$. Using the equation from Bono et al. (2001) $$\log (M_{\mathrm p}/M_\odot)=-0.09(\pm0.03)+0.48(\pm0.03)\log
(R/R_\odot)$$ we derived that the pulsational mass for AU Pegasi is$M_{\mathrm p}=3.64\pm0.43~M_\odot$. Combining this result with the mass function we can calculate the mass of the companion as a function of inclination (see Fig. \[jurkovic\_fig10\]). The inclination angle must be smaller than $87.5^\circ$, because there is no evidence of eclipses in the photometric data, and should not be smaller than $30^\circ$ because smaller values of inclination would result in physically unrealistic orbital radial velocity amplitude for this binary system.
![Prediction for the mass of the companion of AU Pegasi according to the mass function $f(m_2)=0.49 ~{\mathrm{M_\odot}}$[]{data-label="jurkovic_fig10"}](jurkovic_fig10.eps){width="83mm" height="62mm"}
Summary
=======
- Following an extended interval characterised by continuously increasing pulsation period, oscillations of AU Pegasi have settled at 2.411 day periodicity;
- Fourier analysis of photometric data showed thatAU Peg is, in fact, a double-mode Cepheid. The frequency ratio of the two excited modes is $f_0/f_1=0.706$, a value typical of double-mode pulsators among Galactic classical Cepheids;
- Spectroscopic data were used to calculate the orbital elements of the binary system.
A more detailed discussion on the behaviour of AU Pegasi, including the details of the period analysis of the individual data sets, the O$-$C diagram, and the results obtained from the analysis of spectroscopic data will be the topic of a forthcoming paper. In view of the unique properties of this Cepheid (shortest known orbital period for a Cepheid, low amplitude beat phenomenon, ambiguity in classification), AU Pegasi deserves a closer attention from observers.
We wish to express our thanks to the organizers of the British–Hungarian–French N+N+N Workshop for Young Researchers for the support given to presenting our work. This work has been supported by the Hungarian OTKA Grants T 042509, TS 049872, and T 046207.
[11]{} Bono, G., Gieren, W.P., Marconi, M., Fouqué, P., Caputo, F.: 2001, ApJ 563, 319 ESA: 1997, The Hipparcos Catalogue, ESA SP-1200 Harris, H.C.: 1980, PhD Thesis, Univ. of Washington Harris, H.C., Olszewski, E.W., Wallerstein, G.: 1979, AJ 84, 1598 Harris, H.C., Olszewski, E.W., Wallerstein, G.: 1984, AJ 89, 119 Gieren, W.P., Fouqué, P., Gómez, M.: 1998, ApJ 496, 17 Gorynya, N.A., Samus, N.N., Rastorguev, A.S., Sachkov, M.E.: 1995, PAZh 22, 198 Gorynya, N.A., Samus, N.N., Sachkov, M.E., et al.: 1998, PAZh 24, 939 Lenz, P., Breger, M.: 2005, CoAst 146, 53 Pojmanski, G.: 2002, AcA 52, 397 Szabados, L.: 1977, Mitt. Sternw. Ung. Akad. Wiss., Budapest, No. 70 Vinkó, J., Szabados, L., Szatmáry, K.: 1993, A&A 279, 410
[^1]:
|
---
abstract: 'We present a method to estimate lighting from a single image of an indoor scene. Previous work has used an environment map representation that does not account for the localized nature of indoor lighting. Instead, we represent lighting as a set of discrete 3D lights with geometric and photometric parameters. We train a deep neural network to regress these parameters from a single image, on a dataset of environment maps annotated with depth. We propose a differentiable layer to convert these parameters to an environment map to compute our loss; this bypasses the challenge of establishing correspondences between estimated and ground truth lights. We demonstrate, via quantitative and qualitative evaluations, that our representation and training scheme lead to more accurate results compared to previous work, while allowing for more realistic 3D object compositing with spatially-varying lighting.'
author:
- |
Marc-André Gardner^\*^, Yannick Hold-Geoffroy^$\dagger$^, Kalyan Sunkavalli^$\dagger$^,\
Christian Gagné^\*^, Jean-François Lalonde^\*^\
^\*^Université Laval, ^$\dagger$^Adobe Research\
[marc-andre.gardner.1@ulaval.ca {holdgeof,sunkaval}@adobe.com]{}\
[{christian.gagne,jflalonde}@gel.ulaval.ca]{}\
[<https://lvsn.github.io/deepparametric/>]{}
bibliography:
- 'bibliography.bib'
title: Deep Parametric Indoor Lighting Estimation
---
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge the financial support of NSERC for the main author PhD scholarship. This work was supported by the REPARTI Strategic Network, the NSERC Discovery Grant RGPIN-2014-05314, MITACS, Prompt-Québec and E Machine Learning. We gratefully acknowledge the support of Nvidia with the donation of the GPUs used for this work, as well as Adobe with generous gift funding.
|
---
abstract: 'In this work we express the partition function of the integrable elliptic solid-on-solid model with domain-wall boundary conditions as a single determinant. This representation appears naturally as the solution of a system of functional equations governing the model’s partition function.'
author:
- 'W. Galleas'
bibliography:
- 'references1.bib'
title: 'Elliptic solid-on-solid model’s partition function as a single determinant'
---
[^1]
Introduction {#intro}
============
Several types of lattice models are *exactly solvable* in the sense that the summation defining their partition function can be expressed as a closed formula without any approximation. This is usually a highly non-trivial task but it has been achieved for certain models enjoying the *gift* of integrability [@Baxter_book]. Two-dimensional lattice models are rather special and among them we find the most notorious exactly solved models of Statistical Mechanics. For instance, the Ising model and the $8$-vertex model. These models are corner stones of the modern theory of integrable systems and, in particular, a series of developments were due to Baxter’s ingenious works on the $8$-vertex model [@Baxter_1973a; @Baxter_1973b; @Baxter_1973c]. In the course of studying eigenvectors of the symmetric $8$-vertex model Baxter has introduced the so called *solid-on-solid* models or <span style="font-variant:small-caps;">sos</span> models for short. They are also refereed to in the literature as *interaction-round-a-face* (<span style="font-variant:small-caps;">irf</span>) models and they differ from vertex models in the way lattice interactions are characterized. While vertex models assign configuration variables to the edges of a rectangular lattice, <span style="font-variant:small-caps;">sos</span> models associate configuration variables with lattice sites. In this way, the <span style="font-variant:small-caps;">sos</span> model dual to Baxter’s $8$-vertex model consists of an Ising type model with four-spin interaction as discussed in [@Baxter_book].
Boundary conditions are among the main ingredients when defining a lattice statistical system and the elliptic <span style="font-variant:small-caps;">sos</span> model with domain-wall boundaries has received special attention recently. This special type of boundary conditions were firstly introduced by Korepin for the $6$-vertex model [@Korepin_1982] and subsequently translated to <span style="font-variant:small-caps;">sos</span> models in [@Korepin_Justin_2000]. Interestingly, for this particular type of boundary conditions the models’ partition functions can be written down explicitly as a closed formula [@Izergin_1987; @Rosengren_2009; @Galleas_2013], in contrast to the case with periodic boundary conditions. For the latter the *solution* still relies on the resolution of Bethe ansatz equations [@Lieb_1967].
The $6$-vertex model with domain-wall boundaries has found several applications, ranging from enumerative combinatorics [@Kuperberg_1996] to the study of gauge theories [@Szabo_2012], and the elliptic <span style="font-variant:small-caps;">sos</span> model is not far behind. A series of works have been devoted to the study of its combinatorial properties [@Rosengren_2011] and relation to special polynomials [@Rosengren_2015]. These results are mainly due to Rosengren’s representation [@Rosengren_2009] for the model’s partition function as a sum of Frobenius type determinants which seem to generalize Izergin’s single determinant representation for the $6$-vertex model [@Izergin_1987].
Although a compact expression for the elliptic <span style="font-variant:small-caps;">sos</span> model’s partition function has been found in [@Galleas_2013], the possibility of expressing such partition function as a single determinant has eluded the researchers of the field so far. This is not only of interest for the computation of physical quantities but providing a definitive answer to this puzzle also shed new light onto the mathematical structure underlying elliptic integrable systems. This is precisely the purpose of this letter and in what follows we show how a single determinant representation can be derived from the analysis of special functional relations originated from the *dynamical Yang-Baxter algebra*.
The model {#MODEL}
=========
Write $\mathscr{L}_n \coloneqq \{ 1, 2, \dots , n \}$ and let $(i,j) \in \mathscr{L}_{L+1} \times \mathscr{L}_{L+1}$ be $2$-tuples describing a two-dimensional square lattice. Hence our lattice is formed by the juxtaposition of $L \times L$ square cells which we shall simply refer to as *faces*. We assign a statistical weight $w_{ij}$ to the face enclosed by the Cartesian coordinates $(i,j)$, $(i,j+1)$, $(i+1,j)$ and $(i+1,j+1)$. The configuration of a given face $w_{ij}$ is characterized by variables $\{ h_{i,j}, h_{i,j+1}, h_{i+1,j}, h_{i+1,j+1} \}$ and the system’s partition function is defined as $$\label{PF}
Z \coloneqq \sum_{\{ h_{i,j} \}} \prod_{i,j=1}^{L+1} w_{ij} \begin{pmatrix} h_{i+1,j} & h_{i+1,j+1} \\ h_{i,j} & h_{i,j+1} \end{pmatrix} \; .$$ The variable $h_{i,j}$ is also referred to as *height function* and here $h_{i,j} \coloneqq \tau + n_{i,j} \gamma$ with $\tau, \gamma \in \mathbb{C}$ and $n_{i,j} \in \mathbb{Z}$. Also, here we consider that $h_{i,j}$ and $h_{i',j'}$ at neighboring sites can only differ by $\pm \gamma$. The set $\{ h_{i,j} \}$ then contains height functions of allowed face configurations. Baxter’s elliptic <span style="font-variant:small-caps;">sos</span> model has six allowed face configurations and the respective statistical weights are given by, $$\begin{aligned}
\label{BW}
w_{ij} \begin{pmatrix} \tau \pm \gamma & \tau \\ \tau & \tau \mp \gamma \end{pmatrix} &=& [x + \gamma] \nonumber \\
w_{ij} \begin{pmatrix} \tau \pm \gamma & \tau \pm 2\gamma \\ \tau & \tau \pm \gamma \end{pmatrix} &=& [\tau \pm \gamma] \frac{[x]}{[\tau]} \nonumber \\
w_{ij} \begin{pmatrix} \tau \pm \gamma & \tau \\ \tau & \tau \pm \gamma \end{pmatrix} &=& [\tau \pm x] \frac{[\gamma]}{[\tau]} \; ,\end{aligned}$$ where $[x] \coloneqq \frac{1}{2} \sum_{n = -\infty}^{+ \infty} (-1)^{n-\frac{1}{2}} p^{(n+\frac{1}{2})^2} e^{-(2n+1) x}$ for $x \in \mathbb{C}$ and fixed elliptic nome $0 < p < 1$. The function $[x]$ corresponds to the Jacobi theta-function $\Theta_1 ({\mathrm{i}}x, \nu)$ with $p = e^{{\mathrm{i}}\pi \nu}$ according to the conventions of [@Whittaker_Watson_book]. In order to completely define the partition function $Z$ we also need to declare the boundary conditions being used. Here we shall consider boundary conditions of domain-wall type which corresponds to the assumptions $h_{1,j} = h_{j,1} = \tau + (L+1-j)\gamma$ and $h_{L+1,j} = h_{j,L+1} = \tau + (j-1)\gamma$.
Algebraic-functional framework {#AF}
==============================
The algebraic structure underlying the statistical weights are nowadays well known. It consists of the elliptic quantum group $\mathcal{E}_{p, \gamma} [ \widehat{\mathfrak{gl}_2} ]$, as described in [@Felder_1994; @Felder_1995], and this enables the so called *dynamical Yang-Baxter algebra* to be used in the study of the partition function $Z$. Here we shall adopt the procedure developed in [@Galleas_2013] which exploits the dynamical Yang-Baxter algebra as a source of functional equations characterizing quantities of physical interest. We now write $Z = Z_{\tau} (x_1, x_2 , \dots , x_L)$ in order to capture the dependence of our partition function with the relevant variables. The variables $x_i \in \mathbb{C}$ will be referred to as *spectral parameters* while $\tau$ will be called *dynamical parameter*. In addition to that $Z$ also depends on *inhomogeneity parameters* $\mu_i \in \mathbb{C}$ ($ 1 \leq i \leq L$) and an *anisotropy parameter* $\gamma \in \mathbb{C}$. The latter are fixed from now on. Using the algebraic-functional framework we have shown in [@Galleas_2013] that the partition function satisfies the following functional equation, $$\label{eqA}
M_0 \; Z_{\tau} (X) + \sum_{i \in \{0,1, \dots, L \}} N_i \; Z_{\tau + \gamma} (X_i^0) = 0 \; ,$$ where $X \coloneqq \{ x_i \in {\mathbb{C}}\mid 1 \leq i \leq L \}$ and $X_i^{\alpha} \coloneqq X \cup \{ x_{\alpha} \} \backslash \{ x_i \}$. The coefficients in explicitly read $$\begin{aligned}
\label{coeffA}
M_0 &\coloneqq& \frac{[\tau + \gamma]}{[\tau + (L+1)\gamma]} \prod_{j=1}^{L} [x_0 - \mu_j] \\
N_0 &\coloneqq& -\frac{[\tau + 2\gamma]}{[\tau + (L+2)\gamma]} \prod_{j=1}^{L} [x_0 - \mu_j + \gamma] \prod_{j=1}^{L} \frac{[x_j - x_0 + \gamma]}{[x_j - x_0]} \nonumber \\
N_i &\coloneqq& \frac{[\tau] [\tau + 2\gamma + x_0 - x_i]}{[\tau + (L+2)\gamma][x_i - x_0]} \prod_{j=1}^{L} [x_i - \mu_j + \gamma] \nonumber \\
&& \qquad \qquad \quad \times \prod_{\substack{j=1 \\ j \neq i}}^L \frac{[x_j - x_i + \gamma]}{[x_j - x_i]} \qquad i = 1,2, \dots , L \; . \nonumber \end{aligned}$$ We refer to as equation *type A* due to its roots within the algebraic-functional method [@Galleas_2013]. Using the same method we can also derive an equation of *type D* reading $$\begin{aligned}
\label{eqD}
\bar{M}_0 \; Z_{\tau + \gamma} (X) + \sum_{i \in \{ \bar{0}, 1, \dots, L \}} \bar{N}_i \; Z_{\tau} (X_i^{\bar{0}}) = 0 \; ,\end{aligned}$$ with coefficients defined as $$\begin{aligned}
\label{coeffD}
\bar{M}_0 &\coloneqq& \prod_{j=1}^{L} [x_{\bar{0}} - \mu_j + \gamma] \nonumber \\
\bar{N}_0 &\coloneqq& - \prod_{j=1}^{L} [x_{\bar{0}} - \mu_j] \prod_{j=1}^{L} \frac{[x_{\bar{0}} - x_j + \gamma]}{[x_{\bar{0}} - x_j]} \\
\bar{N}_i &\coloneqq& \frac{[\gamma] [\tau + (L+1)\gamma + x_{\bar{0}} - x_i]}{[x_{\bar{0}} - x_i] [\tau + (L+1)\gamma]} \prod_{j=1}^{L} [x_i - \mu_j] \nonumber \\
&& \qquad \qquad \quad \times \prod_{\substack{j=1 \\ j \neq i}}^L \frac{[x_i - x_j + \gamma]}{[x_i - x_j]} \qquad i = 1,2, \dots , L \; . \nonumber \end{aligned}$$ Although each equation and can individually determine the partition function , as shown in [@Galleas_2013], here we shall demonstrate how a determinant representation for $Z_{\tau}$ follows from a particular combination of equations type A and D.
Determinant representation {#DET}
==========================
In the recent paper [@Galleas_2016] we have shown how functional equations with structure similar to and can be solved in terms of determinants. However, in order to tackle our present equations, namely and , we need to generalize the mechanism of [@Galleas_2016]. The reason for that is the dependence of and with the dynamical parameter $\tau$. Hence we shall look for a suitable combination of our equations such that $\tau$ no longer plays the role of variable. Now consider under permutations $x_0 \leftrightarrow x_l$ for $0 \leq l \leq L$. This operation leaves us with a set of $L+1$ equations involving $Z_{\tau} (X^0_l)$ and $Z_{\tau+\gamma} (X^0_i)$ for $0 \leq i \leq L$. Thus we can solve the resulting system of equations and, in particular, express $Z_{\tau+\gamma} (X)$ as a combination of terms $Z_{\tau} (X^0_l)$. By substituting the result of this procedure in we are then left with the functional relation $$\label{eqAD}
\mathcal{M}_0 \; Z_{\tau} (X) + \sum_{i=1}^L \mathcal{N}_i \; Z_{\tau} (X_i^0) + \sum_{i=1}^L \bar{\mathcal{N}}_i \; Z_{\tau} (X_i^{\bar{0}}) = 0 \; .$$ The coefficients of explicitly read $$\begin{aligned}
\label{coeffADneu}
\mathcal{M}_0 &\coloneqq& \prod_{j=1}^L \frac{[x_0 - x_j + \gamma] [x_0 - \mu_j] [x_{\bar{0}} - \mu_j + \gamma]}{[x_0 - x_j] [x_0 - \mu_j + \gamma]} \nonumber \\
&& - \prod_{j=1}^L \frac{[x_{\bar{0}} - x_j + \gamma] [x_{\bar{0}} - \mu_j]}{[x_{\bar{0}} - x_j ]} \nonumber \\
\mathcal{N}_i &\coloneqq& - \frac{[\gamma] [x_0 - x_i + \tau + (L+1)\gamma]}{[\tau + (L+1)\gamma] [x_0 - x_i]} \nonumber \\
&& \quad \times \prod_{j=1}^L \frac{[x_i - \mu_j] [x_{\bar{0}} - \mu_j + \gamma]}{[x_0 - \mu_j + \gamma]} \prod_{\substack{j=1 \\ j \neq i}}^L \frac{[x_i - x_j + \gamma]}{[x_i - x_j]} \nonumber \\
\bar{\mathcal{N}}_i &\coloneqq& \frac{[\gamma] [x_{\bar{0}} - x_i + \tau + (L+1)\gamma]}{[\tau + (L+1)\gamma] [x_{\bar{0}} - x_i]} \prod_{j=1}^L [x_i - \mu_j] \nonumber \\
&& \times \prod_{\substack{j=1 \\ j \neq i}}^L \frac{[x_i - x_j + \gamma]}{[x_i - x_j]} \; .\end{aligned}$$ Compared to equations and , we can readily see that $\tau$ no longer plays the role of variable and we fix it from this point on. Next we notice that permutations $x_0 \leftrightarrow x_l$ for $1 \leq l \leq L$, $x_{\bar{0}} \leftrightarrow x_m$ for $1 \leq m \leq L$ and simultaneous permutations $x_0 \leftrightarrow x_l$, $x_{\bar{0}} \leftrightarrow x_m$ for $1 \leq l < m \leq L$ yields new equations with extra terms of the form $Z_{\tau} ( X_{i,j}^{0, \bar{0}} )$ where $X_{i,j}^{\alpha, \beta} \coloneqq X \cup \{x_{\alpha}, x_{\beta} \} \backslash \{x_i, x_j \}$, in addition to the ones already present in . The precise form of these new equations are not enlightening at the moment but it is important to remark that they form a closed system containing one term $Z_{\tau} (X)$, $L$ terms of type $Z_{\tau} (X_i^0)$, $L$ terms of type $Z_{\tau} (X_i^{\bar{0}})$ and $\frac{L(L-1)}{2}$ terms of form $Z_{\tau} ( X_{i,j}^{0, \bar{0}} )$. On the other hand, we have $L$ equations produced by permutations $x_0 \leftrightarrow x_l$ ($1 \leq l \leq L$), another $L$ equations obtained from $x_{\bar{0}} \leftrightarrow x_m$ ($1 \leq m \leq L$) and $L(L-1)/2$ originated from the simultaneous permutations $x_0 \leftrightarrow x_l, x_{\bar{0}} \leftrightarrow x_m$ for $1 \leq l < m \leq L$. Thus we can solve the resulting system of equations and express each element $Z_{\tau} (X_i^0)$, $Z_{\tau} (X_i^{\bar{0}})$ and $Z_{\tau} ( X_{i,j}^{0, \bar{0}} )$ in terms of $Z_{\tau} (X)$. The proportionality factor will be a ratio of determinants according to Cramer’s rule. For instance, using this approach we can write $Z_{\tau} (X_i^0) = \left( \text{det}(A_i) / \text{det}(B) \right) Z_{\tau} (X)$ with given matrices $A_i$ and $B$ of dimension $d_L \coloneqq L(L+3)/2$. Although the matrices $A_i$ and $B$ exhibit local dependence with the variable $x_{\bar{0}}$, the ratio $\text{det}(A_i) / \text{det}(B)$ is globally independent of $x_{\bar{0}}$ since it corresponds to $Z_{\tau} (X_i^0) /Z_{\tau} (X)$. Interestingly, this same feature has already made its appearance in a different context. This is precisely the property allowing for the computation of path integrals through the *localization method* [@Duistermaat_Heckman_1982] (see also [@Szabo_book] for a review). Therefore, we can choose $x_{\bar{0}}$ at our convenience and the inspection of the entries of $A_i$ and $B$ shows significant simplifications with the choice $x_{\bar{0}} = \mu_1 - \gamma$. Next we proceed with separation of variables and notice that the ratio $\text{det}(B) / \text{det}( \left. B \right|_{\tau = -\gamma})$ is independent of $x_0$. Hence, we can conclude that $Z_{\tau} (X) \sim \text{det}(B) / \text{det}( \left. B \right|_{\tau = -\gamma})$ and again the variable $x_0$ can be chosen at our will [@Galleas_tba]. Here we shall fix $x_{0} = \mu_1 - 2\gamma$ and the proportionality factor can be obtained from the asymptotic behavior presented in [@Galleas_2013].
Following the above described procedure we are left with the solution
$$\begin{aligned}
\label{Z}
Z_{\tau} (X) &=& (-1)^L \left(\frac{[(L+1)\gamma]}{[\tau + (L+2)\gamma]}\right)^{d_{L-1}} \left(\frac{[\tau + (L+1)\gamma]}{[L\gamma]}\right)^{d_L} \prod_{i,j = 1}^L [x_i - \mu_j] \prod_{k=1}^L \frac{[k \gamma]}{[\tau + k \gamma]} \nonumber \\
&& \times \frac{[\sum_{l=1}^L (x_l - \mu_l) + (L+1)\gamma ]}{[\sum_{l=1}^L (x_l - \mu_l) + \tau + (L+2)\gamma ]} \text{det} \left( \Omega \; \Omega_{red}^{-1} \right) \; ,\end{aligned}$$
where $\Omega_{red} \coloneqq \left. \Omega \right|_{\tau = - \gamma}$. In its turn the matrix $\Omega$ can be conveniently depicted as $$\Omega = \begin{pmatrix} \mathcal{F} & \mathcal{I} & \mathcal{G} \\ \bar{\mathcal{I}} & \mathcal{K} & \mathcal{J} \\ \bar{\mathcal{F}} & \bar{\mathcal{J}} & \bar{\mathcal{G}} \end{pmatrix} \; ,$$ where $\mathcal{F}$, $\bar{\mathcal{F}}$, $\mathcal{G}$ and $\bar{\mathcal{G}}$ are sub-matrices of dimension $L\times L$, $\mathcal{I}$ and $\bar{\mathcal{J}}$ are of dimension $L \times L(L-1)/2$, $\bar{\mathcal{I}}$ and $\mathcal{J}$ are of dimension $L(L-1)/2 \times L$, while the dimension of $\mathcal{K}$ is $L(L-1)/2 \times L(L-1)/2$. The matrices $\mathcal{F}$, $\bar{\mathcal{F}}$ and $\mathcal{G}$ are diagonal with non-null entries given by $$\begin{aligned}
\label{FFbG}
\mathcal{F}_{a,a} &\coloneqq& [2\gamma] \prod_{k=2}^L [\mu_1 - \mu_k - \gamma] \prod_{\substack{k=1 \\ k \neq a}}^L \frac{[x_k - \mu_1]}{[x_k - \mu_1 + \gamma]} \nonumber \\
\bar{\mathcal{F}}_{a,a} &\coloneqq& \frac{[\tau + L\gamma]}{[\tau + (L+1)\gamma]} \prod_{k=1}^L [x_a - \mu_k + \gamma] \nonumber \\
&& \qquad\qquad \qquad \qquad \times \prod_{\substack{k=1 \\ k \neq a}}^L \frac{[x_k - \mu_1]}{[x_k - \mu_1 + \gamma]} \nonumber \\
\mathcal{G}_{a,a} &\coloneqq& \frac{[\tau + (L+2) \gamma]}{[\tau + (L+1)\gamma]} \prod_{k=1}^L [\mu_1 - \mu_k - 2\gamma] \nonumber \\
&& \qquad\qquad \qquad \qquad \times \prod_{\substack{k=1 \\ k \neq a}}^L \frac{[x_k - \mu_1 + \gamma]}{[x_k - \mu_1 + 2\gamma]} . \nonumber \\\end{aligned}$$ On the other hand, the matrix $\bar{\mathcal{G}}$ is a full-matrix with entries $$\begin{aligned}
\label{Gb}
\bar{\mathcal{G}}_{a,b} \coloneqq \begin{cases}
- \frac{[x_a - \mu_1 + 2 \gamma]}{[x_a - \mu_1 + \gamma]} \prod_{k=1}^L [x_a - \mu_k] \prod_{\substack{k=1 \\ k \neq a}}^L \frac{[x_a - x_k + \gamma]}{[x_a - x_k]} \\
\hfill b = a \nonumber \\
\frac{[\gamma] [x_a - x_b + \tau + (L+1)\gamma]}{[\tau + (L+1)\gamma] [x_a - x_b]} \frac{[x_b - \mu_1 + 2\gamma]}{[x_b - \mu_1 + \gamma]} \prod_{k=1}^L [x_b - \mu_k] \\
\hfill \times \prod_{\substack{k=1 \\ k \neq a,b}}^L \frac{[x_b - x_k + \gamma]}{[x_b - x_k]} \qquad \text{otherwise}
\end{cases} . \\\end{aligned}$$ As for the remaining matrices it is convenient to introduce an index $n \colon \mathbb{Z}_{> 0} \times \mathbb{Z}_{> 0} \to \mathbb{Z}_{> 0}$. More precisely, we define $n_{r,s} \coloneqq s + L (r-1) - \frac{r(r+1)}{2}$ for $1 \leq r < s \leq L$, and in this way we have $$\begin{aligned}
\label{I}
\mathcal{I}_{a, n_{r,s}} \coloneqq \begin{cases}
\frac{[\gamma] [\mu_1 - x_s + \tau + L\gamma] [\mu_1 - x_s - 3\gamma]}{[\tau + (L+1)\gamma] [\mu_1 - x_s - \gamma] [\mu_1 - x_s - 2\gamma]} \\
\hfill \times \prod_{k=1}^L [x_s - \mu_k] \prod_{\substack{k=1 \\ k \neq r,s}}^L \frac{[x_s - x_k + \gamma]}{[x_s - x_k]} \qquad a = r \nonumber \\
\frac{[\gamma] [\mu_1 - x_r + \tau + L\gamma] [\mu_1 - x_r - 3\gamma]}{[\tau + (L+1)\gamma] [\mu_1 - x_r - \gamma] [\mu_1 - x_r - 2\gamma]} \\
\hfill \times \prod_{k=1}^L [x_r - \mu_k] \prod_{\substack{k=1 \\ k \neq r,s}}^L \frac{[x_r - x_k + \gamma]}{[x_r - x_k]} \qquad a = s \nonumber \\
0 \hfill \text{otherwise}
\end{cases} \\\end{aligned}$$ and $$\begin{aligned}
\label{Jb}
\bar{\mathcal{J}}_{a, n_{r,s}} \coloneqq \begin{cases}
\frac{[\gamma] [\mu_1 - x_s + \tau + (L-1)\gamma]}{[\tau + (L+1)\gamma] [x_s - \mu_1 + \gamma]} \prod_{\substack{k=1 \\ k \neq r,s}}^L \frac{[x_s - x_k + \gamma]}{[x_s - x_k]} \\
\hfill \times \prod_{k=1}^L \frac{[x_a - \mu_k + \gamma] [x_s - \mu_k]}{[\mu_1 - \mu_k - \gamma]} \qquad a = r \nonumber \\
\frac{[\gamma] [\mu_1 - x_r + \tau + (L-1)\gamma]}{[\tau + (L+1)\gamma] [x_r - \mu_1 + \gamma]} \prod_{\substack{k=1 \\ k \neq r,s}}^L \frac{[x_r - x_k + \gamma]}{[x_r - x_k]} \\
\hfill \times \prod_{k=1}^L \frac{[x_a - \mu_k + \gamma] [x_r - \mu_k]}{[\mu_1 - \mu_k - \gamma]} \qquad a = s \nonumber \\
0 \hfill \text{otherwise}
\end{cases}. \\\end{aligned}$$ Next we turn our attention to the matrices $\bar{\mathcal{I}}$ and $\mathcal{J}$. The entries of $\bar{\mathcal{I}}$ are then given by $$\begin{aligned}
\label{Ib}
\bar{\mathcal{I}}_{n_{l,m} , b} \coloneqq \begin{cases}
\frac{[2 \gamma] [x_m - \mu_1 + \tau + (L+2)\gamma]}{[\tau + (L+1)\gamma] [x_m - \mu_1 + \gamma]} \prod_{k=1}^L [\mu_1 - \mu_k - \gamma] \\
\hfill \times \prod_{\substack{k=1 \\ k \neq l, m}}^L \frac{[x_k - \mu_1]}{[x_k - \mu_1 + \gamma]} \qquad b=l \nonumber \\
- \frac{[2 \gamma] [x_l - \mu_1 + \tau + (L+2)\gamma]}{[\tau + (L+1)\gamma] [x_l - \mu_1 + \gamma]} \prod_{\substack{k=1 \\ k \neq l, m}}^L \frac{[x_k - \mu_1]}{[x_k - \mu_1 + \gamma]} \\
\hfill \times \prod_{k=1}^L \frac{[\mu_1 - \mu_k - \gamma] [x_m - \mu_k + \gamma]}{[x_l - \mu_k + \gamma]} \qquad b=m \\
0 \hfill \text{otherwise}
\end{cases} , \\\end{aligned}$$ while $\mathcal{J} \coloneqq \mathbf{0}$ is a null-matrix. Lastly, we have the following expression for the entries of $\mathcal{K}$,
$$\begin{aligned}
\label{K}
\mathcal{K}_{n_{l,m} , n_{r,s}} \coloneqq \begin{cases}
\frac{[x_l - \mu_1 + 3\gamma]}{ [x_l - \mu_1 + \gamma]} \prod_{k=1}^L \frac{[x_l - \mu_k] [x_m - \mu_k + \gamma]}{[x_l - \mu_k + \gamma]} \prod_{\substack{k=1 \\ k \neq l, m}}^L \frac{[x_l - x_k + \gamma]}{[x_l - x_k]} \\
\hfill - \frac{[x_m - \mu_1 + 3\gamma]}{ [x_m - \mu_1 + \gamma]} \prod_{k=1}^L [x_m - \mu_k] \prod_{\substack{k=1 \\ k \neq l, m}}^L \frac{[x_m - x_k + \gamma]}{[x_m - x_k]} \qquad l=r, m=s \nonumber \\
\frac{[\gamma] [x_m - x_s + \tau + (L+1)\gamma] [x_s - \mu_1 + 3\gamma]}{[\tau + (L+1)\gamma] [x_m - x_s] [x_s - \mu_1 + \gamma]} \prod_{k=1}^L [x_s - \mu_k] \prod_{\substack{k=1 \\ k \neq l, m, s}}^L \frac{[x_s - x_k + \gamma]}{[x_s - x_k]} \hfill l=r, m \neq s \nonumber \\
\frac{[\gamma] [x_m - x_r + \tau + (L+1)\gamma] [x_r - \mu_1 + 3\gamma]}{[\tau + (L+1)\gamma] [x_m - x_r] [x_r - \mu_1 + \gamma]} \prod_{k=1}^L [x_r - \mu_k] \prod_{\substack{k=1 \\ k \neq l, m, r}}^L \frac{[x_r - x_k + \gamma]}{[x_r - x_k]} \hfill l=s, m \neq r \nonumber \\
\frac{[\gamma] [x_l - x_s + \tau + (L+1)\gamma] [x_s - \mu_1 + 3\gamma]}{[\tau + (L+1)\gamma] [x_s - x_l] [x_s - \mu_1 + \gamma]} \prod_{k=1}^L \frac{[x_s - \mu_k] [x_m - \mu_k + \gamma]}{[x_l - \mu_k + \gamma]} \prod_{\substack{k=1 \\ k \neq l, m, s}}^L \frac{[x_s - x_k + \gamma]}{[x_s - x_k]} \qquad m=r, l \neq s \nonumber \\
\frac{[\gamma] [x_l - x_r + \tau + (L+1)\gamma] [x_r - \mu_1 + 3\gamma]}{[\tau + (L+1)\gamma] [x_r - x_l] [x_r - \mu_1 + \gamma]} \prod_{k=1}^L \frac{[x_r - \mu_k] [x_m - \mu_k + \gamma]}{[x_l - \mu_k + \gamma]} \prod_{\substack{k=1 \\ k \neq l, m, r}}^L \frac{[x_r - x_k + \gamma]}{[x_r - x_k]} \qquad m=s, l \neq r \nonumber \\
0 \hfill \text{otherwise}
\end{cases} . \\\end{aligned}$$
The set of relations - defines an explicit single determinant formula for the partition function . Although the entries of $\Omega_{red}$ consist of straightforward simplifications of -, it is worth remarking that for practical computations it might be more convenient to write $\text{det} \left( \Omega \; \Omega_{red}^{-1} \right)$ as the ratio $\text{det} \left( \Omega \right) / \text{det} \left( \Omega_{red} \right)$. In this way one avoids computing the inverse of $\Omega_{red}$ and evaluating the product $\Omega \; \Omega_{red}^{-1}$.
Concluding remarks {#CONCL}
==================
In the present paper we have obtained a novel representation for the partition function of the elliptic <span style="font-variant:small-caps;">sos</span> model in terms of a single determinant. This result addresses a long standing question in the field and confirms the existence of such representations. In the limit $p \to 0$, where $[x]$ degenerates into a trigonometric function, followed by the limit $\tau \to \infty$; the partition function reduces to that of the six-vertex model with domain-wall boundaries studied in [@Korepin_1982; @Izergin_1987]. In contrast to the $L \times L$ matrix determinant representation found in [@Izergin_1987], our solution consists of a determinant of a $L(L+3)/2 \times L(L+3)/2 $ matrix. However, it is also important to notice that the determinant of [@Izergin_1987] is taken over a full-matrix, while in our case we have a sparse matrix. In this way one might expect that is still liable to simplifications. Another interesting aspect of the representation is related to the possibility of taking the homogeneous limit. In our case the partial homogeneous limit $\mu_i \to \mu$ can be obtained trivially in contrast to Izergin’s representation for the six-vertex model.
Here we have singled out one particular possibility of determinant representation originated from the algebraic-functional framework. Alternative determinantal representations are also possible and we plan to investigate them in a future publication [@Galleas_tba]. Moreover, it is quite remarkable the similarity between the roles played by the variables $x_0$ and $x_{\bar{0}}$ here and the mechanism employed in the localization method for the evaluation of path integrals [@Szabo_book]. This point certainly deserves further studies and we hope to address it in a future publication.
Numerical checks
================
From definition we find $Z_{\tau} = [\gamma] [\tau + \gamma - \mu_1 + x_1]/[\tau + \gamma]$ for $L=1$. This is precisely the result obtained from our representation with the help of summation formulae for Jacobi theta-functions. For $L > 1$ we can easily compare numerically the value of the partition function computed from the definition with the one obtained from our representation . This provides extra support for the validity of our results. Numerical evaluations have been performed with `Mathematica` and in Table \[tab: sets\] one can find two sets of randomly chosen values for the model’s parameters. Tables \[tab: set1\] and \[tab: set2\] contain numerical comparisons using Set $1$ and Set $2$ respectively for $2 \leq L \leq 5$.
Parameter Set $1$ Set $2$
--------------- ------------------- -------------------
$x_1$;$\mu_1$ $0.4327$;$0.6745$ $0.8919$;$2.5449$
$x_2$;$\mu_2$ $1.0715$;$0.4129$ $0.7233$;$1.8734$
$x_3$;$\mu_3$ $1.7481$;$3.3385$ $0.1519$;$1.2745$
$x_4$;$\mu_4$ $2.2738$;$3.1245$ $0.4388$;$2.0178$
$x_5$;$\mu_5$ $2.1415$;$1.9715$ $2.6662$;$3.0089$
$\gamma$ $0.6512$ $0.1219$
$\tau$ $0.1743$ $0.2759$
$p$ $0.3116$ $0.4421$
: \[tab: sets\] Two sets of numerical values for the parameters.
$L$ Definition Representation
----- ------------------------------------ ------------------------------------
$2$ $0.00057111882715$ $0.00057111882715$
$3$ $6.07562588434218 \; {\mathrm{i}}$ $6.07562588434047 \; {\mathrm{i}}$
$4$ $6195.98835867588$ $6195.98835851194$
$5$ $139.817171384552 \; {\mathrm{i}}$ $139.817171384640 \; {\mathrm{i}}$
: \[tab: set1\] Numerical comparison using Set $1$.
$L$ Definition Representation
----- ------------------------------------- -------------------------------------
$2$ $0.230323036097808$ $0.230323036097803$
$3$ $0.202679526300975 \; {\mathrm{i}}$ $0.202679526300981 \; {\mathrm{i}}$
$4$ $2.659105034549285$ $2.659105034415262$
$5$ $1478.397210835060 \; {\mathrm{i}}$ $1478.397210823134 \; {\mathrm{i}}$
: \[tab: set2\] Numerical comparison using Set $2$.
The author thanks H. Rosengren for valuable discussions and F. Rühle for help with numerical checks.
[^1]: The work of W.G. is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676: Particles, Strings and the Early Universe.
|
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abstract: 'Neutron star models in perturbative $f(R)$ gravity are considered with realistic equations of state. In particular, we consider the FPS, SLy and other equations of state and a case of piecewise equation of state for stars with quark cores. The mass-radius relations for $f(R)=R+R(e^{-R/R_{0}}-1)$ model and for $R^2$ models with logarithmic and cubic corrections are obtained. In the case of $R^2$ gravity with cubic corrections, we obtain that at high central densities ($\rho>10\rho_{ns}$, where $\rho_{ns}=2.7\times 10^{14}$ g/cm$^{3}$ is the nuclear saturation density), stable star configurations exist. The minimal radius of such stars is close to $9$ km with maximal mass $\sim 1.9 M_{\odot}$ (SLy equation). A similar situation takes place for AP4 and BSK20 EoS. Such an effect can give rise to more compact stars than in General Relativity. If observationally identified, such objects could constitute a formidable signature for modified gravity at astrophysical level. Another interesting result can be achieved in modified gravity with only a cubic correction. For some EoS, the upper limit of neutron star mass increases and therefore these EoS can describe realistic star configurations (although, in General Relativity, these EoS are excluded by observational constraints).'
author:
- 'Artyom V. Astashenok$^{1}$, Salvatore Capozziello$^{2,3}$, Sergei D. Odintsov$^{4,5,6}$'
title: 'Further stable neutron star models from $f(R)$ gravity'
---
Introduction
============
The current accelerated expansion of the universe has been confirmed by several independent observations. Standard candles and distance indicators point out an accelerated expansion which cannot be obtained by ordinary perfect fluid matter as source for the cosmological Friedmann equations [@Perlmutter; @Riess1; @Riess2]. This evidence gives rise to difficulties in order to explain the evolution of large scale structures. Furthermore observations of microwave background radiation (CMBR) anisotropies [@Spergel], of cosmic shear through gravitational weak leasing surveys [@Schmidt] and, finally, data on Lyman alpha forest absorption lines [@McDonald] confirm the picture of an accelerated Hubble fluid.
In particular, the discrepancy between the amount of luminous matter revealed from observations and the critical density needed to obtain a spatially flat universe could be solved if one assumes the existence of a non-standard cosmic fluid with negative pressure, which is not clustered in large scale structure. In the simplest scenario, this [*dark energy*]{}, can be addressed as the Einstein Cosmological Constant and would contribute about 70% to the global energy budget of the universe. The remaining 30%, clustered in galaxies and clusters of galaxies, should be constituted for about 4% by baryons and for the rest by cold dark matter (CDM), whose candidates, at fundamental level, could be WIMPs (Weak Interacting Massive Particles), axions or other unknown particles [@Krauss].
From an observational viewpoint, this model has the feature to be in agreement with data coming from observations. It could be assumed as the first step towards a new standard cosmological model and it is indicated as Concordance Lambda Cold Dark Matter ($\Lambda$CDM) Model [@Bancall]. In summary, the observed universe could be self-consistently described once we admit the presence of a cosmological constant (70% of the total content), which would give rise to the observed acceleration of the Hubble fluid, and the presence of dark matter (at least 25%), which would explain the large scale structure. Despite of the agreement with observations, the $\Lambda$CDM model presents incongruences from a theoretical viewpoint. If the cosmological constant constitutes the “vacuum state” of the gravitational field, we have to explain the 120 orders of magnitude between its observed value at cosmological level and the one predicted by any quantum gravity [@Weinberg]. This inconsistency, also known as the [*cosmological constant problem*]{}, is one of the most fundamental problems of cosmology.
A very straightforward approach is to look for explanations for dark matter and dark energy within the realm of known physics. On the other hand, an alternative is that General Relativity is not capable of describing the universe at scales larger than Solar System, and dark components (energy + matter) could be the observable effect of such an inadequacy.
Assuming this point of view, one can propose alternative theories of gravity extending the Einstein theory (in this sense one deals with modified gravity), keeping its positive results, without requiring dark components, up to now not detected at experimental level. In this perspective, it can be shown that the accelerated expansion can be obtained without using new fundamental ingredients but enlarging the gravitational sector (see for example [@Capozziello1; @Capozziello2; @Odintsov1; @Turner; @Odintsov-3; @Capozziello3; @Capozziello_book; @Capozziello4; @Cruz]).
In particular, it has been recently shown that such theories give models able to reproduce the Hubble diagram derived from SNela surveys [@Capozziello3; @Demianski] and the anisotropies observed for CMBR [@Perrotta; @Hwang].
However, also this approach needs new signatures or some [*experimentum crucis*]{} in order to be accepted or refuted. In particular, exotic astrophysical structures, which cannot be addressed by standard gravity, could constitute a powerful tool to address this problem. In particular, strong field regimes of relativistic astrophysical objects could discriminate between General Relativity and its possible extensions.
The study of relativistic stars in modified gravity could have very interesting consequences to address this issue. In fact, new theoretical stellar structures emerge and they could have very important observational consequences constituting the signature for the Extended Gravity (see e.g. [@Laurentis; @Laurentis2]). Furthermore, strong gravitational regimes could be considered if one assume General Relativity as the weak field limit of some more complicated effective gravitational theory [@Dimitri-rev]. In particular, considering the simplest extension of General Relativity, namely the $f(R)$ gravity, some models can be rejected because do not allow the existence of stable star configurations [@Briscese; @Abdalla; @Bamba; @Kobayashi-Maeda; @Nojiri5]. On the other hand, stability can be achieved in certain cases due to the so called [*Chameleon Mechanism*]{} [@Tsujikawa; @Upadhye-Hu]. Another problem is that the possibility of existence of stable star configuration may depend on the choice of equation of state (EoS). For example, in [@Babichev1; @Babichev2], a polytropic EoS is used in order to solve this issue although the adopted EoS does not seem realistic to achieve reliable neutron stars.
In this paper, we start from the fact that $f(R)$ gravity models introduce a new scalar degree of freedom that must be considered into dynamics, then we study the structure of neutron stars in perturbative $f(R)$ gravity where the scalar curvature $R$ is defined by Einstein equations at zeroth order on the small parameter, i.e. $R\sim T$, where $T$ is the trace of energy-momentum tensor.
In this framework, we investigate several $f(R)$ models, namely $f(R)=R+\beta R (\exp(-R/R_{0})-1)$, $R^2$ model with logarithmic \[$f(R)=R+\alpha R^{2} (1+\beta ln (R/\mu^{2})$\] and cubic \[$f(R)=R+\alpha R^{2}(1+\gamma R)$\] corrections. [In particular, we consider the FPS and SLy equations of state for exponential modified gravity and a case of piecewise EoS for neutron stars with quark cores for logarithmic model. For models with a cubic term correction, the cases of realistic EoS such as SLy, AP4 and BSK20 are considered.]{} One of the results is that, if cubic correction term, at some densities, is comparable with the quadratic one, stable star configurations exist at high central densities. The minimal radius of such stars is close to $9$ km for maximal mass $\sim 1.9 M_{\odot}$ (SLy equation) or to $8.5$ km for mass $\sim 1.7M_{\odot}$ (FPS equation). [It is interesting to note that, in the case of simple cubic gravity, the maximal mass of stable configurations may be greater than the maximal limit of mass in the case of General Relativity. Due to this effect, some EoS, which are ruled out by observational constraints in GR, can lead to realistic results in the context of modified gravity. Moreover, these EoS describe some observational data with better precision than in General Relativity.]{} Clearly, such objects cannot be achieved in the context of General Relativity [@werner] so their possible observational evidences could constitute a powerful probe for modified gravity [@felicia; @antoniadis; @freire].
The paper is organized as follows. In Section II, we present the field equations for $f(R)$ gravity. For spherically symmetric solutions of these equations, we obtain the modified Tolman–Oppenheimer–Volkoff (TOV) equations. These equations are numerically solved by a perturbative approach considering realistic equations of state in Section III. In this context, new stable structures, not existing in General Relativity, clearly emerge. Discussion of the results and conclusions are reported in Sec. IV.
Modified TOV equations in $f(R)$ gravity
========================================
Let us start from the action for $f(R)$ gravity. Here the Hilbert-Einstein action, linear in the Ricci curvature scalar $R$, is replaced by a generic function $f(R)$: $$\label{action}
S=\frac{c^4}{16\pi G}\int d^4x \sqrt{-g}f(R) + S_{{\rm matter}}\quad ,$$ where $g$ is determinant of the metric $g_{\mu\nu}$ and $S_{\rm matter}$ is the action of the standard perfect fluid matter. The field equations for metric $g_{\mu\nu}$ can be obtained by varying with respect to $g_{\mu\nu}$. It is convenient to write function $f(R)$ as $$\label{fR}
f(R)=R+\alpha h(R),$$ where $h(R)$ is, for now, an arbitrary function. In this notation, the field equations are $$\label{field}
(1+\alpha h_{R})G_{\mu \nu }-\frac{1}{2}\alpha(h-h_{R}R)g_{\mu \nu
}-\alpha (\nabla _{\mu }\nabla _{\nu }-g_{\mu \nu }\Box )h_{R}=\frac{8\pi
G}{c^4} T_{\mu \nu }\,.$$ Here $G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}$ is the Einstein tensor and ${\displaystyle h_R=\frac{dh}{dR}}$ is the derivative of $h(R)$ with respect to the scalar curvature. We are searching for the solutions of these equations assuming a spherically symmetric metric with two independent functions of radial coordinate, that is: $$\label{metric}
ds^2= -e^{2\phi}c^2 dt^2 +e^{2\lambda}dr^2 +r^2 (d\theta^2
+\sin^2\theta d\phi^2).$$ The energy–momentum tensor in the r.h.s. of Eq. (\[field\]) is that of a perfect fluid, i.e. $T_{\mu\nu}=\mbox{diag}(e^{2\phi}\rho c^{2}, e^{2\lambda}P, r^2P, r^{2}\sin^{2}\theta P)$, where $\rho$ is the matter density and $P$ is the pressure. The components of the field equations can be written as $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\frac{ -8\pi G}{c^2} \rho &=& -r^{-2} +e^{-2\lambda}(1-2r\lambda')r^{-2}
+\alpha h_R(-r^{-2} +e^{-2\lambda}(1-2r\lambda')r^{-2}) \nonumber \\
&& -\frac12\alpha(h-h_{R}R) +e^{-2\lambda}\alpha[h_R'r^{-1}(2-r\lambda')+h_R''] \label{f-tt},\\
\frac{8\pi G}{c^4} P &=& -r^{-2} +e^{-2\lambda}(1+2r\phi')r^{-2}
+\alpha h_R(-r^{-2} +e^{-2\lambda}(1+2r\phi')r^{-2}) \nonumber \\
&& -\frac12\alpha(h-h_{R}R) +e^{-2\lambda}\alpha h_R'r^{-1}(2+r\phi'), \label{f-rr}\end{aligned}$$ where prime denotes derivative with respect to radial distance, $r$. For the exterior solution, we assume a Schwarzschild solution. For this reason, it is convenient to define the change of variable [@Stephani; @Cooney] $$\label{mass}
e^{-2\lambda}=1-\frac{2G M}{c^2 r}\,.$$ The value of parameter $M$ on the surface of a neutron stars can be considered as a gravitational star mass. The following relation $$\label{dMa/dr}
\frac{G dM}{c^{2}dr}=\frac{1}{2}\left[1-e^{-2\lambda}(1-2r\lambda']\right)\,,$$ is useful for the derivative $dM/dr$.
The hydrostatic condition equilibrium can be obtained from the Bianchi identities which give conservation equation of the energy-momentum tensor, $\nabla^\mu T_{\mu\nu}=0$, that, for a perfect fluid, is $$\label{hydro}
\frac{dP}{dr}=-(\rho
+P/c^2)\frac{d\phi}{dr}\,,.$$ The second TOV equation can be obtained by substitution of the derivative $d\phi/dr$ from (\[hydro\]) in Eq.(\[f-rr\]). Then we use the dimensionless variables defined according to the substitutions $$M=m M_{\odot},\quad r\rightarrow r_{g}r, \quad \rho\rightarrow\rho M_{\odot}/r_{g}^{3},\quad P\rightarrow p M_{\odot}c^{2}/r_{g}^{3}, \quad R\rightarrow {R}/r_{g}^{2}.$$ Here $M_{\odot}$ is the Sun mass and $r_{g}=GM_{\odot}/c^{2}=1.47473$ km. In terms of these variables, Eqs. (\[f-tt\]), (\[f-rr\]) can be rewritten, after some manipulations, as $$\label{TOV-1}
\left(1+\alpha r_{g}^{2} h_{{R}}+\frac{1}{2}\alpha r_{g}^{2} h'_{{R}} r\right)\frac{dm}{dr}=4\pi{\rho}r^{2}-\frac{1}{4}\alpha r^2 r_{g}^{2}\left(h-h_{{R}}{R}-2\left(1-\frac{2m}{r}\right)\left(\frac{2h'_{{R}}}{r}+h''_{{R}}\right)\right),$$ $$\label{TOV-2}
8\pi p=-2\left(1+\alpha r_{g}^{2}h_{{R}}\right)\frac{m}{r^{3}}-\left(1-\frac{2m}{r}\right)\left(\frac{2}{r}(1+\alpha r_{g}^{2} h_{{R}})+\alpha r_{g}^{2} h'_{{R}}\right)({\rho}+p)^{-1}\frac{dp}{dr}-$$ $$-\frac{1}{2}\alpha r_{g}^{2}\left(h-h_{{R}}{R}-4\left(1-\frac{2m}{r}\right)\frac{h'_{{R}}}{r}\right),$$ where $'=d/dr$. For $\alpha=0$, Eqs. (\[TOV-1\]), (\[TOV-2\]) reduce to $$\frac{dm}{dr}=4\pi\tilde{\rho} r^{2}$$ $$\frac{dp}{dr}=-\frac{4\pi p r^{3}+m}{r(r-2m)}\left(\tilde{\rho}+p\right),$$ i.e. to ordinary dimensionless TOV equations. These equations can be solved numerically for a given EoS $p=f({\rho})$ and initial conditions $m(0)=0$ and ${\rho}(0)={\rho}_{c}$.
For non-zero $\alpha$, one needs the third equation for the Ricci curvature scalar. The trace of field Eqs. (\[field\]) gives the relation $$3\alpha\square h_{R}+\alpha h_{R}R-2\alpha h-R=-\frac{8\pi G}{c^{4}}(-3P+\rho c^{2}).$$ In dimensionless variables, we have $$\label{TOV-3}
3\alpha r_{g}^{2}\left(\left(\frac{2}{r}-\frac{3m}{r^{2}}-\frac{dm}{rdr}-\left(1-\frac{2m}{r}\right)\frac{dp}{(\rho+p)dr}\right)\frac{d}{dr}+
\left(1-\frac{2m}{r}\right)\frac{d^{2}}{dr^{2}}\right)h_{{R}}+\alpha r_{g}^{2} h_{{R}}{R}-2\alpha r_{g}^{2} h-{R}=-8\pi({\rho}-3p)\,.$$ One has to note that the combination $\alpha r_{g}^{2} h(R)$ is a dimensionless function. We need to add the EoS for matter inside star to the Eqs. (\[TOV-1\]), (\[TOV-2\]), (\[TOV-3\]). For the sake of simplicity, one can use the polytropic EoS $p\sim \rho^{\gamma}$ although a more realistic EoS has to take into account different physical states for different regions of the star and it is more complicated. With these considerations in mind, let us face the problem to construct neutron star models in the context of $f(R)$ gravity.
Neutron star models in $f(R)$ gravity
=====================================
The solution of Eqs. (\[TOV-1\])-(\[TOV-3\]) can be achieved by using a perturbative approach (see [@Arapoglu; @Alavirad] for details). For a perturbative solution the density, pressure, mass and curvature can be expanded as $$p=p^{(0)}+\alpha p^{(1)}+...,\quad \rho=\rho^{(0)}+\alpha \rho^{(1)}+...,$$ $$m=m^{(0)}+\alpha m^{(1)}+...,\quad R=R^{(0)}+\alpha R^{(1)}+...,$$ where functions $\rho^{(0)}$, $p^{(0)}$, $m^{(0)}$ and $R^{(0)}$ satisfy to standard TOV equations assumed at zeroth order. Terms containing $h_{R}$ are assumed to be of first order in the small parameter $\alpha$, so all such terms should be evaluated at ${\mathcal O}(\alpha)$ order. We have, for the $m=m^{(0)}+\alpha m^{(1)}$, the following equation $$\frac{dm}{dr}=4\pi\rho r^2-\alpha r^{2}\left(4\pi \rho^{(0)}h_{R}+\frac{1}{4}\left(h-h_{R}R\right)\right)+\frac{1}{2}\alpha\left(\left(2r-3m^{(0)}-4\pi\rho^{(0)}r^{3}\right)\frac{d}{dr}+r(r-2m^{(0)})\frac{d^{2}}{dr^{2}}\right)
h_{R}$$ and for pressure $p=p^{(0)}+\alpha p^{(1)}$ $$\frac{r-2m}{\rho+p}\frac{dp}{dr}=4\pi r^2 p+\frac{m}{r}-\alpha r^2\left(4\pi p^{(0)}h_{R}+\frac{1}{4}\left(h-h_{R}R\right)\right)-
\alpha \left(r-3m^{(0)}+2\pi p^{(0)}r^{3}\right)\frac{dh_{R}}{dr}.$$ The Ricci curvature scalar, in terms containing $h_{R}$ and $h$, has to be evaluated at ${\mathcal O}(1)$ order, i.e. $$R \thickapprox R^{(0)}=8\pi(\rho^{(0)}-3p^{(0)})\,.$$ In this perturbative approach, we do not consider the curvature scalar as an additional degree of freedom since its value is fixed by this relation.
[We can consider various EoS for the description of the behavior of nuclear matter at high densities.]{} [It is convenient to use analytical representations of these EoS. For example the SLy [@SLy] and FPS [@FPS] equation have the same analytical representation:]{} $$\label{FPS}
\zeta=\frac{a_{1}+a_{2}\xi+a_{3}\xi^3}{1+a_{4}\xi}f(a_{5}(\xi-a_{6}))+(a_{7}+a_{8}\xi)f(a_{9}(a_{10}-\xi))+$$ $$+(a_{11}+a_{12}\xi)f(a_{13}(a_{14}-\xi))+(a_{15}+a_{16}\xi)f(a_{17}(a_{18}-\xi)),$$ where $$\zeta=\log(P/\mbox{dyn} \mbox{cm}^{-2})\,, \qquad \xi=\log(\rho/\mbox{g}\mbox{cm}^{-3})\,, \qquad f(x)=\frac{1}{\exp(x)+1}\,.$$ The coefficients $a_{i}$ for SLy and FPS EoS are given in [@Camenzind]. [In [@Potekhin] the parametrization for three EoS models (BSK19, BSK20, BSK21) is offered. Also analytical parametrization for wide range of various EoS (23 equations) can be found in [@Eksi].]{}
Furthermore, we can consider the model of neutron star with a quark core. The quark matter can be described by the very simple EoS: $$\label{EoSQM}
p_{Q}=a(\rho-4B),$$ where $a$ is a constant and the parameter $B$ can vary from $\sim 60$ to $90$ Mev/fm$^{3}$. For quark matter with massless strange quark, it is $a=1/3$. We consider $a=0.28$ corresponding to $m_{s}=250$ Mev. For numerical calculations, Eq. (\[EoSQM\]) is used for $\rho \geq \rho_{tr}$, where $\rho_{tr}$ is the transition density [for which the pressure of quark matter coincides with the pressure of ordinary dense matter. ]{} For example for FPS equation, the transition density is $\rho_{tr}=1.069\times 10^{15}$ g/cm$^{3}$ ($B=80$ Mev/fm$^{3}$), for SLy equation $\rho_{tr}=1.029\times 10^{15}$ g/cm$^{3}$ ($B=60$ Mev/fm$^{3}$). These parameters allow to set up neutron star models according to given $f(R)$ gravity models.
**Model 1**. Let’s consider the simple exponential model $$\label{EXP}
f(R)=R+\beta R(\exp(-R/R_{0})-1),$$ where $R_{0}$ is a constant. Similar models are considered in cosmology, see for example [@EXP]. We can assume, for example, $R=0.5 r_{g}^{-2}$. For $R<<R_{0}$ this model coincides with quadratic model of $f(R)$ gravity. The neutron stars models in frames of quadratic gravity is investigated in detail in [@Arapoglu]. It is interesting to consider the model (\[EXP\]) for the investigations of higher order effects.
For neutron stars models with quark core, one can see that there is no significant differences with respect to General Relativity. For a given central density, the star mass grows with $\beta $. The dependence is close to linear for $\rho\sim 10^{15} \mbox{g/cm}^{3}$. For the piecewise equation of state (we consider the FPS case for $\rho<\rho_{tr}$) the maximal mass grows with increasing $\beta$. For $\beta=-0.25$, the maximal mass is $1.53M_{\odot}$, for $\beta=0.25$, $M_{max}=1.59M_{\odot}$ (in General Relativity, it is $M_{max}=1.55M_{\odot}$). With an increasing $\beta$, the maximal mass is reached at lower central densities. Furthermore, for $dM/d\rho_{c}<0$, there are no stable star configurations. A similar situation is observed in the SLy case but mass grows with $\beta$ more slowly. It is interesting to stress that the $\beta$ parameter affects also the Jeans instability of any process that from self-gravitating systems leads to stellar formation as reported in [@Laurentis].
For the simplified EoS (\[FPS\]), other interesting effects can occur. For $\beta\sim -0.15$ at high central densities ($\rho_{c}\sim 3.0 - 3.5\times 10^{15}\mbox{g/cm}^{3}$), we have the dependence of the neutron star mass from radius (Figs. 1, 3) and from central density central density (see Figs. 2, 4). For $\beta<0$ for high central densities we have the stable star configurations ($dM/d\rho_{c}>0$).
[Of course the model with FPS EOS is ruled out by recent observations [@antoniadis; @Demorest]. For example the measurement of mass of the neutron star PSR J1614-2230 with $1.97\pm0.04$ $M_{\odot}$ provides a stringent constraint on any $M-R$ relation. The model with SLy equation is more interesting: in the context of model (\[EXP\]), the upper limit of neutron star mass is around $2M_{\odot}$ and there is second branch of stability star configurations at high central densities. This branch describes observational data better than the model with SLy EoS in GR (Fig. 3).]{}
Although the applicability of perturbative approach at high densities is doubtful, it indicates the possibility of a stabilization mechanism in $f(R)$ gravity. This mechanism, as can be seen from a rapid inspection of Figs. 1 - 4, leads to the existence of stable neutron stars [(the stability means that $dM/d\rho_{c}$)]{} which are more compact objects than in General Relativity. In principle, the observation of such objects could be an experimental probe for $f(R)$ gravity.
**Model 2**. Let us consider now the model of quadratic gravity with logarithmic corrections in curvature [@OdintsovLOG]: $$\label{LOG}
f(R)=R+\alpha R^2(1+\beta \ln (R/\mu^2)),$$ where $|\alpha|<1$ (in units $r_{g}^{2}$) and the dimensionless parameter $|\beta|<1$. This model is considered in [@Alavirad] for SLy equation. However it is not valid beyond the point $R=0$ and we cannot apply our analysis for stars with central density for example $\rho_{c}>1.72\times 10^{15}$ g/cm$^{3}$ (for SLy equation) and at $\rho_{c}>2.35\times 10^{15}$ g/cm$^{3}$ (for FPS equation). The similar situation take place for another. The maximal mass of neutron star at various values $\alpha$ and $\beta$ is close to the corresponding one in General Relativity at these critical densities (for FPS - $1.75M_{\odot}$, for SLy - $1.93M_{\odot}$). On the other hand, for model with quark core, the condition $R=8\pi(\rho^{(0)}-3p^{(0)})>0$ is satisfied at arbitrary densities. The analysis shows that maximal mass is decreasing with growing $\alpha$. By using a piecewise EoS (FPS+quark core) one can obtain stars with radii $\sim 9.5 $ km and masses $\sim 1.50M_{\odot}$. In contrast with General Relativity, the minimal radius of neutron star for this equation is $9.9$ km. Using the piecewise (softer) EoS decreases the upper limit of neutron star mass in comparison with using only one EoS for matter within star. For various EoS, this limit is smaller than $\sim 2M_{\odot}$. Therefore the model with logarithmic corrections does not lead to new effects in comparison with GR.
**Model 3**. It is interesting to investigate also the $R^2$ model with a cubic correction: $$\label{CUB}
f(R)=R+\alpha R^{2}(1+\gamma R)\,.$$ The case where $|\gamma R| \sim {\mathcal O}(1)$ for large $R$ is more interesting. In this case the cubic term comparable with quadratic term. Of course we consider the case when $ \alpha R^{2}(1+\gamma R)<<R$. In this case the perturbative approach is valid although the cubic term can exceed the value of quadratic term. For small masses, the results coincides with $R^2$ model. For narrow region of high densities, we have the following situation: the mass of neutron star is close to the analogue mass in General Relativity with $dM/d\rho_{c}>0$. This means that this configuration is stable. For $\gamma=-10$ (in units $r_{g}^{2}$) the maximal mass of neutron star at high densities $\rho>3.7\times 10^{15}$ g/cm$^{3}$ is nearly $1.88M_{\odot}$ and radius is about $\sim 9$ km (SLy equation). For $\gamma=-20$ the maximal mass is $1.94M_{\odot}$ and radius is about $\sim 9.2$ km (see Figs. 5 - 8). In the General Relativity, for SLy equation, the minimal radius of neutron stars is nearly 10 km. Therefore such a model of $f(R)$ gravity can give rise to neutron stars with smaller radii than in General Relativity. Therefore such theory can describe (assuming only the SLy equation), the existence of peculiar neutron stars with mass $\sim 2M_{\odot}$ (the measured mass of PSR J1614-2230 [@Ozel]) and compact stars ($R\sim 9$ km) with masses $M\sim 1.6-1.7M_{\odot}$ (see [@Ozel-2; @Guver; @Guver-2]).
We also investigate the cases of BSK20 (Figs. 9 - 12) and AP4 EoS (Figs. 13 - 16). The results are similar to the case of SLy EoS: in the context of gravity with cubic corrections, the existence of more compact neutron stars (in comparison with GR) at high central densities is possible. At the same time, the maximal neutron star mass satisfies the observational constraints. The parameters of these stable configurations and maximal neutron star mass in model (\[CUB\]) for these EoS are given in Table I.
Some considerations are needed about the validity of the perturbative approach. For example, for $\gamma=-20$, $\alpha=10\times 10^{9}$ cm$^{2}$, the relation $\Delta=|\alpha R^{2}(1+\gamma R)/R|$ reaches the maximal value $~0.15$ only at center of star (for the three above mentioned EoS) and only for maximal densities $\rho_{c}$ at which the stable configurations can exist. For more conservative $\gamma=-10$, $\alpha=5\times 10^{9}$ cm$^{2}$, it is $\Delta_{max}\thickapprox 0.1$. This allows to apply the perturbative approach with good precision.
[For smaller values of $\gamma$ the minimal neutron star mass (and minimal central density at which stable stars exist) on second branch of stability decreases.]{}
EoS $M/M_{\odot}$ $R$, km $\rho_{c15}$ $M_{max}/M_{\odot}$ $\rho_{cmax15}$
------- --------------- ------------- ------------------------ --------------------- ----------------- --
SLy $1.75<M<1.88$ $9.1<R<9.5$ $3.34<\rho_{c15}<3.91$ 2.05 2.86
BSK20 $1.83<M<2.0$ $9.3<R<9.8$ $2.97<\rho_{c15}<3.48$ 2.16 2.64
AP4 $1.95<M<2.07$ $9.4<R<9.9$ $2.64<\rho_{c15}<3.09$ 2.20 2.75
: The parameters of stability star configurations at high central densities (second branch of stability) for $\gamma=-10$, $\alpha=5\times 10^{9}$ cm$^{2}$ ($\rho_{15}\equiv \rho/10^{15}$g/cm$^{3}$). The maximal mass of neutron star in model (\[CUB\]) for various EoS is given also. The maximal central density for stability star configurations in General Relativity for corresponding EoS is given in last column for comparison.
In conclusion we consider the case of cubic modified gravity ($f(R)\approx R+\epsilon R^{3}$). It is interesting to note that for negative and sufficiently large values of $\epsilon$, the maximal limit of neutron star mass can exceed the limit in General Relativity for given EoS (the stable stars exist for higher central densities). Therefore some EoS which ruled out by observational constraints in General Relativity can describes real star configurations in frames of such model of gravity. For example, for BSK19 EoS, the maximal neutron star mass is around $1.86 M_{\odot}$ in General Relativity, but for negative values $|\epsilon|\sim 8$ the maximal mass is around $\sim2M_{\odot}$ (see Fig. 17). The same result holds in the case of FPS, WFF3 and AP2 EoS (at $|\epsilon|\sim 10-12$). For these values of $\epsilon$, these models describe the observational data by [@Ozel] with good precision and provide the acceptable upper limit of neutron star mass. One has to note that the upper limit in this model of gravity is achieved for smaller radii than in General Relativity for acceptable EoS. Therefore the possible measurement of neutron star radii with $M\sim 2M_{\odot}$ can give evidences supporting (or not) this model of gravity.
Perspectives and Conclusions
============================
In this paper, we have studied neutron star structures in some $f(R)$ gravity models assuming realistic equations of state.
In particular, we have considered the mass-radius relations for neutron stars in a gravity models of the form $f(R)=R+R(e^{-R/R_{0}}-1)$ and for the $R^2$ models with logarithmic and cubic corrections. We also investigated the dependence of the maximal mass from the central density of the structure. In the case of quadratic gravity with cubic corrections, we found that, for high central densities ($\rho>10\rho_{ns}$, where $\rho_{ns}=2.7\times 10^{14}$ g/cm$^{3}$ is the nuclear saturation density) stable star configurations exist. In other words, we have a second “branch” of stability with respect to the one existing in General Relativity. The minimal radius of such stars is close to $9$ km with maximal mass $\sim 1.9 M_{\odot}$ (SLy equation), $\sim 2.0 M_{\odot}$ (BSK20 equation), $\sim 2.07 M_{\odot}$ (AP4 equation). This effect gives rise to more compact stars than in General Relativity and could be extremely relevant from an observational point of view. In fact, it is interesting to note that using an equation of state in the framework of $f(R)$ gravity with cubic term gives rise to two important features: the existence of an upper limit on neutron star mass ($\sim2M_{\odot}$) and the existence of neutron stars with radii $R\sim 9 \div 9.5$ km and masses $\sim 1.7M_{\odot}$. These facts could have a twofold interest: from one side, the approach could be useful to explain peculiar objects that evade explanation in the framework of standard General Relativity (e.g. the magnetars [@magnetar]) and, from the other side, it could constitute a very relevant test for alternative gravities. Another interesting result can be realized in cubic modified gravity. Some EoS, ruled out in General Relativity according to observational data, satisfy the observational constraints in this model and give a realistic description of $M$-$R$ relation and acceptable upper limit of neutron star mass.
A.V.A. would like to thank A.V. Yaparova for useful discussion [and Prof. K.Y. Ekşi for providing the observational constraints data]{}. S.C. and S.D.O. acknowledge the support of the visitor program of the Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (Nagoya, Japan). S.C. acknowledges the support of INFN (iniziativa specifica TEONGRAV). S.D.O. acknowledges the support of visitor program of Baltic Federal University and useful discussions with Prof. A. Yurov. This work has been supported in part by MINECO (Spain), FIS2010-15640, AGAUR (Generalitat de Catalunya), contract 2009SGR-345, and MES project 2.1839.2011 (Russia) (S.D.O.).
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![The mass-radius diagram for neutron stars in $f(R)$ model (\[EXP\]) in comparison with General Relativity by using a FPS equation of state. [The constraints derived from observations of three neutron stars from [@Ozel] is depicted by the dotted contour (hereinafter). For negative values of $\beta$ for high central densities we have the possibility of existence of stable star configurations ($dM/d\rho_{c}>0$). If $\beta=-0.3$ the masses of these configurations are $0.85M_{\odot}<M<1.48M_{\odot}$ ($8.1<R<8.8$ km)]{}](1FPS.eps "fig:")\
.
![The dependence of neutron star mass from central density in $f(R)$ model (\[EXP\]) in comparison with General Relativity for a FPS equation of state. In narrow interval of central densities $3.76\times 10^{15}<\rho_{c}<5.25\times 10^{15}$ g/cm$^{-3}$ the stability star configurations ($dM/d\rho_{c}>0$) exist. For comparison the upper limit of central density in GR is $3.48\times10^{15}$ g/cm$^{3}$ for FPS EoS.](1FPS1.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model (\[EXP\]) in comparison with General Relativity for a SLy equation of state. [For $\beta=-0.3$ in this model there are stable stars with masses $1.25_{\odot}<M<1.72M_{\odot}$ and radii $8.8<R<9.7$ km. Therefore the description of observational constraints is more better than in GR for SLy EoS.]{}](1SLy.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[EXP\]) in comparison with General Relativity for Sly equation of state. [The stable stars don’t exist in GR for $\rho_{c}>2.86\times 10^{15}$ g/cm$^{3}$ while for model (\[EXP\]) with $\beta=-0.3$ such possibility can take place at $2.97\times10^{15}<\rho_{c}<4.23\times10^{15}$ g/cm$^{3}$.]{}](1SLy1.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model with cubic corrections (\[CUB\])($\gamma=-10$) in comparison with General Relativity assuming a SLy EoS. The notation $\alpha_{9}$ means $\alpha_{9}=\alpha/10^{9}$ cm$^{2}$.](3SLy1.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[CUB\]) ($\gamma=-10$) for SLy EoS.](3SLy11.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model with cubic corrections (\[CUB\])($\gamma=-20$) for SLy EoS.](3SLy2.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[CUB\]) ($\gamma=-20$) for SLy EoS.](3SLy21.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model with cubic corrections (\[CUB\])($\gamma=-10$) for BSK20 EoS.](3BSK1.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[CUB\]) ($\gamma=-10$) for BSK20 EoS.](3BSK11.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model with cubic corrections (\[CUB\])($\gamma=-20$) for BSK20 EoS.](3BSK2.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[CUB\]) ($\gamma=-20$) for BSK20 EoS.](3BSK21.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model with cubic corrections (\[CUB\])($\gamma=-10$) for AP4 EoS.](3AP41.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[CUB\]) ($\gamma=-10$) for AP4 EoS.](3AP411.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)$ model with cubic corrections (\[CUB\])($\gamma=-20$) for AP4 EoS.](3AP42.eps "fig:")\
![The dependence of neutron star mass from central density in $f(R)$ model (\[CUB\]) ($\gamma=-20$) for AP4 EoS.](3AP421.eps "fig:")\
![The mass-radius diagram for neutron stars in $f(R)\approx R+\epsilon R^{3}$ model for AP2 and BSK19 EoS (thick lines) in comparison with General Relativity (bold lines). The parameter $\epsilon=-10$ (in units of $r_{g}^{4}$. In frames of model of modified gravity these EoS give the upper limit of neutron star mass $\sim 2M_{\odot}$ (the corresponding radius is $R~8.6-8.8$ km and describe observational data from [@Ozel] with acceptable precision. With increasing $|\epsilon|$ the upper limit of mass increases. The similar effect one can seen for WFF3 and FPS EoS at slightly larger $|\epsilon|$.](3AP21.eps "fig:")\
|
---
abstract: 'This document presents a studies of the stochastic behavior of D-Wave qubits, qubit cells, and qubit chains. The purpose of this paper is to address the algorithmic behavior of execution rather than the physical behavior, though they are related. The measurements from an actual D-Wave adiabatic quantum computer are compare with calculated measurements from a theoretical adiabatic quantum computer running with an effective temperature of zero. In this way the paper attempts to shed light on how the D-Wave’s behavior effects how well it minimizes its objective function and why the D-Wave performs as it does.'
author:
- |
John E. Dorband\
Department of Computer Science and Electrical Engineering\
University of Maryland, Baltimore County\
Maryland, USA\
`dorband@umbc.edu`
bibliography:
- 'QubitChains.bib'
title: 'Stochastic Characteristics of Qubits and Qubit chains on the D-Wave 2X'
---
Introduction {#sec:intro}
============
The D-Wave[@Dwave13] is an adiabatic quantum computer[@Farhi00; @Giuseppe08]. The problem class that it addresses is based on the objective function: $$\label{eq:obfunc}
min\left({\sum\limits_i a_i q_i + \sum\limits_i \sum\limits_j b_{ij} q_i q_j}\right)$$ where $q_i$ are the qubit values returned by the D-Wave, $a_i$ and $b_{ij}$ are the coefficients given to the D-Wave associated with the qubits and the qubit couplers respectively. The D-Wave returns the set of qubit values which minimize the above objective function. Theoretically there is only one global minimum value, even if there are multiple global minima. This minimum value corresponds to the ground state of the D-Wave with the given coefficients. The D-Wave often returns a non-minimum energy state due to inherent noise in the system, and the closeness of a large number of slightly higher energy active states near the ground state. This leads to the question: if the D-Wave does not always return qubit values corresponding to the ground state what are the properties of the D-wave that can be depended upon to perform useful computations. The purpose of this paper is to present the stochastic characteristics of the D-Wave that may be useful in understanding quantum adiabatic algorithmic computations.
The D-Wave architecture is base on a non-complete graph for coupling, the chimera graph (see figure \[fig:chimera\]). Thus it is necessary to organize physical qubits into virtual qubits that have greater connectivity than the chimera graph provides for physical qubits. This concept is used throughout algorithmic development for the D-Wave. Virtual qubits arranged as physical qubit pairs are presented in a paper about solving the Map Coloring problem with the D-Wave[@Dahl15]. Also virtual qubits organized as physical qubit chains were used to implement on the D-Wave a large complete bipartite graph with higher degree of connectivity than is currently available in the D-Wave[@Adachi15]. We will present the results of our study of the stochastic properties of virtual qubits, as qubit chains and qubit cells in this paper.
Measuring Stochastic Properties {#sec:metrics}
===============================
In this section the stochastic property of a qubit entity whether it is a qubit, a qubit cell or a qubit chain is measured as the probabalility that an individual qubit of the entity 1) is on ($P(q)$), 2) has an up spin ($P(\uparrow)$), or 3) has a value of one ($P(q=1)$).
Properties of a Qubit {#sec:qubitmetric}
---------------------
To measure the stochastic property of the D-Wave qubit, the D-Wave was run 10000 times with all the coupler coefficients, $b_{ij}$, set to zero ($C_{c}=0.0$) and all the qubit coefficients, $a_{i}$, set to a specific value, $C_q$, in the range \[-1,1\]. This was done for 129 values over the \[-1,1\] range. Figure \[fig:qubitprob\] shows the qubit properties of System6 (BS6) and System13 (BS13) at Burnaby, BC, and C12 (AC12) at NASA Ames Research Center, CA.
The probabilities displayed in figure \[fig:qubitprob\] can be modeled as a function of effective temperature[@Adachi15; @Benedetti15]: $$\label{eq:effectivetemp}
P(q=1)=\frac{e^{-f(C_q,T_e)}}{e^{f(C_q,T_e)}+e^{-f(C_q,T_e)}}$$ where $T_e$ is the effective temperature and $f(C_q,T_e)=\frac{C_q}{T_e}$. The theoretical behavior of a perfect quantum qubit that would minimize the objective function (Eq. \[eq:obfunc\]) can be represented by $T_e=0$ or a step function at $C_q=0.0$ as in figure \[fig:qubitprob\]. None of the D-Wave qubit characteristics are as good as the theoretical qubit, since no quantum system can be perfectly isolated from it’s environment. Note however that the properties of the BS13 and the AC12 are much better than the older SB6. This is probably due to improvements in the newer D-Waves in isolating it from its quantum environment (eg. lower running temperature). The equation \[eq:effectivetemp\] can be reduced to: $$\label{eq:sigmoid}
P(q=1)=\frac{1}{1+e^{kC_q}}$$ where $k={2}/{T_e}$. Figure \[fig:sigBS6AC12\] is the comparison of the theoretical model with the measure data where $k=7$ is used for SB6 and $k=24$ is used for AC12.
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The effects of the environment can also be seen in figure \[fig:qubitSTD\]. Figure \[fig:qubitTstd\] is the standard deviation of the probabilities over time. Time here is represented by 10000 D-Wave samples. The samples were divided into 10 partitions and the standard deviation was computed across the partitions using the average probabilities over all samples and qubits for each partition. Figure \[fig:qubitQstd\] is the standard deviation of the probabilities over space (or qubits). The average probability was computed for each qubit over all time partitions and the standard deviation was computed across the qubits of the D-Wave. Note that the peak temporal standard deviation ($1.13x10^{-4}$) is much lower than the peak spatial standard deviation ($1.45x10^{-3}$). This would indicate the properties of a qubit vary much less with time than amongst the qubits. Note also that the temporal STD of SB13 is higher than AC12 or SB6 and has a broader and noisier temporal STD than AC12. It can also be seen the the spatial STD of SB6 is broader and noisier than SB13 or AC12 while SB13’s peak spatial STD is higher than either SB6 or AC12. Since SB6 is an older version than SB13 or AC12 and SB13 was an early test prototype of AC12 this is all quite understandable. The important point is that the quantum environmental isolation of the D-Wave is improving.
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Properties of a Qubit Chain {#sec:chainmetric}
---------------------------
Since the D-Wave architecture is based on a Chimera graph rather than a completely connected graph (clique), it is necessary to form virtual qubits out of sets of physical qubit. The idea is to configure the qubits and couplers to act as a single qubit that can be coupled to more qubits which are more widely dispersed across the D-Wave. The premise behind the virtual qubit (qubit chain) is that the physical qubits making up the virtual qubit should all return the same value. That is if one qubit of the group returns a value of one all the qubits of the group return values of one or if one qubit returns a value of zero all the qubits of the group return values of zero. This however is seldom the case, even if the value of the virtual qubit should be one (all the qubits of the chain should return one) some of the qubits may return a value of zero.
We have performed experiments with groups of qubits formed from chains of qubits. Thirty chains were selected from the working qubits of AC12 for performing the measurement of stochastic behavior of virtual qubits. Each virtual qubit consisted of a chain of 12 physical qubits. For the measurements every coefficient of the qubits of the qubit chain are set to $C_q$ and every coupler coefficient of the qubit chain are set to $C_c$. All other coefficients are set to zero. No two qubit chains have any couplers in common. The probability of a virtual qubit to be one is calculated by counting all qubits that have a value of one divided by the number of qubits in the chain. For a given $C_q$ and $C_c$ the D-Wave is run 1000 times and the virtual qubit probability is averaged over all the runs and all virtual qubits (30) per run.
Figures \[fig:CHisingCQ\]-\[fig:CHquboQC\] are plots of families of characteristic curves that show the stochastic properties of 12 qubit virtual qubits. Ensembles were run with a fixed value of $C_q$ between 2 and -2 and a fixed value of $C_c$ between 1 and -1. Each ensemble was an average over 1000 D-Wave runs of 30 virtual qubits (qubit chains). The ensembles were run using the Ising model (where qubits can have a value of -1 or 1) and the QUBO model (where qubits can have a value of 0 or 1). The D-Wave is based on the Ising model but can be coerced into running as a QUBO model through algebraic manipulation of the qubit and coupler coefficients.
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Figure \[fig:CHisingCQ\] is based on the Ising model. Each line in the plot is for a specific value of $C_c$. There are 17 plots for different values of $C_c$ in the range \[-1,1\]. Figure \[fig:CHquboCQ\] is similar to \[fig:CHisingCQ\] using the QUBO model instead of Ising.
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Figure \[fig:CHisingQC\] is based on the Ising model. Each line in the plot is a for a specific value of $C_q$. There are 17 plots with different values of $C_q$ in the range \[-2,2\]. Figures \[fig:CHquboQC\] is similar to \[fig:CHisingQC\] using the QUBO model instead of Ising.
Figures \[fig:CHisingDcq\], \[fig:CHquboDcq\], \[fig:CHisingDqc\], and \[fig:CHquboDqc\] are plotted from measurement taken from AC12. While figures \[fig:CHisingTcq\], \[fig:CHquboTcq\], \[fig:CHisingTqc\], and \[fig:CHquboTqc\] are plots of what a theoretically perfect machine would produce if given the same coefficients that were given the AC12. This theoretically perfect machine($T_e=0$) is based on a machine which would always return a set of values of qubits which minimize the objective function, equation \[eq:obfunc\], thus, it represents the stochastic behavior of an errorless/noiseless D-Wave. These theoretic predictions were calculated by computing the value of equation \[eq:obfunc\] for all possible cases of 12 qubits (4096) for each pair of values of $C_q$ and $C_c$. The resultant theoretical probability($P(q=1)$) for each value pair is the average probability over all global minimum states. Fundamentally the D-Wave plots are very similar to the theoretical result, but differ in the sharpness of the curves as was the case in figure \[fig:qubitprob\]. This will effect how probable it is that the resultant value of the objective function will be the global minimum or how close that value is to the global minimum. Note that qubit chains only behave properly as describe previously while using the QUBO model and $C_q$ is within the range \[0,1\] and $C_c$ is within the range \[-1,0\].
Properties of a Qubit Cell {#sec:cellmetric}
--------------------------
The D-Wave 2X consists of 12x12 or 144 8 qubit cells. Each cell consists of a 4x4 qubit bipartite graph. The AC12 has 108 such complete cells, having 8 working qubits. Though the qubit cell is not as significant as the qubit chain in algorithm design for the D-Wave, it still presents interesting conformation of the properties of the D-Wave. Figures \[fig:CEisingDcq\], \[fig:CEquboDcq\], \[fig:CEisingDqc\], and \[fig:CEquboDqc\] corespond to figures \[fig:CHisingDcq\], \[fig:CHquboDcq\], \[fig:CHisingDqc\], and \[fig:CHquboDqc\], using cells of qubits rather than chains. Figures \[fig:CEisingTcq\], \[fig:CEquboTcq\], \[fig:CEisingTqc\], and \[fig:CEquboTqc\] are the theoretical versions of cell behavior coresponding to figures \[fig:CHisingTcq\], \[fig:CHquboTcq\], \[fig:CHisingTqc\], and \[fig:CHquboTqc\] for chains.
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Conclusion {#sec:conclusion}
==========
The purpose of this paper was to show the properties of the D-Wave that facilitate application development. In the process it was necessary to show what are the stable properties as well as the properties for which there is need for improvement. When starting this paper it appeared that the D-Wave was not performing as it was portrayed or at least as well as expected, that is inexplicably unpredictable and erratic. This begged for an explanation as to why this seemed to be the case. The first step was to make simple measurements of a simple configuration, how does a qubit behave. When stochastic measurements of the probabilities of a qubit, $P(q=1)$, given different coefficients (figure \[fig:qubitprob\]) was made, it was clear that the qubit was behaving very predictably. But how well was it contributing to the minimization of the objective function (equation \[eq:obfunc\])? Assuming that the behavior of the qubit needs to either match or be near the theoretically perfect behavior, figure \[fig:qubitprob\] show that AC12 is much closer to the theoretically perfect behavior than SB6. This paper has not studied if the AC12 behavior is good enough. Next, stochastic measurements of qubit chains, needed to implement virtual qubits, were performed. Figures \[fig:CHisingCQ\] thru \[fig:CHquboQC\] present that study. It was thought that chains longer than a half a dozen qubits do not give very good results. The plots of qubits chains in this paper indicate this problem. The sharpness of the theoretical results in contrast to the D-Wave measurements would indicate why these qubit chains did not perform as desired. The simple conclusion is that the D-Wave qubit properties need to be improved by improving it’s quantum environment isolation. In time this will happen as with all new technologies. In the present it is necessary for there to be work done in discovering more robust algorithms, that will adjust to the current and future frailties of this type of computational architecture. Classical ones that can utilize what is returned by the D-Wave to enhance the classical solution either in speed or accuracy, a hybrid solution integrating classical algorithms with quantum ones.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author would like to thank Michael Little and Marjorie Cole of the NASA Advanced Information Systems Technology Office for their continued support for this research effort under grant NNX15AK58G and to the NASA Ames Research Center for providing access to the D-Wave quantum annealing computer. In addition, the author thanks the NSF funded Center for Hybrid Multicore Productivity Research and D-Wave Systems for their support and access to their computational resources.
|
---
abstract: 'In this work, we review and expand recent theoretical proposals for the realization of electronic thermal diodes based on tunnel-junctions of normal metal and superconducting thin films. Starting from the basic rectifying properties of a single hybrid tunnel junction, we will show how the rectification efficiency can be largely increased by combining multiple junctions in an asymmetric chain of tunnel-coupled islands. We propose three different designs, analyzing their performance and their potential advantages. Besides being relevant from a fundamental physics point of view, this kind of devices might find important technological application as fundamental building blocks in solid-state thermal nanocircuits and in general-purpose cryogenic electronic applications requiring energy management.'
author:
- Antonio Fornieri
- 'María José Martínez-Pérez'
- Francesco Giazotto
title: Electronic heat current rectification in hybrid superconducting devices
---
Introduction
============
A thermal rectifier [@Starr; @LiPRL; @RobertsRev] can be defined as a device that connects asymmetrically two thermal reservoirs: the heat current transmitted through the diode depends on the sign of the temperature bias imposed to the reservoirs. This non-linear device represents the thermal counterpart of the well-known electric diode, which helped the extraordinary evolution of modern electronics together with the transistor. The realization of efficient thermal rectifiers would represent a giant leap for the control of heat currents at the nanoscale, [@GiazottoRev; @Dubi; @LiRev] boosting emerging fields such as coherent caloritronics,[@GiazottoNature; @MartinezNature; @MartinezRev] nanophononics,[@LiRev] and thermal logic.[@LiRev] Furthermore, a great number of nanoscience fields, including solid state cooling,[@GiazottoRev] ultrasensitive cryogenic radiation detection[@GiazottoRev; @GiazottoHeikkila] and quantum information,[@NielsenChuang; @Spilla] might strongly benefit from the possibility of releasing the dissipated power to the thermal bath in an unidirectional way.
A highly efficient thermal diode should provide differences of at least one order of magnitude between the heat current transmitted in the forward temperature-bias configuration, $J_{\rm fw}$, and that generated upon temperature bias reversal, $J_{\rm rev}$. This is equivalent to say that the rectification efficiency $$\mathcal{R}=\frac{J_{\rm fw}}{J_{\rm rev}}$$ must be $\gg 1$ or $\ll 1$.
Since the first theoretical proposal,[@LiPRL] several groups have put a great effort into envisioning different designs for thermal rectifiers dealing with phonons,[@Terraneo; @Li; @Segal; @Segal2] electrons[@SanchezPRB; @Ren; @Ruokola1; @Kuo; @Ruokola2; @Chen] and photons.[@BenAbdallah] Alongside these theoretical works, promising experimental results were obtained in systems that exploited phononic[@Chang; @Kobayashi; @Tian] and electronic[@Scheibner] thermal currents. Nevertheless, a maximum $\mathcal{R}\sim 1.4$ has been reported[@Kobayashi] and more efficient rectification mechanisms are required to realize competitive thermal diodes.
In this article, we will focus on the rectification performance of devices based on tunnel-junctions between normal metal (N) and superconducting (S) thin films at low temperatures. First, we will review the intrinsic properties of single NIN, NIS and S$_1$IS$_2$ junctions[@MartinezAPL; @GiazottoBergeret] (where I stands for an insulating barrier and S$_1$ and S$_2$ represent two different superconducting electrodes). Then, we will analyze possible improvements of the rectification efficiency when these structures are combined together forming an asymmetric chain of tunnel-coupled islands.[@FornieriAPL; @MartinezArxiv] In particular, as we shall argue, an asymmetric coupling to the phonon bath can strongly enhance the efficiency of the device, providing outstanding values of $\mathcal{R}$ with realistic parameters. Very recently, the latter mechanism has been experimentally demonstrated in a hybrid device exhibiting a maximum rectification efficiency of $\sim 140$.[@MartinezArxiv]
NIN and NIS junctions {#NISsec}
=====================
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We shall start, first of all, by describing the equations governing heat transport in simple NIN and NIS junctions. In these opening sections, we focus on the electronic heat currents only, and neglect possible contributions by lattice phonons. The latter will be discussed in the analysis of chain diodes.
If we consider two N electrodes residing at electronic temperatures $T_{\rm hot}$ and $T_{\rm cold}$ (with $T_{\rm hot}\geq T_{\rm cold}$) coupled by means of a tunnel junction, the stationary electronic thermal current flowing through the junction can be written as:[@GiazottoBergeret] $$J_\mathrm{NIN}(T_{\rm hot},T_{\rm cold})=\frac{k_{\mathrm{B}}^2 \pi^2}{6e^2 R_\mathrm{T}}(T_{\rm hot}^2-T_{\rm cold}^2),\label{JNIN}$$ where $R_\mathrm{T}$ is the contact resistance, $e$ is the electron charge and $k_\mathrm{B}$ is the Boltzmann’s constant. Equation (\[JNIN\]) clearly shows that no rectification is possible in a full normal-metal tunnel junction, since $\vert J_\mathrm{NIN}(T_{\rm hot},T_{\rm cold})\vert = \vert J_\mathrm{NIN}(T_{\rm cold},T_{\rm hot})\vert$.
However, if we substitute the second N electrode with a superconductor, the thermal current flowing through the NIS junction becomes:[@GiazottoRev; @MakiGriffin] $$\begin{aligned}
\begin{aligned}
J_\mathrm{NIS}(T_{\rm hot},T_{\rm cold})=\frac{2}{e^2 R_{\rm T}} \int_0^{\infty} & dE E \mathcal{N}(E,T_{\rm cold})\\
\times &[f(E,T_{\rm hot})-f(E,T_{\rm cold})].\label{JNIS}
\end{aligned}\end{aligned}$$ Here, $ \mathcal{N}(E , T)=\left| \Re \left[ E+i \Gamma/ \sqrt{(E+i \Gamma)^2- \Delta^2(T)} \right] \right|$ is the smeared (by non-zero $\Gamma$) normalized Bardeen-Cooper-Schrieffer (BCS) density of states (DOS) in the superconductor,[@Dynes; @PekolaPRL2004; @PekolaPRL2010] $\Delta(T)$ is the temperature-dependent superconducting energy gap with a critical temperature $T_{\rm c}=\Delta(0)/(1.764 k_{\rm B})$ and $f(E,T)=[1+\textrm{exp}(\frac{E}{k_{\rm B}T})]^{-1}$ is the Fermi-Dirac distribution function. In the following, we will assume $\Gamma \sim 10^{-4}\Delta(0)$, which describes realistic NIS junctions.[@MartinezArxiv; @PekolaPRL2004; @PekolaPRL2010] We also define $J_{\rm fw}=\vert J_{\rm NIS}(T_{\rm hot},T_{\rm cold})\vert$ and $J_{\rm rev}=\vert J_{\rm NIS}(T_{\rm cold},T_{\rm hot})\vert$ for the forward and the reverse configuration, respectively. Figure \[Fig1\](a) shows the influence of the superconducting DOS on $J$: at a first glance, $J_{\rm fw}$ immediately appears different from $J_{\rm rev}$, thanks to the temperature dependence of $\Delta(T)$, which breaks the thermal symmetry of the tunnel junction. For $T_{\rm hot}>0.4T_{\rm c}$ the value of $\Delta(T)$ in the reverse configuration starts to decrease and $J_{\rm rev}$ gets significantly larger than $J_{\rm fw}$, leading to $\mathcal{R}<1$, as shown in Fig. \[Fig1\](b). On the contrary, if $T_{\rm hot}$ is raised above $T_{\rm c}$, heat flux from N to S becomes preferred, resulting in $\mathcal{R}>1$. As $T_{\rm cold}$ is increased, the reverse rectification regime gets gradually suppressed, while the forward regime becomes more efficient. We can highlight this behavior by introducing the optimal rectification efficiency $\mathcal{R}_{\rm opt}$, defined as that corresponding to the maximum value between $\mathcal{R}$ and $1/\mathcal{R}$ at a given $T_{\rm cold}$. This quantity provides the best working point of the diode, which can occur in the forward (if $\mathcal{R}_{\rm opt}>1$) or in the reverse (if $\mathcal{R}_{\rm opt}<1$) regime of rectification. As displayed in the inset of Fig. \[Fig1\](b), for $T_{\rm cold}<0.5 T_{\rm c}$ the reverse regime is the most efficient, while at higher temperatures the forward regime is favored. Finally, at $T_{\rm cold}=T_{\rm c}$ the system turns into a NIN junction and no rectification is possible.
It is also worth noting that at low values of $T_{\rm hot}$ and $T_{\rm cold}$, $J_{\rm NIS}$ is suppressed by a factor $\Gamma$ with respect to $J_{\rm NIN}$ and does not depend on the sign of the temperature bias \[see inset of Fig. \[Fig1\](a)\], indicating the effectiveness of $\Delta(T)$ as a thermal bottleneck. On the other hand, when $T_{\rm hot}$ and $T_{\rm cold}$ increase, $f(E,T_{\rm hot})-f(E,T_{\rm cold})$ becomes significantly different from zero at energies higher than the smeared $\Delta(T)$: this corresponds to a rapid increase of both $J_{\rm fw}$ and $J_{\rm rev}$, as indicated by the arrow in the inset. Thus, the thermal bottleneck effect generated by the superconducting energy gap decays quickly by raising temperatures. Although this change of behavior is not relevant for the rectifying properties of a single NIS junction, it will be important to understand the operation of a NINISIN chain.
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S$_1$IS$_2$ junctions
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We now move to the case in which both the tunnel-coupled electrodes are superconductors, i.e., a S$_1$IS$_2$ junction.[@MartinezAPL] The latter can be thermally biased by setting the quasiparticle temperature of one electrode to $T_{\rm hot}$ and that of the other to $T_{\rm cold}$. Moreover, we assume that S$_1$ and S$_2$ are characterized by different superconducting energy gaps $\Delta_1$ and $\Delta_2$ and by a phase difference $\varphi=\varphi_2-\varphi_1$. Then, the forward and reverse total heat currents flowing through the Josephson junction read:[@MakiGriffin; @GiazottoAPL12] $$\begin{aligned}
\begin{aligned}
J_{\rm S_1IS_2,fw(rev)}=\;\; & J_{\rm qp}[ T_{\textrm{hot(cold)}}, T_{\textrm{cold(hot)}}] \\
&-J_{\rm int}[T_{\textrm{hot(cold)}}, T_{\textrm{cold(hot)}}]\cos\varphi. \label{JSIS}
\end{aligned}\end{aligned}$$ Here, the term $J_{\rm qp}$ accounts for the energy carried by quasiparticles[@MakiGriffin; @Golubev; @Guttman; @Guttman2; @Zhao1; @Zhao2] and is equivalent to Eq. \[JNIS\] for the present system: $$\begin{aligned}
\begin{aligned}
J_{\rm qp}(T_{\rm hot}, T_{\rm cold}) = \frac{ 2 }{e^2 R_{\textrm{T}} } \int_0 ^\infty dE E \mathcal{N}_1(&E, T_{\rm hot}) \mathcal{N}_2 (E, T_{\rm cold})\\ \times & [f(T_{\rm hot}) - f(T_{\rm cold}) ],
\label{Jquasiparticles}
\end{aligned}\end{aligned}$$ where $\mathcal{N}_1$ and $\mathcal{N}_2$ are the smeared normalized superconducting DOS of S$_1$ and S$_2$, respectively. On the other hand, $J_{\rm int}$ is the phase-dependent component of the heat current:[@MakiGriffin; @Guttman; @Guttman2; @Zhao1; @Zhao2] $$\begin{aligned}
\begin{aligned}
J_{int}(T_{\rm hot}, T_{\rm cold}) = \frac{ 2}{e^2 R_{\textrm{T}} } \int _0 ^\infty dE E \mathcal{M}_1(&E , T_{\rm hot}) \mathcal{M}_2 (E , T_{\rm cold}) \\ \times & [f(T_{\rm hot}) - f(T_{\rm cold}) ],
\label{int}
\end{aligned}\end{aligned}$$ where $\mathcal{M}_{k}(E , T) = \left| \Im \left[ -i\Delta_{k}(T)/\sqrt{ (E+i\Gamma_{k})^2 -\Delta_{k}^2(T)} \right] \right|$ is the Cooper pair BCS density of states in S$_{k}$ at temperature $T_{k}$,[@GiazottoBergeret2] $\Gamma_{k}\sim 10^{-4}\Delta_{k}(0)$ and $k=1,2$. This term is peculiar to the Josephson effect (so it vanishes if one or both of the superconducting energy gaps are null) and originates from tunneling processes through the junctions involving both Cooper pairs and quasiparticles.[@MakiGriffin; @Guttman] Depending on $\varphi$, $J_{\rm int}$ can flow in opposite direction with respect to that imposed by the thermal gradient, but the total heat current still flows from the hot to the cold reservoir, preserving the second principle of thermodynamics. This was experimentally demonstrated in Ref. .
In the following, we shall assume, for clarity, that $\delta=\Delta_2/\Delta_1\leq 1$. Figure \[Fig2\](a) shows the behavior of $J_{fw}$ and $J_{rev}$ vs. $T_{\rm hot}$ for $T_{\rm cold}=0.01 T_{\rm c1}$ ($T_{\rm c1}$ being the critical temperature of S$_1$) and $\delta=0.75$. The difference between the heat currents is particularly evident for $\varphi=\pi$: in the forward configuration, $J_{\rm fw}$ exhibits a sharp peak at $T_{\rm hot}\simeq 0.77 T_{\rm c1}$, due to the matching of singularities in superconducting DOSs when $\Delta_1(T_{\rm hot})=\Delta_2(T_{\rm cold})$. At higher values of $T_{\rm hot}$, $\Delta_1(T_{\rm hot})<\Delta_2(T_{\rm cold})$ and the energy transmission through the junction is reduced, leading to a negative thermal differential conductance region, which is further enhanced by the gradual suppression of $J_{\rm int}$ as $T_{\rm hot}$ approaches $T_{\rm c1}$. In the reverse configuration, instead, $J_{\rm rev}$ presents just one cusp caused by the vanishing of $J_{\rm int}$ when $T_{\rm hot}=T_{\rm c2}=0.75 T_{\rm c1}$. Then, the rectification efficiency is maximum when the aforementioned points are perfectly aligned, i.e., $R_{\rm opt}\simeq 7.5$ for $\delta=0.75$ and $T_{\rm hot}\simeq 0.77 T_{\rm c1}$ at $T_{\rm cold}=0.01 T_{\rm c1}$, as shown in Figs. \[Fig2\](b) and \[Fig2\](c). As $T_{\rm cold}$ is increased, $R_{\rm opt}$ decreases monotonically to $\sim 1.3$ at $T_{\rm cold}=0.7 T_{\rm c1}$, then it becomes $<1$ and finally reaches the identity for $T_{\rm cold}=T_{\rm c1}$ (data not shown). More interestingly, the rectification efficiency can be finely tuned by controlling the phase bias across the junction. As a matter of fact, if $\varphi =0$, $\mathcal{R}$ is not only reduced by one order of magnitude but it also inverts the favored rectification regime, as displayed by Fig. \[Fig2\](d). This effect can be readily understood by observing the trend of $J_{fw}$ and $J_{rev}$ vs. $T_{\rm hot}$ for $\varphi=0$ \[see Fig. \[Fig2\](a)\]. In this case, the negative sign in front of $J_{\rm int}$ in Eq. \[int\] perfectly cancels the singularity-matching peak in the forward configuration.
To conclude the analysis of the intrinsic properties of NIS and S$_1$IS$_2$ junctions, it is worthwhile to summarize the conditions necessary to achieve heat rectification: (1) two *different* DOSs must be tunnel coupled (i.e., $\Delta_1\neq \Delta_2$) and (2) at least one of them must be strongly temperature dependent in the range of operation. Furthermore, a fairly large temperature gradient is needed, since heat rectification is absent in the linear response regime, i.e., when $\delta T=T_{\rm hot}-T_{\rm cold}\ll \overline{T}=(T_{\rm hot}+T_{\rm cold})/2$.[@MartinezAPL]
NININ chain
===========
The rectifying performance of the discussed tunnel junctions can be largely improved if they are combined in an asymmetric chain of tunnel-coupled electrodes that can exchange energy with an independent phonon bath residing at temperature $T_{\rm bath}$. We stress that we are concerned with the heat current carried by electrons only. We assume that lattice phonons present in the entire structure are thermalized, with substrate phonons residing at $T_{\rm bath}$, and are therefore not responsible for any heat transport. This assumption is expected to hold because the Kapitza resistance between thin metallic films and the substrate is vanishingly small at low temperatures.[@GiazottoNature; @MartinezNature; @MartinezArxiv; @Wellstood]
We shall begin with a simple N$_1$IN$_2$IN$_3$ chain,[@FornieriAPL] which, as we will show, represents a fundamental element to understand the working principles of chain diodes. N$_1$ and N$_3$ act as thermal reservoirs and are used to impose a temperature gradient across the device. Since $\mathcal{R}$ is defined under the condition of equal temperature bias, these electrodes must be identical and equally coupled to the phonon bath. Thus, $T_{\rm bath}$ now plays the role embodied by $T_{\rm cold}$ in the single junction analysis. N$_2$ is connected to N$_1$ and N$_3$ by means of two tunnel junctions characterized by resistances $R_1$ and $R_2$, respectively. For simplicity, we assume $R_1$ equal to 1 k$\Omega$ and consider only the parameter $r=R_2/R_1\geq 1$, accounting for the asymmetry of the chain. N$_2$ is the core of the diode and controls the heat flow from one reservoir to the other by releasing energy to the phonon bath. In the forward configuration, the electronic temperature of N$_1$ is set to $T_{\rm hot}>T_{\rm bath}$, leading to electronic thermal currents (see Eq. \[JNIN\]) $J_{\rm 2,fw}$ and $J_{\rm fw}$ flowing into N$_2$ and N$_3$, respectively. On the contrary, in the reverse configuration the electronic temperature of N$_3$ is set to $T_{\rm hot}$, generating heat currents $J_{\rm 2,rev}$ and $J_{\rm rev}$ flowing into N$_2$ and N$_1$, respectively.
Besides electronic thermal currents, we must take into account the heat exchanged by electrons in the metal with lattice phonons:[@MartinezNature; @Maasilta] $$J_{\mathrm{N,ph}}(T,T_\mathrm{bath})=\Sigma \mathcal{V} (T^n-T_\mathrm{bath}^n).\label{Jqpph}$$ Here $\Sigma$ is the material-dependent quasiparticle-phonon coupling constant, $\mathcal{V}$ is the volume of the electrode and $n$ is the characteristic exponent of the material. In this work we will consider two materials that are commonly exploited to realize N electrodes in nanostructures, i.e., copper (Cu) and manganese-doped aluminum (AlMn). The former is typically characterized by $\Sigma_{\rm Cu}=3\times 10^9$ WK$^{-5}$m$^{-3}$ and $n_{\rm Cu}=5$,[@GiazottoRev; @GiazottoNature] while the latter exhibits $\Sigma_{\rm AlMn}=4\times 10^9$ WK$^{-6}$m$^{-3}$ and $n_{\rm AlMn}=6$.[@MartinezNature; @MartinezArxiv; @Maasilta] Furthermore, we assume that all the electrodes of the chain have a volume $\mathcal{V}_{\rm N}= 1 \times 10^{-20}$ m$^{-3}$.
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Equations \[JNIN\] and \[Jqpph\] can be used to elaborate a thermal model accounting for heat transport through the device. The model is sketched in Fig. \[Fig3\](a) and describes the forward temperature bias configuration, in which the electrodes of the chain reside at temperatures $T_{\rm hot}> T_{\rm 2,fw}> T_{\rm fw}>T_{\rm bath}$. Here, $T_{\rm 2,fw}$ and $T_{\rm fw}$ represent the electronic temperatures of N$_2$ and N$_3$, respectively. The terms $J_{\rm 2,fw}=J_{\rm NIN}(T_{\rm hot}, T_{\rm 2,fw})$ and $J_{\rm fw}=J_{\rm NIN}(T_{\rm 2,fw}, T_{\rm fw})$ account for the heat transferred from N$_1$ to N$_3$. The reservoirs can release energy to the phonon bath by means of $J_{\rm 1,out}$ and $J_{\rm 3,out}$. Photon-mediated thermal transport, [@MeschkeNature; @Schmidt; @Pascal] owing to poor impedence matching, as well as pure phononic heat currents are neglected in our analysis.[@GiazottoNature; @MartinezNature; @MartinezArxiv] We can now write a system of energy-balance equations that account for the detailed thermal budget in N$_2$ and N$_3$ by setting to zero the sum of all the incoming and outgoing heat currents: $$\begin{aligned}
&J_\mathrm{2,fw}(T_\mathrm{hot},T_\mathrm{2,fw})-J_{\rm fw}(T_\mathrm{2,fw},T_{\rm fw})-J_\mathrm{2,out}(T_\mathrm{2,fw},T_\mathrm{bath})=0 \notag \\
&J_{\rm fw}(T_\mathrm{2,fw},T_{\rm fw})-J_\mathrm{3,out}(T_{\rm fw},T_\mathrm{bath})=0\label{eqs}.\end{aligned}$$
Here $J_{\rm 2,out}(T_\mathrm{2,fw},T_\mathrm{bath})$ is the heat current that flows from N$_2$ to the phonon bath. In Eqs. \[eqs\] we can set $T_{\rm hot}$ and $T_{\rm bath}$ as independent variables and calculate the resulting $T_{\rm 2,fw}$ and $T_{\rm fw}$. Another system of energy-balance equations can be written and solved for the reverse configuration,[@reverse] in which N$_2$ and N$_1$ reach electronic temperatures $T_{\rm 2,rev}$ and $T_{\rm rev}$, respectively. Finally, we can extract the values of $J_{\rm fw}$ and $J_{\rm rev}$, thereby obtaining $\mathcal{R}$.
In order to grasp the essential physics underlying this device, it is instructive to consider an ideal case, in which N$_1$, N$_2$ and N$_3$ can release energy to the phonon bath only through three tunnel-coupled N electrodes, acting as thermalizing cold fingers[@FornieriAPL] and labeled as F$_1$, F$_2$ and F$_3$, respectively. We assume that electrons in F$_1$, F$_2$ and F$_3$ are strongly coupled to the phonon bath, so they reside at $T_{\rm bath}$. As we shall argue, F$_2$ is fundamental to achieve a high rectification performance and is characterized by a resistance $R_{\rm F,2}$, while F$_1$ and F$_3$ must have the same tunnel junction resistance $R_{\rm F,1}$. We ignore at this stage contributions due to the direct quasiparticle-phonon coupling (see Eq. \[Jqpph\]) in N$_1$, N$_2$ and N$_3$, in order to obtain simple analytic results for the rectification efficiency. As a matter of fact, this ideal device can be easily described by substituting $J_\mathrm{2,out}(T_\mathrm{2,fw},T_\mathrm{bath})=J_{\rm NIN}(T_{\rm 2,fw}, T_{\rm bath})$ and $J_\mathrm{3,out}(T_{\rm fw},T_\mathrm{bath})=J_{\rm NIN}(T_{\rm fw}, T_{\rm bath})$ in Eqs. \[eqs\]. Remarkably, the resulting expression for the rectification efficiency does not depend on $T_{\rm hot}$ nor $T_{\rm bath}$, but only on the resistances of the tunnel junctions defining our NININ chain: $$\mathcal{R}=\frac{R_1(R_2+R_{\rm F,2})+R_2(R_{\rm F,1}+R_{\rm F,2})+R_{\rm F,1}R_{\rm F,2}}{R_{\rm F,2}(R_2+R_{\rm F,1})+R_1(R_2+R_{\rm F,1}+R_{\rm F,2})}.\label{Rideal}$$ Equation \[Rideal\] is particularly transparent in two limit cases: first, if $R_{\rm F,2}\rightarrow\infty$, $\mathcal{R}\rightarrow 1$ for every value of $r$. This clearly demonstrates that no rectification is achievable if the central island of the chain is not coupled to the phonon bath, regardless the asymmetry of the device. This result holds true for a general design of a NININ chain, provided that N$_1$ and N$_3$ are identically coupled to the phonon bath.[@FornieriAPL]
On the other hand, if $R_{\rm F,1}\rightarrow\infty$, Eq. \[Rideal\] is simplified as follows: $$\mathcal{R}_{\rm no-ph}=\frac{R_2+R_{\rm F,2}}{R_1+R_{\rm F,2}}\label{Rasympt}.$$ This expression summarizes the conditions needed to obtain an highly-efficient NININ diode: (1) the thermal symmetry of the device must be broken ($r\gg1$) and (2) N$_2$ must be very well coupled to the phonon bath with respect to N$_1$ and N$_3$. A natural trade off arising from the fulfilment of these conditions is the reduction of the thermal efficiency of the device $\eta=J_{\rm fw}/J_{\rm 2,fw}\leq 1$, i.e., the fraction of energy that is actually transferred from one reservoir to the other in the transmissive regime.[@FornieriAPL] Nevertheless, the device parameters can be designed in order to maximize the required performance in terms of the global efficiency $\mathcal{R}\eta$,[@FornieriAPL] which can range from 0 to $\infty$, since $\mathcal{R}$ has no upper limit. Finally, this ideal case demonstrates that such a thermal rectifier can operate efficiently also in the regime for $T_{\rm hot}\rightarrow T_{\rm bath}$.
In a more realistic case, every electrode in the chain can exchange energy with the phonon bath by means of the quasiparticle-phonon coupling, which introduces a temperature dependence in the rectification efficiency. It is then useful to rewrite $\mathcal{R}\gg 1$ as a general condition for temperatures: $$\mathcal{R}=\frac{\overline{T}_{\rm fw} \delta T_{\rm fw}}{r \overline{T}_{\rm rev}\delta T_{\rm rev}}\gg 1,\label{rect}$$ where $\delta T_{\rm fw (rev)}= T_{\rm 2, fw (rev)}-T_{\rm fw (rev)}$ and the mean temperatures $\overline{T}_{\rm fw (rev)}=(T_{\rm 2,fw (rev)}+T_{\rm fw (rev)})/2$. In order to maximize $\mathcal{R}$, in the reverse configuration the electronic temperatures of N$_1$ and N$_2$ must be similar and close to the lowest temperature in the system, i.e., $T_{\rm bath}$. This can be easily done by setting $r>1$ and by coupling N$_2$ to the phonon bath. On the other hand, the coupling between N$_2$ and the phonon bath should have a limited impact on heat transport in the forward configuration. From Eq. \[Jqpph\], it appears evident that $J_{\rm N,ph}$ is not well-suited to satisfy the aforementioned requirements, since it is not much effective at low temperatures, while it becomes strongly invasive at high temperatures. Thus, it turns out that the most efficient design for a NININ chain diode is obtained by coupling just one thermalizing cold finger F$_2$ to N$_2$ and by minimizing the impact of the quasiparticle-phonon coupling in all the electrodes of the chain.[@FornieriAPL] Then, we can set $J_\mathrm{2,out}(T_{\rm 2,fw},T_\mathrm{bath})=J_{\rm N,ph}(T_{\rm 2,fw},T_\mathrm{bath})+J_{\rm NIN}(T_{\rm 2,fw},T_\mathrm{bath})$ and $J_\mathrm{3,out}(T_{\rm fw},T_\mathrm{bath})=J_{\rm N,ph}(T_{\rm fw},T_\mathrm{bath})$ in the energy-balance equations. Figure \[Fig3\](b) shows $\mathcal{R}$ as a function of $T_{\rm hot}$ for different values of $R_{\rm F,2}$ at $T_{\rm bath}=50$ mK. Remarkably, an optimal rectification efficiency[@ropt] $\mathcal{R}_{\rm opt}\sim 3000$ is obtained for $R_{\rm F,2}=500$ $\Omega$ and $r=100$. This configuration generates a maximum global efficiency $[\mathcal{R}\eta]_{\rm max}\simeq 8$, which can be optimized up to a value of about 9.6 by setting $R_{\rm F,2}=1.2$ k$\Omega$. We can also notice that for $T_{\rm hot}\rightarrow T_{\rm bath}$, $\mathcal{R}$ approaches $\mathcal{R}_{\rm no-ph}$, which represents an asymptotic value for the rectification efficiency at low temperatures, i.e., when the quasiparticle-phonon coupling is almost uneffective.
Figure \[Fig3\](c) displays the rectifier’s output temperatures $T_{\rm fw}$ and $T_{\rm rev}$ vs. $T_{\rm hot}$ for $r=100$ and $R_{\rm F}=500$ $\Omega$ at different values of $T_{\rm bath}$. The results point out a maximum difference of $\sim 200$ mK between the forward and reverse configurations at $T_{\rm bath}=50$ mK. Additionally, the behavior of output temperatures clearly highlights the effectiveness of F$_2$ in keeping $T_{\rm rev}$ close to $T_{\rm bath}$, while $J_{\rm N,ph}$ strongly limits the increasing rate of $T_{\rm fw}$ as $T_{\rm hot}$ and $T_{\rm bath}$ become larger.
Finally, Fig. \[Fig3\](d) confirms the negative effect of the quasiparticle-phonon coupling on the performance of the diode. As a matter of fact, the behavior of $R_{\rm opt}$ as a function of $T_{\rm bath}$ shows that the $T^5$ dependence of $J_{\rm N,ph}$ in Cu is extremely detrimental in the forward configuration and can reduce the rectification efficiency up to a factor of 10.
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NINISIN chain
=============
Despite its conceptual simplicity and potential insensitivity to magnetic fields, the NININ chain diode might be demanding from a fabrication point of view because of the required high asymmetry between $R_1$ and $R_2$. A good alternative is represented by a N$_1$IN$_2$ISIN$_3$ chain, in which all the tunnel-junction normal-state resistances $R_1$, $R_2$ and $R_3$ are equal and the thermal symmetry is broken by the presence of the energy gap in the superconducting density of states. At low temperatures, the latter plays a role equivalent to the one embodied by the largest tunnel-junction resistance in a NININ chain. In the following analysis, we will consider an S electrode made of aluminum with a critical temperature $T_{\rm c}=1.4$ K.
Emulating the most efficient design for a NININ chain, also in this case the N$_2$ electrode can be coupled to the phonon bath by means of a cold finger F residing at $T_{\rm bath}$. On the other hand, it is worthwhile to note that a simple NISIN chain would not represent an efficient design because $\Delta(T)$ would prevent the central electrode to release energy to F in an efficient way, especially at low temperatures \[see Fig. \[Fig1\](a)\]. Figure \[Fig4\](a) sketches the thermal model accounting for thermal transport through the device in the forward configuration, characterized by $T_{\rm hot}>T_{\rm N,fw}>T_{\rm S,fw}>T_{\rm fw}>T_{\rm bath}$. Here, $T_{\rm N,fw}$ and $T_{\rm S,fw}$ label the electronic temperatures of N$_2$ and S, respectively. We assume that all the electrodes have the same volume $\mathcal{V}=1\times 10^{-20}$ m$^{-3}$ and $R_1=R_2=R_3=1$ k$\Omega$. Heat is transferred from N$_1$ to N$_3$ by means of the heat currents $J_{\rm NIN}(T_{\rm hot},T_{\rm N,fw})$, $J_{\rm NIS}(T_{\rm N,fw},T_{\rm S,fw})$ and $J_{\rm fw}=J_{\rm NIS}(T_{\rm S,fw},T_{\rm fw})$. Electrode N$_2$ can release energy to the phonon bath via $J_{\rm 2,out}=J_{\rm NIN}(T_{\rm N,fw},T_{\rm bath})+J_{\rm N,ph}(T_{\rm N,fw},T_{\rm bath})$, while N$_3$ is coupled to $T_{\rm bath}$ through $J_{\rm N,ph}(T_{\rm fw},T_{\rm bath})$. Finally, S is affected by the quasiparticle-phonon coupling as well, which can be written as:[@Timofeev] $$\begin{aligned}
&J_{\mathrm{S,ph}}(T,T_\mathrm{bath})=-\frac{\Sigma \mathcal{V}}{96 \zeta (5)k_\mathrm{B}^5}\int_{-\infty}^{\infty} dE E \int_{-\infty}^{\infty} d\mathrm{\epsilon}\mathrm{\epsilon}^2 \mathrm{sgn}(\mathrm{\epsilon}) \notag \\
&\times L_{E,E+\mathrm{\epsilon}}\left[\mathrm{coth}\left(\frac{\mathrm{\epsilon}}{2k_\mathrm{B}T_\mathrm{bath}}\right)(f_E^{\mathrm{(1)}}-f_{E+\mathrm{\epsilon}}^{\mathrm{(1)}})-f_E^{\mathrm{(1)}}f_{E+\mathrm{\epsilon}}^{\mathrm{(1)}}+1\right], \label{superph}\end{aligned}$$ where $f_E^\mathrm{(1)}=f(-E,T)-f(E,T)$ and the phonons are assumed to be in thermal equilibrium with occupation $n(\mathrm{\epsilon},T_\mathrm{bath})=[\mathrm{exp}(\frac{\mathrm{\epsilon}}{k_\mathrm{B}T_\mathrm{bath}})-1]^{-1}$. The factor $L_{E,E'}=\mathcal{N}(E,T)\mathcal{N}(E',T)\left[1-\frac{\mathrm{\Delta^2(T)}}{EE'} \right]$. In our case, we assume $\Sigma_\mathrm{Al}=$ 0.3 $\times$ 10$^9$ WK$^{-5}$m$^{-3}$.[@GiazottoRev] With all these ingredients we can write and solve the energy balance equations for the forward and reverse configuration, thus allowing us to analyze the operation of the diode vs. $T_{\rm hot}$ and $T_{\rm bath}$ for different parameters.
Figure \[Fig4\](b) displays the behavior of $\mathcal{R}$ as a function $T_{\rm hot}$ for different values of the resistance $R_{\rm F}$ between F and N$_2$ at $T_{\rm bath}=50$ mK. At low values of $T_{\rm hot}$ the rectification efficiency can reach values as high as 5000 for $R_{\rm F}=0.5$ k$\Omega$, but for increasing temperatures it dramatically drops and approaches unity for $T_{\rm hot}>1$ K. This trend can be easily understood by first focusing on the case of $R_{\rm F}>10$ k$\Omega$. The upper panel of Fig. \[Fig4\](c) shows the diode’s output currents $J_{\rm fw (rev)}$ vs. $T_{\rm hot}$ for $R_{\rm F}=50$ k$\Omega$ and directly explains the $\mathcal{R}$ curve (solid line) reported in the lower panel. As already noticed at the end of Sect. \[NISsec\], for sufficiently high thermal bias an S electrode loses its effectiveness as a thermal bottleneck, as can be noticed from the change in the derivative of the output currents for $T_{\rm hot}\gtrsim 0.2$ K in both the forward and reverse configurations. This change of behavior produces a maximum in the rectification efficiency at $T_{\rm hot}\simeq 0.15$ K, after which $\mathcal{R}$ rapidly decreases. Then, the first part of the curve can be compared to the efficiency (dashed line) of an equivalent NINININ chain with identical volumes of electrodes and a resistance asymmetry $r=R_3/R_1=12000$, which is about the value of $\Delta(0)/\Gamma=10^4$. On the other hand, for $R_{\rm F}<10$ k$\Omega$, $T_{\rm S,rev}$ can be significantly higher than $T_{\rm S,fw}$ at the same value of $T_{\rm hot}$. In particular, for $R_{\rm F}=0.5$ k$\Omega$ in the reverse configuration S loses its properties of thermal bottleneck for $T_{\rm hot}\gtrsim 0.15$ K, while in the forward configuration $\Delta$ blocks efficiently the thermal flux up to $T_{\rm hot}\simeq 0.27$ K. This discrepancy in the “breaking points” of the S electrode is reflected in the behavior of $\mathcal{R}$, which exhibits a shoulder after the first maximum \[see Fig.\[Fig4\](b)\]. Furthermore, it is worth noting that for $R_{\rm F}>2.5$ k$\Omega$ the device can operate also in the reverse regime of rectification at sufficiently high $T_{\rm hot}$, reaching values of $\mathcal{R}\simeq 0.8$, as expected for a simple NIS junction (see Sect. \[NISsec\]).
As $T_{\rm bath}$ is increased, the maximum value of $\mathcal{R}$ in the forward regime becomes strongly suppressed due to the invasive enhancement of the quasiparticle-phonon coupling. This can be indirectly observed in Fig.\[Fig4\](d), which shows the diode’s output temperatures $T_{\rm fw (rev)}$ vs. $T_{\rm hot}$ for $R_{\rm F}=0.5$ k$\Omega$ at three selected values of $T_{\rm bath}$. At $T_{\rm bath}=50$ mK we obtain a maximum temperature difference $\delta T=T_{\rm fw}-T_{\rm rev}$ of about 110 mK, while at higher $T_{\rm bath}$ it rapidly decreases in conjunction with the derivative of $T_{\rm fw (rev)}$.
Remarkably, the global efficiency of this device at $T_{\rm bath}=50$ mK is comparable to the NININ case and can be optimized to $[\mathcal{R}\eta]_{\rm max}\simeq 7.5$ by setting $R_{\rm F}=7.5$ k$\Omega$.
Finally, it is also possible to use a different superconductor with a larger $\Delta$ (such as vanadium or niobium) to implement the S electrode of the chain. This can significantly increase the range of $T_{\rm hot}$ where the superconductor can act as a large thermal resistance, thus providing $\mathcal{R}\gg 1$. Nevertheless, the performance of the device vs. $T_{\rm bath}$ will not be significantly improved, since it is mainly determined by the disruptive effect of the quasiparticle-phonon coupling in N electrodes.
The effectiveness of this kind of device has been recently proven in the experiment reported in Ref. .
NINIS$_2$IS$_1$IN chain
=======================
{width="80.00000%"}
In previous works,[@MartinezAPL; @MartinezRev] we analyzed theoretically the performance of a S$_1$IS$_2$ junction tunnel-coupled to two N electrodes acting as thermal reservoirs. This thermal diode exhibits a remarkable phase-modulation of the rectification efficiency for $T_{\rm c1}=1.4$ K and $\delta=0.75$ at $T_{\rm bath}=10$ mK: the best performance is reached at $T_{\rm hot}=1.15$, where $\mathcal{R}$ can be varied from a maximum of $\sim 4.4$ (if $\varphi= \pi$) to a minimum of $\sim 0.7$ (if $\varphi=0$).[@MartinezAPL] Thus, the behavior of a NIS$_1$IS$_2$IN chain can be fully tuned by varying the phase difference between S$_1$ and S$_2$, allowing to invert the direction of the rectification regime. Nevertheless, this device has some experimental drawbacks, since its optimal working point requires a large bias temperature, close to the critical temperature of aluminum. As a consequence, superconducting thermometers used to detect $T_{\rm hot}$, $T_{\rm fw}$ and $T_{\rm rev}$ (as in Refs. ) should be realized with a superconductor S$_3$, characterized by an energy gap $\Delta_3 \gg \Delta_1,\Delta_2$, thus complicating the experimental design of the rectifier. Therefore, we can think of employing an additional N electrode connected to the phonon bath by means of a cold finger in order to improve $\mathcal{R}_{\rm opt}$ and to move the optimal working point of the device towards lower values of $T_{\rm hot}$. As we will argue, the proposed design offers the possibility to increase $\mathcal{R}_{\rm opt}$ by more than 3 orders of magnitude with respect to the NIS$_1$IS$_2$IN chain diode and to demonstrate experimentally the phase modulation of $\mathcal{R}$ in a relatively simpler way. However, it prevents us inverting the rectification regime through the phase modulation.
We consider a N$_1$IN$_2$IS$_2$IS$_1$IN$_3$ chain, in which N$_2$ is tunnel-coupled to a cold finger F residing at $T_{\rm bath}$, S$_1$ has a critical temperature $T_{\rm c1}=1.4$ K and $\delta=\Delta_2/\Delta_1=0.75$. This design results to be the one which maximizes the effect of a $\varphi$ variation on the rectification efficiency. Figure \[Fig5\](a) displays the thermal model accounting for thermal transport through the device in the forward configuration, characterized by $T_{\rm hot}>T_{\rm N,fw}>T_{\rm S_2,fw}>T_{\rm S_1,fw}>T_{\rm fw}>T_{\rm bath}$. Here, $T_{\rm N,fw}$, $T_{\rm S_1,fw}$ and $T_{\rm S_2,fw}$ label the electronic temperatures of N$_2$, S$_1$ and S$_2$, respectively. We assume that all the electrodes have the same volume $\mathcal{V}=1\times 10^{-20}$ m$^{-3}$ and all the tunnel junction have a normal-state resistance $R_1=R_2=R_3=R_4=1$ k$\Omega$. Heat is transferred from N$_1$ to N$_3$ by means of the heat currents $J_{\rm NIN}(T_{\rm hot},T_{\rm N,fw})$, $J_{\rm NIS}(T_{\rm N,fw},T_{\rm S_2,fw})$, $J_{\rm S_1IS_2}(T_{\rm S_2,fw},T_{\rm S_1,fw})$ and $J_{\rm fw}=J_{\rm NIS}(T_{\rm S_1,fw},T_{\rm fw})$. Electrode N$_2$ can release energy to the phonon bath via $J_{\rm 2,out}=J_{\rm NIN}(T_{\rm N,fw},T_{\rm bath})+J_{\rm N,ph}(T_{\rm N,fw},T_{\rm bath})$, while N$_3$ is coupled to $T_{\rm bath}$ through $J_{\rm N,ph}(T_{\rm fw},T_{\rm bath})$. Finally, S$_1$ and S$_2$ are affected by the quasiparticle-phonon coupling $J_{\rm S_{1,2},ph}(T_{\rm S_{1,2}fw},T_{\rm bath})$. In general, the junction S$_2$IS$_1$ can be phase biased through supercurrent injection or by applying an external magnetic flux.[@MartinezAPL] In order to achieve a complete modulation of $\mathcal{R}$, $\varphi$ must vary between 0 and $\pi$. This can be obtained by using a radio frequency superconducting quantum interference device (rf SQUID), but this would again require the use of a clean contact between the S$_2$IS$_1$ junction and a third superconductor S$_3$ (with $\Delta_3 \gg \Delta_1,\Delta_2$) in order to suppress heat losses.[@MartinezAPL; @MartinezRev] A good alternative is represented by an asymmetric direct current SQUID formed by different Josephson junctions (JJs), of which one corresponds to the S$_2$IS$_1$ junction. The latter can be biased up to $\varphi=\pi$ if its Josephson inductance[@Tinkham] is the largest in the SQUID, i.e., its Josephson critical current is much lower than those of the other JJs.
As done in previous cases, we can now write and solve the energy-balance equations accounting for the thermal budget in every electrode of the chain. Thus, we can obtain the behavior of the thermal diode vs. $T_{\rm hot}$, $T_{\rm bath}$ and $\varphi$ for different values of the parameters.
Figure \[Fig5\](b) shows $\mathcal{R}$ vs. $T_{\rm hot}$ for different values of the cold finger resistance $R_{\rm F}$ at $T_{\rm bath}=50$ mK. Solid lines represent the rectification efficiency for $\varphi=0$, while dashed lines stand for that obtained in the case of $\varphi=\pi$. The values of $\mathcal{R}$ provided here are comparable to those obtained with the NINISIN chain, but the phase dependence of $J_{\rm S_1IS_2}$ represents an additional ingredient that strongly influences the performance of the diode over a large part of the $T_{\rm hot}$ range. In particular, for $R_{\rm F}=0.5$ k$\Omega$ we can reach a maximum relative variation $[\mathcal{R}(0)-\mathcal{R}(\pi)]/\mathcal{R}(0)\simeq 77$% at $T_{\rm hot}=0.13$ K. From the point of view of output temperatures, though, the phase tuning of the device is clearly detectable only for $T_{\rm hot}>0.2$ K. The effect of the phase biasing is especially visible on $T_{\rm rev}$, which is affected by a maximum variation of $\sim$ 60 mK at $T_{\rm hot}\simeq$ 0.6 K. At this temperature also the phase modulation of $\delta T=T_{\rm fw}-T_{\rm rev}$ exhibits the largest amplitude (of $\sim 30$ mK), as shown in Fig. \[Fig5\](d).
We finally notice that the global efficiency of a N$_1$IN$_2$IS$_2$IS$_1$IN$_3$ chain is comparable to those of other chain diodes, with a maximum $[\mathcal{R}\eta]_{\rm max} \sim 6$ for $R_{\rm F}=8$ k$\Omega$.
Summary and final remarks
=========================
In summary, we first reviewed the intrinsic properties of superconducting hybrid junctions as thermal rectifiers. NIS and S$_1$IS$_2$ junctions can reach an optimal rectification efficiency of $\sim 0.8$[@MartinezAPL; @GiazottoBergeret] and $\sim 7.5$,[@MartinezAPL; @MartinezRev] respectively. On the other hand, even though a simple NIN junction cannot provide heat rectification, it is possible to obtain $\mathcal{R}_{\rm opt}$ as high as 3000 by realizing an asymmetric NININ chain with the central island coupled to the phonon bath by means of a cold finger. The latter provides an efficient channel through which the diode can release energy in the non-transmissive temperature bias configuration.[@FornieriAPL] This design can be used also in a NINISIN chain, which offers the opportunity to boost the performance of a single NIS junction and to overcome the fabrication complexity imposed by the high resistance asymmetry required in a NININ chain. The superconducting energy gap plays the role of a thermal bottleneck at low temperatures, while for higher temperatures it can generate an inversion of the rectification regime, as experimentally proven in Ref. . Finally, we analyzed a NINIS$_2$IS$_1$IN chain as a testing ground for the experimental demonstration of a phase-tunable thermal diode, providing a larger rectification efficiency and requiring lower bias temperatures with respect to those obtained in previous theoretical proposals.[@MartinezAPL; @MartinezRev] These thermal rectifiers could be easily implemented by standard nanofabrication techniques [@MartinezArxiv] and, combined with heat current interferometers, [@GiazottoNature; @MartinezNature] might become building blocks of coherent caloritronic nanocircuits.[@GiazottoNature; @MartinezNature; @MartinezRev] Moreover, they could be connected to electronic coolers and radiation detectors,[@GiazottoRev] offering the possibility to evacuate dissipated power in an unidirectional way. Finally, this technology might also have a potential impact in general-purpose cryogenic electronic microcircuitry, e.g., solid-state quantum information architectures.[@NielsenChuang]
Acknowledgments
===============
The European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 615187-COMANCHE is acknowledged for partial financial support.
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abstract: 'We present the Josephson junction intersected superconducting transmission line resonator. In contrast to the Josephson parametric amplifier, Josephson bifurcation amplifier and Josephson parametric converter we consider the regime of few microwave photons. We review the derivation of eigenmode frequencies and zero point fluctuations of the nonlinear transmission line resonator and the derivation of the eigenmode Kerr nonlinearities. Remarkably these nonlinearities can reach values comparable to Transmon qubits rendering the device ideal for accessing the strongly correlated regime. This is particularly interesting for investigation of quantum many-body dynamics of interacting particles under the influence of drive and dissipation. We provide current profiles for the device modes and investigate the coupling between resonators in a network of nonlinear transmission line resonators.'
address: '1 Technische Universit[ä]{}t M[ü]{}nchen, Physik Department, James-Franck-Str., D-85748 Garching, Germany'
author:
- M Leib$^1$ and M J Hartmann$^1$
title: Many Body Physics with Coupled Transmission Line Resonators
---
Introduction
============
The theory of interacting quantum many-body systems is a vibrant discipline in physics full of unsolved riddles like high temperature superconductivity and promising technological prospects such as topologically protected quantum states, novel schemes for quantum error correction or one way quantum computing. Minimal Hamiltonians are considered for both situations: to explain quantitatively phenomena in nature or to provide prescriptions how to build artificial systems for technological applications.
There exists a constantly growing amount of analytical and numerical tools to investigate the properties of many body Hamiltonians. However the analytical tools typically provide answers only in limited regimes while the numeric approaches are constrained by the exponential growth of the Hilbert space with the number of particles. Dating back to an idea of Feynman [@feynman] who proposed to use an computer that uses the principles of quantum mechanics itself to investigate quantum mechanical many-body Hamiltonians the idea of quantum simulation [@quantSim] is gaining a lot of attention nowadays. A quantum simulator is a well controllable quantum systems that emulates the physics of other systems which are less amenable to experimental investigation. Therefore quantum simulators are typically strongly enlarged versions of the “real world” system we try to investigate. This provides us with the possibility of tuning parameters of the Hamiltonian and with individual addressability for measurements because of larger lattice constants. Quantum many-body Hamiltonians can for example be simulated with cold atoms trapped by laser fields in various shapes and dimensions [@review_Bloch_2008], in ion traps [@Friedenauer:2008fk; @Kim2010] or arrays of cavity quantum electrodynamics (QED) systems [@Hartmann:2006kx; @hartmann-2008-2].
Massless photons in cavities don’t interact. However by strongly coupling the photons to nonlinear systems, joint excitations emerge, called polaritons [@Hartmann:2006kx; @hartmann-2008-2; @1367-2630-10-3-033011; @PhysRevA.81.021806; @Hartmann2010], and the nonlinearity of the onsite spectrum for the polaritons can be interpreted as an interaction [@brandao; @leib]. Cavities can be coupled in arbitrary topologies thereby generating networks for moving photons. The polaritons inherit this abiltiy from the photons. The description of the fundamental excitations of the quantum simulator in terms of polaritons is only valid in the strong coupling regime where the coupling between the cavity and the nonlinear system exceeds the decay rates of both. In circuit quantum electrodynamics (circuit QED), the microwave realization of cavity quantum electrodynamics, one can easily achieve the strong coupling regime. In circuit QED superconducting wires (transmission line resonators) that confine microwave photons are coupled to Josephson junction based lumped element circuits (Josephson qubits) [@PhysRevA.69.062320; @Wallraff:2004rz; @Deppe:2008kx]. Strong zero point fluctuations of the field due to the restricted dimensionality of the transmission line resonator and the artificial coupling to the Josephson qubit result in exceptionally high coupling rates [@Niemczyk] while the superconducting gap ensures low dissipation. The polariton approach has two major flaws. Firstly, the ratio of nonlinearity to hopping strength is changed by changing the polariton from purely photonic to Josephson qubit excitation. Therefore nonlinearity and hopping are mutually exclusive. Secondly, we always have two polariton species separated in energy space by the coupling strength which provides us with unwanted but unavoidable crosstalk.
In this work, we propose making the resonator itself nonlinear by intersecting it with a Josephson junction [@Mallet:2009fk; @leib2; @Bourassa]. We show that the modes of such a device can achieve amounts of nonlinearity comparable to transmon qubits [@koch:042319; @Majer:2007sf] for realistic circuit parameters and that the mode separation in frequency space is sufficient to suppress the crosstalk that is introduced by the nonlinearity. Moreover the coupling of these resonators is independent of the nonlinearity. Notably, a set of experimental analysis methods for the quantum statistics of resonator networks have been demonstrated in recent years [@Menzel; @LangEichler; @Devoret]. These properties make the nonlinear transmission line resonator an ideal building block for a Bose Hubbard quantum simulator.
The article is organized as follows. First we review the derivation of frequencies and zero point fluctuations of the eigenmodes of the nonlinear transmission line resonator [@leib2]. Then we derive the individual Kerr nonlinearities for the eigenmodes in a low energy and rotating wave approximation. Finally we provide the current profiles including current through the inductive part of the Josephson junction and current through the transmission line resonator and discuss capacitive coupling of the eigenmodes between adjacent transmission line resonators.
Nonlinear transmission line resonator
=====================================
We consider a transmission line resonator intersected in the middle by a Josephson junction. In this section we briefly review the decomposition of the microscopic Lagrangian of the nonlinear transmission line resonator into independent modes, we provide some physical intuition for the modes and calculate their coupling strength in a network of nonlinear transmission line resonators.
spectrum
--------
The microscopic Lagrangian of the whole setup reads, $$\mathcal{L}=\mathcal{L}_l+\mathcal{L}_r+\mathcal{L}_{JJ}\,,$$ with the Lagrangian of the transmission line to the left and the right of the Josephson junction, $$\mathcal{L}_{l/r}=\int\limits_{-\frac{L}{2}/0}^{0/\frac{L}{2}} \frac{c}{2}(\dot{\phi}_{l/r})^2-\frac{1}{2l}(\partial_x\phi_{l/r})^2 dx\,,$$ with the flux field $\phi(x)=\int_{-\infty}^tV(x,t')dt'$ [@fluctuation] and the capacitance and inductance per unit length of transmission line resonator $c$ and $l$. The Lagrangian of the Josephson junction $\mathcal{L}_{JJ}$ depends on the flux drop at the Josephson junction $\delta\phi=\phi_l|_{x=0}-\phi_r|_{x=0}$, $$\mathcal{L}_{JJ}=\frac{C_J}{2}\delta\dot{\phi}^2+E_J\cos(\frac{2\pi}{\phi_0}\delta\phi)\,,$$ where $C_J$ and $E_J$ are the capacitance and Josephson energy of the junction respectively. $\phi_0=h/2e$ is the quantum of flux. We proceed by separating the linear from the purely nonlinear parts of the Lagrangian. This way we can exactly diagonalize the linear parts and get the normal modes of the system. We then reintroduce the nonlinear parts as an perturbation to the oscillator modes of the system. It suffices to consider the nonlinear cosine inductance of the Josephson junction when separating the nonlinear parts from the linear ones , $$E_J\cos(\frac{2\pi}{\phi_0}\delta\phi)=-\frac{1}{2L_J}\delta\phi^2+E_J\left(1+\sum\limits_{n=2}\frac{-1^n}{(2n)!}(\frac{2\pi}{\phi_0}\delta\phi)^{2n}\right)\,,$$ where $L_J=\phi_0^2/(4\pi^2E_J)$ is the Josephson inductance. Considering only the linear part of the Lagrangian, the Euler Lagrange equations provide us with the wave equation in the two arms of the transmission line resonator, $$\partial_t^2\phi_{l/r}=v^2\partial_x^2\phi_{l/r}\,,$$ where the phase velocity is given by the inductance and capacitance per unit length $v=1/\sqrt{cl}$. Boundary conditions for the flux field can either be derived by Kirchhoffs rules or also via the Euler Lagrange equations. The current in the transmission line resonator is proportional to the spatial derivative of the flux field $I=\frac{1}{l}\partial_x\phi$ and has to vanish at the ends of the transmission line, $$\partial_x \phi_{l/r}|_{x=\mp\frac{L}{2}}=0.$$ We also know that the current flowing into the Josephson junction from one side has to exit on the opposite side, $$\partial_x\phi_l|_{x=0}=\partial_x\phi_r|_{x=0}\,.$$ The current flowing through the junction can either flow into the shunting capacitor or the Josephson junction. The current flowing through the junction is related to the flux drop across the junction in the Josephson constitutive relation $I_{J}=I_c\sin((2\pi/\phi_0)\delta\phi)$. The current into the capacitor in turn reads, $I_{cap}=C_J\delta\ddot{\phi}$. For the linearized Lagrangian the condition for the current flowing through the Josephson junction reads, $$-\frac{1}{l}\partial_x \phi_l=C_J\delta\ddot{\phi}+\frac{1}{L_J}\delta\phi\,.$$ To find the spatial eigenmodes of the nonlinear transmission line resonator we first make an ansatz with separation of variables, $\phi(x,t)=f(x)g(t)$ with $f_l(x)=A_l\cos(k(x+L/2))$ for the transmission line to the left of the Josephson junction and $f_r(x)=A_r\cos(k(x-L/2))$ for the right part of the transmission line. With this ansatz the necessity of vanishing currents at the ends of the transmission line is automatically satisfied and the wave equation is fulfilled provided that $\omega=k v$, where $\omega$ is the frequency of the temporal part of the flux function $\ddot{g}+\omega^2 g=0$. There are two distinct possibilities to fulfill the boundary conditions at the Josephson junction: we could either choose $A_l=A_r$ which provides us with the symmetric modes of the resonator or we could choose $A_l=-A_r$ to get the antisymmetric modes. The symmetric modes show no flux drop at the Josephson junction and consequently there is no current flowing through the junction. Therefore the symmetric modes of the transmission line resonator are completely unaffected by the presence of the Josephson junction and we omit them in the further discussion of the system dynamics. For the antisymmetric modes we choose the amplitudes to be $A_l=-A_r=1$ and get the following transcendental equation for the mode frequencies $\omega_n=k_n v$, $$\frac{\omega_n}{v}=\cot\left(\frac{\omega_n}{v}\frac{L}{2}\right)\frac{2l}{L_J}\left(1-\frac{\omega_n^2}{\omega_p^2}\right)\,,$$ where $\omega_p=1/\sqrt{L_JC_J}$ is the Josephson plasma frequency. The solution of the above transcendental equation is the spectrum of the system, not taking into account the symmetric modes, presented in a) for generic parameters given in . We plotted the spectrum as a function of the critical current $I_c$ of the Josephson junction. The resonator modes without the Josephson junction do not depend on the cricital current of the Josephson junction but the plasma frequency $\omega_p=\sqrt{I_c/(\phi_0 C_J)}$ does. If the plasma frequency and the frequency of the antisymmetric modes of the bare transmission line resonator (without Josephson junction) are detuned the frequencies of the combined system are the frequencies of its constituents, Josephson junction or transmission line resonator. But if the plasma frequency is resonant with one of the symmetric modes we get an anticrossing of the two frequencies which is of the order of the mode frequency itself accounting for the large coupling between Josephson junction and transmission line resonator.
![a) Numerically calculated frequencies of the linear part of the transmission line resonator intersected by a Josephson junction. Anti-crossings show up where the plasma frequency of the Josephson junction $\omega_p$ (dashed ascending graph) matches one of the frequencies of the odd modes of the transmission line resonator (dashed horizontal graphs). b) Nonlinearity parameter of the different modes $U_n$. If the frequency of the mode is near the plasma frequency of the Josephson junction, the nonlinearity $e^2/(4 C_s)$ is inherited. c) Coupling between different antisymmetric modes of adjacent nonlinear transmission line resonators. Comparison with the mode frequencies reveals the connection of coupling strength $g_n=(C_c\omega_n)/(4\eta_n)$ and frequency of the respective mode. Dashed vertical lines mark the values of the critical current $I_c$ where we computed the current profile of the mode cf. []{data-label="fig:specNonlin"}](specNonlin.pdf)
[@rll]{}\
characteristic impedance & $ Z_0 $ & $\unit{50}{\ohm}$\
phase velocity & $v$ & $\unit{0.94 \cdot 10^8}{\meter\per\second}$\
inductance per length & $l$ & $\unit{5\cdot 10^{-7}}{\henry\per\meter}$\
capacitance per length & $c$ & $\unit{2 \cdot 10^{-10}}{\farad\per\meter}$\
\
shunting capacitance&$C_J$& $\unit{1.9 \cdot 10^{-12}}{\farad}$\
We deduce the following generalized “scalar product” between the spatial modes with the help of the transcendental equation of the mode frequencies, $$c\int_{-\frac{L}{2}}^{\frac{L}{2}}f_n(x)f_m(x)dx+C_s\delta f_n \delta f_m = \delta_{n,m} \eta_n\,,$$ with, $$\eta_n=c\left(\frac{L}{2}+\frac{\delta f_n^2}{k_n^2}\frac{l}{2L_J}\left(1+\frac{\omega_n^2}{\omega_p^2}\right)\right)\,,$$ where $\delta f_n = f_{n,l}|_{x=0}-f_{n,r}|_{x=0}$ is the normalized flux drop of the spatial modes. We use this “scalar product” to decompose the linearized Lagrangian into independent oscillators of frequencies $\omega_n$ and masses $\eta_n$. After a Legendre transformation and omitting constant terms we get the Hamiltonian, $$\begin{aligned}
H&=&\sum_{n=1}^{\infty}\left(\frac{\pi_n^2}{2\eta_n}+\frac{1}{2}\eta_n \omega_n^2g_n^2\right)-E_J\sum\limits_{n=2}\frac{-1^n}{(2n)!}(\frac{2\pi}{\phi_0}\delta\phi)^{2n}\\
\pi_n&=&\eta_n\dot{g}_n\,. \end{aligned}$$ Up to here everything we did was classical physics, now we proceed by quantizing the Hamiltonian in the usual way by introducing independent raising and lowering operators for the different modes, $$\begin{aligned}
\hat{\pi}_n&=&-i\sqrt{\frac{\eta_n\omega_n}{2}}(a_n-a_n^{\dag})\\
\hat{g}_n&=&\frac{1}{\sqrt{2\eta_n\omega_n}}(a_n+a_n^{\dag})\,,\end{aligned}$$ with $\left[a_n,a_m^{\dag}\right]=\delta_{n,m}$. Finally, we arrive at the following Hamilton operator, $$\begin{aligned}
H&=&\sum_{n=1}^{\infty}\omega_n(a_n^{\dag}a_n+\frac{1}{2})+H^{\text{nonlin}}\nonumber\\
H^{\text{nonlin}}&=&-E_J\left(\cos\left(\delta\tilde{\phi}\right)+\frac{1}{2}\delta\tilde{\phi}^2\right)\,,\label{eq:unHarmHam}\end{aligned}$$ with $$\delta\tilde{\phi}=\sum_{n=1}^{\infty}\lambda_n\left(a_n+a_n^{\dag}\right)\quad \text{and,} \quad
\lambda_n=\frac{2\pi\delta f_n}{\Phi_0\sqrt{2\eta_n \omega_n}}\,.$$ The nonlinearity again reintroduces interactions between the modes and nonlinearities for the individual modes. As we will show below these are all much smaller than the differences between the mode frequencies. They only play a role in strong driving scenarios like the Josephson parametric converter. We are only interested in few microwave photon physics and therefore may omit the residual coupling in a rotating wave approximation. The individual mode nonlinearities however can not be omitted. For their derivation we start to decompose the cosine term with a product-to-sum formula which is applicable in this case because all individual flux operators commute, $$\begin{aligned}
\cos\left(\sum\limits_{n=1}^{\infty} \lambda_n(a_n+a_n^{\dag})\right)&=&\cos\left(\lambda_1(a_1+a_1^{\dag})\right)\cos\left(\sum\limits_{n=2}^{\infty} \lambda_n(a_n+a_n^{\dag})\right)\\
&&-\sin\left(\lambda_1(a_1+a_1^{\dag})\right)\sin\left(\sum\limits_{n=2}^{\infty} \lambda_n(a_n+a_n^{\dag})\right)\,.\end{aligned}$$ The sine terms only contain rotating operator products and therefore can be neglected in the framework of the rotating wave approximation. If we further proceed this way we may transform the cosine of the sum of all mode flux operators into the product of the cosines of all mode flux operators, $$\cos\left(\sum_n \lambda_n (a_n+a_n^{\dag})\right)\to \prod_n \cos\left(\lambda_n(a_n+a_n^{\dag})\right)\,.$$ The cosines of the mode flux operators may be rewritten in a sum of normal ordered products of mode operators. We may further also omit all operator products containing a different amount of raising and lowering operators, again because of the rotating wave approximation. Finally we get as a rotating wave, low energy approximation for the cosine of a mode flux operator, $$\cos\left(\lambda_n(a_n+a_n^{\dag})\right)\to e^{-\frac{\lambda_n^2}{2}}\left(1-\lambda_n^2 a_n^{\dag}a_n+\frac{\lambda_n^4}{4} a_n^{\dag} a_n^{\dag}a_na_n+\dots\right)\,.$$ We concentrate on the fundamental mode and consider all the other modes to be in the vacuum state which leads us to an Hamiltonian only describing the fundamental antisymmetric mode of the transmission line resonator coupled to the plasma mode of the Josephson junction, $$H_1=\left(\omega_1-\delta \omega\right) a_1^{\dag}a_1-U a_1^{\dag}a_1^{\dag}a_1a_1\,,$$ with $$\begin{aligned}
\delta \omega &=& E_J \lambda_1^2\left(1-\prod\limits_{n=1}^{\infty} e^{-\frac{\lambda_n^2}{2}}\right)\\
U &=& E_J \frac{\lambda_1^4}{4}\prod\limits_{n=1}^{\infty} e^{-\frac{\lambda_n^2}{2}}\,,\end{aligned}$$ where $\delta\omega$ is a small renormalization of the fundamental mode frequency due to the nonlinearity. The Kerr nonlinearity $U$ is plotted in b) as a function of the Josephson junctions critical current $I_c$. Direct comparison with the spectrum of the system reveals the origin of the nonlinearity. For values of $I_c$ where the respective frequency is defined by the plasma frequency of the Josephson junction, the mode inherits the full nonlinearity of the Josephson junction.
modes
-----
We now decomposed the linear part of the Lagrangian into independent harmonic oscillators with frequencies $\omega_n$ and mode capacitances $\eta_n$. The latter only make sense in combination with the normalization convention we chose for the spatial mode functions $f_n(x)$. In order to get the correct physical intuition of the oscillation mode, we have to revert to the well known quantities of oscillating circuits, currents and charges or rather coulomb potential. We start by calculating the current through the Josephson junction. The observable for the current through the Josephson junction is, $$\hat{I}_J=I_c\sin(\delta\tilde{\Phi})\,.$$ Lets suppose that all modes except for the fundamental mode are in their vacuum state and the fundamental mode is in a Fock state $\left|n \,0\dots 0\right\rangle$. In this state the mean value of current flowing through the Josephson junction vanishes comparable to the vanishing displacement of a quantum harmonic oscillator in a Fock state. The variance of the current however doesn’t vanish and provides us with an estimation of the mean current flowing through the Josephson junction, $$\Delta I_J=I_c\sqrt{\left\langle n\,0\dots0\right|\sin(\delta\tilde{\phi})^2\left|n\,0\dots 0\right\rangle}\,.$$ With the same arguments as presented above for the derivation of the Kerr parameter, we calculate the variance, which is exact up to the second Fock state, $$\Delta I_J=I_c \sqrt{\frac{1}{2}\left(1-\prod_k e^{-2\lambda_k^2}\left(1-4\lambda_1^2 n + 4\lambda_1^4 n (n-1)\right)\right)}\,.$$ The observable for the current in the transmission line to the left and the right of the Josephson junction is, $$\hat{I}_r(x)=\frac{1}{l}\partial_x \hat{\phi}(x)=\sum_n\frac{\partial_x f_n(x)}{l\sqrt{2 \eta_n \omega_n}} (a_n+a_n^{\dag})\,.$$ Here again the mean current in the transmission line resonator for a Fock state vanishes and we evaluate the variance of the current, $$\Delta I_r=\frac{1}{l}\sqrt{ \frac{\left|\partial_x f_1(x)\right|^2}{2 \eta_1 \omega_1}( 2n + 1)+\sum\limits_{k=2}^{n_{cutoff}}\frac{\left|\partial_x f_k(x)\right|^2}{2 \eta_k \omega_k}}\,.$$ shows the variance of the current in the left half of the nonlinear transmission line resonator and the inductive part of the Josephson junction. The variance of the current in the right part is for symmetry reasons the mirror image of its counterpart in the left half.
![Variance of the current in the transmission line resonator and Josephson junction for the first Fock state of the fundamental mode.[]{data-label="fig:Current"}](Current.pdf)
coupling
--------
Next we are calculate the strength of the coupling between two neighboring transmission line resonators in a network. We only consider capacitive coupling of resonators where the respective ends of the central lines of adjacent transmission line resonators are either close to each other or connected by interdigitated capacitors. To integrate the coupling into our theoretical model we have to include the energy of the coupling capacitor, with coupling capacitance $C_c$, into the Lagrangian of the nonlinear transmission line resonator, $$\mathcal{L}_c=\frac{C_c}{2}\left(\dot{\phi}_1|_{x=\frac{L}{2}}-\dot{\phi}_2|_{x=-\frac{L}{2}}\right)^2\,,$$ where $\phi_{1/2}$ is the flux field of adjacent nonlinear transmission line resonators. After a Legendre transformation to get the corresponding energy term in the Hamiltonian we would get different conjugate momenta $\pi_n$. We neglect this effect because the coupling capacitance $C_c$ is very small compared to the overall capacitance of the nonlinear transmission line resonator and get $\dot{\phi}_{1/2}=-i\sum_n\sqrt{\omega_n/(2\eta_n)}f_n(x)\left(a_{1/2,n}-a_{1/2,n}^{\dag}\right)$. The coupling term in the Lagrangian therefore provides us with many different effects: We get renormalizations of the different resonator mode frequencies, exchange couplings between different modes of adjacent resonators and on the same resonator, which we neglect in a rotating wave approximation, and exchange coupling of the same mode of adjacent resonators, $$H_{n,c}=-\frac{C_c}{4}\frac{\omega_n}{\eta_n}\left(a_{1,n}^{\dag}a_{2,n}+a_{1,n}a_{2,n}^{\dag}\right)\,.$$ The dimensionless capacitance of the fundamental modes of the nonlinear transmission line resonator is not affected by a change of the Josephson junction’s critical current for generic parameters (cf. ) of our setup . The coupling $g_n=(C_c \omega_n)/(4 \eta_n)$ of the same modes in adjacent resonators is therefore determined by the frequency of the modes cf. c), where we plotted the coupling of the first three antisymmetric modes of the nonlinear transmission line resonator. This in turn enables us to increase the nonlinearity while increasing the coupling, provided we do not use the fundamental mode of the nonlinear transmission line resonator.
Summary
=======
We reviewed the derivation of the eigenmode frequencies and respective zero point fluctuations for the Josephson junction intersected transmission line resonator. We derived the Kerr nonlinearities of the individual eigenmodes and calculated the current in the transmission line resonator and the Josephson junction. Finally we showed that the coupling depends on the eigemode frequency and is thereby independent of the nonlinearity. This lack of mutual exclusivity of nonlinearity and coupling makes quantum simulators with nonlinear transmission line resonators superior to polariton approaches.
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abstract: 'We present the phase diagram and dynamical correlation functions for the Holstein-Hubbard model at half filling and at zero temperature. The calculations are based on the Dynamical Mean Field Theory. The effective impurity model is solved using Exact Diagonalization and the Numerical Renormalization Group. Excluding long-range order, we find three different paramagnetic phases, metallic, bipolaronic and Mott insulating, depending on the Hubbard interaction $U$ and the electron-phonon coupling $g$. We present the behaviour of the one-electron spectral functions and phonon spectra close to the metal insulator transitions.'
address:
- 'Department of Mathematics, Imperial College, London SW7 2AZ, U.K.'
- 'Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan'
author:
- 'W. Koller'
- 'D. Meyer'
- 'A. C. Hewson'
- 'Y. Ōno'
title: 'Phase diagram and dynamic response functions of the Holstein-Hubbard model'
---
strongly correlated electrons, electron-phonon coupling, Mott transition
Electron-phonon effects in strongly correlated electron systems are expected to be significant but have sofar received little theoretical attention. We study these effects here using the Holstein–Hubbard model[@KMH04pre; @KMOH04] where a local Einstein phonon mode couples linearily to local charge fluctuations. The Hamiltonian is given by $$\begin{aligned}
H =&
\sum_{\vec{k}\sigma}
\epsilon(\vec{k})\, {c^{\dagger}}_{\vec{k}\sigma} {c^{\phantom{\dagger}}}_{\vec{k}\sigma} +
U \sum_{i} n_{i\uparrow} n_{i\downarrow} \\ &+
\omega_0 \sum_i {b^{\dagger}}_i {b^{\phantom{\dagger}}}_i +
g\sum_i ({b^{\dagger}}_i + {b^{\phantom{\dagger}}}_i) \big(n_{i{\uparrow}}+ n_{i{\downarrow}}-1 \big) \:.
\end{aligned}$$ We use a semi-elliptical band of width $W=4$ and focus on the particle-hole symmetric case at zero temperature. The phonon frequency is fixed to $\omega_0=0.2$. We calculate the phase diagram, in the absence of long-range order, using a number of local approximations (DMFT-NRG/ED [@KMOH04], DIA [@Pot03]).
{width="98.00000%"}
All methods agree on the phase diagram shown in the figure. It consists of a metallic region surrounded by two distinct gapped phases, a Mott insulator for large $U$ and a bipolaronic phase, when the electron-phonon coupling $g$ dominates. The transition to the Mott insulating phase is always continuous, as indicated by the grey boundary curve; the critical value of $U$ is largely independent of $g$. In contrast, the transition to the bipolaronic phase is first order for $U\gtrsim 3$, as indicated by the full black line, but is also continuous for smaller values of $U$.
More of the physics of the interplay of the electron-phonon and electron-electron interactions is revealed in the spectra of the dynamic response functions. These were calculated using the DMFT-NRG [@MHB02]. Electron and phonon spectra are shown within the figure along different scans in the phase diagram. The results for the corresponding dynamic spin and charge susceptibilities are discussed in [@KMH04pre].
The two upper insets show the electron spectra $\rho_G(\omega)$. The left-hand scan for $U=1$ and $g=0.0, 0.4, 0.46, 0.5$ shows the continuous narrowing of the central resonance with increasing $g$ and the subsequent opening of a gap at $g_c\approx 0.47$. Contrasting behaviour is seen in the upper right inset ($U=5, g=0.0, 0.6, 0.7,
0.75$) with an initial broadening of the central peak followed by its sudden disappearance at a critical $g\approx 0.71$.
The two insets below the electron spectra show the behaviour of the corresponding phonon spectra $\rho_d(\omega) = -\frac{1}{\pi}\textrm{Im} {{ \big<\!\big< {{b^{\phantom{\dagger}}}_i;{b^{\dagger}}_i} \big>\!\big> }}_\omega$. For $U=1$ we see with increasing $g$ a complete softening of the phonon peak as the transition to the bipolaronic phase is approached. The increase of negative spectral weight for $\omega<0$ indicates a growing number excited phonons. At $g_c\approx 0.47$ we observe a two-peak structure whose low energy peak vanishes and whose high-energy peak hardens back to $\omega_0$ after the continuous transition. For $U=5$ the softening of the phonons is much weaker due to suppression of charge fluctuations. There is no two-peak structure at the transition and again the peak hardens to $\omega_0$ in the gapped phase.
Similar features can also be seen in the scan for a fixed $g=0.45$ and increasing $U$, shown in the right-hand inset. The transition from the bipolaronic to the metallic phase is reflected by a two-peak structure. Increasing $U$ further suppresses charge fluctuations and hardens the completely softened phonon peak until the Mott transition is approached and the peak arrives at the bare frequency $\omega_0$.\
[**Acknowledgements:**]{} We wish to thank the EPSRC (Grant GR/S18571/01) for financial support. One of us (YŌ) was supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology. We also thank M. Aichhorn, R. Bulla, D. Edwards and M. Potthoff for helpful discussions.
[00]{}
W. Koller, D. Meyer, A. C. Hewson, [**cond-mat/0404328**]{}
W. Koller, D. Meyer, Y. Ōno, A. C. Hewson, Europhys. Lett. [**66**]{}, 559 (2004)
M. Potthoff, Eur. Phys. J. B [**32**]{}, 429 (2003)
D. Meyer, A. Hewson, R. Bulla, Phys. Rev. Lett. [**89**]{}, 196401 (2002)
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---
abstract: 'Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces of genus $g\geq 1$ with (pseudo)-Anosov elements. That is, we consider a closed surface $S$ of genus $g\geq 1$, and homeomorphisms $f, h: S \to S$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$. It is known that $f$ (or some power of $f$) must be homotopic to the identity. Suppose that $h$ is (pseudo)-Anosov with stretch factor $\lambda >1$. We show that $\langle f,h \rangle$ is not a faithful representation of $BS(1, n)$ if $\lambda > n$. We also show that there are no faithful representations of $BS(1, n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $\lambda$).'
author:
- 'Juan Alonso, Nancy Guelman and Juliana Xavier'
title: 'Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements'
---
[Introduction]{}
Baumslag-Solitar groups $\operatorname{BS}(m,n)= \langle a,b : a b^m a ^{-1} = b ^n\rangle$, $m,n \in {\mathbb{Z}}$, were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non-Hopfian groups (see [@bsp]). These groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory, geometric group theory, topology and dynamical systems (see for example [@bv], [@fmo]). In particular, the groups $\operatorname{BS}(1,n)$, $n \geq 2$, are the simplest examples of infinite non abelian solvable groups. Actions of solvable groups on one-manifolds have been studied by many authors (see, for example, [@plante] and [@navas]; specifically for $BS(1,n)$-actions see [@glp1]). Results in dimension two appear in [@glp]; in particular several examples of $BS(1,n)$-actions on ${\mathbb{T}}^2$ are exhibited.
The groups $\operatorname{BS}(1,n)$ also provide examples of distortion elements, which are related to Zimmer’s conjecture and dynamical aspects in general (see [@fh] and [@zimmer]). In particular, J. Franks and M.Handel proved that on a surface $S$ of genus greater than one, any distortion element in the group $\operatorname{Diff}^1_0(S,area)$ is a torsion element, and therefore there are not faithful representations of $\operatorname{BS}(1,n)$ in $\operatorname{Diff}^1_0(S,area)$ ([@fh] ). It is then natural to wonder what dynamics are allowed for Baumslag-Solitar group actions.
Throughout this paper $S$ will be a closed connected surface of genus $g\geq 1$, and $f, h: S \to S$ homeomorphisms satisfying the Baumslag-Solitar ($\operatorname{BS}$) equation $$h f h^{-1} = f^n,$$for some $ n\geq 2$. We will also assume that $f$ is isotopic to the identity, which yields no loss of generality due to the following result:
\[gl\] Let $ \langle f, h \rangle$ be an action of $\operatorname{BS}(1, n)$ on a closed surface $S $. If $n\geq 2$, there exists $k\geq 1$ such that $f^k$ is isotopic to the identity. Moreover, if the action is faithful on ${\mathbb{T}}^2$, $f^
k$ has a lift whose rotation set is the single point $\{(0,0)\}$ and the set of $f^k$- fixed points is non-empty.
This is consequence of Theorem 1.2 of [@flm] that claims that every element of infinite order in the mapping class group has linear growth. So, there are no distortion elements of infinite order in the mapping class group. Since $f$ is a distortion element of $BS(1,n)$, it follows that $[f]$ has finite order. The statement on faithful actions on ${\mathbb{T}}^2$ is the content of Theorem 3 in [@glp].
If the action is faithful, the group $\langle f^k, h \rangle$ is isomorphic to $BS(1, n)$. Then, the previous theorem allows us to restrict our study to the case where $f$ is isotopic to identity. Moreover, if $S= {\mathbb{T}}^2$ we may assume that $f$ has a lift $\tilde f$ to the universal covering space such that the rotation set is $\{(0,0)\}$ and that $\operatorname{Fix}(f)\neq \emptyset$. We say that $\tilde f$ is the [*irrotational lift*]{} for $f$.\
Once the homotopy class of $f$ has been stablished, we seek to understand the possibilities for the homotopy class of the conjugating generator $h$. We exhibit examples of faithful actions $ \langle f, h \rangle$ of $BS(1, n)$ where $h$ is isotopic to a (pseudo)-Anosov map (see Section \[ex\]). However, these examples are somewhat trivial, in the sense that the action is supported in an invariant subset which is collapsed by a semiconjugacy taking $h$ to its pseudo-Anosov representative (the construction is detailed in Section \[ex\]). This led us to consider the case where $h$ is a (pseudo)-Anosov homeomorphism (and not just isotopic to one). It is known that the centralizer of a (pseudo)-Anosov map is virtually cyclic and so there are no faithful actions of ${\mathbb{Z}}^2$ on surfaces with pseudo-Anosov elements (see [@mccarthy], [@rocha] and also [@py]). The present work is a natural generalization of this kind of results. In the more general scope, our work lies in the context of trying to understand the nature of the obstructions to the existence of (faithful) group actions on certain phase spaces (a survey of these ideas can be found in [@fisher]). Our first result is the following:
\[teo1\] Let $ \langle f, h \rangle$ be an action of $BS(1, n)$ on a closed surface $S$, where $f$ is isotopic to the identity, and $h$ is a (pseudo)-Anosov homeomorphism with stretch factor $\lambda > n$. Then $f= \operatorname{Id}$.
It follows that there are no faithful representations of $BS(1, n)$ by surface homeomorphisms with $h$ a (pseudo)- Anosov map with stretch factor $\lambda > n$.
As was already pointed out, the last theorem is false if we ask only that $h$ is [*isotopic*]{} to a (pseudo)-Anosov map as the examples in Section \[ex\] show. Moreover, an example of G. Mess [@fh] shows that the theorem is no longer true with a slight generalization on the acting group: if we set $h$ to be a linear Anosov map on ${\mathbb{T}}^2$ with eigenvalue $\lambda$ and $f(x) = x+w$, where $w \neq 0$ is an unstable eigenvector of $h$, then $hfh^{-1} (x) = x +\lambda w$.
When the surface is a torus and $f$ is area preserving, we are able to remove the hypothesis $\lambda > n$:
\[toro\] Let $ \langle f, h \rangle$ be an action of $BS(1, n)$ on ${\mathbb{T}}^2$, where $f$ is an area-preserving homeomorphism isotopic to the identity and $h$ an Anosov homeomorphism. Then, $f= id$.
So, there are no faithful representations of $BS(1,n)$ by torus homeomorphisms with $h$ an Anosov map and $f$ area preserving (regardless of the value of $\lambda$). This result is a nice application of the work of A. Koropecki and F. Tal in [@korotal].
The idea behind these results is making the structures of identity isotopies and of (pseudo)-Anosov maps interact. Namely, we use the transverse measures of the foliations of the (pseudo)-Anosov to measure the trajectories $(f_t (x))_{t\in [0,1]}$ of points defined by the identity isotopy $(f_t)_{t\in [0,1]}$.
Finally, we point out that several questions remain unanswered. Is it possible to extend Theorem \[teo1\] for $\lambda < n$? Does there exist a $BS(1,n)$ action with $h$ isotopic to a (pseudo)-Anosov map that is not semi-conjugate to a non-faithful one? Is it possible to classify the dynamics of the action according to the isotopy class of $h$?
[Preliminaries]{}
[Definitions and notations]{}
For the remainder of the paper $S$ will be a closed surface, $f: S \to S$ will be a homeomorphism isotopic to the identity and $h: S \to S$ will be a (pseudo)-Anosov homeomorphism (see \[pa\] for the definition). If $g:S\to S$ is any map, $\operatorname{Fix}(g)$ is the set of fixed points of $g$, $\operatorname{Per}(g)$ is the set of periodic points of $g$, and $\operatorname{Per}^p (g)$ is the set of periodic points of $g$ of period $p$.
If $X$ is a topological space, we say that any continuous function $\gamma: [0,1]\to X$ is an arc in $X$. If $\gamma_1, \gamma _2: [0,1]\to X$ are arcs, we note $\gamma _1 \cdot \gamma_2$ the standard concatenation of arcs ($\gamma _1 \cdot \gamma_2: [0,1]\to X$). If $\gamma _1 \ldots \gamma _n$ are arcs, we define $\prod _{i=0}^n \gamma _i= \gamma_ 1 \cdot \ldots \cdot \gamma _n $.
[Identity isotopies]{}
For $f: S \to S$ as before, fix an isotopy $(f_t)_{t\in [0,1]}$ such that $f_0 = \operatorname{Id}$ and $f_1 = f$. For all $x\in S$ we note $\gamma _x$ the arc $t\to f_t (x), t\in [0,1]$. More generally, for $x\in S$ and $m\geq 1$ define $\gamma ^m_x = \prod _{i= 0} ^{m-1}
\gamma _{f^i (x)}$.
If $S$ is hyperbolic, the homotopy class of $\gamma _x$ with fixed endpoints is independent of the isotopy $(f_t)_{t\in [0,1]}$; it is determined by $f$. In this case, a homeomorphism $f: S \to S$ admits a unique lift to the universal covering that commutes with every element of the group of covering transformations. We call this lift the [*cannonical lift*]{} of $f$; it is the lift determined by lifting the isotopy $(f_t)_{t\in [0,1]}$ starting in the identity. In fact, on a compact surface of genus $g\geq 2$, two isotopies joining the identity to a homeomorphism $f$ are homotopic (see [@ham]).
If $S = {\mathbb{T}}^2$, however, the homotopy class of $\gamma _x$ with fixed endpoints depends on the isotopy $(f_t)_{t\in [0,1]}$. In this case, we will further assume that $f$ has a lift $\tilde f$ whose rotation set is $\{(0,0)\}$, and that the isotopy is such that the lift $(\tilde f_t)_{t\in [0,1]}$ to ${\mathbb{R}}^2$ starting at the identity verifies $\tilde f _ 1 = \tilde f$. (This assumption can be made if $\langle f, h\rangle$ is a faithful action of $BS(1,n)$ on ${\mathbb{T}}^2$, by Proposition \[gl\]).
We say that $x\in \operatorname{Fix}(f)$ is [*contractible*]{} if the loop $\gamma _x$ is null-homotopic. Analogously, if $x\in \operatorname{Per}^p (f)$, we say that $x$ is contracible, if the loop $\gamma ^p_x$ is null-homotopic.
[pseudo-Anosov homeomorphisms]{}\[pa\]
We say that $h: S \to S$ is a [*pseudo-Anosov*]{} homeomorphism if there is a pair of transverse measured singular foliations $({\cal F}^u, \nu ^u)$ and $({\cal F}^s, \nu ^s)$ on $S$ and a number $\lambda >1$ such that $$h \cdot ({\cal F}^u, \nu ^u) = ({\cal F}^u, \lambda \nu ^u) \textrm{ and } h \cdot ({\cal F}^s, \nu ^s) = ({\cal F}^s, \lambda ^{-1}\nu ^s).$$The measured foliations $({\cal F}^u, \nu ^u)$ and $({\cal F}^s, \nu ^s)$ are called the unstable foliation and the stable foliation, respectively, and the number $\lambda$ is called the [*stretch factor*]{} of $h$. We recall that the action of $h$ on $({\cal F}, \nu)$ is given by $$h\cdot ({\cal F}, \nu) = (h({\cal F}),h_*( \nu)),$$where $h_*( \nu)(\gamma)$ is defined as $\nu (h^{-1} (\gamma))$ for any arc $\gamma$ transverse to $h ({\cal F})$. So, in particular, $$\label{eq1} \nu ^s (h(\gamma)) = \lambda \nu ^s (\gamma)$$ for any arc $\gamma$ transverse to the stable foliation, and
$$\label{eq2} \nu ^u (h(\gamma)) = \lambda^{-1} \nu ^u (\gamma),$$
for any arc $\gamma$ transverse to the unstable foliation. Note that this corresponds to the statement that $h$ shrinks the leaves of the stable foliation and stretches the leaves of the unstable foliation.\
We can extend the measures $\nu^s$ and $\nu^u$ to arcs not necessarily transverse to the foliations. We do so as follows
$$\nu(\gamma) = \inf \{ \sum_i\nu(\alpha_i): \alpha = \alpha_1\cdots\alpha_k \sim \gamma \mbox{ rel endpoints, and } \alpha_i \mbox{ transverse to } {\cal F} \}$$
where $\nu$ (and ${\cal F}$) stand for either $\nu^s$ or $\nu^u$ (respectively ${\cal F}^s$ or ${\cal F}^u$). Notice that equations \[eq1\] and \[eq2\] still hold for a general arc $\gamma$. By construction, $\nu^s$ and $\nu^u$ are invariant under homotopy with fixed endpoints, and they are subadditive, that is: $$\nu(\alpha\cdot\beta) \leq \nu(\alpha) + \nu(\beta).$$
We will use several properties of pseudo-Anosov homeomorphisms that we state in this section. The reader not familiar with this theory should refer to Chapter Fourteen in [@fm].
\[pap\] The following holds:
1. Let $({\cal F}, \nu)$ be the stable or unstable measured foliation of a pseudo-Anosov homeomorphism on a compact surface $S$. Then, $\nu (\alpha) > 0$ for every essential simple closed curve $\alpha\in S$.
2. Moreover, $\nu(\alpha) = 0$ if and only if $\alpha$ is homotopic with fixed endpoints to a leaf segment of ${\cal F}$.
3. For any leaf segment $l$ of ${\cal F}^u$ and any $\epsilon>0$, there is $m\geq 0$ so that $h^m(l)$ is $\epsilon$-dense in $S$.
4. For any pseudo-Anosov homeomorphism $h$ of a compact surface $S$, the periodic points of $h$ are dense.
5. A pseudo-Anosov homeomorphism $h$ of a compact surface $S$ has a dense orbit.
6. The functions $x\to \nu ^s (\gamma _x)$ and $x\to \nu ^u (\gamma _x)$ are continuous, and thus bounded on $S$.
Orientation covers for foliations {#oc}
---------------------------------
A measured singular foliation $({\cal F}, \nu)$ on a surface $S$ may not be orientable. It is easy to see that this is the case if the foliation has a singularity with an odd number of prongs. Moreover, the stable and unstable foliations for a pseudo-Anosov homeomorphism can fail to be orientable (see Corollary 14.13 in [@fm]). However, if ${\cal F}$ is not orientable, there exists a connected twofold branched cover $$p: \tilde S \to S,$$called the [*orientation cover*]{} of $S$ for ${\cal F}$ such that there is an induced measured foliation $(\tilde{\cal F}, \tilde \nu)$ on $\tilde S$ that is orientable and such that $p$ maps leaves of $\tilde {\cal F}$ to leaves of ${\cal F}$ and $p_* (\tilde \nu) = \nu$. The branch points of the cover are exactly the preimages under $p$ of the singularities of ${\cal F}$ with an odd number of prongs.
The construction of the orientation cover is similar to the standard construction of the orientation double cover of a non-orientable manifold. We refer the reader to page 403 on [@fm] for details.
[The Baumslag-Solitar equation]{}
The following is an easy consequence of the group relation $hfh^{-1}=f^n$ that we will need later on.
\[bsp\] Let $f,h$ be such that $hfh ^{-1} = f ^n$. Then:
1. For all $m\geq 1$, $h^m f h ^{-m} = f ^{n^m} $;
2. $h^{-1} (\operatorname{Fix}(f))\subset \operatorname{Fix}(f)$
[Examples of actions with (pseudo)- Anosov classes]{}\[ex\]
In this section we provide examples of homeomorphisms $f,h:S\to S$ defining a faithful action of $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ on a surface $S$, where $h$ is isotopic to a (pseudo)- Anosov map.
Let $S$ be a closed surface of genus $g\geq 1$ and $h_1:S\to S$ a (pseudo)- Anosov homeomorphism with a fixed point $x\in S$.
[*Example 1:*]{} Consider the standard action of $BS(1,n)$ on $S^2 = {\mathbb{C}}\cup\{\infty \}$ generated by the Möbius transformations $f_0(z) = z+1$ and $h_0(z) = nz$. It is known that this action is faithful. Moreover, every orbit is free except for the global fixed point at $\infty$. Blow up this fixed point to a circle to obtain a faithful action $\hat f_0,\hat h_0 : D \to D$ of $BS(1,n)$ on the disk $D$, fixing every point in the boundary $\partial D$.
On the other hand, blow up $x\in S$ to a circle, obtaining a homeomorphism $\hat h_1: \hat S\to \hat S$ on the surface with boundary $\hat S$, fixing every point in $\partial\hat S$.
Glue $D$ and $\hat S$ by their boundaries, obtaining a surface $S'$ homeomorphic to $S$. Define $f$ and $h$ on $S'$ so that $f|_D = \hat f_0$, $f|_{\hat S} = id_{\hat S}$, $h|_D=\hat h_0$ and $h|_{\hat S} = \hat h_1$. Clearly they agree on the identified boundaries. The resulting action is faithful, since the action on $D$ is. Identifying $S$ and $S'$, we have that $h$ is isotopic to $h_1$. This example is somewhat trivial because the faithful action only happens on a disk whose complement contains all the homology of $S$. Moreover, it is semi-conjugate to the non-faithful action on $S$ defined by letting $b$ act as the identity and $a$ as $h_1$. The semi-conjugacy is $P:\hat S \to S$ the map that collapses $D$.
[*Example 2:*]{} Let $y\in S$ and $x_j = h_1^j(y)$ for $j\in {\mathbb{Z}}$. Blow up each $x_j$ to a circle, and glue back a disk $D_j$ to obtain a surface $S'$ homeomorphic to $S$ as before. To picture it,imagine that the disk $D_j$ has area $2^{-|j|}$. Define $\hat S = S'-\bigcup D_j$ and $\hat h_1: \hat S \to \hat S$ the blow-up map of $h_1$.
Let $\varphi_j:D\to D_j$ be an identification of each $D_j$ with the disk $D$ in the previous example. Also, let $\hat f_0$ and $\hat h_0$ be as before. Define $f:S'\to S'$ by $f|{\hat S} = id_{\hat S}$ and $f|_{D_j} = \varphi_j \hat f_0 \varphi_j^{-1}$. That is, like $\hat f_0$ on each disk and the identity everywhere else. Define $h:S'\to S'$ by $h|_{\hat S} = \hat h_1$ and $h|_{D_j} = \varphi_{j+1}\hat h_0 \varphi_j^{-1}$.
This action is also faithful, and $h$ is still isotopic to $h_1$ (when regarding $S'$ as $S$). We can pick $y$ to have a dense orbit under $h_1$, and in that case the action is free on an open dense set. The complement of this set, i.e. $\hat S$, still contains all the homology of $S$. And the action is again semi-conjugate to the action by the identity and $h_1$, by collapsing each disk $D_j$.
[Actions of $BS(1,n)$ with (pseudo)- Anosov elements]{}
Recall that if $S$ is hyperbolic, $f$ has a cannonical lift (the unique lift that commutes with the group of covering transformations). If $S= {\mathbb{T}}^2$ by the cannonical lift we mean the irrotational lift (see Proposition \[gl\]). In both cases, we denote $\tilde S$ the universal covering space of $S$.
\[up\] Let $\tilde f: \tilde S \to \tilde S$ be the cannonical lift of $f$, and $\tilde h:\tilde S\to \tilde S$ be any lift of $h$. Then, the $\operatorname{BS}$ equation holds for $\tilde f$ and $\tilde h$, that is $$\tilde h \tilde f \tilde h ^{-1} = \tilde f ^n .$$
If $S$ is hyperbolic any two isotopies from the identity to a given homeomorphism $g: S \to S$ are homotopic (see [@ham]). Both isotopies $(h f_t h ^{-1} )_{t\in[0,1]}$ and $\prod_{i= 0} ^{n-1} f_t $ join the identity to $f^{n}$ (the product stands for concatenation of arcs in the space $\operatorname{Homeo}(S)$). In particular, the arcs $\gamma ^{n}_x$ and $h (\gamma_{h^{-1}(x)})$ are homotopic with fixed endpoints. This means that $\tilde f ^{n} (\tilde x) = \tilde h \tilde f \tilde h ^{-1} (\tilde x)$ for all $\tilde x \in \tilde S$ as we wanted.
If $S = {\mathbb{T}}^2$, as $\tilde f$ is the irrotational lift for $f$, we have that $f^p (x) = x$ implies $\tilde f ^p (\tilde x) = \tilde x$ for any lift $\tilde x$ of $x$. We know that $\tilde h \tilde f$ and $\tilde f ^n \tilde h$ are both lifts of the same map $hf$. Moreover, by Proposition \[gl\] $f$ has a fixed point $x$ and $\tilde f (\tilde x) = \tilde x$ for any lift $\tilde x$ of $x$. So, $\tilde h \tilde f \tilde x = \tilde h \tilde x$. Besides, $f^n h (x) = h f (x) = h (x)$ implies $\tilde f ^n \tilde h (\tilde x) = \tilde h (\tilde x)$. So, the lifts $\tilde h \tilde f$ and $\tilde f ^n \tilde h$ of $hf$ coincide over $\tilde x$, which implies they are identical, that is $$\tilde h \tilde f
\tilde h ^{-1} = \tilde f ^n$$
\[arcsh\] For any $x\in S$ and $m\geq 0$, the arcs $\gamma ^{n^m}_x$ and $h^m (\gamma_{h^{-m}(x)})$ are homotopic with fixed endpoints.
The lemma above implies that for all $m\geq 0$, $\tilde h^m \tilde f \tilde h ^{-m} = \tilde f^{n^m}$, where $\tilde h$ is any lift of $h$ and $\tilde f$ is the cannonical lift for $f$ (see Lemma \[bsp\] item 1). The lift $\tilde \gamma ^{n^m} (x)$ of $\gamma ^{n^m} (x)$ starting at $\tilde x$ ends at $\tilde f ^{n^m} (x)$, and the lift $\tilde \gamma _{h^{-m}(x)}$ of $\gamma _{h^{-m}(x)}$ starting at $\tilde h ^{-m} (\tilde x)$ ends at $\tilde f \tilde h ^{-m} (x)$. So, $\tilde h^m \tilde \gamma_{h^{-m} (x)}$ has the same endpoints that $\tilde \gamma ^{n^m} (x)$, finishing the proof.
\[arcs\] For any $x\in S$ and $m\geq 0$, the following equations hold: $$\nu ^s (\gamma ^{n^m}_x) = \lambda ^m \nu ^s (\gamma_{h^{-m}(x)}),$$ $$\nu ^u (\gamma ^{n^m}_x) = \lambda ^{-m} \nu ^u (\gamma_{h^{-m}(x)}).$$
We know that $\nu ^s$ and $\nu ^u$ are invariant by homotopy with fixed endpoints. So, by Corollary \[arcsh\] we obtain: $$\nu ^s (\gamma ^{n^m}_x) = \nu ^s (h^m (\gamma_{h^{-m}(x)})) = \lambda ^m \nu ^s (\gamma_{h^{-m}(x)}),$$ $$\nu ^u (\gamma ^{n^m}_x) = \nu ^u (h^m (\gamma_{h^{-m}(x)})) = \lambda ^{-m} \nu ^u (\gamma_{h^{-m}(x)}).$$
\[fixcont\] Every $f$- fixed point is contractible.
If $x\in \operatorname{Fix}(f)$, then $\gamma _x$ is a loop. So, for all $m\geq 0$, $\nu ^u (\gamma _x ^{n^m}) = n^m\nu ^u (\gamma _x)$. We want to show that $\gamma _x$ is homotopically trivial for every $x\in \operatorname{Fix}(f)$. Recall that $\nu^u (h(\gamma)) = \lambda ^{-1} \nu^u (\gamma)$. By Corollary \[arcs\], $$\lambda ^{-m}\nu ^u (\gamma_{ h^{-m}(x)}) = \nu ^u (\gamma _x ^{n^m}) = n^m\nu ^u (\gamma _x).$$ We obtain $(\frac{1}{\lambda n})^m \nu ^u(\gamma_{h^ {-m}(x)}) = \nu ^u (\gamma _x)$ for all $m\in {\mathbb{N}}$. As $\nu^u (\gamma_{h^ {-m}(x)}) $ is bounded, this implies that $\nu ^u (\gamma _x) = 0$, and therefore that $\gamma _x$ is a trivial loop by Lemma \[pap\] item 1.
\[nus0\] If $\lambda > n$, then $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$.
By Lemma \[arcs\], $$\nu ^s (\gamma ^{n^m}_x) = \lambda ^m \nu ^s (\gamma_{h^{-m}(x)}) .$$ So, $$\nu ^s (\gamma _{h ^{-m}(x)}) = \lambda ^{-m} \nu ^s (\gamma_x^{n^m}) \leq \lambda ^{-m} \sum _{i=0}^{n^m-1} \nu ^s (\gamma _{f^i (x)})\leq \left( \frac{n}{\lambda}\right) ^m C ,$$where $C$ is a bound for the function $y\to \nu ^s (\gamma _y)$ on $S$. Since this inequality holds for all $x\in S$, we can apply it to $x_m = h^m(y)$ for any $y\in S$, so we get that $$\nu^s (\gamma_y) \leq \left( \frac{n}{\lambda}\right) ^m C$$ for all $y\in S$ and $m$ a positive integer. Since we are supposing that $\lambda > n$, this implies the lemma.
\[fixFs\] If $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$, then $f$ preserves each leaf and semi-leaf of ${\cal F}^s$.
That $f$ preserves the stable leaves is a direct consequence of the hypothesis, since any curve with endpoints in different leaves of ${\cal F}^s$ must have positive $\nu^s$ measure. To show that it preserves semi-leaves, let $x\in S$ be a singularity of ${\cal F}^s$. Consider a lift of the action to the universal cover $\tilde S = {\mathbb{D}}$ (it exists by Lemma \[up\]). Let $\tilde f$ and $\tilde h$ be the respective lifts of $f$ and $h$. Also, let $\tilde x \in {\mathbb{D}}$ be a preimage of $x$, and $(\tilde {\cal F}^s,\tilde \nu^s)$ be the measured lamination on ${\mathbb{D}}$ that covers $({\cal F}^s,\nu^s)$. Notice that $\tilde f$ must preserve the leaves of $\tilde {\cal F}^s$. Now recall that $\tilde {\cal F}^s(\tilde x)$ is homeomorphic to a star of center $\tilde x$, and whose prongs are the semi-leaves of $\tilde {\cal F}^s(\tilde x)$. Since $\tilde f$ is an homeomorphism preserving $\tilde {\cal F}^s(\tilde x)$, it must fix the center $\tilde x$ and permute the legs. Moreover, each semi-leaf is a ray from $\tilde x$ that converges to a point in $\partial {\mathbb{D}}$. Since $f$ is isotopic to the identity, $\tilde f$ extends continuously to $\bar {\mathbb{D}}$ as the identity on $\partial {\mathbb{D}}$, and so it preserves each semi-leaf of $\tilde {\cal F}^s(\tilde x)$. We obtain the lemma by projecting to $S$.
\[fixh\] If $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$, and $x$ is a singularity of ${\cal F }^s$, then $ x\in \operatorname{Fix}(f)$.
Direct from Lemma \[fixFs\].
Let $p: \tilde S \to S$ be the orientation cover for ${\cal F}^s$ and $\tilde{\cal F}^s$ the induced oriented measured foliation on $\tilde S$ (see Section \[oc\]).
The action lifts to $\tilde S$. That is, there exist homeomorphisms $\tilde f, \tilde h : \tilde S \to \tilde S$ such that $p\tilde f = f p$, $p\tilde h = h p$ and $\tilde h \tilde f \tilde h ^{-1}=\tilde f ^n$. Moreover, $\tilde f$ is isotopic to the identity and $\tilde h$ is pseudo-Anosov with both stable and unstable foliations orientable.
Let $B$ be the set of branch points of $p$, so that $p|_{\tilde S \backslash B}:\tilde S \backslash B\to S\backslash p(B)$ is a covering map. Then, by the lifting criterion any map $g:S \to S$ preserving $p(B)$ lifts to a map $\tilde g: \tilde S \to \tilde S$ (see Proposition 1.33 in [@hatcher]). Moreover, there are two possible lifts for such a map; one preserving the orientation of $\tilde{\cal F}^s$ and the other one reversing it. For any such map $g:S \to S$, we will call $\tilde g$ (and $-\tilde g$) to the lift of $g$ preserving (resp. reversing) the orientation of $\tilde{\cal F}^s$. Recall that $p(B)$ are the singularities of ${\cal F}^s$ with an odd number of prongs. So clearly $h$ preserves $p(B)$. Moreover, corollary \[fixh\] tells us that $f$ also preserves $p(B)$. Notice that $\tilde h \tilde f \tilde h ^{-1}$ and $\tilde f ^n$ are both lifts of the same map $f^n$. So, either $\tilde h \tilde f \tilde h ^{-1}=\tilde f ^n$, or $\tilde h \tilde f \tilde h ^{-1}=- \tilde {f ^n}$. As $\tilde f$ preserves the orientation of $\tilde{\cal F}^s$, so does $\tilde f ^n$, and so $\tilde f ^n$ is the lift of $f^n$ preserving orientation of $\tilde{\cal F}^s$. On the other hand, as both $\tilde h$ and $\tilde f$ are the orientation preserving lifts, $\tilde h \tilde f \tilde h ^{-1}$ preserves the orientation of $\tilde{\cal F}^s$. So, $\tilde h \tilde f \tilde h ^{-1}=\tilde f ^n$, as we wanted.
Note that $\tilde f$ is isotopic to the identity because $f$ is. To see that $\tilde h$ is pseudo-Anosov with oriented measured foliations, note that the unstable measured foliation $({\cal F}^u, \nu ^u)$ lifts to a singular measured foliation $(\tilde{\cal F}^u,\tilde \nu ^u)$ on $\tilde S$ that is transverse to $\tilde{\cal F}^s$. So, we can use the orientation on $\tilde{\cal F}^s$ to induce an orientation on $\tilde{\cal F}^u$ by imposing, for instance, that the stable foliation always crosses the unstable foliation locally from left to right.
The preceeding lemma allows us to restrict ourselves to the case where both measured foliations of $h$ are oriented. Note that in the case where $S = {\mathbb{T}}^2$ and $h$ is an Anosov homeomorphism, this assumption is fulfilled. We devote the next section to proving our results in this setting.
[The orientable case]{}
Throughout this section, we will assume that the measured foliations of $h$ are globally oriented. We say that an arc $\gamma : I \to S$ is positively transverse to an oriented foliation ${\cal F}$ on $S$ if for any $t_0 \in I$ there exists a neighborhood $U (\gamma (t_0))$ in $S$ and an orientation preserving homeomorphism $g$ between $U$ and an open set $V\subset {\mathbb{R}}^2$ sending the foliation ${\cal F}$ onto the vertical foliation, oriented with increasing $y$ coordinate, such that the mapping $t \to p_1 (g(\gamma (t)))$ is strictly increasing in a neighborhood of $t_0$, where $p_1$ denotes projection over the first coordinate.
In this case, we can define signed transverse measures $\hat \nu^s$ and $\hat \nu ^u$, assigning positive measure to positively transverse arcs, and negative measures to negatively transverse arcs. This implies that one has the nice property of additivity, that is $\hat \nu (\alpha \cdot \beta) = \hat \nu (\alpha ) + \hat \nu (\beta)$, where $\hat \nu$ is any of the transverse measures.
These signed transverse measures can be extended to any arc in the same fashion as we did for the positive ones $\nu^s$ and $\nu^u$. In this case $\hat \nu(\gamma)$ is invariant under homotopy with fixed endpoints, so no infimum needs to be taken. Moreover, it can be shown that $\hat\nu$ comes from integration of a closed $1$-form on $S$.
Notice that equations \[eq1\] and \[eq2\] hold for $\hat\nu^s$ and $\hat\nu^u$, and in both cases $|\hat\nu(\gamma)|\leq \nu(\gamma)$.
We define the functions $\varphi _s , \varphi _u : S \to {\mathbb{R}}$ by the equations $$\varphi _s (x) = \hat \nu ^s (\gamma_x), \ \varphi _u (x) = \hat \nu ^u (\gamma_x).$$Notice that $\varphi _s$ and $\varphi _u$ are continuous (thus bounded) by continuity of the stable and unstable foliations of $h$. By Corollary \[arcs\], together with additivity, we get that:
$$\label{ecu1}\lambda ^m \varphi_s (h^{-m}(x)) = \hat \nu ^s (\gamma _x ^{n^m}) = \sum _{k=0}^{n^m-1} \varphi _s (f^k (x)).$$
Analogously,
$$\label{ecu2} \lambda ^{-m} \varphi _u (h^{-m}(x)) = \hat \nu ^u (\gamma _x ^{n^m}) = \sum _{k=0}^{n^m-1} \varphi _u (f^k (x)).$$
\[fijos\] If $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$, then $x\in \operatorname{Fix}(f)$ if and only if $\varphi _u (x) = 0$.
By Lemma \[fixFs\], $f$ preserves each semi-leaf of ${\cal F}^s$. For $x\in S$, let $\alpha_x$ be a stable semi-leaf segment between $x$ and $f(x)$. Then $\alpha_x$ is transverse to ${\cal F}^u$, and we have that $|\hat \nu^u(\alpha_x)|=\nu^u(\alpha_x)$ and $\hat \nu^u(\alpha_x)=0$ if and only if $\alpha_x$ is a point, i.e. $x\in\operatorname{Fix}(f)$. If $x\in \operatorname{Fix}(f)$, then $x$ is contractible, wich imples $\varphi _u (x) = 0$ as $\gamma_x$ is homotopically trivial (see Lemmas \[fixcont\] and \[pap\] item 1).
\[sign\] If $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$, then the sign of the real-valued function $\varphi _u$ is constant.
Let $U = \varphi _ u ^{-1} ((0, +\infty))$ and $V = \varphi _ u ^{-1} ((-\infty, 0))$. Suppose that $\varphi_u$ changes sign, so that $U$ and $V$ are both non-empty. Observe they are disjoint open sets with boundaries $\partial U = \partial V = \operatorname{Fix}(f)$ as Lemma \[fijos\] gives $\operatorname{Fix}(f) = \varphi _u ^{-1} (0)$. Take $x\in U$ and $B= B (x, \epsilon)\subset U$. Then, for all $m\geq 0$, $h^m (B)$ contains a leaf segment of the unstable foliation of length $\lambda ^m \epsilon$. Moreover, for sufficiently big $m$ $h^m (B)$ intersects both $U$ and $V$ because the $h$- iterates of unstable leaf segments accumulate all over $S$. As $h^m (B)$ is connected, we obtain $h^m (B)\cap \operatorname{Fix}(f)\neq \emptyset$. Now, recall that $\operatorname{Fix}(f)$ is $h^{-1}$- invariant (Lemma \[bsp\] item 2.). Then, $B\cap \operatorname{Fix}(f)\neq \emptyset$, a contradiction, as $B\subset U$.
We are now ready to prove Theorem \[teo1\] in the orientable case:
If $\lambda>n$, Lemma \[nus0\] gives us $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$. Now, by Lemma \[fijos\], $\operatorname{Fix}(f) = \varphi _u ^{-1} (0)$. Therefore, we have to prove $\varphi _u \equiv 0$. By Lemma \[sign\], we may assume $\varphi _u \geq 0$ (the case $\varphi _u \leq 0$ is analogous). Then, for all $x\in S$ the series $\sum _{k=0}^\infty \varphi _u (f^k(x))$ is of positive terms and either converges or diverges. Its limit can be computed as the limit of any subsequence of partial sums. In particular, by equation \[ecu2\],$$\sum _{k=0}^\infty \varphi _u (f^k(x)) = \lim _{m\to \infty} \sum _{k=0}^{n^m -1} \varphi _u (f^k(x)) =
\lim_{m\to \infty} \lambda ^{-m} \varphi _ u (h^{-m} (x)) = 0.$$Then, every term must be zero, that is $\varphi _ u \equiv 0$, proving the theorem.
\[pimba\] If $ \nu ^s (\gamma _x ) = 0$ for all $x\in S$, then $f= \operatorname{Id}$.
Note that in the previous proof we only needed $ \nu ^s (\gamma _x ) \equiv 0$ to conclude.
[The conservative case in the torus]{}
In this section we prove Theorem \[toro\]. This result is a consequence of the work of A. Koropecki and F. Tal in [@korotal].
We say that a closed subset $K\subset T ^2$ is [*fully essential*]{} if any loop in ${\mathbb{T}}^2 \backslash K$ is null- homotopic.
\[kt\] Let $f: {\mathbb{T}}^2 \to {\mathbb{T}}^2$ be an irrotational homeomorphism preserving a Borel probability measure of full support, and let $\tilde f$ be its irrotational lift. Then one of the following holds:
1. $\operatorname{Fix}(f)$ is fully essential;
2. Every point in ${\mathbb{R}}^2$ has bounded $\tilde f$- orbit;
3. $\tilde f$ has uniformly bounded displacement in a rational direction; i.e. there is a nonzero $v\in {\mathbb{Z}}^2$ and $M>0$ such that $$|<\tilde f ^n (z)-z, v>|\leq M$$for all $z\in {\mathbb{R}}^2$ and $n\in {\mathbb{Z}}$.
We assume in this section that $h : {\mathbb{T}}^2 \to T ^2$ is an Anosov homeomorphism, $f : {\mathbb{T}}^2 \to T ^2$ is isotopic to the identity preserving a Borel probability measure of full support, and $hfh ^{-1} = f ^n$ for some $n\geq 2$.
\[unacomp\] $T^2 \backslash \operatorname{Fix}(f)$ is connected and $h^{-1}$ - invariant.
By Lemma \[bsp\] item 2, $T^2 \backslash \operatorname{Fix}(f)$ is $h^{-1}$ invariant. Suppose that $T^2 \backslash \operatorname{Fix}(f)$ has two different connected components $U$ and $V$ and take a segment of unstable leaf $\delta\subset U$. Then, if $m$ is big enough $h^m(\delta)\cap \operatorname{Fix}(f)\neq \emptyset$ because $h^m (\delta)$ is connected and intersects both $U$ and $V$. But then, $\delta \cap \operatorname{Fix}(f)\neq \emptyset$, which contradicts that $\delta\subset U\subset T^2 \backslash \operatorname{Fix}(f)$.
\[fixess\] $T^2 \backslash \operatorname{Fix}(f)$ contains an essential simple closed curve.
Let $D = T^2 \backslash \operatorname{Fix}(f)$. Then, the previous lemma implies that $D$ is connected and $h^{-1}$- invariant. By Lemma \[pap\] item 4., there exists $x\in D\cap \operatorname{Per}(h)$. Take a segment $\delta \subset W^u_ h (x)\cap D$ through $x$ and a flow box of the unstable foliation $U\subset D$. Follow the segment of stable leaf from $x$ until it hits $\delta$ again. Note that this whole segment is contained in $D$ and that is transverse to ${\cal F}^u$. We can modify this segment slightly, just inside $U$, to obtain a loop that is transverse to ${\cal F}^u$. As the foliations of an Anosov toral homeomorphism are non-singular, it follows that this loop is homotopically non trivial. This loop is the desired essential closed curve.
\[racd\] If $\{\tilde f ^{n^m}(z): m\in {\mathbb{Z}}\}$ is not bounded, then for any nonzero $v\in {\mathbb{Z}}^2$ $|<\tilde f ^{n^m} (z)-z, v>|$ is not bounded.
Note that $$\lim _{m\to \infty} \nu ^u (\gamma_x ^{n^m})= \lim _{m\to \infty} \lambda ^{-m} \nu ^u (\gamma _{h^{-m}(x)}) = 0$$(see Corollary \[arcs\]). So, if $\{\tilde f ^{n^m}(z): m\in {\mathbb{Z}}\}$ is not bounded it has an irrational asymptotical direction (that of the unstable eigenvector of $\tilde h$).
\[bounded\] If $\{\tilde f ^{n^m}(z): m\in {\mathbb{Z}}\}$ is bounded for all $z\in {\mathbb{R}}^2$, then $\nu ^s(\gamma _ x) \equiv 0$.
Let $x\in \operatorname{Per}^m(h)$, and let $ \widetilde {h ^m}$ be the lift of $h^m$ fixing a lift $\tilde x$ of $x$. Let $\Phi := \widetilde {h ^m}$, so that $\Phi (\tilde x ) = \tilde x$. Lemma \[up\] gives us $\Phi \tilde f \Phi ^{-1} = \tilde f ^{n^m}$. Note that $$\Phi^k \tilde f (\tilde x) = \Phi^k \tilde f \Phi^{-k} (\tilde x) = \tilde f^{(n^m)^k},$$and that we are assuming that $\{\tilde f ^{{n^m}^k} (\tilde x): k\in {\mathbb{Z}}\}$ is bounded. This implies that $\tilde f (x)\in W^s_{\Phi} (\tilde x)$. Furthermore, $W^s_{\Phi} (\tilde x)$ projects to $W^s_{h^m} (x) \subset W ^s _h (x)$. So, $f(x)\in W^s _h (x) $ and $\nu ^s (\gamma _ x)= 0$ over the set $\operatorname{Per}(h)$. As this set is dense (Lemma \[pap\] item 4.), one obtains $\nu ^s(\gamma _ x) \equiv 0$.
We are now ready to prove Theorem \[toro\]:
Recall that Proposition \[gl\] states that $f$ is irrotational. We may now apply Theorem \[kt\]. Lemma \[fixess\] states that item 1 does not hold. If item 2 holds, then Lemma \[bounded\] tells us that $\nu ^s(\gamma _ x) \equiv 0$. Then, $f = \operatorname{Id}$ by Corollary \[pimba\]. If item 3 holds Lemma \[racd\] tells us that $\{\tilde f ^{n^m}(z): m\in {\mathbb{Z}}\}$ is bounded for all $z\in {\mathbb{R}}^2$, and we are done by re-applying Lemma \[bounded\].
[99]{}
Available on http://www.math.utah.edu/∼bestvina
Bull. Amer. Math. Soc. , 689 (1962) pp. 199-201
(2011), 72-157.
Volume 131, Number 3 (2006), 397-591
Duke Math. Jour. [Vol. 106, No. 3 (2001), 581-597.]{}
Princeton University Press (2011)
Invent. Math. , 131 (1998) pp. 419-451
Ill. J. Math. [10 (1996), 563-573.]{}
arXiv:1207.5573
Preprint.
Bull. Braz. Math. Soc. (N.S.) 35 (2004) 13-50
Trans. Amer. Math. Soc. 278 (1983) 401-414
Ann. Sc. ENS [22,1 (1989), 99-108.]{}
Aequ. math. 2008, Volume 76, Issue 1-2, pp 105-111.
Proc. Internat. Congr. Math. (Berkeley 1986), Vol 2
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---
abstract: 'In this work we present a method of decomposition of arbitrary unitary matrix $U\in{\mathbf{U}}(2^k)$ into a product of single-qubit negator and controlled-$\sqrt{\mbox{NOT}}$ gates. Since the product results with negator matrix, which can be treated as complex analogue if bistochastic matrix, our method can be seen as complex analogue of Sinkhorn-Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively and resulting circuit consists of $O(4^k)$ entangling gates, which is proved to be optimal. An example of such transformation is presented.'
author:
- Adam Glos
- Przemysław Sadowski
bibliography:
- 'negatorBib.bib'
title: Constructive quantum scaling of unitary matrices
---
Introduction
============
Scaling a real matrix $O$ with non-negative entries means finding diagonal matrices $D_1,D_2$ such that $B=D_1OD_2$ is bistochastic. Sinkhorn theorem presents a necessary and sufficient condition for existence of the decomposition of a matrix. Moreover, the iterative Sinkhorn-Knopp algorithm finds the bistochastic matrix $B$ [@sinkhorn1967concerning]. Such decomposition can be used for ranking web pages [@knight2008sinkhorn], preconditioning sparse matrices [@livne2004scaling] and understanding traffic circulation [@knight2012fast].
Since unitary matrices are complex analogue of orthogonal matrices, it is natural to ask whether there exist a counterpart of Sinkhorn theorem for them. De Vos and De Baerdemacker considered whether it is possible, that for arbitrary unitary matrix $U\in{\mathbf{U}}(n)$ there exist two unitary diagonal matrices $U_1,U_2$ such, that matrix $U_1UU_2$ has all lines sums equal to 1. Such decomposition exists for arbitrary unitary matrix and an algorithm for finding it approximately was presented [@vos2014scaling]. Matrices called *negators* were treated as quantum counterpart of bistochastic matrices and form a group ${\mathbf{XU}}(n)$ under multiplication. Idel and Wolf propose an application of the quantum scaling in quantum optics [@idel2015sinkhorn].
Algorithm converges for arbitrary unitary matrix $U$ [@de2015two]. Similar decomposition of unitary matrices $U\in{\mathbf{U}}(2m)$ called $bZbXbZ$ decomposition was presented [@fuhr2015biunimodular]. They show, that there always exist matrices $A,B,C,D\in{\mathbf{U}}(m)$ such that $$U = \begin{bmatrix}
A & 0 \\ 0 & B
\end{bmatrix}
\frac{1}{2}\begin{bmatrix}
{\mathrm{I}}+C & {\mathrm{I}}-C \\ {\mathrm{I}}-C & {\mathrm{I}}+C
\end{bmatrix}
\begin{bmatrix}
{\mathrm{I}}& 0 \\ 0 & D
\end{bmatrix},$$ where ${\mathrm{I}}$ is identity matrix. Matrix in the middle is a block-negator matrix (which is also a negator matrix), while left and right matrices are block diagonal matrices. In [@de2015sinkhorn] an algorithm of finding such decomposition was presented.
Group ${\mathbf{XU}}(2^n)$ is isomorphic to ${\mathbf{ U}}(2^n-1)$ and can be generated by single-qubit negator and controlled-$\sqrt{\mbox{NOT}}$ gates [@de2013negator]. However, the proof is non-constructive since a decomposition designed for generating random matrices was used [@pozniak1998composed]. Although it is proved that it exists for any unitary matrix, obtaining such a decomposition is a very complex task. Therefore another approach is needed for efficient decomposition procedure.
In this article, using similar method to presented by de Vos and de Baerdemacker [@de2013negator], we demonstrate an implementation of arbitrary $k$-qubit unitary operation using one-qubit ancilla with controlled-$ \sqrt{\textrm{NOT}}$ and single-qubit negator gates. Since product of these basic negator gates is still a negator matrix, our result can be seen as quantum analogue of scaling matrix. More precisely we prove, that for arbitrary matrix $U\in {\mathbf{U}}(2^k)$, which is performed on system ${\mathcal{H}}$, there exist a negator $N\in{\mathbf{XU}}(2^{k+1})$ such that for arbitrary state ${| \psi \rangle}\in {\mathcal{H}}$ we have $$U{| \psi \rangle} = \Psi(N \Phi({| \psi \rangle})).$$ Here $\Phi$ denotes the operation of extending the system with an ancilla register in ${| - \rangle}$ state and $\Psi$ denotes partial trace over the ancilla system. Since after performing operations $\Phi$ and $N$ the state is of the form ${| - \rangle}{\otimes}U{| \psi \rangle}$, the partial trace is simply removing the ancilla system giving a pure state $U{| \psi \rangle}$. We describe an efficient algorithm that for given $U$ returns explicit and exact form of $N$ with decomposition into a sequence of single-qubit negator and controlled-$\sqrt{\textrm{NOT}}$ gates only in contrast to results of de Vos and de Baerdemacker [@de2015sinkhorn; @de2013negator].
In Section 2 we recall basic facts. In Section 3 we show how to perform such transformation efficiently and demonstrate the cost in term of controlled-$
\sqrt{\textrm{NOT}}$ gates. To illustrate the transformation method, a transformation of Grover’s search algorithm is presented step by step in Section 4.
Basic facts
===========
Negator gates of dimension 2 were introduced by de Vos and de Baerdemacker [@de2013negator] as unitary matrices $N\in{\mathbf{U}}(2)$ which are also a convex combination of identity matrix and NOT gate. Simple calculation shows, that they are of the form $$N(\theta)=\frac{1}{2} \begin{bmatrix}
1 + e^{i\theta} & 1 - e^{i\theta} \\
1 - e^{i\theta} & 1 + e^{i\theta}
\end{bmatrix},$$ where $\theta\in[0,2\pi]$. Negators form a subgroup of single-qubit unitary operations, i.e. $N(\phi)N(\psi)=N(\phi+\psi)$ for any values of $\phi$ and $\psi$. In the following we will also use a 2-qubit negator operation controlled-$\sqrt{\textrm{NOT}}$ gate (which is also controlled-$N(\frac{\pi}{2})$ gate) $$\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \frac{1+i}{2} & \frac{1-i}{2} \\
0 & 0 & \frac{1-i}{2} & \frac{1+i}{2} \\
\end{bmatrix}.$$ As these gates are used as basic operators, we will use a simplified notation in circuit, respectively
-- ----- -- -- -- -- --
and
-- ----- -- -- -- -- --
.
These two kinds of unitary matrices will be called NCN gates (*Negators-Controlled-Negator*).
In Section 3 decomposition of single-qubit unitary gates will be needed. Every unitary matrix $U\in
{\mathbf{U}}(2)$ can be presented as a product of global phase, two $z$-rotators and one $y$-rotator [@nielsen2010quantum] $$\label{eq:unitary2Dec}
\begin{split}
U &= e^{i\phi_0}
\begin{bmatrix}
\cos \frac{\phi_1}{2} e^{i\phi_2} & \sin\frac{\phi_1}{2} e^{i\phi_3}\\
-\sin\frac{\phi_1}{2} e^{-i\phi_3} & \cos \frac{\phi_1}{2} e^{-i\phi_2}
\end{bmatrix} \\
&=
e^{i\phi_0}
\begin{bmatrix}
e^{i\frac{\phi_2+\phi_3}{2}} & 0 \\
0 & e^{-i\frac{\phi_2+\phi_3}{2}}
\end{bmatrix}
\begin{bmatrix}
\cos \frac{\phi_1}{2} & \sin\frac{\phi_1}{2}\\
-\sin\frac{\phi_1}{2} & \cos \frac{\phi_1}{2}
\end{bmatrix}
\begin{bmatrix}
e^{i\frac{\phi_2-\phi_3}{2}} & 0 \\
0 & e^{-i\frac{\phi_2-\phi_3}{2}}
\end{bmatrix} \\
&= e^{i\phi_0} R_z(-\phi_2-\phi_3)R_y(\phi_1)R_z(\phi_3-\phi_2).
\end{split}$$ Since global phase is not measurable, we can simplify this representation without loss of information $$U \cong R_z(\gamma)R_y(\beta)R_z(\alpha),
\label{eq:unitary2DecSimp}$$ where ‘$\cong$’ means equality up to a global phase. The same applies in the case of global phase change on one of the registers of a bigger system $$U_1{\otimes}e^{i\phi} U_2 {\otimes}U_3 = e^{i\phi}(U_1{\otimes}U_2{\otimes}U_3) \cong U_1{\otimes}U_2{\otimes}U_3 .$$ Using these two facts we can say that in any situation we can ignore global phase change on any register.
While it may lead to a conclusion that our transformation is mainly applied to group ${\mathbf{SU}}(n)$, we decided to stay with the unitary matrices formalism, since negator gates are not special unitary matrices. The result may be written using the special matrices, however then the negators gates column and row sums will equal $e^{i\theta}$ in general.
Circuit transformation method
=============================
In this section we provide complete description of the transformation method. We recall a sketch of a proof of universality theorem between quantum gates and negator gates from the work of de Vos and de Baerdemacker [@de2013negator]. Next we present transformation method of arbitrary single-qubit gate into NCN product. Then we provide a method of decomposition for arbitrary $k$-qubit circuit, based on the single qubit case. Finally, we analyse the cost of presented transformation.
Universality theorem
--------------------
De Vos and de Baerdemacker proved a universality theorem: group ${\mathbf{XU}}(2^k)$ generated by negators and controlled-$ \sqrt{\textrm{NOT}}$ is isomorphic to ${\mathbf{U}}(2^k-1)$ [@de2013negator]. The proof consists of several steps:
1. Every matrix $U\in {\mathbf{U}}(2^k-1)$ can be decomposed into a product of $m$ gates $U_1U_2\dots U_m$, where matrices $U_i\in {\mathbf{U}}(2^k-1)$ are of some special forms [@pozniak1998composed].
2. Group ${\mathbf{U}}(2^k-1)$ is isomorphic to group $${\mathbf{^{1}U}}(2^k) = \left \{\begin{bmatrix} 1 & \mathbf 0 ^T
\\ \mathbf 0 & U\end{bmatrix}: U\in {\mathbf{U}}(2^k-1) \right \},$$ because of the isomorphism $h: {\mathbf{U}}(2^k-1) \to {\mathbf{^{1}U}}(2^k) $ $$h(U) = \begin{bmatrix}
1 & \mathbf 0 \\
\mathbf 0 & U
\end{bmatrix}.$$
3. Function $f : {\mathbf{^{1}U}}(2^k) \to {\mathbf{XU}}(2^k)$ of the form $f(U)=(H\kron{\mathrm{I}}_{2^k})U(H\kron{\mathrm{I}}_{2^k})$ is an isomorphism.
4. Decomposition of every $f(h(U_i))$ into a product of NCN gates is possible, where $U_i$ comes from point 1.
The proof used the decomposition presented in the work of Poźniak, Życzkowski and Kuś [@pozniak1998composed], because it is proven that the decomposition exists for any unitary matrix. However obtaining such decomposition is a very complex task. Therefore we need to choose a different decomposition in order to find an efficient decomposition procedure.
Obviously, group ${\mathbf{U}}(2^k)$ is isomorphic to some subgroup of ${\mathbf{XU}}(2^{k+1})$. In other words, with ancilla (one additional qubit) every unitary matrix can be replaced with a sequence of NCN gates. For our purpose we choose function $g:{\mathbf{U}}(2^k) \to {\mathbf{XU}}(2^{k+1})$ $$g(U) = \frac{1}{2}H{\otimes}{\mathrm{I}}({| 0 \rangle \langle 0 |}{\otimes}{\mathrm{I}}+
{| 1 \rangle \langle 1 |}{\otimes}U)H{\otimes}{\mathrm{I}}= \frac{1}{2}
\begin{bmatrix}
{\mathrm{I}}& {\mathrm{I}}\\ {\mathrm{I}}& -{\mathrm{I}}\end{bmatrix}
\begin{bmatrix}
{\mathrm{I}}& \mathbf 0\\
\mathbf 0 & U
\end{bmatrix}\begin{bmatrix}
{\mathrm{I}}& {\mathrm{I}}\\ {\mathrm{I}}& -{\mathrm{I}}\end{bmatrix}. \label{eq:isomorphism}$$ Using the function $g$, every gate $U$ changes into controlled-$U$. Using circuit notation we can present this fact as
Note that if we assume that the first qubit is set to ${| - \rangle}$, the control qubit does not influence the result (the condition is always ‘true’).
Single-qubit gate transformation
--------------------------------
Now we aim at decomposition of arbitrary single-qubit gate into NCN gates. With Eq. (\[eq:unitary2DecSimp\]) for any $U\in {\mathrm{L}}({\mathbb{C}}^2)$ there exist real parameters $\alpha,\beta,\gamma$ such that $$U \cong R_z(\gamma)R_y(\beta)R_z(\alpha).$$ Therefore after applying function $g$ we have
We change the rotators with neighbouring Hadamard gates into NCN gates as in Fig. (\[fig:basicGatesDecomposition\])
Let us note that the symbols of controlled-NOT, controlled-$\sqrt{\textrm{NOT}}^\dagger$ and controlled-negator used in the decomposed circuit do not mean that these gates cannot be transformed. We use these symbols as a simplified notation for its decomposition with use of controlled-$\sqrt{\textrm{NOT}}$ gates as shown in Fig. (\[fig:NotsNot\]).
\
\
\
\
General transformation method
-----------------------------
Now we consider transformation of arbitrary $k$-qubit circuit. Let us assume that we have a circuit which consists of unitary operations $U\in{\mathrm{L}}({\mathbb{C}}^{2^k})$, generalized measurement $\mathbf M =
\{M_a\in{\mathrm{L}}({\mathbb{C}}^{2^k}):a\in\Sigma\}$, where $\Sigma$ is a set of classical outputs of measurement, and starting state ${| \phi_0 \rangle}$
In order to construct a decomposition of unitary $U$ into a sequence of negator gates we begin with obtaining a decomposition of $U$ into controlled-NOT and single-qubit gates
here denoted by a sequance of gates $U=V_m\cdots V_1$. Contrary to the decomposition presented in the work of Poźniak, Życzkowski and Kuś, there exist efficient methods for constructing such circuit [@mottonen_quantum_2004]. Next we need to add an additional qubit, transform $V_i$ gates into controlled-$V_i$ gates and add Hadamard gates as below (since $HH={\mathrm{I}}$)
Let us note that product $H\cdot \textrm{controlled-}V_j\cdot H$ is an image of homomorphism presented in Eq. (\[eq:isomorphism\]) on $V_j$. Next we replace the product with the sequence of NCN gates (here denoted by $\mathbf N_j$) as in previous subsection (if $V_j$ is controlled-NOT, then we choose Toffoli gate transformation from Fig. (\[fig:basicGatesDecomposition\]))
For the sake of simplicity we may change the starting state and resulting state on the first wire
Now we have an equivalent circuit which consists of negators and controlled-$ \sqrt{\textrm{NOT}}$ gates only.
Transformation cost
-------------------
Now we consider upper bound of cost of decomposition into negator circuit. Two kinds will be discussed: memory complexity and number of single and two-qubit gates. In the first case for arbitrary $k$-qubit circuit transformation requires one additional qubit.
Let $c_{\textrm{CNOT}}(k) $ and $c_s(k)$ denote upper bound of the number of respectively controlled-NOT and single qubit-gates needed for the implementation of an arbitrary $k$-qubit circuit. Using the operation presented above we need $17c_{\textrm{CNOT}}(k) + 64c_s(k)$ controlled-$
\sqrt{\textrm{NOT}}$ gates and $11c_{\textrm{CNOT}}(k)+34c_s(k)$ negators to implement an equivalent circuit (up to global phase).
Any circuit which consists of controlled-NOT and single-qubit gates can be simplified in such a way, that $c_s(k)\leq 2c_{\textrm{CNOT}}(k)+k$. This estimation is based on the worst case, when there are two single-qubit gates between every controlled-NOT gate. Taking this into account we can express the previous result in terms of $c_{\textrm{CNOT}}$ only, because only $17c_{\textrm{CNOT}}(k)+ 64c_s(k) \leq 145 c_{\textrm{CNOT}}(k)+64k$ controlled-$ \sqrt{\textrm{NOT}}$ gates are needed. In fact, if $c_{\textrm{CNOT}} = O(4^k)$, then so is the number of controlled-$
\sqrt{\textrm{NOT}}$ gates.
Step by step transformation example
===================================
To illustrate the introduced decomposition we will present Grover’s algorithm for $k=2$ qubits as NCN circuit. The original circuit for this algorithm is presented in Fig. (\[fig:originalGroverAlgorithm\]), where $\omega$ denotes the searched state.
As in the previous section, we will add one qubit, change every $H$ and $G$ gate into controlled-$H$ and controlled-$G$ respectively and add Hadamard gates on the ancilla register. Former steps of the decomposition are explicitly presented in Fig. (\[fig:GroverDecomposition\]). The following facts were used
- the decomposition of Hadamard gate is $H\cong
R_z(\pi)R_y(\frac{\pi}{2})R_z(0)=R_z(\pi)R_y(\frac{\pi}{2})$,
- the decomposition of NOT gate is $\mbox{NOT}\cong
R_z(\pi)R_y(\pi)R_z(0)=R_z(\pi)R_y(\pi)$,
- for any $U,V\in{\mathrm{L}}({\mathbb{C}}^2)$ we have
- Grover’s diffusion operator can be decomposed in the following way
Decomposition of $U_\omega$ depends strictly on the value of $\omega$, therefore it is not presented in the example. The full decomposition is presented in Fig. (\[fig:GroverDecomposition\]).
Concluding remarks
==================
In the presented work we provide a constructive method of scaling arbitrary unitary matrices $U\in {\mathbf{U}}(2^k)$. More precisely we proved that for arbitrary unitary matrix $U\in {\mathbf{U}}(2^k)$ there exists unitary negator matrix $N\in {\mathbf{XU}}(2^{k+1})$ such that for arbitrary state ${| \psi \rangle}$ we have $$U{| \psi \rangle} = \Psi(N \Phi({| \psi \rangle})).$$ Here $\Phi$ denotes the operation of extending the system with an ancilla register in ${| - \rangle}$ state and $\Psi$ denotes partial trace over the ancilla system. We described efficient algorithm of decomposing $N$ into product of single-qubit negator and controlled-$\sqrt{\mbox{NOT}}$ gates. Our decomposition consists of $O(4^k)$ entangling gates which is proved to be optimal and needs one qubit ancilla.
Our result can be seen as complex analogue of Sinkhorn-Knopp algorithm, which is known to have wide applications. The result is in contrast to the previous results [@de2013negator], which could be only used to prove the existence of such decomposition. Moreover, our transformation is exact and can be found constructively. In contrast to [@de2015sinkhorn], our transformation consists only of negator gates. The main difference is that transformation needs one-qubit ancilla.
Acknowledgements {#acknowledgements .unnumbered}
================
The work was supported by the Polish National Science Centre: A. Glos under the research project number DEC-2011/03/D/ST6/00413, P. Sadowski under the research project number 2013/11/N/ST6/03030.
|
---
abstract: 'We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: We construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.'
author:
- Benjamin Brück
bibliography:
- 'mybibliography.bib'
title: Maximal hyperbolic towers and weight in the theory of free groups
---
\[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Fact]{} \[thm\][Proposition]{}
\[thm\][Definition]{} \[thm\][Remark]{} \[thm\][Example]{}
Introduction
============
Around 1945, Tarski asked the question whether all non-abelian free groups share the same first order theory. The affirmative was given independently by Kharlampovich and Myasnikov ([@MyasnikovElementary]) and Sela ([[@SelaDiophantine]]{}). However, being a free group is not a first order property. This means that in addition to the free groups, there are also non-standard models of their theory $T_{fg}$ (also called *elementary free groups*), i.e. groups that share the same theory as free groups but are not free themselves. Sela gave a geometric description of all finitely generated models of $T_{fg}$ by introducing the notion of a hyperbolic tower. He showed that the following is true (see [@SelaDiophantine Theorem 6] and the comments on it in [@LPSTowers]):
\[introduction trivial tower implies T\_fg\] Let $G$ be a finitely generated group. Then $G$ is a model of $T_{fg}$ if and only if $G$ is non-abelian and admits a hyperbolic tower structure over the trivial subgroup.
Furthermore and more surprisingly, Sela showed in [@SelaCommonTheory] that the common theory $T_{fg}$ of non-abelian free groups is stable. This provided a new and rich example of a group that is, on the one hand, a classical and complex structure but, on the other hand, tame enough in the model theoretic sense to allow the application of the various tools developed in stability theory. Conversely, the study of free groups in algebra and topology has brought forth many geometric methods that can now be used to refine stability-theoretic analysis.
This is the context in which this article is set. Motivated by model theoretic ideas, we seek to gain a better understanding of hyperbolic towers by applying geometric tools that include Whitehead graphs, Bass-Serre theory, and covering spaces.
If $G$ is a non-abelian, finitely generated group, we call a free subgroup $H\leq G$ a *maximal free ground floor*, if $G$ admits a hyperbolic tower structure over $H$, but not over any other free subgroup in which $H$ is a free factor. From the perspective of model theory, a basis of $H$ now plays a similar role for the group $G$ that a basis plays for an arbitrary free group, meaning that both such sets have the same type and are maximal independent with respect to forking independence over $\emptyset$. This is clear if $G$ is a free group itself, but much more interesting if $G$ is a non-standard model of $T_{fg}$ where we have a priori no notion of a basis. Our main result in the first part of this article is:
\[Theorem differences in sizes\] For each $n\in \mathbb{N}$, there is a finitely generated group that has one hyperbolic tower structure over a maximal free ground floor of basis length $2$ and another tower structure over a maximal free ground floor of basis length $n+2$.
We explicitely construct these different tower structures, building on ideas of Louder, Perin and Sklinos (see [@LPSTowers]).
Closely related to this is the weight of the type $p_0$ of a primitive element in a free group. Pillay showed in [@PilForking] that $p_0$ is the unique generic type over the empty set in $T_{fg}$. In general, if a type $p$ has finite weight, its weight bounds the ratio of the sizes of maximal independent sets of realisations of $p$. Hence, Theorem \[Theorem differences in sizes\] can also be seen as an alternative proof for the infinite weight of $p_0$, a fact already proven by Pillay ([@PilGenweight]) and Sklinos ([@Rizinfweight]).
In the last section of this article, we extend Sklinos’ techniques in order to show the following:
[thmletter]{}[Theoreminfiniteweight]{} \[Theoreminfiniteweight\] In $T_{fg}$, every non-algebraic (1-)type over the empty set that is realised in a free group has infinite weight.
The organisation of the article is as follows: We start in Section \[Section Bass Serre and towers\] with a short account of Bass-Serre theory and surface groups before presenting the definition of a hyperbolic tower. In Section \[Section model theory\], we give some model-theoretic basics. Afterwards, we present a criterion for a subgroup to be a maximal free ground floor in Section \[Section maximal ground floors\] and use this to proof Theorem \[Theorem differences in sizes\] in Section \[Section build the towers\]. Finally, Section \[Section infinite weight\] contains more details about weight and introduces Whitehead graphs in order to proof Theorem \[Theoreminfiniteweight\].
The results of this article are taken from the author’s master thesis. Many thanks are due to Rizos Sklinos and Tuna Altinel for all their help, time and patience during the creation of this work. I would also like to thank Katrin Tent for her helpful advise especially on the final presentation of this article. Furthermore, I am grateful for Chloé Perin’s comments on coverings of surfaces that made it possible to state Theorem \[maximaltowers\] in a more general form and to simplify its proof.
Bass-Serre theory and hyperbolic towers {#Section Bass Serre and towers}
=======================================
In this section, we collect some notions from geometric group theory needed for this article, define hyperbolic towers and give the results about them that we will use later. It follows [@LPSTowers Section 3].
Bass-Serre theory
-----------------
We begin with Bass-Serre theory and will only give the ideas and most important definitions. For more details, the reader is referred to [@Trees].
A *graph of groups* is a connected graph $\Gamma$, together with two collections of groups, ${\ensuremath{ \lbrace G_v \rbrace}}_{v\in V(\Gamma)}$ (the *vertex groups*) and ${\ensuremath{ \lbrace G_e \rbrace}}_{e\in E(\Gamma)}$ (the *edge groups*), and, for each edge $e\in E(\Gamma)$ that has endpoints $v_1$ and $v_2$, two embeddings $\alpha_e:G_e\hookrightarrow G_{v_1}$ and $\omega_e:G_e\hookrightarrow G_{v_2}$. We denote such a graph of groups by $(\mathbb{G},\Gamma)$. To a graph of groups we can associate its *fundamental group* $\pi_1(\mathbb{G},\Gamma)$. It is defined by $$\pi_1(\mathbb{G},\Gamma){\coloneqq}
\left\langle{
\begin{array}{c@{\hspace{0.5ex}:\hspace{0.5ex}}l|c@{\hspace{0.5ex}:\hspace{0.5ex}}l}
G_v &v\in V(\Gamma),& t_e^{-1}\alpha_e(g) t_e=\omega_e(g) &e\in E(\Gamma), g\in G_e, \\
t_e&e\in E(\Gamma)\,&t_e=1 &e\in E(\Gamma_0)\\
\end{array}}\right\rangle,$$ where $\Gamma_0\subseteq \Gamma$ is a maximal tree in $\Gamma$. So this fundamental group consists of the elements of the vertex groups of $(\mathbb{G},\Gamma)$, together with new so-called *Bass-Serre elements* $t_e$ which are introduced for each edge $e$ of $\Gamma$. The relations inside the vertex groups stay as before. Relations between elements of different vertex groups are defined by identifying images of the given embeddings and conjugating with the corresponding $t_e$. Furthermore, whenever $e$ takes part of a fixed maximal tree $\Gamma_0$, the corresponding element $t_e$ is made trivial. The remaining non-trivial Bass-Serre elements are called *Bass-Serre generators*. This means that $\pi_1(\mathbb{G},\Gamma)$ is derived from the vertex groups by a series of amalgamated products or HNN-extensions where the stable letter is the corresponding Bass-Serre generator. One can show that the isomorphism class of this fundamental group does not dependent on the choice of $\Gamma_0$. However, taking another maximal subtree changes the presentation of $\pi_1(\mathbb{G},\Gamma)$ and the choice of Bass-Serre generators.
Whenever we have a graph of groups decomposition (or *splitting*) of a group $G$ (i.e. a graph of groups with fundamental group $G$), we can find a canonical action of $G$ on a simplicial tree $T$ whose quotient $G{\ensuremath{\backslash}}T$ is isomorphic to $\Gamma$. On the other hand, whenever $G$ acts on a simplicial tree $T$ without inversions, we get a graph of groups decomposition of $G$ with underlying graph isomorphic to $G{\ensuremath{\backslash}}T$. In both cases we know that vertex (respectively edge) groups of the graph of groups are conjugate to the stabilisers of the vertices (respectively edges) of the action on $T$. In this situation, an element or a subgroup of $G$ fixing a point in $T$ is called *elliptic*.
The easiest example of this is the case where $G=H\ast R$ is the free product of subgroups $H$ and $R$. In this case, the corresponding graph of groups has one edge connecting two vertices, one with vertex group $H$, the other with vertex group $R$. The edge group is trivial. If we take the same setting with a non-trivial edge group, we get an amalgamated product of $H$ and $R$.
Given an action on a tree, there several different corresponding graph of groups decompositions corresponding to a choice of “presentation” that is defined as follows:
Let $G$ be a group acting on a tree $T$ without inversions, denote by $(\mathbb{G},\Gamma)$ the associated graph of groups and by $p$ the quotient map $p:T\to \Gamma$. A *Bass-Serre presentation* for $(\mathbb{G},\Gamma)$ is a pair $(T^1,T^0)$ consisting of
- a subtree $T^1$ of $T$ which contains exactly one edge of $p^{-1}(e)$ for each edge $e$ of $\Gamma$;
- a subtree $T^0$ of $T^1$ which is mapped injectively by $p$ onto a maximal subtree $\Gamma_0$ of $\Gamma$.
Surface groups
--------------
In the whole text, we assume all surfaces to be connected and compact.
It is a standard fact from the classification of surfaces that every surface $\Sigma$ is determined up to homeomorphism by its orientability, its Euler characteristic $\chi (\Sigma)$ and the number of its boundary components $b(\Sigma)$. The sphere has Euler characteristic $2$, the torus has characteristic $0$. Puncturing a surface decreases its Euler characteristic by $1$. If we decompose a surface $\Sigma$ into two surfaces $\Sigma_1$ and $\Sigma_2$, we have $\chi(\Sigma)=\chi(\Sigma_1)+\chi(\Sigma_2)$.
If $\Sigma$ is a surface with non-empty boundary, each of its boundary components has a cyclic fundamental group, which gives rise to a conjugacy class of cyclic subgroups in $\pi_1(\Sigma)$. They are called *maximal boundary subgroups*. A *boundary subgroup* of $\pi_1(\Sigma)$ is a non-trivial subgroup of a maximal boundary subgroup.
Let $\Sigma$ be an orientable surface with $r$ boundary components, and let $s_1,\ldots, s_r$ be generators of non-conjugate maximal boundary subgroups. Then $\pi_1(\Sigma)$ has a presentation of the form $$\begin{gathered}
{\ensuremath{\left\langle y_1,\ldots , y_{2m},s_1,\ldots ,s_r | [y_1,y_2]\ldots [y_{2m-1},y_{2m}]=s_1\ldots s_r\right\rangle}}\end{gathered}$$ where $\chi(\Sigma)=-(2m-2+r)$. In particular, if $\Sigma$ has non-empty boundary, we can apply a Tietze transformation by removing one of the $s_i$’s and the relation and thus get another presentation of $\pi_1(\Sigma)$ which shows that it is a free group of rank $1-\chi(\Sigma)$. This is true for non-orientable surfaces as well.
Let $\Sigma$ be a surface with non-empty boundary and $P{\coloneqq}\pi_1(\Sigma)$ its fundamental group. Let $\mathcal{C}$ be a set of 2-sided disjoint simple closed curves on $\Sigma$ that allows a collection ${\ensuremath{ \lbrace T_c|c\in \mathcal{C} \rbrace}}$ of disjoint open neighbourhoods of the curves in $\mathcal{C}$ with homeomorphisms $c\times (-1,1)\to T_c$ sending $c\times{\ensuremath{ \lbrace 0 \rbrace}}$ onto $c$. Assume in addition that no component of $\Sigma{\ensuremath{\backslash}}\cup\, \mathcal{C}$ has trivial fundamental group. Then we get a splitting of the group $P$ that we call the *decomposition of $P$ dual to $\mathcal{C}$*. It is defined as follows: For each connected component $\Sigma_k$ of $\Sigma{\ensuremath{\backslash}}\bigcup_{c\in \mathcal{C}}T_c$ we get a vertex whose vertex group is $\pi_1(\Sigma_k)$. For each curve in $\mathcal{C}$ separating the components $\Sigma_k$ and $\Sigma_{k'}$, we get an edge $e_c$ with infinite cyclic edge group between the vertices corresponding to $\Sigma_k$ and $\Sigma_{k'}$ (we allow $k=k'$). Using functoriality of $\pi_1$, the inclusion maps $c\hookrightarrow \Sigma_k$ induces the embeddings of the edge groups. Such a decomposition is called the *decomposition of $P$ dual to $\mathcal{C}$*. Note that here, all boundary subgroups are elliptic and edge groups are infinite cyclic. The following lemma gives a converse for this. Originally being [[@MSValuations Theorem III.2.6.]]{}, this version is a slight variation given in [@LPSTowers] as Lemma 3.2.
\[dual decomposition\] Let $\Sigma$ be a surface with non-empty boundary and $P{\coloneqq}\pi_1(\Sigma)$ be its fundamental group. Suppose that $P$ admits a graph of groups decomposition $(\mathbb{G},\Gamma)$ in which edge groups are cyclic and boundary subgroups are elliptic. Then there exists a set $\mathcal{C}$ of disjoint simple closed curves on $\Sigma$ such that $(\mathbb{G},\Gamma)$ is the graph of groups decomposition dual to it.
Hyperbolic floors and towers {#Hyperbolic towers}
----------------------------
\[Definition graph of groups with surfaces\] A *graph of groups with surfaces* is a graph of groups $(\mathbb{G},\Gamma)$ together with a subset $V_S$ of the vertex set $V(\Gamma)$, such that any vertex $v$ in $V_S$ satisfies:
- there exists a surface $\Sigma$ with non-empty boundary, such that the vertex group $G_v$ is the fundamental group $\pi_1(\Sigma)$ of $\Sigma$;
- for each edge $e$ that has endpoint $v$, the embedding $G_e\hookrightarrow G_v$ maps $G_e$ onto a maximal boundary subgroup of $\pi_1(\Sigma)$;
- this induces a bijection between the set of edges adjacent to $v$ and the set of conjugacy classes of maximal boundary subgroups in $\pi_1(\Sigma)$.
The vertices of $V_S$ are called *surface (type) vertices* and, with a slight abuse of language, the vertex groups associated to surface type vertices are called *surface (type) groups*. The surfaces associated to the vertices of $V_S$ are called the surfaces of $(\mathbb{G},\Gamma)$.
\[Definition hyperbolic floor\] Let $(G,G',r)$ be a triple consisting of a group $G$, a subgroup $G' \leq G$ and a retraction $r$ from $G$ onto $G'$ (i.e. $r$ is a morphism $G\to G'$ which restricts to the identity on $G'$). We say that $(G,G',r)$ is a *hyperbolic floor*, if there exists a graph of groups with surfaces $(\mathbb{G},\Gamma)$ with associated fundamental group $\pi_1(\mathbb{G},\Gamma)=G$ and a Bass-Serre presentation $(T^1,T^0)$ of $(\mathbb{G},\Gamma)$ such that:
1. all the surfaces of $(\mathbb{G},\Gamma)$ are either once punctured tori or have Euler characteristic at most $-2$;
2. $G'$ is the free product of the stabilisers of the non-surface type vertices of $T^0$;
3. every edge of $\Gamma$ joins a surface type vertex to a non-surface type vertex;
4. either the retraction $r$ sends surface type vertex groups of $(\mathbb{G},\Gamma)$ to non-abelian images, or the subgroup $G'$ is cyclic and there exists a retraction $r':G\ast \mathbb{Z} \to G'\ast \mathbb{Z}$ that does this.
Let $G$ be a non-cyclic group and $H\leq G$ a subgroup. We say that $G$ is a *hyperbolic tower* over $H$ (or *admits a hyperbolic tower structure* over $H$), if there is a sequence of subgroups $G=G_0\geq G_1 \geq \ldots \geq G_m\geq H$ satisfying the following conditions:
- for each $0\leq i\leq m-1$, there exists a retraction $r_i:G_i\to G_{i+1}$ such that the triple $(G_i, G_{i+1}, r_i)$ is a hyperbolic floor and $H$ is contained in one of the non-surface type vertex groups of the corresponding graph of groups decomposition;
- $G_m=H\ast F\ast S_1\ast\ldots\ast S_p$ where $F$ is a (possibly trivial) free group, $p\geq 0$ and each $S_i$ is the fundamental group of a closed surface of Euler characteristic at most $-2$.
![A hyperbolic tower over $H$ consisting of two floors[]{data-label="Starting example"}](startingexample.png)
It is helpful to have in mind the following image of hyperbolic towers: If $G$ admits a hyperbolic tower structure over $H$, we can see $G$ as the fundamental group of a topological space $X_0$ that is derived from a space $X_H$ having fundamental group $H$ in several steps. We start with a space $X_m$ that is the disjoint union of $X_H$, a graph $X_F$ and closed surfaces $\Sigma_1,\ldots ,\Sigma_p$ of Euler characteristic at most $-2$. When $X_{i+1}$ is constructed, we get $X_{i}$ by gluing surfaces along their boundary components to $X_{i+1}$ such that there exist suitable retractions.
An example of this is shown in Figure \[Starting example\]. The nested boxes mark the sequence of subgroups of $G$. Note that although a surface represents a surface type vertex in the corresponding graph of groups, we did not mark the non-surface type vertices. The ends of the edges starting at the punctured surfaces represent the (here unspecified) points to which their boundary components are glued.
As mentioned in the introduction, hyperbolic towers gain importance for the study of $T_{fg}$ by Fact \[introduction trivial tower implies T\_fg\] which we restate here in the following way:
\[trivial tower implies T\_fg\] Let $G$ be a finitely generated group. Then $G$ is a model of $T_{fg}$ if and only if $G$ is non-abelian and admits a hyperbolic tower structure over a free subgroup.
Here again, we consider the trivial group to be a free group as well. That this is equivalent to the formulation given above follows immediately from the definitions.
Model-theoretic basics {#Section model theory}
======================
In this section, we will give some model theoretic basics. This will be done very briefly because although those model theoretic ideas motivate the constructions given later on, they are not needed to understand them as the definition of hyperbolic towers translates those model theoretic problems in the language of geometric group theory. For a general introduction to model theory, see for example [@TentZiegler], for details about stability theory, see [@PilGeomStabTheory].
As already mentioned, we know that the common first order theory $T_{fg}$ of non-abelian free groups is stable. Stable theories enjoy a model theoretic notion of independence between elements in a given model which is called forking independence. It can be seen as a generalisation of linear independence in vector spaces and algebraic independence in algebraically closed fields, which are also basic examples for this. From now on, whenever we talk about independence, we mean forking independence. If two elements are not forking independent, we say that they “fork” with each other. By the results of Sela, we now can ask whether a set of elements in a free group or another model of $T_{fg}$ is independent or not.
Recall that an $m$-type over a set $A$ of a first order theory $T$ is a maximal consistent set of formulas with parameters in $A$ and at most $m$ free variables. If $G$ is a model of $T$ and $a\in G$, then the *type of $a$ over $A$*, denoted by $tp(a/A)$, is the set of all formulas with parameters in $A$ satisfied by the element $a$. An important property in stable theories is the existence of *generic types* over any set of parameters. A definable set $X$ of a stable group $G$ is said to be *generic*, if finitely many left- (or equally right-) translates of $X$ cover $G$. A formula $\psi(x)$ is called generic, if it defines a generic set. Finally we say that a type is generic, if it contains only generic formulas. Hence we can imagine a generic type to be a type with a “big” set of realisations. By results of Poizat, we know that in the theory of free groups, there is a unique generic (1-)type $p_0\in S_1(T_{fg})$ over the empty set (see [@PilForking]), namely the type of a primitive element in a free group.
This type is especially interesting because of the following Fact:
In a finitely generated, non-abelian free group $F$, a set is a maximal independent set of realisations of $p_0$ if and only if it forms a basis of $F$.
This means that at first glance, maximal independent sets of realisations of $p_0$ in non-standard models of $T_{fg}$ could be seen as analogues to bases in free groups. This is due to the fact that they look the same from the perspective of first order logic, meaning that both such sets satisfy exactly the same first order formulas and both are maximal independent with respect to forking independence. However, those sets do not necessarily generate the groups that they are taken from. Furthermore, there is no fixed size of such a ”basis“ in a non-standard model, as we will show by proving Theorem \[Theorem differences in sizes\].
Towers with maximal ground floors {#Section maximal ground floors}
=================================
In this section, we will define maximal free ground floors and give an instruction on how to attain such floors by proving Theorem \[maximaltowers\]. Afterwards, we will provide a model-theoretic approach to these maximal tower structures.
Maximal free ground floors
--------------------------
Let $G$ be a finitely generated model of $T_{fg}$. A subgroup $H\leq G$ is called a *maximal free ground floor (in $G$)* if $H$ is free and $G$ admits a hyperbolic tower structure over $H$ but not over any other free subgroup $K\leq G$ in which $H$ is a free factor $K=H\ast H'$.
Bearing in mind Fact \[trivial tower implies T\_fg\], the fact that $G$ admits a tower structure over $H$ already implies that it is a model of $T_{fg}$.
For the proof of Theorem \[maximaltowers\], we begin by collecting some lemmas about graphs of groups. The following is part of the statement of [[@MSValuations Theorem III.2.6.]]{} and the comments after it.
\[Lemma subtree of action\] Let $\Sigma$ be a surface, possibly with boundary, such that $P{\coloneqq}\pi_1(\Sigma)$ acts on a tree $T$ in a way that all edge stabilisers are cyclic and boundary subgroups act elliptically. Then there is a subtree $T_0$ of $T$ that is invariant under the action of $P$ such that all edge stabilisers of the action of $P$ on $T_0$ are non-trivial, thus infinite cyclic.
Using this, we deduce:
\[indecomposabilitysurfacegroups\] Let $A_1,\ldots ,\,A_k$ be any groups and let $P\leq A_1\ast\ldots\ast A_k$ be a subgroup of their free product. Assume in addition that $P$ is the fundamental group of a surface with boundary. If every boundary subgroup of $P$ can be conjugated into some $A_i$, the group $P$ can be conjugated into one of those factors as well. If we know in addition that $P\cap A_j\not= {\ensuremath{ \lbrace 1 \rbrace}}$, we get $P\leq A_j$.
In this situation, we know that $P$ acts on $T$, the tree associated to the free product $A_1\ast\ldots\ast A_k$, in a way that all boundary subgroups act elliptically and all edge stabilisers are trivial. So all conditions of the preceding lemma are fulfilled and we know that there is a subtree $T_0$ which is invariant under the action. If $P$ cannot be conjugated into one of the factors, this subtree cannot be trivial, so it contains at least one edge of $T$. Furthermore, the stabiliser of this edge has to be infinite cyclic which is a contradiction. Hence we know that $P\leq A_i^x$ for some $x\in A_1\ast\ldots\ast A_k$. The second part is an immediate consequence of the free product structure.
\[acylindricity\] Let $(\mathbb{G,}\Gamma)$ be a graph of groups with surfaces decomposition of a group $G$ that comes from a hyperbolic floor structure $(G, G',r)$ and let $T$ be the corresponding tree. Then the canonical action of $G$ on $T$ is *1-acylindrical around surface type vertices*. That is, no element $g\in G{\ensuremath{\backslash}}{\ensuremath{ \lbrace 1 \rbrace}}$ fixes more than one non-surface type vertex.
If we have $g\in G$ that fixes two non-surface type vertices, it also fixes the shortest path between them. Since every edge of $T$ joins a surface type vertex to a non-surface type vertex, this means that $g$ fixes a segment of the tree consisting of a surface type vertex and two different edges adjacent to it. Suppose this surface type vertex is given by the coset $hP$ of the surface type vertex group $P$. With this notation, the element $g'{\coloneqq}h^{-1}g h$ fixes the vertex $(1\cdot)P$ and two different edges adjacent to it. Inspecting the structure of $T$, one can see that this implies that $g'\in C_e^{p_e}\cap C_{e'}^{p_{e'}}$ where $e,e'\in E(\Gamma)$ are edges in $\Gamma$ which are adjacent to the vertex corresponding to $P$. The groups $C_e, C_{e'}\leq P$ are the maximal boundary subgroups corresponding to those edges and $p_{e},p_{e'}\in P$ are elements of the surface group $P$. Since $g'\not=1$, this means that $$C_e^{p_e p_{e'}^{-1}}\cap C_{e'}\not= {\ensuremath{ \lbrace 1 \rbrace}}.$$ Now there are two possibilities to consider: If $e=e'$, we have $C_e^{p}\cap C_{e}\not= {\ensuremath{ \lbrace 1 \rbrace}}$ for cyclic subgroups $C_e$ and an element $p$ of $P$. The fact that $P$ is a free group implies that $C_e^{p}= C_{e}$ which is a contradiction to the assumption that the two edges in $T$, that are fixed by $g$, are distinct.
If on the other hand $e\not= e'$, we can conclude that the maximal boundary subgroups $C_e$ and $C_{e'}$ are conjugate in $P$. This is a contradiction, as the definition of a graph of groups with surfaces demands that different edges correspond to different conjugacy classes of maximal boundary subgroups.
With this we can proof the following theorem which we want to use to construct examples of maximal free ground floors in Section \[Section build the towers\]. We denote by $F_n$ the free group in $n$ generators.
\[maximaltowers\] Suppose that a finitely generated group $G$ admits a hyperbolic tower structure over $H\cong F_n$ with the associated sequence of subgroups $G=G_0\geq G_1 \geq \ldots\geq G_m=H$ subject to the following conditions:
1. The graph of groups corresponding to the floor $(G_{i}, G_{i+1},r_i)$ consists of two vertices. One of them has vertex group $G_{i+1}$, the other one is a surface type vertex with vertex group $P_i{\coloneqq}\pi_1(\Sigma_i)$.
2. For all $i$, the surface $\Sigma_i$ is either a once punctured torus, a four times punctured sphere or a thrice punctured projective plane.
Then $H$ is a maximal free ground floor in $G$.
Suppose that $G$ admits a second tower structure over a free subgroup $K=H\ast H'$ and take the associated sequence of subgroups to be $G=G'_0\geq G'_1 \geq \ldots\geq G'_l\geq K$. After dividing the graph of groups decompositions of this tower structure and taking free products, we may assume that for each hyperbolic floor in this structure, the associated graph of groups consists of only one surface type and one non-surface type vertex that are connected by at least one edge. We denote by $T_j'$ the associated tree of the graph of groups decomposition corresponding to the floor $(G'_j, G'_{j+1}, r'_j)$.
First look at the top floor $(G'_0=G, G'_1, r'_{0})$. As $P_{m-1}$, the surface group that comes with the ground floor of the first tower, is a subgroup of $G$, it acts on $T_0'$. Every maximal boundary subgroups of $P_{m-1}$ can be conjugated into $H$ by its corresponding Bass-Serre element $t$ and thus acts elliptically, fixing a vertex corresponding to the coset $tG'_1$. Therefore, this action induces a splitting of $P_{m-1}$ that is by Lemma \[dual decomposition\] dual to a set $\mathcal{C}$ of disjoint simple closed curves on $\Sigma_{m-1}$. We can assume this set of curves to be essential, i.e. no component of $\Sigma_{m-1}{\ensuremath{\backslash}}\cup\, \mathcal{C}$ is homeomorphic to a disc with one or no puncture.
This means that $\Sigma_{m-1}$ is decomposed into subsurfaces $(\Sigma''_k)_k$ whose fundamental groups all act elliptically on $T_0'$.
If for no $k$, the fundamental group $\pi_1(\Sigma''_k)$ stabilises a surface type vertex of $T_0'$, we know that $P_{m-1}$ is conjugate to a subgroup of $G'_1$. Using acylindricity and the fact that at least one boundary subgroup of $P_{m-1}$ is identified with a subgroup of $H\leq G_1'$, one can show that $P_{m-1}\leq G'_1$. Thus, it acts on the next floor $(G'_1, G'_2, r'_1)$. Furthermore, if $P_{m-1}\leq G'_j$ and $t$ is a Bass-Serre generator arising in the graph of groups associated to the hyperbolic floor $(G_{m-1},G_m,r_{m-1})$, we have $t\in G'_j$ as well. Iterating this process we see that either $P_{m-1}\leq G'_l$, or for some $0\leq j < l$ and a subsurface $\Sigma''$ of $\Sigma_{m-1}$, the fundamental group $\pi_1(\Sigma'')$ is not included in $G'_{j+1}$ and thus fixes a surface type vertex of $T'_j$.
Assume $P_{m-1}\leq G'_l$. As the second tower structure of $G$ is over $K$, we know that $G'_l=H\ast F\ast S_1\ast\ldots\ast S_p$ for a free group $F$ and surface groups $S_i$. All boundary subgroups of $P_{m-1}$ can be conjugated into subgroups of $G_m=H$ by their corresponding Bass-Serre element. As all those Bass-Serre elements take part of $G_l'$ as well, Lemma \[indecomposabilitysurfacegroups\] now implies $P_{m-1}\leq H$ which is impossible.
So we know that for some $j<l$, the group $\pi_1(\Sigma'')$ fixes a surface type vertex of $T_j'$. After changing the Bass-Serre presentation of $T_j'$, we can assume that $\pi_1(\Sigma'')\leq P_j'$ where $P_j'=\pi_1(\Sigma'_{j})$ is the surface group arising in the graph of groups decomposition of the floor $(G'_j, G'_{j+1}, r'_j)$.
If $\pi_1(\Sigma'')$ is an infinite index subgroup of $P_j'$, we know by [@PerThesis Lemma 3.10] that $\pi_1(\Sigma'')=C_1\ast \ldots \ast C_m\ast F$ where $F$ is a (possibly trivial) free group, each $C_j$ is a boundary subgroup of $P'_j$ and any boundary element of $P'_j$ contained in $\pi_1(\Sigma'')$ can be conjugated into one of the groups $C_j$ by an element of $\pi_1(\Sigma'')$. Because the subsurface $\Sigma''\subseteq \Sigma_{m-1}$ comes from the graph of groups decomposition corresponding to the action of $P_{m-1}$ on the tree $T_j'$, we know that $\pi_1(\Sigma'')$ embeds into $P_j'$ as a surface group with boundaries. I.e. the boundary subgroups of $\pi_1(\Sigma'')$ are given by the boundary subgroups of $P_j'$ that lie in $\pi_1(\Sigma'')$. By Lemma \[indecomposabilitysurfacegroups\], this implies that $\pi_1(\Sigma'')$ has to be included completely in a boundary subgroup of $P'_j$ which is a contradiction.
Hence, the index $n{\coloneqq}[P_j':\pi_1(\Sigma'')]$ is finite. Now, by the study of covering spaces from topology, we know that there is a covering map $p:\Sigma''\to \Sigma'_j$ of degree $n$ such that $\chi(\Sigma'')=n\cdot\chi(\Sigma'_j)$. As $\Sigma''$ is a subsurface of a once punctured torus, a four times punctured sphere or a thrice punctured projective plane, it has Euler-characteristic $-2$ or $-1$, so the index $n$ is either $1$ or $2$.
Assume that $n=2$. This can only be true if $\Sigma''$ has Euler characteristic $\chi (\Sigma'')=-2$ and $\Sigma_{j}'$ has Euler characteristic $\chi(\Sigma_{j}')=-1$. Since we assumed that the set of curves dividing $\Sigma_{m-1}$ is essential, one can deduce that in this case, $\Sigma''=\Sigma_{m-1}$ is no proper subsurface. The only surface with Euler characteristic $-1$ allowed in a hyperbolic tower structure is a once punctured torus, so we know that $\Sigma_{j}'$ has exactly one boundary component. On the other hand, it quickly follows from the definition of a covering map that $$1=b(\Sigma_{j}')\leq b(\Sigma_{m-1})\leq n\cdot b(\Sigma_j')=2.$$ As we know that $b(\Sigma_{m-1})\in{\ensuremath{ \lbrace 3,4 \rbrace}}$, this is a contradiction.
Thus we know that $n=1$, which implies that $\Sigma'_j\cong\Sigma_{m-1}$ and $P'_j=P_{m-1}$ seen as subgroups of $G$. Reordering the floors of the second tower, we may assume that $j=l-1$ which means that $G'_{l-1}$ is derived from $G'_{l}$ by gluing $\Sigma'_{j}=\Sigma_{m-1}$ to $H\leq G'_{l}$ in the same way as in the first tower.
Continuing with the action of $P_{m-2}$ on the second tower, we can apply almost the same arguments. The only thing that one needs to think about is why $P_{m-2}\leq G'_{l-1}$ is impossible. However, in the last paragraph, we assumed that $G'_{l-1}=G_{m-1}\ast F\ast S_1\ast\ldots\ast S_p$. As all boundary subgroups of $P_{m-2}$ can be conjugated into subgroups of $G_{m-1}$, we can again apply Lemma \[indecomposabilitysurfacegroups\] to get a contradiction.
In the end of this induction process, we see that $$\begin{gathered}
G=G'_0= G_0\ast F\ast S_1\ast\ldots\ast S_p =G\ast F\ast S_1\ast\ldots\ast S_p\, ,\end{gathered}$$ so in particular, $F$ is trivial and we have shown the maximality.
In fact, the proof shows that the second tower can be changed into the first one by permuting floors and dividing floors with several surface vertices into floors with only one surface vertex each. So up to those changes, there is only one hyperbolic tower structure of $G$ over $H$.
Model theoretic formulation
---------------------------
The motivation to look at maximal free ground floors comes from the next statement:
\[independent realizations because of tower\] Let $G$ be a non-abelian finitely generated group. Then $k$ elements $u_1,\ldots ,u_k$ of $G$ form an independent set of realisations of $p_0$ if and only if $H{\coloneqq}\langle u_1,\ldots ,u_k \rangle\leq G$ is free of rank $k$ and $G$ admits a hyperbolic tower structure over $H$.
This immediately implies the following corollary:
\[Corollary maximal ground floors and p\_0\] A subgroup $H\cong F_n$ of a finitely generated group $G$ is a maximal free ground floor in $G$ if and only if each basis of $H$ is a maximal independent set of realisations of $p_0$.
So the generators of maximal free ground floors are exactly the analogues to bases mentioned at the end of Section \[Section model theory\]. However, in contrast to free groups, not all such “bases” of a fixed non-standard model of $T_{fg}$ have the same cardinality. The fact that the ratios between the basis lengths of two such subgroups can even get arbitrarily large is what we will prove in the next section.
Constructing such towers {#Section build the towers}
========================
In this section, we give examples of models of $T_{fg}$ that each contain maximal free ground floors of different basis lengths.
The ideas of this are taken from [@LPSTowers Proposition 5.1] which now can be seen as the special case $n=1$ of Theorem \[alternative proof for infinite weight\].
A special case with pictures {#2s5s}
----------------------------
To start with and in order to explain the idea, we will construct a group that contains one maximal free ground floor of basis length $2$ and one of basis length $5$. Doing this, we will emphasise the geometric motivation and give a more technical proof in the general case afterwards.
At first, we look at two hyperbolic floors that we will use during the construction. Let $H$ be any non-abelian group, $\Sigma$ a four times punctured sphere and $\Sigma'$ a once punctured torus. We describe the hyperbolic floors by their decompositions as graphs of groups with surfaces. In both cases, we have one non-surface type vertex with vertex group $H$ and one surface type vertex with vertex group $\pi_1(\Sigma)$ (respectively $\pi_1(\Sigma')$). As required by the definitions, the edge groups of these graphs of groups with surfaces are identified with maximal boundary subgroups.
### Gluing $\Sigma$ to $H$ {#gluing 3 holes}
We know that there is a presentation $$\pi_1(\Sigma)={\ensuremath{\left\langle s_1,s_2,s_3,s_4 | s_1s_2s_3s_4=1\right\rangle}} ,$$ where the $s_i$’s are generators of non-conjugate maximal boundary subgroups of $\pi_1(\Sigma)$. As there are four conjugacy classes of maximal boundary subgroups, the two vertices of the graph of groups we describe are connected by four edges. Thus, we get three Bass-Serre generators $t_1, t_2, t_3$. The embeddings of the edge groups into $H$ are given by identifying $$t_1^{-1}s_1t_1=w_1,\, t_2^{-1}s_2t_2=w_1^{-1},\, t_3^{-1}s_3t_3=w_2,\, s_4=w_2^{-1}$$ for any two non-commuting elements $w_1,w_2\in H$. The result is the group $$\begin{gathered}
G{\coloneqq}{\ensuremath{\left\langle H, t_1,t_2,t_3 | t_1 w_1 t_1^{-1}t_2 w_1^{-1} t_2^{-1}=[w_2,t_3]\right\rangle}}. \end{gathered}$$ If one looks at the retraction given by $$\begin{aligned}
\nonumber
r:G & \to & H \\
t_1,t_2,t_3 & \mapsto & 1
\nonumber\, ,\end{aligned}$$ $\pi_1(\Sigma)$ is sent to $\langle w_1, w_2\rangle\leq H$. Thus, the tuple $(G,H,r)$ is a hyperbolic floor.
### Gluing $\Sigma'$ to $H$ {#gluing one hole}
Writing $$\pi_1(\Sigma')=\langle y_1, y_2, s | [y_1,y_2]=s\rangle ,$$ the element $s$ is a generator of a maximal boundary subgroup. Identifying $s$ with the commutator $[w_1,w_2]$ for any non-commuting elements $w_1,w_2\in H$, we get the group $$G'{\coloneqq}{\ensuremath{\left\langle H, y_1, y_2 | [y_1,y_2]=[w_1,w_2]\right\rangle}}.$$ By adding the retraction $$\begin{aligned}
\nonumber
r':G' & \to & H \\
\nonumber
y_1 & \mapsto & w_1\\
\nonumber
y_2 & \mapsto & w_2,\end{aligned}$$ we get a hyperbolic floor $(G',H,r')$.
We now use these two kinds of floors to construct a group with different maximal tower structures.
\[Theorem 2 and 5\] The group $$\begin{gathered}
G{\coloneqq}
\left\langle{
\begin{array}{c|l@{\hspace{0.5ex}=\hspace{0.5ex}}l}
a_1,a_2,t_1,t_2,t_3, & t_1 a_1 t_1^{-1}t_2 a_1^{-1} t_2^{-1}&[a_2,t_3], \\
t_4,t_5,t_6, & t_4 a_2 t_4^{-1}t_5 a_2^{-1} t_5^{-1}&[t_1^{-1},t_6], \\
t_7,t_8,t_9 & t_7 t_1^{-1} t_7^{-1}t_8 t_1 t_8^{-1}&[t_4^{-1},t_9] \\
\end{array}} \right\rangle\end{gathered}$$ is a model of $T_{fg}$ and contains maximal free ground floors of basis lengths $2$ and $5$.
By Fact \[trivial tower implies T\_fg\], the fact that $G$ contains some maximal free ground floor already implies that $G$ has the same theory as a free group, so it suffices to describe such tower structures of $G$.
![The tower structures of $G$ and $G'$; on the left the basepoint $\star$ matching the isomorphism $f:G\to G'$ is marked[]{data-label="Firststep in example"}](4timessphrewith.png)
We begin by observing that $G$ has a hyperbolic tower structure over $\langle a_1, a_2\rangle\cong F_2$ consisting of three floors of the form $G=G_0\geq G_1\geq G_2\geq G_3=\langle a_1, a_2\rangle$. It is illustrated on the left of Figure \[Firststep in example\]. In all of the three floors, the corresponding graph of groups decomposition of $G_i$ consists of two vertices: one vertex with vertex group $G_{i+1}$ and one surface type vertex where the surface $\Sigma_i$ that is added is a four times punctured sphere which is glued along its boundary to $G_{i+1}$ as described in \[gluing 3 holes\]. Firstly, $G_2=\langle a_1, a_2, t_1, t_2, t_3\rangle$ is derived from $G_3$ by gluing the boundary components of $\Sigma_2$ to $a_1, a_1^{-1}, a_2$ and $a_2^{-1}$ (i.e. choosing $w_1=a_1$ and $w_2=a_2$) and adding Bass-Serre generators $t_1, t_2$ and $t_3$ for the first three gluings. From this, $G_1= {\left\langle}a_1, a_2, t_1, \ldots ,t_6 {\right\rangle}$ is derived by gluing the boundary components of $\Sigma_1$ to $a_2, a_2^{-1}, t_1^{-1}$ and $t_1$. Here, the Bass-Serre generators $t_4,t_5$ and $t_6$ are added. Lastly, we obtain $G=G_0$ from $G_1$ by gluing $\Sigma_0$ to $G_1$, identifying the sphere’s maximal boundary subgroups with the groups generated by $t_1^{-1},t_1, t_4^{-1}$ and $t_4$ and adding Bass-Serre generators $t_7,t_8$ and $t_9$. So the sequence of subgroups is given by the different lines in the presentation above. Although here, it is quite easy to believe that in all floors, the “gluing points” do not commute, this will be less obvious in the general case, so we check it now to explain how one can verify this. For the first floor $G_2\geq G_3$, it is clear that $a_1$ and $a_2$ do not commute as they form a basis of $G_3$. We will show later that the other floors fulfil this condition as well.
Clearly, the conditions of Theorem \[maximaltowers\] are satisfied. Thus, we know that ${\ensuremath{ \lbrace a_1,a_2 \rbrace}}$ is a maximal independent set of realisations of $p_0$. To get such sets of other sizes, we will step by step change the geometric interpretation of the hyperbolic floors.
At first, we observe that the first floor $G_2\geq G_3$ can be interpreted as a decomposition of a double torus with one arc. We imagine the handles of this double torus to be cut such that $G_3$ can be seen as the fundamental group of two loops connected by the arc and $G_2$ is derived from this by gluing the rest of the double torus (a four times punctured sphere) to it (see Figure \[Firststep in example\] on the left). On the other hand, there is another decomposition of this object that can be interpreted as a hyperbolic floor $G_2'\geq G_3'$: Here we cut the double torus between the two handles such that we gain a once punctured torus with an arc whose fundamental group is $G_3'$. What remains is another once punctured torus whose fundamental group is the surface type vertex group in this hyperbolic floor (as shown on the right of Figure \[Firststep in example\]). This implies that $G$ admits as well a presentation $G'$ of the following form: $$G'{\coloneqq}
\left\langle{
\begin{array}{c|l}
b_1,b_2,b_3,y_1,y_2, & [y_1,y_2]=[b_1,b_3], \\
t_4,t_5,t_6, & t_4 b_1 t_4^{-1}t_5 b_1^{-1} t_5^{-1}=[b_2,t_6], \\
t_7,t_8,t_9 & t_7 b_2 t_7^{-1}t_8 b_2^{-1} t_8^{-1}=[t_4^{-1},t_9] \\
\end{array}}\right\rangle.$$ The isomorphism $f:G\to G'$ is given by $$\begin{aligned}
\nonumber
f:G & \to & G' \\
\nonumber
a_1 & \mapsto & b_2y_1b_2^{-1}\\
\nonumber
a_2 & \mapsto & b_1\\
\nonumber
t_1 & \mapsto & b_2^{-1}\\
\nonumber
t_2 & \mapsto & y_2b_2^{-1}\\
\nonumber
t_3 & \mapsto & b_3\end{aligned}$$ and the identity on the other generators. This isomorphism $f$ sends the images of our gluing points for $\Sigma_1$ in the original tower structure of $G$ to $f(a_2)=b_1$ and $f(t_1^{-1})=b_2$. As $b_1$ and $b_2$ take part of a basis of $G'_3$, we see that $a_2$ and $t_1^{-1}$ do not commute in $G$.
![The reinterpreted tower structure of $G'$ and the one of $G''$[]{data-label="Second step in example"}](Example2-5secondstepwith)
Now we change the geometric interpretation of $G_3'={\left\langle}b_1, b_2, b_3{\right\rangle}$ and see it as the fundamental group of two loops which are connected by an arc and have corresponding generators $b_1$ and $b_2$ together with a third loop represented by $b_3$ (see Figure \[Second step in example\] on the left). Since the four times punctured sphere of the floor $G'_1\geq G_2'$ is now, similar to the case above, glued to the first two loops, we can apply the same procedure to get an isomorphism $f':G'\to G''$ onto the group $$\begin{gathered}
G''{\coloneqq}
\left\langle{
\begin{array}{c|l}
c_1,c_2,c_3,c_4, y_1,y_2, & [y_1,y_2]=[c_2y_3c_2^{-1},c_3], \\
y_3,y_4, & [y_3,y_4]=[c_1,c_4], \\
t_7,t_8,t_9 & t_7 c_1 t_7^{-1}t_8 c_1^{-1} t_8^{-1}=[c_2,t_9] \\
\end{array}}\right\rangle\end{gathered}$$ (see Figure \[Second step in example\]). The images of the gluing points for $\Sigma_0$ are $f'(f(t_1^{-1}))=c_1$ and $f'(f(t_4^{-1}))=c_2$. So we see that $t_1^{-1}$ and $t_4^{-1}$ do not commute in $G$.
Doing the same reinterpretation process a third time, we see that $G$ is isomorphic to $$\begin{gathered}
G'''{\coloneqq}
\left\langle{
\begin{array}{c|l}
d_1,d_2,d_3,d_4,d_5, y_1,y_2, & [y_1,y_2]=[d_1 y_3 d_1^{-1},d_3], \\
y_3,y_4, & [y_3,y_4]=[d_2y_5d_2^{-1},d_4], \\
y_5,y_6 & [y_5,y_6]=[d_1,d_5] \\
\end{array}}\right\rangle
.\end{gathered}$$ $G'''$ now has a hyperbolic tower structure $G'''=G'''_0\geq G'''_1\geq G'''_2\geq G'''_3=\langle d_1,\ldots d_5 ,\rangle$ with $$\begin{aligned}
G'''_1=\langle d_1, \ldots ,d_5, y_3,y_4,y_5,y_6\rangle \leq G''',& & G'''_2=\langle d_1, \ldots ,d_5, y_5, y_6 \rangle\leq G'''.\end{aligned}$$ Here, for all $i$, the corresponding graph of groups decomposition of $G'''_i$ consists of two vertices, one with vertex group $G'''_{i+1}$ and one surface type vertex where the surface is a once punctured torus. All those tori are glued to the floor below as described in \[gluing one hole\], their gluing points are $$\begin{aligned}
[d_1 y_3 d_1^{-1},d_3]&\in G'''_1,&[d_2y_5d_2^{-1},d_4]&\in G'''_2&\text{and}&&[d_1,d_5]&\in G'''_3.\end{aligned}$$ The only thing that one has to check is whether all those commutators are non-trivial. But this can be shown in the same way as it was done for the gluing points of the four times punctured spheres.
Hence, Theorem \[maximaltowers\] tells us that $\langle d_1,\ldots ,d_5\rangle$ is a maximal free ground floor in $G'''$ and taking its preimage, we find such a subgroup in $G$, too.
The general case
----------------
Now, we generalise the result of the last subsection to arbitrarily large ratios between the basis lengths of the ground floors. We start with the following technical proposition:
The group \[presentations\] $$\sbox0{\ensuremath{
\begin{array}{c|c}
a_1,a_2,t_1,t_2,t_3, & t_1 w_1 t_1^{-1}t_2 w_1^{-1} t_2^{-1}=[w_2,t_3], \\
t_4,t_5,t_6, & t_4 w_3 t_4^{-1}t_5 w_3^{-1} t_5^{-1}=[w_4,t_6], \\
\vdots& \vdots \\
t_{3n-2},t_{3n-1},t_{3n} & t_{3n-2} w_{2n-1} t_{3n-2}^{-1}t_{3n-1} w_{2n-1}^{-1} t_{3n-1}^{-1}=[w_{2n},t_{3n}] \\
\end{array}}}
\mathopen{G^n{\coloneqq} \resizebox{1.2\width}{1.1\ht0}{$\Bigg\langle$}}
\raisebox{2pt}{\usebox{0}}
\mathclose{\resizebox{1.2\width}{1.1\ht0}{$\Bigg\rangle$}}$$ with $$\begin{gathered}
w_1{\coloneqq}a_1,\, w_2=w_3{\coloneqq}a_2, \nonumber \\
w_{2i+2}=w_{2i+3}{\coloneqq}t^{-1}_{3i-2} \text{ for $i\geq 1$,}
\nonumber\end{gathered}$$ admits a presentation of the form $$\sbox0{\ensuremath{
\begin{array}{c|c}
e_1,\ldots , e_{n+2}, y_1,y_2, & [y_1,y_2]=[w'_1,w'_2], \\
y_3,y_4, & [y_3,y_4]=[w'_3,w'_4], \\
\vdots & \vdots \\
y_{2n-1},y_{2n} & [y_{2n-1},y_{2n}]=[w'_{2n-1},w'_{2n}] \\
\end{array}}}
\mathopen{{\tilde{G}^n}= \resizebox{1.2\width}{1.1\ht0}{$\Bigg\langle$}}
\raisebox{2pt}{\usebox{0}}
\mathclose{\resizebox{1.2\width}{1.1\ht0}{$\Bigg\rangle$}}$$ where for all $1\leq j\leq n$, the words $w'_{2j-1}$ and $w'_{2j}$ are elements in the subgroup generated by $e_1,\ldots ,e_{n+2},y_{2j+1},\ldots , y_{2n} $.
We give a sequence of isomorphisms $(f^{(i)}:G^{(i)}\to G^{(i+1)})_{0\leq i\leq n-1}$ where $G^{(0)}=G^n$, $G^{(n)}=\tilde{G}^n$ and, for $0< i < n$, the group $G^{(i)}$ is defined by the following presentation: $$\sbox0{${
\begin{array}{c|c}
a^{(i)}_1,\ldots , a^{(i)}_{i+2}, y_1,y_2, & [y_1,y_2]=[f_{i-1}(w_2),a^{(i)}_3], \\
y_3,y_4, & [y_3,y_4]=[f_{i-1}(w_4),a^{(i)}_4], \\
\vdots & \vdots \\
y_{2i-1},y_{2i},& [y_{2i-1},y_{2i}]=[f_{i-1}(w_{2i}),a^{(i)}_{i+2}], \\
t_{3i+1},t_{3i+2},t_{3i+3},& t_{3i+1} \mathbf{a^{(i)}_1} t_{3i+1}^{-1}t_{3i+2} \mathbf{(a^{(i)}_1)^{-1}} t_{3i+2}^{-1}=[\mathbf{a^{(i)}_2},t_{3i+3}],\\
t_{3i+4},t_{3i+5},t_{3i+6}, & t_{3i+4} \mathbf{a^{(i)}_2} t_{3i+4}^{-1} t_{3i+5} \mathbf{(a^{(i)}_2)^{-1}} t_{3i+5}^{-1}=[\mathbf{t^{-1}_{3i+1}},t_{3i+6}],\\
t_{3i+7},t_{3i+8},t_{3i+9}, &
t_{3i+7} \mathbf{t^{-1}_{3i+1}} t_{3i+7} t_{3i+8} \mathbf{t_{3i+1}} t_{3i+8}^{-1}=[\mathbf{t_{3i+4}^{-1}},t_{3i+9}],\\
\vdots & \vdots \\
t_{3n-2},t_{3n-1},t_{3n} & t_{3n-2} w_{2n-1} t_{3n-2}^{-1}t_{3n-1} w_{2n-1}^{-1} t_{3n-1}^{-1}=[w_{2n},t_{3n}]
\end{array}}$
}
\mathopen{\resizebox{1.2\width}{1.13\ht0}{$\Bigg\langle$}}
\raisebox{9pt}{\usebox{0}}
\mathclose{\resizebox{1.2\width}{1.13\ht0}{$\Bigg\rangle$}}$$ where $f_i{\coloneqq}f^{(i)}\circ f^{(i-1)}\circ\ldots \circ f^{(0)}$ and bold letters mark some images of the $w_j$’s that are important to understand this step. The isomorphisms are defined by $$\begin{aligned}
\nonumber
f^{(i)}:G^{(i)}&\to &G^{(i+1)} \\
\nonumber
a^{(i)}_1 & \mapsto & a^{(i+1)}_2 y_{2i+1} (a^{(i+1)}_2)^{-1}\\
\nonumber
a^{(i)}_2 & \mapsto & a^{(i+1)}_1\\
\nonumber
a^{(i)}_3 & \mapsto & a^{(i+1)}_3\\
\nonumber
&
\vdots&\\
\nonumber
a^{(i)}_{i+2} & \mapsto & a^{(i+1)}_{i+2}\\
\nonumber
t_{3i+1} & \mapsto & (a^{(i+1)}_2)^{-1}\\
\nonumber
t_{3i+2} & \mapsto & y_{2i+2} (a^{(i+1)}_2)^{-1}\\
\nonumber
t_{3i+3} & \mapsto & a^{(i+1)}_{i+3}\end{aligned}$$ and the identity on the remaining generators. (In the cases $i=0$ respectively $i=n-1$, we take $a^{(0)}_j{\coloneqq}a_j$ and $a^{(n)}_j{\coloneqq}e_j$.) Since we find a preimage for every generator of $G^{(i+1)}$, the map $f^{(i)}$ is surjective. We have $$\begin{gathered}
f^{(i)}(t_{3i+1} a^{(i)}_1 t_{3i+1}^{-1}t_{3i+2} (a^{(i)}_1)^{-1} t_{3i+2}^{-1})=[y_{2i+1},y_{2i+2}], \nonumber\\
f^{(i)}(t_{3i+1}^{-1})=(a^{(i+1)}_2),\nonumber\end{gathered}$$ and $f^{(i)}$ fixes all $t_j$ with $j>3i+3$. This shows that each relation in $G^{(i+1)}$ corresponds to exactly one relation in $G^{(i)}$, so $f^{(i)}$ is a well-defined homomorphism and injective.
Defining $$\begin{aligned}
w'_{2j}&{\coloneqq}f_{n-1}(t_{3j})=e_{j+2} ,\\
w'_{2j-1}&{\coloneqq}f_{n-1}(w_{2j}) ,\end{aligned}$$ it follows that $f_{n-1}=f^{(n-1)}\circ f^{(n-2)}\circ\ldots \circ f^{(0)}$ is an isomorphism between $G^n$ and $\tilde{G}^n$.
It remains to show that $w'_{2j-1}$ and $w'_{2j}$ lie in the subgroup generated by $e_1, \ldots , e_{n+2}$, $y_{2j+1}, \ldots , y_{2n}$. A short computation shows that $f_{j-1}(w_{2j})
=a_1^{(j)}$, so the smallest index of any instance of $y_k$ appearing in $f_{n-1}(w_{2j})$ is $k=2j+1$. As we already know that $w'_{2j}=e_{j+2}$, this finishes the proof.
Using this proposition, we can finally show the following which proves Theorem \[Theorem differences in sizes\]:
\[alternative proof for infinite weight\] The group $G^n$ as defined in Proposition \[presentations\] is a model of $T_{fg}$ that contains maximal free ground floors of basis lengths $2$ and $n+2$.
We will describe two hyperbolic tower structures of $G^n$ over free subgroups. As in the special case of Theorem \[Theorem 2 and 5\], the existence of such structures immediately implies that $G^n$ is a model of $T_{fg}$.
The first structure is over $\langle a_1, a_2\rangle\cong F_2$ and consists of $n$ floors. The associated sequence of subgroups is given by $G^n=G_0\geq G_1\geq \ldots\geq G_n=\langle a_1, a_2\rangle$ where $G_j$ is generated by $a_1,a_2,t_1,\ldots , t_{3(n-j)}$. For all floors, the corresponding graph of groups decomposition of $G_j$ consists of two vertices: one vertex with vertex group $G_{j+1}$ and one surface vertex where the surface $\Sigma_{j}$ added is a four times punctured sphere that is glued to $G_{j+1}$ as in \[gluing 3 holes\]. That is, the maximal boundary subgroups of $\pi_1(\Sigma_j)$ are identified with $w_{2(n-j)-1},\,w_{2(n-j)-1}^{-1},\, w_{2(n-j)}$ and $w_{2(n-j)}^{-1}$, which are all elements of $G_{j+1}$. Doing so, we have to add the Bass-Serre generators $t_{3(n-j)-2},t_{3(n-j)-1}$ and $t_{3(n-j)}$. As in the proof of Theorem \[Theorem 2 and 5\], we can deduce from the proof of Proposition \[presentations\] that $w_{2(n-j)-1}$ and $w_{2(n-j)}$ do not commute. This shows that these decompositions describe hyperbolic floors that satisfy all the conditions of Theorem \[maximaltowers\]. Consequently, we know that $\langle a_1,a_2\rangle$ is a maximal free ground floor.
On the other hand, Proposition \[presentations\] tells us that $G^n\cong\tilde{G}^n$ and $\tilde{G}^n$ admits a hyperbolic tower structure over $\langle e_1,\ldots,e_{n+2}\rangle\cong F_{n+2}$. Just like the first decomposition, it consists of $n$ floors where each associated graph of groups has one non-surface type vertex and one surface type vertex. Here, all the surfaces are once punctured tori denoted by $\tilde{\Sigma}_j$ and they are glued to the floors below as described in \[gluing one hole\]. The corresponding sequence of subgroups is $\tilde{G}^n=\tilde{G}_0\geq \tilde{G}_1\geq \ldots\geq \tilde{G}_n=\langle e_1,\ldots,e_{n+2}\rangle$ where $\tilde{G}_j$ is the subgroup generated by $e_1,\ldots ,\,e_{n+2},\,y_{2j+1},\ldots ,\, y_{2n} $. In the floor $\tilde{G}_j\geq\tilde{G}_{j+1}$, a maximal boundary subgroup of $\pi_1(\tilde{\Sigma}_j)$ is identified with the commutator $[w'_{2j+1},w'_{2j+2}]$ that takes by Proposition \[presentations\] part of $\tilde{G}_{j+1}$. This tower satisfies all conditions of Theorem \[maximaltowers\], so we know that $\langle_1,\ldots,e_{n+2}\rangle$ is a maximal free ground floor as well.
In fact the proof of Proposition \[presentations\] shows that $G^n$ even contains maximal free ground floors of all basis lengths between $2$ and $n+2$ because for each $0\leq i\leq n$, the group $G^{(i)}$ admits a hyperbolic tower over $F_{i+2}$ that fulfils the conditions of Theorem \[maximaltowers\].
Weight and Whitehead graphs {#Section infinite weight}
===========================
In this last section, we want to take a closer look at the model theoretic meaning of the results presented so far and proof Theorem \[Theoreminfiniteweight\]. These model theoretic questions were the point of departure for this article.
The analogies of forking independence to classical independence notions as linear independence or algebraic independence lead to the idea of comparing the sizes of maximal independent sets of realisations of a fixed type. That is the motivation for introducing the so-called weight of a type which bounds the ratio of the sizes of different such sets.
The *preweight* of a type $p(\bar{x}){\coloneqq}tp(\bar{a}/A)$ is the supremum of the set of cardinals $\kappa$ for which there exists a set ${\ensuremath{ \lbrace \bar{b}_i|i<\kappa \rbrace}}$ independent over $A$, such that $\bar{a}$ forks with $\bar{b}_i$ over $A$ for all $i$. It is denoted by $\text{prwt}(q)$. The *weight* $wt(p)$ of a type $p$ is the supremum of $${\ensuremath{ \lbrace \text{prwt}(q)|q \text{ a non-forking extension of }p \rbrace}}.$$
In fact, for every type $p$ in a theory $T$, the weight $wt(p)$ is smaller or equal to the cardinality of $T$. In our case, where we consider the countable theory $T_{fg}$ of free groups, the weight of all types is bounded by $\omega$.
The mentioned bound to the ratio of maximal independent sets is given by the following:
\[fact about weight\] Let $T$ be a complete theory. Suppose $p$ is a type in $T$ such that $wt(p)\leq n$ for a natural number $n\in\mathbb{N}$. Then we can find no model of $T$ in which there exist two maximal independent sets of realisations of $p$, such that one has size $k$ while the other one has size greater than $k\cdot n$.
In particular, if $wt(p)=1$, we know that all such sets have the same size.
Baring in mind that each basis of a maximal free ground floor forms a maximal independent set of realisations of the generic type $p_0$ (see Corollary \[Corollary maximal ground floors and p\_0\]), one can also see Theorem \[alternative proof for infinite weight\] as a proof that $p_0$ has infinite weight. This is a fact that was already shown by Pillay ([@PilGenweight]) and Sklinos ([@Rizinfweight]).
Extending the methods of Sklinos’ proof, we now want to generalise this result to arbitrary types realised in free groups. More precisely, we show that any non-algebraic (1-)type over the empty set which is realised in a free group has infinite weight. The condition on the types to be non-algebraic is no great restriction as in $T_{fg}$, all (1-)types over the empty set but the one of the neutral element are non-algebraic.
For this, we will use the following strong answer to the Tarski problem given by Sela:
\[Answer Tarski problem\] For any $2\leq m\leq n$, the natural embedding of $F_m$ in $F_n$ is elementary.
From now on, we will denote by $F_n$ the free group generated by the set $X{\coloneqq}{\ensuremath{ \lbrace e_1, \ldots , e_n \rbrace}}$. We call a word $w=u_1 u_2 \ldots u_k$ with $u_i\in X\cup X^{-1}$ *reduced*, if it contains no subword of the form $u u^{-1}$. We say that $w$ is *cyclically reduced*, if it cyclically contains no such subword, that is neither $w$ nor any cyclical permutation of its letters contain a subword $u u^{-1}$. With this notation, $F_n$ can be seen as the set of reduced words over $X$ with multiplication given by concatenation of words followed by reductions.
Let $A\subseteq F_n$ be a set of elements in $F_n$. Then $A$ is called *separable*, if there exists a non-trivial free decomposition $F_n=G\ast H$, such that each element of $A$ can be conjugated either into $G$ or into $H$. This means that for each $a\in A$, there exists $x\in F_n$ such that $xax^{-1}\in G\cup H$.
The connection between separability and independence in free groups is established by the following result from [@ForkingandJSJ] that characterises independence in free groups by the possibility to find proper free decompositions.
\[separability and independence\] Let $\bar{u}, \bar{v}$ be tuples of elements in the free group with $n$ generators and let $S$ be a free factor of $F_n$. Then $\bar{u}$ and $\bar{v}$ are independent over $S$ if and only if $F_n$ admits a free decomposition $F_n=G\ast S\ast H$ with $\bar{u}\in G\ast S$ and $\bar{v}\in S\ast H$.
For our purposes, it will suffice to look at the case in which $S={\ensuremath{ \lbrace 1 \rbrace}}$ is trivial and $u$ and $v$ are elements of $F_n$. Regarding Fact \[separability and independence\], we see that independence of $u$ and $v$ over the empty set implies that the set ${\ensuremath{ \lbrace u,v \rbrace}}$ is separable. So if ${\ensuremath{ \lbrace u,v \rbrace}}$ is not separable, we know as well that $u$ and $v$ fork over the empty set.
Let $A$ be a set of words over $X$ representing elements in the free group $F_n=\langle e_1,\ldots ,e_n\rangle$. The *Whitehead graph* of $A$, which we denote by $W_A$, is the graph with set of vertices $V(W_A)={\ensuremath{ \lbrace e_1,\ldots , e_n,e_1^{-1},\ldots , e_n^{-1} \rbrace}}$, and edges joining the vertices $u$ and $v^{-1}$ if and only if one of the words in $A$ cyclically contains the subword $uv$.
Let $W$ be a graph. A vertex $u\in V(W)$ is called a *cut vertex*, if removing $u$ and its adjacent edges leaves the graph disconnected.
Whitehead graphs occur as projections of closed paths in certain 3-dimensional manifolds and were first introduced by Whitehead in [@WhiteheadOnCertainSets]. Using this topological picture , Stallings showed the following fact which is crucial for our method to show that certain elements in free groups fork with each other.
\[separable -> cut vertex\] Let $A$ be a set of cyclically reduced words representing elements in $F_n$. If $A$ is separable in $F_n$, the Whitehead graph $W_A$ has a cut vertex.
Now we have all necessary tools to construct an independent sequence that witnesses the infinite weight of $p_0$.
\[my sequence\] The following sequence is independent over the empty set: $$\begin{gathered}
(b_i)_{i< \omega}{\coloneqq}(e_2 e_1 e_2,\, e_3 e_2 e_1 e_2^2 e_3,\, \ldots ,\,e_{i+1} e_i \ldots e_2 e_1 e_2^2 e_3^2 \ldots e_i^2 e_{i+1},\, \ldots)\, .\end{gathered}$$
One can easily see that $$\begin{gathered}
\langle e_2 e_1 e_2,\, e_3 e_2 e_1 e_2^2 e_3,\,\ldots ,\,e_{n+1} e_n \ldots e_2 e_1 e_2^2 e_3^2 \ldots e_n^2 e_{n+1},\, e_{n+1}\rangle =F_{n+1} .\end{gathered}$$ This suffices because as a special case of \[separability and independence\], we know that each basis of a free group forms an independent set.
For the proof of Theorem \[Theoreminfiniteweight\], we will make use of the high level of connection in the Whitehead graphs of the elements $b_i$ (see Figure \[b\_i Whitehead\]).
![The Whitehead graphs of $b_1=e_2 e_1 e_2$ and $b_i$. In both graphs, $e_2$ and $e_2^{-1}$ are the only cut vertices.[]{data-label="b_i Whitehead"}](Whiteheadb_iswider.png)
Let $p(x)$ be a type over the empty set with a non-trivial realisation $a\in F{\ensuremath{\backslash}}{\ensuremath{ \lbrace 1 \rbrace}}$ in some free group $F$. Fix a basis $X={\ensuremath{ \lbrace e_1,e_2,\ldots \rbrace}}$ of $F$. Permuting the elements of $X$ induces an automorphism of $F$ and thus does not change the type of $a$ over the empty set. So we may assume $a\in F_n$ for some $n\in \mathbb{N}$. As conjugating with an element of $F$ is also an automorphism, we can as well assume that $a$ is cyclically reduced.
Now take $(b_i)_{i<\omega}$ as defined in Lemma \[my sequence\]. Using Whitehead graphs, we show that after leaving out the first elements of this sequence, the remaining sequence $(b_i)_{n\leq i<\omega}$ witnesses the infinite weight of $p$. By the last lemma, we already know that $(b_i)_i$ is an independent sequence. It remains to show that $a$ forks with $b_i$ over the empty set for all $i\geq n$. To do this, we show that the Whitehead graph $W_A$ of the set $A{\coloneqq}{\ensuremath{ \lbrace a,b_i \rbrace}}$ has no cut vertex in the free group $F_{i+1}=\langle e_1,\ldots , e_{i+1}\rangle$. In this situation, we can apply Fact \[separable -> cut vertex\] to see that there is no decomposition $$F_{i+1}=G\ast H$$ such that $a\in G$ and $b_i\in H$. This implies that $a$ forks with $b_i$ in $F_{i+1}$ and thus, as the embedding $F_{i+1}\hookrightarrow F$ is elementary (see Fact \[Answer Tarski problem\]), they fork in $F$ as well and we are finished.
Permuting $X$ another time, we may assume that $a$ contains the letter $e_1$. That means that in $W_A$, the vertices $e_1$ and $e_1^{-1}$ are each connected by an edge to at least one other vertex. If the only edge starting at $e_1^{\pm 1}$ ends at $e_1^{\mp 1}$, we have $a=e_1^k$ for some $k\in \mathbb{Z}$. In this case, we can easily derive the infinite weight of $p$ from the infinite weight of $p_0$ because by [@PilForking Corollary 2.7], we have $p_0=tp(e_1/\emptyset)$. So without loss, at least one of the vertices $e_1,e_1^{-1}$ is connected to a vertex $e_k^{\pm 1}$ with $1<k\leq n$. It follows from the definition of Whitehead graphs that in this case, they are in fact both connected to at least one other vertex. By the same arguments as above, we may assume that $e_1$ is either connected to a vertex $e_k$ or $e_k^{-1}$ and that $e_1^{-1}$ is connected to $e_{k'}$ or $e_{k'}^{-1}$ where both $k$ and $k'$ are greater than 2 (we do not assume that those vertices are distinct). So $W({\ensuremath{ \lbrace a \rbrace}})$ contains at least the following two edges:
On the other hand, we already know that the Whitehead graph $W_{{\ensuremath{ \lbrace b_i \rbrace}}}$ is of the form shown in Figure \[b\_i Whitehead\]. Since $W_A$ is the union of $W_{{\ensuremath{ \lbrace a \rbrace}}}$ and $W_{{\ensuremath{ \lbrace b_i \rbrace}}}$, one sees that it has no cut vertex, because removing $e_2^{\pm 1}$ no longer disconnects $e_1^{\mp 1}$ from the rest of the graph. This finishes the proof.
Benjamin Brück\
<span style="font-variant:small-caps;">Fakultät für Mathematik\
Universität Bielefeld\
Postfach 100131\
D-33501 Bielefeld\
Germany</span>\
`benjamin.brueck@uni-bielefeld.de`
|
---
abstract: 'Using Gutzmer’s formula, due to Lassalle, we characterise the images of Soblolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.'
address: |
Department of Mathematics\
Indian Institute of Science\
Bangalore 560 012, India [*E-mail :*]{} [veluma@math.iisc.ernet.in]{}
author:
- 'S. Thangavelu'
title: |
Holomorphic Sobolev spaces associated to\
.5em compact symmetric spaces\
1.5em [By]{}
---
[Dedicated to the memory of Mischa Cotlar]{}
Introduction
============
In 1994 Brian Hall \[11\] studied the Segal-Bargmann transform on a compact Lie group $ G.$ For $ f \in L^2(G) $ let $ f*h_t $ be the convolution of $ f $ with the heat kernel $ h_t $ associated to the Laplacian on $ G.$ The Segal-Bargmann transform, also known as the heat kernel transform, is just the holomorphic extension of $ f*h_t $ to the complexification $ G_\C $ of $ G .$ The main result of Hall is a characterisation of the image of $ L^2(G)
$ as a weighted Bergman space. This extended the classical results of Segal and Bargmann \[4\] where the same problem was considered on $ \R^n.$ Later in \[19\] Stenzel treated the case of compact symmetric spaces obtaining a similar characterisation. Recently some surprising results came out on Heisenberg groups (see Kroetz-Thangavelu-Xu \[15\] ) and Riemannian symmetric spaces of noncompact type ( see Kroetz- Olafsson-Stanton \[16\]).
In 2004 Hall and Lewkeeratiyutkul \[13\] considered the Segal-Bargmann transform on Sobolev spaces $ \H^{2m}(G) $ on compact Lie groups. They have shown that the image can be characterised as certain holomorphic Sobolev spaces. The problem of treating the Segal-Bargmann transform on Sobolev spaces defined over compact symmetric spaces remains open. Our aim in this article is to characterise the image of $ \H^{m}(X) $ under the Segal-Bargmann transform as a holomorphic Sobolev space when $ X $ is a compact symmetric space.
Using an interesting formula due to Lassalle \[17\], called the Gutzmer’s formula, Faraut \[6\] gave a nice proof of Stenzel’s result. In this article we show that his arguments can be extended to treat Sobolev spaces as well. For the proof of our main theorem we need some estimates on derivatives of the heat kernel on a noncompact Riemannian symmetric space. This is achieved by using a result of Flensted-Jensen \[7\]. We also remark that the image of the Sobolev spaces turn out to be Bergman spaces defined in terms of certain weight functions. These weight functions are not necessarily nonnegative. Nevertheless, they can be used to define weighted Bergman spaces. This is reminiscent of the case of the heat kernel transform on the Heisenberg group. However, if we do not care about the isometry property of the Segal-Bargmann transform, then the images can be characterised as weighted Bergman spaces with nonnegative weight functions. Further, the isometry property of the heat kernel transform can be regained either by changing the original Sobolev norm into a different but equivalent one or by equiping the weighted Bergman space (with the positive weight function) with the previously defined norm (with the oscillating weight function)(see Theorems 3.3 and 3.5). That the weight function can be chosen to be nonnegative follows easily when the complexification of the noncompact dual of the compact symmetric space is of complex type. We use a reduction technique due to Flensted-Jensen to treat the general case.
In Section 4 we characterise the image of $ C^\infty(X) $ under the heat kernel transform. By using good estimates on the heat kernel on noncompact Riemannian symmetric spaces, recently proved by Anker and Ostellari \[3\], we obtain necessary and sufficient conditions on a holomorphic function to be in the image of $ C^\infty(X) .$ This extends the result of Hall and Lewkeeratiyutkul \[13\] to all comapct symmetric spaces. We also characerise the image of distributions under the heat kernel transform settling a conjecture stated in \[13\]. The results in Section 4 depend on the characterisation of holomorphic Sobolev spaces in terms of the holomorphic Fourier coefficients of a function. This in turn depends on the duality between Sobolev spaces $ \H_t^m(X_\C) $ of positive order and $ \H_t^{-m}(X_\C) $ of negative order. The latter spaces are easily shown to be Bergman spaces with non-negative weights.
The plan of the paper is as follows. We set up notation and collect relevant results on compact symmetric spaces and their complexifications in Section 2. We also indicate how Gutzmer’s formula is used to study the image of $ L^2 $ under the Segal-Bargmann transform. In Section 3 we introduce and obtain various characterisations of holomorphic Sobolev spaces $ \H_t^s(X_\C).$ Finally, in Section 4 we charactrise the images of $ C^\infty $ functions and distributions on $ X.$
Compact Riemannian symmetric spaces:\
Notations and Preliminaries
=====================================
The aim of this section is to set up notation and recall the main results from the literature which are needed in the sequel. The general references for this section are the papers of Lassalle \[17\], \[18\] and Faraut \[6\]. See also Helgason \[14\] and Flensted-Jensen \[7\].
Compact symmetric spaces and their duals
-----------------------------------------
We consider a compact Riemannian symmetric space $ X = U/K $ where $ (U, K) $ is a compact symmetric pair. By this we mean the following: $ U $ is a connected compact Lie group and $ (U^\theta)_0 \subset K \subset U^\theta $ where $ \theta $ is an involutive automorphism of $ U $ and $ (U^\theta)_0 $ is the connected component of $ U^\theta = \{ g\in U:
\theta(g) = g \}$ containing the identity. We may assume that $ K $ is connected and $ U $ is semisimple. We denote by $ \bu
$ and $ \bk $ the Lie algebras of $ U $ and $ K $ respectively so that $ \bk = \{ Y \in \bu: d\theta(Y) = Y \}.$ The base point $ eK \in X $ will be denoted by $o.$
Let $ \bp = \{ Y \in \bu : d\theta(Y) = -Y \} $ so that $ \bu = \bk
\oplus \bp.$ Let $ \ba $ be a Cartan subspace of $\bp$. Then $ A =
\exp \ba $ is a closed connected abelian subgroup of $ U.$ Every $ g \in U
$ has a decomposition $ g = k \exp H , k \in K, H \in \bp $ which in general is not unique. The maximal torus of the symmetric space $ X = U/K $ is defined by $ A_0 = \{ \exp H.o : H \in \ba \} $ which can be identified with the quotient $ \ba/\Gamma $ where $ \Gamma = \{ H \in \ba : \exp H \in K \}.$
Let $ U_\C $ (resp. $K_\C $)be the universal complexification of $ U $ (resp. $ K$). As $ U $ is compact we can identify $ U_\C $ as a closed subgroup of $ GL(N,\C) $ for some $ N.$ The group $ K_\C $ sits inside $ U_\C $ as a closed subgroup. We may then consider the complex homogeneous space $ X_\C =
U_\C/K_\C $ which is a complex variety and gives the complexification of the symmetric space $ X = U/K.$ The Lie algebra $ \bu_\C $ of $ U_\C $ is the complexified Lie algebra $ \bu_\C = \bu +i\bu.$ For every $ g \in U_\C $ there exists $ u \in U $ and $ X \in \bu $ such that $ g = u \exp iX.$
We let $ G = K \exp i\bp $ which forms a closed subgroup of $ U_\C $ whose Lie algebra is given by $ \bg = \bk +i\bp.$ It can be shown that $ G $ is a real linear reductive Lie group which is semisimple whenever $ U $ is and $ (G,K) $ forms a noncompact symmetric pair relative to the restriction of the involution $ \theta $ to $ G.$ The symmetric space $ Y = G/K $ is called the noncompact dual of the compact symmetric space $ X.$ The set $ i\ba $ is a Cartan subspace for the symmetric space $ G/K.$ Let $ \Sigma = \Sigma(\bg,i\ba)
$ be the system of restricted roots. It is then known that $ \Sigma(\bg,i\ba)
= \Sigma(\bu_\C,\ba_\C).$ Let $ \bt $ be a Cartan subalgebra of $ \bu $ containing $ \ba $ and let $ \Sigma(\bu_\C,\bt_\C) $ be the corresponding root system for the complex semisimple Lie algebra $ \bu_\C.$ Then the elements of $\Sigma(\bg,i\ba) $ are precisely the roots in $ \Sigma(\bu_\C,\bt_\C) $ that have a nontrivial restriction to $ \ba_C $ which explains the terminology ’restricted roots’.
We need the following integration formulas on $ X, X_\C $ and $ Y.$ A general reference for these formulas is Helgason \[14\] ( Chap.I, Section 5.2). We choose a positive system $ \Sigma^+ $ and denote by $ (i\ba)_+ = \{
H \in i\ba : \alpha(H) > 0, \alpha \in \Sigma^+ \} $ a positive Weyl chamber. Define $ J_0(H) = \large{\Pi}_{\alpha \in \Sigma^+} (\sin(\alpha,iH))^{m_\alpha} $ where $ m_\alpha $ is the dimension of the root space $ \bg_\alpha.$ Let the $ U-$invariant measure on $ X $ be denoted by $ dm_0.$ Then integration on $ X $ is given by the formula $$\int_X f(x)dm_0(x) = c_0 \int_K \int_{\ba/\Gamma} f(k\exp H.0)J_0(H)dkDH.$$ For a proof of this formula see Faraut \[6\] (Theorem 1V.1.1). We have a similar formula on the complexification.
Each point $ z \in X_\C $ can be written as $ z = g \exp(H).o $ where $ g \in U $ and $ H \in i\ba.$ If $ g_1 \exp(H_1).o = g_2 \exp(H_2).o $ then there exists $ w \in W $ such that $ H_2 = w.H_1.$ If we choose $ H \in
\overline{i\ba_+} $ then $ H $ is unique. Let $ dm $ be the $ U_\C $ invariant measure on $ X_\C.$ Then we have $$\int_{X_\C} f(z)dm(z) = c \int_U \int_{(i\ba)_+} f(g\exp H.o)J(H)dgdH$$ where $ J(H) = \Pi_{\alpha \in \Sigma^+} (\sinh 2(\alpha,H))^{m_\alpha}.$ (see Theorem IV.2.4 in Faraut \[6\]; the powers $ m_\alpha $ are missing in the formula for $ J(H)$). Finally we also need an integration formula on the noncompact dual $ Y = G/K.$ If $ dm_1 $ is the $ G $ invariant measure on $ Y $ then $$\int_Y f(y)dm_1(y) = c_1 \int_K \int_{i\ba}f(k\exp(H).o)J_1(H) dk dH$$ where $ J_1(2H) = J(H) $ defined above.
Gutzmer’s formula
------------------
For results in this section we refer to the papers of Lassalle \[17\],\[18\] and the article by Faraut \[6\]. We closely follow the notations used in Faraut \[6\].
Given an irreducible unitary representation $ (\pi,V) $ of $ U $ and a function $ f \in L^1(U) $ we define $$\hat{f}(\pi) = \int_U f(g)\pi(g) dg$$ where $ dg $ is the Haar measure on $ U.$ When $ f $ is a function on $ X $ so that it can be considered as a right $ K $ invariant function on $ U $ it can be shown that $ \hat{f}(\pi) =
0 $ unless the representation $ (\pi,V) $ is spherical which means that $ V $ has a unique $ K $ invariant vector. When $ (\pi,V) $ is spherical and $u $ is the unit invariant vector then $ \hat{f}(\pi)v = (v,u)\hat{f}(\pi)u.$ This means that $ \hat{f}(\pi) $ is of rank one. Let $ \hat{U}_K $ be the subset of the unitary dual $ \hat{U} $ containing spherical representations (also called class one representations). Then $\hat{U}_K $ is in one to one correspondence with a discrete subset $ \CP^+ $ of $ \ba^* $ called the set of restricted dominant weights.
For each $ \lambda \in \CP^+ $ let $ (\pi_\lambda, V_\lambda) $ be a spherical representation of $ U $ of dimension $ d_\lambda.$ Let $ \{v_j^\lambda, 1 \leq j \leq d_\lambda \} $ be an orthonormal basis for $ V_\lambda $ with $ v_1^\lambda $ being the unique $ K$-invariant vector. Then the functions $$\varphi_j^\lambda(g) =(\pi_\lambda(g)v_1^\lambda,v_j^\lambda)$$ form an orthogonal family of right $ K $ invariant analytic functions on $ U.$ Note that each $\varphi_j^\lambda(g) $ is right $K$-invariant and hence they can be considered as functions of the symmetric space. When $ x = g.o \in X $ we simply denote by $ \varphi_j^\lambda(x) $ the function $ \varphi_j^\lambda(g.o).$ The function $\varphi_1^\lambda(g)$ is $ K $ biinvariant called an elementary spherical function. It is usually denoted by $ \varphi_\lambda.$
For $ f \in L^2(X) $ we define its Fourier coefficients $ \hat{f}_j(\lambda) , 1 \leq j \leq d_\lambda $ by $$\hat{f}_j(\lambda) = \int_X f(x)\overline{\varphi_j^\lambda(x)}dm_0(x).$$ The Fourier series of $ f $ is written as $$f(x) = \sum_{\lambda \in \CP} d_\lambda \sum_{j=1}^{d_\lambda}
\hat{f}_j(\lambda)\varphi_j^\lambda(x)$$ and the Plancherel theorem reads as $$\int_X |f(x)|^2 dm_0(x) = \sum_{\lambda \in \CP} d_\lambda
\sum_{j=1}^{d_\lambda} |\hat{f}_j(\lambda) |^2 .$$ Defining $ A_\lambda(f) = d_\lambda^{-\frac{1}{2}}
\hat{f}(\pi_\lambda) $ the Plancherel formula can be put in the form $$\int_X |f(x)|^2 dm_0(x) = \sum_{\lambda \in \CP} d_\lambda
\|A_\lambda(f)\|^2.$$
Let $ \Omega $ be an $ U $ invariant domain in $ X_\C $ and let $ \CO(\Omega)$ stand for the space of holomorphic functions on $ \Omega.$ The group $ U $ acts on $ \CO(\Omega)$ by $ T(g)f(z) = f(g^{-1}z).$ For each $ \lambda \in
\CP^+ $ the matrix coefficients $ \varphi_j^\lambda $ extend to $ X_\C $ as holomorphic functions. When $ f \in \CO(\Omega) $ it can be shown that the series $$\sum_{\lambda \in \CP} d_\lambda \sum_{j=1}^{d_\lambda} \hat{f}_j(\lambda)
\varphi_j^\lambda(z)$$ converge uniformly over compact subsets of $ \Omega.$ Thus we have the expansion $$f(z) = \sum_{\lambda \in \CP} d_\lambda \sum_{j=1}^{d_\lambda}
\hat{f}_j(\lambda) \varphi_j^\lambda(z)$$ called the Laurent expansion of $ f.$ The following formula known as Gutzmer’s formula is very crucial for our main result.
(Gutzmer’s formula) For every $ f \in \CO(X_\C) $ and $ H \in i\ba $ we have $$\int_U |f(g.\exp(H).o)|^2 dg = \sum_{\lambda \in \CP^+} d_\lambda
\|A_\lambda(f)\|^2 \varphi_\lambda(\exp(2H).o).$$
This theorem is due to Lasalle; we refer to \[17\] and \[18\] for a proof. See also Faraut \[6\]. Polarisation of the above formula gives $$\int_U f(g.\exp(H).o)\overline{h(g.\exp(H).o)}dg$$ $$= \sum_{\lambda \in \CP^+} d_\lambda \left( \sum_{j=1}^{d_\lambda}
\hat{f}_j(\lambda)\overline{\hat{h}_j(\lambda)} \right)
\varphi_\lambda(\exp(2H).o)$$ for any two $ f, h \in \CO(X_\C). $
Segal-Bargmann transform
-------------------------
We now turn our attention to the Segal-Bargmann or heat kernel transform on $ X.$ Let $ D $ stand for the Laplace operator on the symmetric space defined in Faraut \[6\]. The functions $ \varphi_j^
\lambda $ turn out to be eigenfunctions of $ D $ with eigenvalues $ \kappa(\lambda) = -(|\lambda|^2+2\rho(\lambda)) $ where $ \rho $ is the half sum of positive roots. We let $ \Delta = D-|\rho|^2 $ so that the eigenvalues of $ \Delta $ are given by $ -|\lambda +\rho|^2.$ Note that our $ \delta $ differs from the standard Laplacian $ D $ by a constant. To avoid further notation we have denoted the shifted Laplacian by the symbol $ \Delta $ which is generally used for the unshifted one.
Given $ f \in L^2(X) $ the function $ u(g,t) $ defined by the expansion $$u(g,t) = \sum_{\lambda \in \CP^+} d_\lambda e^{-t|\lambda+\rho|^2}\sum_{j=1}
^{d_\lambda}\hat{f}_j(\lambda) \varphi_j^\lambda(g)$$ solves the heat equation $$\partial_t u(g,t) = \Delta u(g,t),~~~~ u(g,0) = f(g).$$ Defining the heat kernel $\gamma_t(g) $ by $$\gamma_t(g) = \sum_{\lambda \in \CP^+}d_\lambda e^{-t|\lambda+\rho|^2}
\varphi_\lambda(g)$$ we can write the solution as $$u(g,t) = f*\gamma_t(g) = \int_U f(h)\gamma_t(h^{-1}g) dh.$$ The heat kernel $ \gamma_t $ is analytic, strictly positive and satisfies $\gamma_t*\gamma_s = \gamma_{t+s}.$ Moreover, it extends to $ X_\C $ as a holomorphic function. It can be shown that for each $ f \in L^2(X) $ the function $ u(g,t) = f*\gamma_t(g) $ also extends to $ X_\C $ as a holomorphic function. The transformation taking $ f $ into the holomorphic function $ u(z,t) = f*\gamma_t(g.o), z = g.o, g \in U_\C $ is called the Segal-Bargmann or heat kernel transform.
The image of $ L^2(X) $ under this transform was characterised as a weighted Bergman space by Stenzel in \[19\] which was an extension of the result of Hall \[11\] for the case of compact Lie groups. Another proof of Stenzel’s theorem was given by Faraut in \[6\] using Gutzmer’s formula. Since we are going to use similar arguments in our characerisations of holomorphic Sobolev spaces it is informative to briefly sketch the proof given by Faraut \[6\].
Let $ \gamma^1_t $ be the heat kernel associated to the Laplace-Beltrami operator $ \Delta_G $ on the noncompact Riemannian symmetric space $ Y = G/K.$ Then $ \gamma^1_t $ is given by $$\gamma^1_t(g) =\int_{i\ba} e^{-t(|\mu|^2+|\rho|^2)}\psi_\mu(g)|c(\mu)|^{-2}
d\mu$$ where $ \psi_\mu $ are the spherical functions of the pair $(G,K).$ This is the standard representation of the heat kernel on a noncompact symmetric space using Fourier inversion. Here $ c(\mu) $ is the $c$-function associated to $ Y = G/K.$ The heat kernel $ \gamma_t^1 $ is characterised by the defining property $$\int_Y \gamma^1_t(g)\psi_{-\mu}(g) dm_1(g) = e^{-t(|\mu|^2+|\rho|^2)},~~~~~
\mu \in i\ba$$ where $ dm_1 $ is the $ G $ invariant measure on $ Y.$ In view of the integration formula mentioned earlier this reads as $$\int_{i\ba} \gamma_t^1(\exp(H).o)\psi_\mu(\exp(H).o)J_1(H) dH =
e^{-t(|\mu|^2+|\rho|^2)}.$$ Note that both sides of the above equation are holomorphic in $ \mu $ and hence the above equation is valid for all $ \mu \in \ba_C.$ In particular, $$\int_Y \gamma^1_t(g)\psi_{-i\mu}(g) dm_1(g) = e^{t(|\mu|^2-|\rho|^2)},~~~~~
\mu \in i\ba .$$ We can now prove the following result which characterises the image of $ L^2(X)
$ under the Segal-Bargmann transform. Define $ p_t(z) $ on $ X_\C $ by $$p_t(z) = p_t(g\exp(H).o)= \gamma_{2t}^1(\exp(2H).o),~~~~~~ g \in U, H \in i\ba.$$
(Stenzel) A holomorphic function $ F \in \CO(X_\C) $ is of the form $ f*\gamma_t $ for some $ f \in L^2(X) $ if and only if $$\int_{X_\C} |F(z)|^2 p_t(z) dm(z) < \infty.$$
The integration formula on $ X_\C $ together with Gutzmer’s formula leads to $$\int_{X_\C} |F(z)|^2 p_t(z) dm(z) = c_1 \sum_{\lambda \in \CP^+}
d_\lambda \|A_\lambda(f)\|^2 \times$$ $$e^{-2t |\lambda+\rho|^2}
\int_{i\ba} \varphi_{\lambda}(\exp(2H).o) \gamma_{2t}^1(\exp(2H).o)J_1(2H) dH.$$ We now make use of the well known relation $$\varphi_\lambda(\exp(H).o) = \psi_{-i(\lambda+\rho)}(\exp(H).o).$$ Using this and recalling the defining relation for $ \gamma_t^1 $ we get $$\int_{i\ba} \varphi_{\lambda}(\exp(2H).o) \gamma_{2t}^1(\exp(2H).o)J_1(2H)
dH = c e^{2t |\lambda+\rho|^2}e^{-2t|\rho|^2}$$ for some constant $ c.$ Hence $$\int_{X_\C} |F(z)|^2 p_t(z) dm(z) = c_t \int_X |f(x)|^2 dm_0(x).$$ This completes the proof of the theorem.
Some estimates for the heat kernel on $ G/K $
---------------------------------------------
The heat kernel $ \gamma_t^1 $ on the noncompact dual $ Y = G/K $ of $ X =
U/K $ is explicitly known only when $ G $ is a complex Lie group, see Gangolli \[8\]. This happens precisely when we are dealing with compact Lie groups as symmetric spaces. In this case we have explicit formulas even for derivatives of the heat kernel and this has been made use of by Hall and Lewkeeratiyutkul \[13\] in their study of holomorphic Sobolev spaces associated to compact Lie groups. In 2003 Anker and Ostellari \[3\] has sketched a proof for the following estimate for the heat kernel $ \gamma_t^1.$ For a fixed $ t > 0 $ their main result says that $ \gamma_t^1(\exp H) $ behaves like $$\Phi(H)^{1/2} e^{-t|\rho|^2} e^{-\frac{1}{4t}|H|^2} ,~~~~ H \in i\ba$$ where $ \Phi $ is the function defined on $ i\ba $ by $$\Phi(H) = \large{\Pi}_{\alpha \in \Sigma^+} \left(
\frac{(\alpha,H)}{\sinh(\alpha,H)}\right)^{m_\alpha}.$$
The following remarks on the $ \Phi $ function are important. Note that the product is taken with respect to all the restricted roots for the pair $ (\bg,i\ba) .$ The product remains unaltered even if we take it over all roots in $ \Sigma(\bu_\C,\bt_\C) $ since $ (\alpha,H) = 0 $ for all elements of $ \Sigma(\bu_\C,\bt_\C) $ which are not in $ (\bg,i\ba) .$ We note that $$\Phi(H) = \Pi_{\alpha \in \Sigma^+} \left(
\frac{(\alpha,H)}{\sinh(\alpha,H)}\right)^{m_\alpha} = J_1(H)^{-1}
\Pi_{\alpha \in \Sigma^+} (\alpha,H)^{m_\alpha}.$$ We make use of these facts later.
Complete proof of the above estimate for the heat kernel is not yet available but we believe the arguments of Anker and Ostellari are sound. The estimates are known to be true in several particular cases by different methods. In an earlier paper Anker \[1\] have established slightly weaker estimates (which are polynomially close to the optimal estimates) whenever $ G $ is a normal real form. These are good enough for some purposes. For example, in the characterisations of the images of smooth fuunctions and distributions the polynomial discrepencies do not really matter. We are thankful to the referee for pointing this out.
For the study of holomorphic Sobolev spaces on $ X_\C $ we also need estimates on the $ t $-derivatives of $ \gamma_t^1.$ We do not have any result available in the literature except when $ G $ is complex or $ G/K $ is of rank one. However, there is a powerful method of reduction to the complex case by Flensted-Jensen using which we can express the heat kernel $ \gamma_t^1 $ on $ G/K $ in terms of the heat kernel $ \Gamma_t $ on $ U_\C/U.$ As the latter heat kernel is known explicitly we can get estimates for $ \gamma_t^1 $ and its derivatives. We recall this result from Flensted-Jensen \[7\] and state the result using our notation. (In \[7\] the group $ G $ stands for a complex Lie group, and $ G_0 $ the real group whose Lie algebra $ \bg_0 $ is a real form of $ \bg.$ This should not cause any confusion. We refer the reader to \[7\] ( Theorem 6.1 and Example on page 131) for details.)
Recall that $ U $ is a maximal compact subgroup of $ U_\C.$ We let $ K_c$ stand for the noncompact group whose Lie algebra is $ \bk +i\bk $, a subalgebra of $ \bu_\C = \bu +i\bu.$ In \[7\] Flensted-Jensen has proved that there is a one to one correspondence between $ K $-biinvariant functions on $ G $ and certain functions on $ U_\C $ that are right $ U $-invariant and left $ K_c $ invariant ( see Theorm 5.2 in \[7\]). This isomorphism is denoted by $ f \rightarrow f^\eta $ and satisfies $ f^\eta(g)
= f(g\theta(g)^{-1}) $ for all $ g \in G.$ Let $ g_t $ and $ G_t $ be the Gauss kernels on $ G/K $ and $ U_\C/U $ respectively as defined by Flensted-Jensen. These are almost the heat kernels $ \gamma_t^1 $ and $
\Gamma_t $ differing from them only by multiplicative constants. The formula connecting $ g_t $ and $ G_t $ is given by $$g_t(x) = \int_{K_c} G_t(hx) dh, ~~~~ x \in G .$$ The above formula has to be interpreted using the isomorphism $ f \rightarrow f^\eta .$
The above formula connecting $ g_t $ and $ G_t $ leads to a similar formula for $ \gamma_t^1 $ and $ \Gamma_t.$ For a reader not familiar with the work of Flensted-Jensen the above formula might appear a bit mysterious. However, the mystery can be unravelled if we recall that $ f^\eta(\exp H) =
f(\exp(2H))$ for $ H \in \bp.$ If we properly take care of the definitions of various inner products and Laplacians, then the final formula connecting the two heat kernels take the form $$\gamma_t^1(\exp H) =
\int_{K_c} \Gamma_{t/4}(h\exp(H/2)) dh, ~~~~ H \in i\ba .$$ It can be directly checked that the function defined by the integral on the right hand side solves the heat equation on $ G/K $ which follows by the invariance of the Laplacian. We are indebted to the referee for this reasoning leading to the correct scaling of the heat kernels in the above formula.
We have the following explicit formula for the heat kernel $ \Gamma_t $ obtained by Gangolli \[8\]: $$\Gamma_t(\exp H) = c(4t)^{-n/2}\Pi\frac{(\alpha,H)}{\sinh(\alpha,H)}
e^{-t|\rho|^2}e^{-\frac{1}{4t}|H|^2}$$ where the product is taken over all positive roots in $ \Sigma(\bu_\C,\bt_C).$ Using this formula and the connection between $ \gamma_t^1 $ and $ \Gamma_t $ we can prove the following estimate.
For every $ s > t, m \in \N $ and $ H \in i\ba $ we have $$|\partial_t^m \gamma_t^1(\exp H)| \leq C_{s,t,m} e^{-\frac{1}{4s}|H|^2}.$$
First consider the case $ m = 0.$ Since $ |\exp H| \leq
|h\exp H | $ (see \[7\], eqn. 6.5) the formula for $ \gamma_t^1 $ in terms of $ \Gamma_t $, gives $$\gamma_t^1(\exp H) e^{\frac{1}{4s}|H|^2} \leq \int_{K_c}
\Gamma_{t/4}(h\exp(H/2))
e^{\frac{1}{4s}|h\exp H|^2} dh.$$ We only need to show that the right hand side is a bounded function of $ H.$ In view of the formula for $ \Gamma_t ,$ we see that $ \Gamma_{t/4}(h\exp(H/2))e^{\frac{1}{4s}|h\exp H|^2} $ is bounded by a constant times $ \Gamma_{r/4}(h\exp(H/2))$ where $ r = (st)/(s-t) .$ Thus, using the Flensted-Jensen formula once again, we see that $ \gamma_t^1(\exp H) e^{\frac{1}{4s}|H|^2} $ is bounded by a constant times $ \gamma_r^1(\exp H) $ which is clearly bounded.
In the case of derivatives we need to show that the function defined by $$\int_{K_c} P_{t,s}(h \exp(H/2) \Gamma_{r/4}(h\exp(H/2)) dh$$ is bounded for any polynomial $ P_{t,s}.$ The spherical Fourier transform of this function on $ G $ can be expressed as the spherical Fourier transform on $ U_\C/U $ of the integrand (evaluated at $ h $ = identity) which can be calculated in terms of derivatives of the spherical Fourier transform of $ \Gamma_{r/4} $ which is a Gaussian. The latter is a Schwartz function, which means that the spherical Fourier transform of the integral is a Schwartz function on $ G $ and hence bounded.
We would like to conclude this proof with a couple of remarks. The above connection between the ’two Fourier transforms’ is stated and proved as Theorem 6.1 in \[7\]. For the case of the Gauss-kernel (alias heat kernel) Flensted-Jensen has explicitly discussed this connection at the end of Section 6 in \[7\] (see Example on page 131). We also take this opportunity to indicate another proof suggested by the referee: the time derivative of $
\Gamma_t $ pulls down a polynomial factor in $ H $, with coefficients that depend on $ t.$ Thus, $$|\partial_t^m \Gamma_t(\exp H)| \leq C_{t,m,\epsilon}e^{\epsilon |H|^2}
\Gamma_t(\exp H).$$ In view of the case $ m = 0 $ this gives us the desired estimate.
Holomorphic Sobolev spaces
==========================
In this section we introduce and study holomorphic Sobolev spaces $ H^s(X_\C)
$ for any $ s \in \R.$ When $ s= -m $ is a negative integer we show that $ H^s(X_\C) $ is a weighted Bergman space. But when $ s = m $ is a positive integer $ H^s(X_\C) $ can be described as the completion of a weighted Bergman space with respect to a smaller norm. Later, using the reduction formula of Flensted-Jensen \[7\] we show that we can choose a positive weight function so that $ H^m(X_\C)$ can be described as a weighted Bergman space in all the cases.
Holomorphic Sobolev spaces
--------------------------
Recall that for each real umber $ s $ the Sobolev space $ \H^{s}(X)
$ of order $ s $ can be defined as the completion of $ C^\infty(X) $ under the norm $ \|f\|_{(s)} = \|(1-\Delta)^{\frac{s}{2}}f\|_2.$ In view of Plancherel theorem a distribution $ f $ on $ X $ belongs to $ H^{s}(X) $ if and only if $$\sum_{\lambda \in \CP^+} d_\lambda (1+|\lambda+\rho|^2)^s
\|A_\lambda(f)\|^2 < \infty.$$ We define $ \H_t^{s}(X_\C) $ to be the image of $ \H^{s}(X) $ under the heat kernel transform. This can be made into a Hilbert space simply by transfering the Hilbert space structure of $ \H^{s}(X) $ to $ \H_t^{s}(X_\C) .$ This means that if $ F = f*\gamma_t, G = g*\gamma_t $ where $ f, g \in \H^s(X) $ then $ (F,G)_{\H_t^{s}(X_\C)} = (f,g)_{\H_t^{s}(X)}.$ Then, it is clear that the heat kernel transform is an isometric isomorphism from $ \H^{s}(X) $ onto $ \H_t^{s}(X_\C).$ We are interested in realising $ \H_t^{s}(X_\C)$ as weighted Bergman spaces.
The spherical functions $ \varphi_j^\lambda, 1 \leq j \leq d_\lambda, \lambda
\in \CP^+ $ form an orthogonal system in $ \H^{s}(X) $ for every $ s \in \R.$ More precisely, $$(\varphi_j^\lambda,\varphi_k^\mu)_{\H^{s}(X)} = \delta_{j,k}~
\delta_{\lambda,\mu}~ d_\lambda^{-1} (1+|\lambda+\rho|^2)^s.$$ From the definition of $ \H_t^s(X_\C) $ it is clear that the holomorphically extended spherical functions $ \varphi_j^\lambda(g \exp(iH).o),
1 \leq j \leq d_\lambda, \lambda \in \CP^+ $ form an orthogonal system in $ \H_t^k(X_\C) $ : $$(\varphi_j^\lambda,\varphi_{k}^{\mu})_{ \H_t^s(X_\C) } = \delta_{j,k}~
\delta_{\lambda \mu}~ d_\lambda^{-1} e^{2t|\lambda+\rho|^2}
(1+|\lambda+\rho|^2)^{s}.$$ Suitably normalised, they form an orthonormal basis for $ \H_t^s(X_\C).$ This motivates us to define the holomorphic Fourier coefficients as follows.
For a holomorphic function $ F $ on $ X_\C $ we define its holomorphic Fourier coefficients by $$\tilde{F}_j(\lambda) = \int_{X_\C} F(z) \overline{\varphi_j^\lambda(z)}
p_t(z) dm(z).$$ Note that the holomorphic Fourier coefficients depend on $ t $ which we have suppressed for the sake of simplicity.(For us $ t $ is fixed throughout). The integration formula on $ X_\C $ shows that $$\tilde{F}_j(\lambda) = \int_{i\ba}\int_U F(g\exp H.o)\overline
{\varphi_j^\lambda(g \exp H.o)} \gamma_{2t}^1(\exp 2H)J_1(2H) dg dH.$$ When $ F = f*\gamma_t $ it follows from the polarised form of the Gutzmer’s formula that $ \tilde{F}_j(\lambda) =
e^{t|\lambda+\rho|^2}\hat{f}_j(\lambda).$ This leads to the following characterisation.
A holomorphic function $ F $ on $ X_\C $ belongs to $\H_t^s(X_\C)$ if and only if $$\sum_{\lambda \in \CP^+} d_\lambda \left(\sum_{j=1}^{d_\lambda}
|\tilde{F}_j(\lambda)|^2 \right)(1+|\lambda+\rho|^2)^{s}
e^{-2t|\lambda+\rho|^2} < \infty.$$
The spaces $ \H_t^s(X_\C) $ and $ \H_t^{-s}(X_\C)$ are dual to each other and the duality bracket is given by $$(F,G) = \int_{X_\C} F(z)\overline{G(z)}p_t(z) dm(z).$$
From the (polarised) Gutzmer’s formula we see that $$\int_{i\ba}\int_{U} F(g\exp H.o)\overline{G(g\exp H.o)}\gamma_{2t}^1
(\exp H) J_1(2H) dg dH$$ $$= \sum_{\lambda \in \CP^+} d_\lambda \left(\sum_{j=1}^{d_\lambda}
\hat{f}_j(\lambda) \overline{\hat{g}_j(\lambda)}\right)
= \sum_{\lambda \in \CP^+} d_\lambda \left(\sum_{j=1}^{d_\lambda}
\tilde{F}_j(\lambda) \overline{\tilde{G}_j(\lambda)}\right)
e^{-2t|\lambda+\rho|^2}$$ where $ F = f*\gamma_t $ and $ G = g*\gamma_t.$ Since $ \H^s(X) $ and $ \H^{-s}(X) $ are dual to each other under the duality bracket $$(f,g) = \sum_{\lambda \in \CP^+} d_\lambda \left(\sum_{j=1}^{d_\lambda}
\hat{f}_j(\lambda) \overline{\hat{g}_j(\lambda)}\right)$$ it follows that the series $$\sum_{\lambda \in \CP^+} d_\lambda \left(\sum_{j=1}^{d_\lambda}
\tilde{F}_j(\lambda) \overline{\tilde{G}_j(\lambda)}\right)
e^{-2t|\lambda+\rho|^2}$$ converges whenever $ F \in \H_t^s(X_\C) $ and $ G \in \H_t^{-s}(X_\C).$ This proves the corollary.
Note that the duality bracket between $ \H_t^s(X_\C) $ and $ \H_t^{-s}(X_\C)$ which can be put in the form $$(F,G) = \int_{i\ba}\int_{U} F(g\exp H.o)\overline{G(g\exp H.o)}
\gamma_{2t}^1(\exp(2H)) J_1(2H) dg dH$$ involves only the heat kernel $ \gamma_{2t}^1 $ but not its derivatives. This fact is crucial for some purposes.
$\H_t^m(X_\C) $ as weighted Bergman spaces
------------------------------------------
In proving Stenzel’s theorem we have made use of the crucial fact $$\int_{i\ba} \gamma_{2t}^1(\exp(2H).o)\varphi_\lambda(\exp(2H))J_1(2H)dH
= c~ e^{2t|\lambda+\rho|^2}$$ for some positive constant $ c.$ Differentiating the above identity $ m $ times with respect to $ t $ we get $$\int_{i\ba} \partial_t^m \gamma_{2t}^1(\exp(2H).o)
\varphi_\lambda(\exp(2H))J_1(2H)dH
= c ~2^m |\lambda+\rho|^{2m} e^{2t|\lambda+\rho|^2} .$$ In view of Gutzmer’s formula the natural weight function for $ \H_t^m(X_\C) $ should be $$w_t^m(z) = (1+\partial_t)^mp_t(z).$$ But unfortunately this weight function need not be positive and hence in defining a Bergman space with respect to $ w_t^m(z) $ we have to be careful.
Let $ \CF_t^m(X_\C) $ be the space of all $ F \in \CO(X_\C) $ such that $$\int_{X_\C} |F(z)|^2 |w_t^m(z)| dm(z) < \infty.$$ We equip $ \CF_t^m(X_\C) $ with the sesquilinear form $$(F,G)_m = \int_{X_\C} F(z)\overline{G(z)}
w_t^m(z) dm(z).$$ We show below that this defines a pre-Hilbert structure on $ \CF_t^m(X_\C) $. Let $ \CB_t^m(X_\C) $ be the completion of $ \CF_t^m(X_\C) $ with respect to the above inner product. We have the following characterisation of $ \H_t^m(X_\C) .$
For every nonnegative integer $ m $ we have $ \H_t^m(X_\C) =
\CB_t^m(X_\C)$ and the heat kernel transform is an isometric isomorphism from $ \H^m(X) $ onto $ \CB_t^m(X_\C)$ upto a multiplicative constant.
We first check that the sesquilinear form defined above is indeed an inner product. Let $ F, G \in \CF_t^m(X_\C).$ In view of the integration formula on $ X_\C $ the sesquilinear form is given by $$(F,G)_m = \int_{i\ba} \int_{U} F(u\exp(H).o) \overline{G(u\exp(H).o)}
J_1(2H) du dH.$$ Then by Gutzmer’s formula we have $$\int_U |F(u \exp(H).o)|^2 du = \sum_{\lambda \in \CP^+} d_\lambda
\|A_\lambda(F)\|^2 \varphi_\lambda(\exp(2H))$$ for all $ H \in i\ba.$ Since the left hand side is integrable with respect to $ |w_t^m(\exp(H).o)|J_1(2H) $ so is the right hand side. By Fubini we get $$\int_{i\ba}\int_U |F(u\exp(H).o)|^2 w_t^m(\exp(H).o) J_1(2H)du dH$$ $$= \sum_{\lambda \in \CP^+} d_\lambda \|A_\lambda(F)\|^2
\int_{i\ba} \varphi_\lambda(\exp(2H))w_t^m(\exp(H).o) J_1(H)du dH.$$ If we use the relation $ \varphi_\lambda(\exp H) =
\psi_{-i(\lambda+\rho)}(\exp H) $ the integral on the right hand side becomes a constant multiple of $$\int_{i\ba} (1+\partial_t)^m \gamma^1_{2t}(\exp H)
\psi_{-i(\lambda+\rho)}(\exp H)J_1(H) dH$$ which is just $ e^{2t|\lambda+\rho|^2}(1+|\lambda+\rho|^2)^m .$ This proves that $$\int_{X_\C} |F(z)|^2 w_t^m(z) dm(z) dz$$ $$=
\sum_{\lambda \in \CP^+} d_\lambda e^{2t(|\lambda+\rho|^2)}
(1+|\lambda+\rho|^2)^m \|A_\lambda(F)\|^2$$ and hence the sesquilinear form is indeed positive definite.
The above calculation also shows that any $ F \in \CF_t^m(X_\C) $ is the holomorphic extension of $ f*\gamma_t $ for some $ f \in \H^{m}(X).$ Indeed, we only have to define $ f $ by the expansion $$f(g.o) = \sum_{\lambda \in \CP^+} d_\lambda e^{t|\lambda+\rho|^2}
\sum_{j=1}^{d_\lambda} \hat{F}_j(\lambda)\varphi_j^\lambda(g.o).$$ Here $ \hat{F}_j(\lambda) $ are the Fourier coefficients of $ F $ defined by $$\hat{F}_j(\lambda) = \int_X F(x) \overline{\varphi_j^\lambda(x)} dm_0(x).$$ Thus we have proved that $ \CF_t^m(X_\C) $ is contained in $ \H_t^{m}(X_\C).$ And also the norms are equivalent. To complete the proof of the theorem, it is enough to show that $ \CF_t^m(X_\C) $ is dense in $ \H_t^{m}(X_\C).$
As we have already observed the functions $ \varphi_j^\lambda $ initially defined on $ X $ have holomorphic extensions to $ X_\C.$ From the manner we have defined the holomorphic Sobolev spaces $ \H_t^{m}(X_\C) $ it follows that the functions $ \varphi_j^\lambda $, after suitable normalisation, form an orthonormal basis for $ \H_t^{m}(X_\C).$ The proof will be complete if we can show that all $ \varphi_j^\lambda $ belong to $ \CF_t^m(X_\C) $ since the finite linear combinations of them forms a dense subspace of $ \H_t^{m}(X_\C).$ As $$\varphi_j^\lambda*\gamma_t(g\exp(H).o) = e^{-t|\lambda+\rho|^2}
\varphi_j^\lambda(g\exp(H).o)$$ by applying Gutzmer’s formula to the functions $ \varphi_j^\lambda(g\exp(H).o) $ we only need to check if $$\int_{i\ba} \varphi_\lambda(\exp(2H).o) |w_t^m(\exp(H).o)|
J_1(2H) dH < \infty.$$ The functions $ \varphi_\lambda $ are known to satisfy the estimate $$\varphi_\lambda(\exp H.o) \leq e^{\lambda(H)}$$ for all $ H \in i\ba $ (see Proposition 2 in Lassalle \[17\]). The weight function $ w_t^m $ involves derivatives of the heat kernel $ \gamma_t^1 $ for which we have the estimates stated in Theorem 2.3 . Using them we can easily see that the above integrals are finite.
A positive weight function for $ \H_t^m(X_\C)$
-----------------------------------------------
In this section we show that the holomorphic Sobolev spaces $ \H_t^{m}(X_\C) $ can be characterised as weighted Bergman spaces with non-negative weight functions. Note that if $ w_t^m $ happens to be positive then $ \CF_t^m(X_\C) = \CB_t^m(X_\C) = \H_t^m(X_\C).$ We show that it is possible to define a new weight function $ w_{t,\delta}^m $ which will be positive and $ \H_t^m(X_\C) $ is precisely the weighted Bergman space defined in terms of $ w_{t,\delta}^m .$ But we lose the isometry property of the heat kernel transform. If we are ready to change the norm on $ \H_t^m(X_\C) $ into another equivalent norm, the isometry property can also be regained.
The case of compact Lie groups $ H $ studied by Hall \[11\] corresponds to the symmetric space $ U/K $ where $ U = H \times H $ and $ K $ is the diagonal subgroup of $ U.$ This is precisely the case for which the subgroup $ G $ of $ U_{\C} $ is a complex Lie group. Therefore, we do not have to use the result of Flensted-Jensen in getting estimates for the heat kernel on $ G/K.$ In this case the weight function $ w_t^m $ can be modified to be positive. In \[13\] Hall and Lewkeeratiyutkul have shown that by a proper choice of $ \delta > 0 $ the kernel $ w_{t,\delta}^m(z) = (\delta +\partial_t^m)p_t(z) $ can be made positive. That this is indeed the case can be easily seen from the explicit formula for $ \gamma_t^1 $ in the complex case. The kernel $ w_{t,\delta}^m(
\exp H.o) $ turns out to be $ (P_t(H)+\delta) \gamma_{2t}^1(\exp(2H)) $ where $ P_t(H) $ is a polynomial. It is then clear that $ \delta $ can be chosen large enough to make $ (P_t(H)+\delta) $ positive. The same is true in the general case also.
Let $ m $ be a non-negative integer. Then $ F \in \H_t^m(X_\C) $ if and only if $$\int_{X_\C} |F(z)|^2 w_{t,\delta}^m(z) dm(z) < \infty.$$ Moreover, the norm on $ \H_t^m(X_\C) $ is equivalent to the above weighted $ L^2 $ norm.
To check the positivity of the weight function we only need to recall that $$w_{t,\delta}^m(\exp H) = \int_{K_c} (\delta +\partial_t^m)\Gamma_{2t}(
h \exp(2H)) dh$$ and the integrand can be made positive by a proper choice of $ \delta.$ By Gutzmer’s formula the integral in the theorem reduces to $$C \sum_{\lambda \in \CP^+} d_\lambda \|A_\lambda(f)\|^2
(\delta +|\lambda+\rho|^{2m})$$ if $ F = f*\gamma_t.$ The above is clearly equivalent to the Sobolev norm on $ \H_t^m(X_\C) .$ If we equip $ \H_t^m(X_\C)$ with this norm instead of the original norm, then it follows that the heat kernel transform is an isometric isomorphism.
Perhaps it is better to state the characterisation of $ \H_t^m(X_\C)$ in the following form. Let us set $ W_t^m(z) = p_t(z)+w_{t,\delta}^m(z) = (1+\delta+
\partial_t^m)p_t(z) $ so that $ W_t^m(z) \geq p_t(z).$ Let $ \CB_t^m(X_\C) $ be the set of all holomorphic functions which are square integrable with respect to $ W_t^m .$ Equip $ \CB_t^m(X_\C) $ with the sesquilinear form $$(F,G)_m = \int_{X_\C} F(z)\overline{G(z)} w_t^m(z) dm(z).$$ This turns out to be a genuine inner product on $ \CB_t^m(X_\C) $ turning it into a Hilbert space which is the same as $ \H_t^m(X_\C)$.
The Segal-Bargmann transform is an isometric isomorphism from $ \H^m(X) $ onto $ \CB_t^m(X_\C) = \H_t^m(X_\C).$
Holomorphic Sobolev spaces of negative order
---------------------------------------------
The problem of characterising $ \H_t^{-s}(X_\C), s > 0 $ as a weighted Bergman space has a simple solution. In this case the weight functions are given by the Riemann-Liouville fractional integrals $$w_t^{-s}(\exp H) = \frac{1}{\Gamma(s)} \int_0^{2t} (2t-r)^{s-1} e^{r}
\gamma_{r}^1(\exp 2H) dr.$$ Note that unlike $ w_t^m $ the weight function $ w_t^{-s} $ are always positive.
Let $ s $ be positive. A holomorphic function $ F $ on $X_\C $ belongs to $ \H_t^{-s}(X_\C) $ if and only if $$\int_{X_\C} |F(z)|^2 w_t^{-s}(z) dm(z) < \infty .$$ Thus we can identify $\H_t^{-s}(X_\C) $ with $ \CF_t^{-s}(X_\C) $ defined using the weight function $ w_t^{-s}.$
Using Gutzmer’s formula we have $$\int_{i\ba}\int_U |F(g \exp(H).o)|^2 w_t^{-s}(\exp H.o) J(H) dg dH$$ $$= \sum_{\lambda \in \CP^+} d_\lambda \|A_\lambda(F)\|^2
\int_{i\ba}w_t^{-s}(\exp H.o)\varphi_\lambda(\exp(2H).o)J_1(2H) dH.$$ Since $$\int_{i\ba} \gamma_{r}^1(\exp 2H) \varphi_\lambda(\exp(2H).o) J_1(2H) dH
= c e^{r|\lambda+\rho|^2}$$ we see that $$\int_{i\ba}\int_U |F(g \exp(H).o)|^2 w_t^{-s}(\exp H.o) J(H) dg dH$$ $$= \sum_{\lambda \in \CP^+} d_\lambda \|A_\lambda(F)\|^2
\frac{1}{\Gamma(s)} \int_0^{2t} (2t- r)^{s-1} e^{r(1+|\lambda+\rho|^2)} dr.$$ We show below that $$c_1 (1+|\lambda+\rho|^2)^{-s} e^{2t(1+|\lambda+\rho|^2)} \leq
\frac{1}{\Gamma(s)} \int_0^{2t} (2t- r)^{s-1} e^{r(1+|\lambda+\rho|^2)} dr$$ $$\leq c_2 (1+|\lambda+\rho|^2)^{-s}e^{2t(1+|\lambda+\rho|^2)}.$$ The theorem follows immediately from these estimates. To verify our claim we look at the integral $$\frac{1}{\Gamma(s)} \int_0^t (t-r)^{s-1} e^{ar} dr =
e^{at} \frac{1}{\Gamma(s)} \int_0^t r^{s-1} e^{-ar} dr.$$ The last integral is nothing but $$e^{at} a^{-s} \left( 1- \frac{1}{\Gamma(s)} \int_{at}^\infty r^{s-1}
e^{-r} dr \right).$$ Since $ \int_{at}^\infty r^{s-1}e^{-r} dr $ goes to $ 0 $ as $ a $ tends to infinity our claim is verified.
The image of $ C^\infty(X)$ under heat kernel transform
=======================================================
In this section we characterise the image of $ C^\infty(X) $ under the heat kernel transform. We are looking for pointwise estimates on a holomorphic function $ F $ on $ X_\C $ that will guarantee that $ F = f*\gamma_t $ for a function $ f \in C^\infty(X).$ We begin with a necessary condition for functions in the Sobolev space $ \H_t^{m}(X_\C).$ Define the function $ \Phi_0 $ on $ \bt_\C $ by $\Phi_0(H) = \Pi_{\alpha \in R^+} \frac{(\alpha,H)}
{\sinh (\alpha,H)} $ where the product is taken over all $ R^+ $ which is the set of all positive roots in $ \Sigma(\bu_\C,\bt_\C).$ Recall that elements of $ \Sigma $ are the elements of $ \Sigma(\bu_\C,\bt_\C) $ having a nontrivial restriction to $ i\ba.$ The roots in $ R^+ $ give rise to elements of $ \Sigma^+ $ and a single $ \alpha \in \Sigma^+ $ may be given by several elements of $ R^+.$ (This number is denoted by $ m_\alpha .$) If we recall the definition of $ \Phi $ which occured in the estimates for $ \gamma_t^1 $ we see that $ \Phi(H) = \Phi_0(H) $ as long as $ H \in i\ba.$ We make use of this in what follows.
Let $ m $ be a non-negative integer. Every $ F \in
\H_t^{m}(X_\C)$ satisfies the estimate $$|F(u\exp H)|^2 \leq C (1+|H|^2)^{-m} \Phi(H) e^{\frac{1}{2t}|H|^2}$$ for all $ u \in U, H \in i\ba.$
: By standard arguments we can show that the reproducing kernel for the Hilbert space $ \H_t^{m}(X_\C) $ is given by $$K_t^{m}(g,h) = \frac{1}{(m-1)!}\int_0^\infty s^{m-1}e^{-s}
\gamma_{2(t+s)}(gh^*) ds$$ where $ h \rightarrow h^* $ is the anti-holomorphic anti-involution of $ U_\C $ which satisfies $ h^* = h^{-1} $ for $ h \in U $ (see e.g. \[12\]). Therefore, every $ F \in \H_t^{m}(X_\C) $ satisfies the estimate $$|F(g)|^2 \leq K_t^{m}(g,g) \|F\|_{m}.$$ When $ g = u\exp H $ it follows that $ gg^* = u \exp(2H) u^{-1} $ and hence we need to estimate $$\frac{1}{(m-1)!}\int_0^\infty s^{m-1} e^{-s}
\gamma_{2(t+s)}(\exp(2H)) ds .$$ In order to estimate the above integral we proceed as follows.
Recall that $ \gamma_t $ is the heat kernel associated to the operator $ \Delta = D-|\rho|^2 $ where $ D $ is the Laplace operator on $ X = U/K.$ Let $ D_U $ be the Laplacian on the group $ U $ and let $ \Delta_U =
D_U-|\rho|^2 .$ Let $ \rho_t(g) $ be the heat kernel associated to $\Delta_U $ which is given by $$\rho_t(g) = \sum_{\pi \in \hat{U}} d_\pi e^{-t\lambda(\pi)^2} \chi_\pi(g)$$ where $ \chi_\pi $ is the character of $ \pi $ and $ \lambda(\pi)^2 $ are the eigenvalues of $ \pi.$ When $ \pi = \pi_\lambda, \lambda \in \CP^+ $ we have $ \lambda(\pi)^2 = |\lambda+\rho|^2.$ We also have $$\gamma_t(g) = \sum_{\lambda \in \CP^+} d_\lambda e^{-t(|\lambda+\rho|^2)}
\varphi_\lambda(g).$$ Moreover, we have the relation $$\int_K \chi_\pi(gk) dk = c_\pi \varphi_\lambda(g)$$ where $ c_\pi = 1 $ if $ \pi = \pi_\lambda $ and $ c_\pi = 0 $ otherwise. Therefore, we have $$\gamma_t(g) = \int_K \rho_t(gk) dk$$ and consequently we need to estimate the integral $$\frac{1}{(m-1)!}\int_0^\infty \left(\int_K \rho_{2(t+s)}(\exp(2H)k) dk
\right) s^{m-1} e^{-s} ds .$$
Written explicitly the above integral is given by the sum $$\sum_{\pi \in \hat{U}}d_\pi (1+\lambda(\pi)^2)^{-m}
e^{-2t\lambda(\pi)^2} \int_K \chi_\pi(\exp(2H)k) dk .$$ Since $ \pi(\exp(2H)) $ is positive definite $ tr \pi(\exp(2H)) = \|\pi(\exp(2H))\|_1 $, the trace norm of $ \pi(\exp(2H)) $. Using the fact that $$\|\pi(\exp(2H))\|_1 = \sup \{ |tr \pi(\exp(2H))V|: V^*V = VV^* = I \}$$ we have the estimate $$|\chi_\pi(\exp(2H)k)| = |tr (\pi(\exp(2H))\pi(k))|$$ $$\leq tr \pi(\exp(2H)) = \chi_\pi(\exp(2H)).$$ Therefore, the sum is bounded by $$C \sum_{\pi \in \hat{U}}d_\pi (1+\lambda(\pi)^2)^{-m}
e^{-2t\lambda(\pi)^2} \chi_\pi(\exp(2H)) .$$ The above sum is related to the reproducing kernel for holomorphic Sobolev spaces on the compact Lie group $ U $ studied by Hall and Lewkeeratiyutkul in \[13\]. In that paper using estimates for the heat kernel $\rho_t $ they have proved that $$\sum_{\pi \in \hat{U}}d_\pi (1+\lambda(\pi)^2)^{-m}
e^{-2t\lambda(\pi)^2} \chi_\pi(\exp(2H))$$ $$\leq C (1+|H|^2)^{-m} \Phi_0(H)
e^{\frac{1}{2t}|H|^2}.$$ ( In \[13\] the authors have defined the heat kernel for the operator $ \frac{1}{2}\Delta_U $ rather than $ \Delta_U$.) This estimate immediately gives the required estimate for our kernel since $
\Phi_0(H) = \Phi(H), H \in i\ba.$ This completes the proof of the theorem.
Finding suitable pointwise estimates on a holomorphic function sufficient for the membership of the Holomorphic Sobolev spaces is a difficult problem as the proof requires good estimates on the derivatives of the heat kernel $ \gamma_t^1 $ on the noncompact dual. Such estimates are not available in the literature. Only recently good estimates on $ \gamma_t^1 $ have been obtained by Anker and Ostellari \[3\] and it is not clear if the same techniques will give us estimates on the derivatives of $ \gamma_t^1.$ So we proceed indirectly to get a sufficient condition. The method avoids estimates on the derivatives but uses only the estimate on $ \gamma_t^1.$ This is done by using Holomorphic Sobolev spaces of negative order.
Let $ n $ be the dimension of the Cartan subspace $ i\ba $ and let $ r $ be the least positive integer for which $ \Pi_{\alpha \in \Sigma^+}
|(\alpha,H)|^{m_\alpha} \leq C (1+|H|)^r.$ Determine $ d $ by the condition that the series $ \sum_{\lambda \in \CP^+} d_\lambda^2 (1+|\lambda+\rho|^2)^{-d+r+n+1} $ converges. ( Such a $ d $ exists since $ d_\lambda $ has a polynomial growth in $ |\lambda|.$)
Let $ F $ be a holomorphic function on $ X_\C $ which satisfies the estimate $$|F(u\exp(H))|^2 \leq C (1+|H|^2)^{-m-d}\Phi(H)
e^{\frac{1}{2t}|H|^2}$$ for all $ u \in U $ and $ H \in i\ba.$ Then $ F \in \H_t^m(X_\C).$
: In view of Theorem 3.1 which characterises holomorphic Sobolev spaces in terms of the holomorphic Fourier series, we have to show that $$\sum_{\lambda \in \CP^+} d_\lambda \left( \sum_{j=1}^{d_\lambda}
|\tilde{F}_j(\lambda)|^2 \right) (1+|\lambda+\rho|^2)^{m}
e^{-2t(|\lambda+\rho|^2)} < \infty .$$ In order to estimate the holomorphic Fourier coefficients $ \tilde{F}_j(\lambda)
$ we make use of the estimates on $ \gamma_t^1 $ proved by Anker and Ostellari \[3\]. They have shown that $$\gamma_t^1(\exp H) \leq C_t P_t(H) e^{-(\rho,H)-\frac{1}{4t}|H|^2}$$ where $ P_t(H) $ is an explicit polynomial (see the equation 3.1 in \[3\] for the exact expressin for $ P_t$). Since $ t $ is fixed we actually have the estimate $$\gamma_t^1(\exp H) \leq C_t (\Phi(H))^{\frac{1}{2}}e^{-\frac{1}{4t}|H|^2}.$$
We also know that the holomorphically extended spherical functions $ \varphi_j^\lambda $ satisfy the estimates $$|\varphi_j^\lambda(u\exp(H))| \leq \varphi_\lambda(\exp(H)).$$ Moreover, $ \varphi_\lambda(\exp(H)) = \psi_{-i(\lambda+\rho)}(\exp(H)) $ for all $ H \in i\ba $ and hence well known estimates on $ \psi_\lambda $ leads to $$|\varphi_j^\lambda(u\exp(H))| \leq C e^{|\lambda+\rho||H|} e^{-(\rho,H)}.$$ We refer to Gangolli-Varadarajan \[9\] ( Section 4.6 ) for these estimates on the spherical functions $ \psi_\lambda.$ We also note that $ \Phi(H)(\Phi(2H))^{-1} \leq C e^{2(\rho,H)}.$
Therefore, making use of the above two estimates, under the hypothesis on $ F $ we see that $ |\tilde{F}_j(\lambda)| $ is bounded by a constant multiple of the integral $$\int_{i\ba} \Phi(2H) e^{\frac{1}{4t}|H|^2}
(1+|H|^2)^{-m-d} e^{|\lambda+\rho||H|} e^{-\frac{1}{2t}|H|^2}
J_1(2H) dH.$$ Recalling the definition of $ J_1(2H) $ we see that $ \Phi(2H) J_1(2H) $ is bounded by a constant multiple of $ (1+|H|)^r.$ Thus the above integral is bounded by $$\int_{i\ba}(1+|H|^2)^{-m-d+r} e^{|\lambda+\rho||H|}e^{-\frac{1}{4t}|H|^2}
dH .$$ The above integral can be easily estimated to give $$|\tilde{F}_j(\lambda)| \leq C_m (1+|\lambda +\rho|^2)^{-m-d+r+n+1}
e^{t|\lambda +\rho|^2} .$$ This proves our claim and completes the proof of sufficiency.
Combining Theorems 4.1 and 4.2 and we obtain the following characterisation of the image of $ C^\infty(X) $ under the Segal-Bargmann transform.
A holomorphic function $ F $ on $ X_\C $ is of the form $ F = f*\gamma_t $ with $ f \in C^\infty(X) $ if and only if it satisfies $$|F(u\exp(H))| \leq C_m (1+|H|^2)^{-m/2}(\Phi(H))^{\frac{1}{2}}
e^{\frac{1}{4t}|H|^2}$$ for all $ u \in U, H \in i\ba $ and for all positive integers $m.$
This theorem follows from the fact that $ C^\infty(X) $ is the intersection of all the Sobolev spaces $ \H^m(X).$
We conclude this section by giving a characterisation of the image of distributions on $ X $ under the heat kernel transform. If $ f $ is a distribution $ f*\gamma_t $ still makes sense and extends to $ X_\C $ as a holomorphic function. We now prove the following theorem which was stated as a conjecture in \[13\].
A holomorphic function $ F $ on $ X_\C $ is of the form $ F = f*\gamma_t $ for a distribution $ f $ on $ X $ if and only if it satisfies the estimate $$|F( u\exp(H))| \leq C (1+|H|^2)^{m/2}(\Phi(H))^{\frac{1}{2}}
e^{\frac{1}{4t}|H|^2}$$ for some positive integer $ m $ for all $ u \in U $ and $ H \in i\ba.$
First we prove the sufficiency of the above condition. If we could show that the holomorphic Fourier coefficients of $ F $ satisfy $$|\tilde{F}_j(\lambda)| \leq A (1+|\lambda+\rho|^2)^{N}
e^{t(|\lambda+\rho|^2)}$$ for some $ N $ then by Theorem 3.1 it would follow that $ F = f*\gamma_t $ for some $ f \in \H^{-d}(X) $ for a suitable $ d.$ Since the union of all the Sobolev spaces is precisely the space of distributions we get the result. In order to prove the above estimate we can proceed as in the previous theorem. We end up with the integral $$\int_{i\ba} \Phi(2H) e^{\frac{1}{4t}|H|^2}
(1+|H|^2)^{m/2} e^{|\lambda+\rho||H|} e^{-\frac{1}{2t}|H|^2}
J(H) dH.$$ As before this leads to the estimate $ A (1+|\lambda+\rho|^2)^{m+r+n+1} e^{t(|\lambda+\rho|^2)} $ proving the sufficiency.
For the necessity: since every distribution belongs to some Sobolev space let us assume $ f \in \H^{-m}(X) $ for a positive integer. Then $ F = f*\gamma_t $ belongs to $ \H_t^{-m}(X_\C) $ whose reproducing kernel is given by $$K_t^{-m}(g,h) = \sum_{\lambda \in \CP} d_\lambda (1+|\lambda+\rho|^2)^m
e^{-2t|\lambda+\rho|^2} \sum_{j=1}^{d_\lambda}\varphi_j^\lambda(g)\overline{
\varphi_j^\lambda(h^*)}.$$ Proceeding as in Theorem 3.1 we need to estimate $$\sum_{\pi \in \hat{U}} d_\pi (1+\lambda(\pi)^2)^m e^{-2t\lambda(\pi)^2}
\chi_\pi(\exp(2H)).$$ To this end we make use of the Poisson summation formula proved by Urakawa \[21\] as in Hall \[12\]. According to this formula $$\sum_{\pi \in \hat{U}} d_\pi e^{-2t\lambda(\pi)^2}\chi_\pi(\exp(2H))
= e^{2t|\rho|^2}(8\pi t)^{-\frac{n}{2}}e^{\frac{1}{2t}|H|^2}\Phi(H)k(t,H)$$ where $ k(t,H) $ is known explicitly (see equation 8 in \[12\]). We need to estimate the $m-$th derivative of $ k(t,H) $ with respect to $ t.$
The above function $ k(t,H) $ has been estimated in \[12\]. There good esimates for all values of $ t $ were needed and consequently the estimation was not easy. Here we just need to estimate the derivative for a fixed $t.$ Observe that any derivative falling on $ e^{\frac{1}{2t}|H|^2} $ brings down a factor of $ |H|^2.$ The function $ k(t,H) $ is given by the sum $$k(t,H) = \sum_{\gamma_0 \in \Gamma \cap \overline{\ba^+}} \epsilon(\gamma_0)
e^{-\frac{1}{8t}|\gamma_0|^2}p_{\gamma_0}(t,H)$$ with $ p_{\gamma_0}(t,H) $ given by the expression $$p_{\gamma_0}(t,H) = \pi(H)^{-1} \sum_{\gamma \in W.\gamma_0}\pi(H-\frac{1}{2i}\gamma)
e^{\frac{i}{t}(H,\gamma)}.$$ In the above, $ \pi(H) = \Pi_{\alpha \in \Delta^+}(\alpha,H), W $ is the Weyl group, $ \overline{\ba^+} $ is the closed Weyl chamber and $ \Gamma $ is the kernel of the exponential mapping for the maximal torus etc. If we can show that any derivative falling on $ k(t,H)
$ in effect brings down a factor of $ |H| $ then the $m-$th derivative can be estimated to give $$K_t^{-m}(g,g^*) \leq C (1+|H|^2)^{2(m+d)} \Phi(H) e^{\frac{1}{2t}|H|^2} .$$ This will then complete the proof of the necessity.
We now give some details of the above sketch of the proof. In \[12\] the author has proved that there is a polynomial $ P $ such that the estimate $$|p_{\gamma_0}(t,H)| \leq P(t^{-1/2}|\gamma_0)|)$$ holds. This has been stated and proved as Proposition 3 in \[12\]. For our proof we need to get estimates for sums of the form $$p_{\gamma_0,j}(t,H) = \pi(H)^{-1} \sum_{\gamma \in W.\gamma_0}
\pi(H-\frac{1}{2i}\gamma) (H,\gamma)^j e^{\frac{i}{t}(H,\gamma)}.$$ We claim that $$| p_{\gamma_0,j}(t,H)| \leq C_{j,t}|H|^j P_j(t,|\gamma_0|)$$ for some polynomials $ P_j(t,.).$ This will give us the required estimate. As in \[12\] we can assume that $ t = 1$. We indicate the proof when $ j =1, $ the general case being very similar.
Consider the operators $ I_\alpha $ defined by ( see \[12\]) $$I_\alpha f(x) = \int_0^\infty f(x-t\alpha)dt$$ which invert the directional derivative operators $ D_\alpha.$ For any distribution supported on a cone over $ \Delta^+ $ we can define $ I_\alpha T $ by duality (cf. Definition 8 in \[12\]). In \[12\] the author has proved that the convex hull of the support of the distribution $ S = I_{\alpha_1}I_{\alpha_2}....
I_{\alpha_k}T $ is contained in the convex hull of the support of $ T $ whenever $ T $ is a compactly supported distribution which is alternating with respect to the action of the Weyl group. (This is proved in Lemma 9 of \[12\].) Let $ \CF $ be the Euclidean Fourier transform. Let $ T $ denote the Fourier transform of the distribution $ \pi(H) p_{\gamma_0,1}(1,H) $ which can be written as $$T = c \sum_{\gamma \in W.\gamma_0} D_\gamma T_\gamma$$ where $ T_\gamma $ is the Fourier transform of $ \pi(H-\frac{1}{2\i}\gamma)
e^{i(H,\gamma)}.$ It is clear that $ T $ is alternating and hence Lemma 9 of \[12\] applies.
As in \[12\] we set $ S_\gamma = I_{\alpha_1}I_{\alpha_2}....I_{\alpha_k}T_\gamma $ and note that $ S_\gamma $ is a finite linear combination of distributions of the form $ (\alpha_{i_1},\gamma)....(\alpha_{i_l},\gamma)I_{\alpha_{i_1}}.....
I_{\alpha_{i_l}}\delta_\gamma.$ Defining $ S = I_{\alpha_1}I_{\alpha_2}....I_{\alpha_k}T $ we get $$S = c \sum_{\gamma \in W.\gamma_0}D_\gamma S_\gamma$$ and therefore, $$\CF^{-1}S(H) = c (\pi(H))^{-1}\CF^{-1}T(H) = c' p_{\gamma_0,1}(1,H).$$ Thus we need to estimate $ \CF^{-1}S(H).$ If $ E $ is the convex hull of the support of $ S $ then by Lemma 9 (of \[12\]) it is contained in the convex hull of $ W.\gamma_0 .$ This follows from the fact that $ T_\gamma $ are linear combinations of $$(\alpha_{i_1},\gamma)....(\alpha_{i_l},\gamma)D_\gamma D_{\alpha_{i_{l+1}}}
.....D_{\alpha_{i_k}}\delta_\gamma.$$
Finally, if $ \varphi $ is any nonnegative $ C_0^\infty $ function supported in a small neighbourhood $ E_\epsilon $ of $ E $ and identically one on another (smaller) neighbourhood of $ E $ then $ \left(S, f \right) =
\sum_{\gamma \in W.\gamma_0}
\left( D_\gamma(\varphi S_\gamma),f\right) $ for any test function $ f $ as can be easily checked. This gives us $$\CF^{-1}S(H) = c \sum_{\gamma \in W.\gamma_0} (H,\gamma)
\CF^{-1}(\varphi S_\gamma)(H)$$ which leads to the estimate $$|\CF^{-1}S(H)| \leq C |H| |\gamma_0| \sum_{\gamma \in W.\gamma_0}
|\CF^{-1}(\varphi S_\gamma)(H)|.$$ The last term is bounded by $$\int \varphi(H) d|S_\gamma| \leq |S_\gamma|(E_\epsilon) .$$ Since $ S_\gamma $ is a linear combination of the positive measures $ (\alpha_{i_1},\gamma)....(\alpha_{i_l},\gamma)I_{\alpha_{i_1}}
.....I_{\alpha_{i_l}}\delta_\gamma $ the measure $ |S_\gamma|(E_\epsilon) $ can be estimated as in \[12\] to give the required estimate $$|\CF^{-1}S(H)| \leq C_\epsilon P_1(1,\gamma_0) |H| .$$ This completes the proof of the theorem.
[**Acknowledgments**]{}
The author wishes to thank the referee for his thorough reading of the previous version of this paper and making several useful remarks. He pointed out several inaccuracies, demanded clarifications of several points and suggested a reorganisation of the paper all of which have considerably improved the readability of the paper. The author wishes to thank E. K. Narayanan for answering several naive questions on the structure theory of semisimple Lie groups. He is also thankful to Bernhard Kroetz for pointing out an error in a previous version of this paper. This work is supported by a grant from UGC under SAP.
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abstract: 'We compare the response of five different models of two interacting electrons in a quantum dot to an external short lived radial excitation that is strong enough to excite the system well beyond the linear response regime. The models considered describe the Coulomb interaction between the electrons in different ways ranging from mean-field approaches to configuration interaction (CI) models, where the two-electron Hamiltonian is diagonalized in a large truncated Fock space. The radially symmetric excitation is selected in order to severely put to test the different approaches to describe the interaction and correlations of an electron system in a nonequilibrium state. As can be expected for the case of only two electrons none of the mean-field models can in full details reproduce the results obtained by the CI model. Nonetheless, some linear and nonlinear characteristics are reproduced reasonably well. All the models show activation of an increasing number of collective modes as the strength of the excitation is increased. By varying slightly the confinement potential of the dot we observe how sensitive the properties of the excitation spectrum are to the Coulomb interaction and its correlation effects. In order to approach closer the question of nonlinearity we solve one of the mean-field models directly in a nonlinear fashion without resorting to iterations.'
author:
- Vidar Gudmundsson
- Sigtryggur Hauksson
- Arni Johnsen
- Gilbert Reinisch
- Andrei Manolescu
- Christophe Besse
- Guillaume Dujardin
title: 'Excitation of radial collective modes in a quantum dot: Beyond linear response'
---
introduction
============
Far-infrared spectroscopy and transport measurements were from early on used to investigate the electronic structure[@Reimann02:1283] of quantum dots of various types. Far-infrared spectroscopy of arrays of quantum dots[@Demel90:788a; @Shikin91:11903a] turned out to be rather insensitive to the exact form of the interaction between the electrons. The reason being that most arrays of quantum dots resulted in almost parabolic confinement of electrons to individual dots in the low energy regime. Soon it was realized that an exact symmetry condition, known as the the extended Kohn’s theorem[@Kohn61:1242] is valid for such systems as long as each dot is much smaller than the wavelength of the dipole radiation, and results in a pure center-of-mass motion of the electrons in each dot, independent of the number of electrons and the nature of the interaction between them. Signatures of deviations from the parabolic confinement where soon discovered in experimental results and interpreted with model calculations based on various approaches to linear response.[@Pfannkuche91:13132; @Gudmundsson91:12098; @Pfannkuche93:6] The Coulomb blockade helped guaranteeing a definite number of electrons in each quantum dot homogeneously in the large arrays that were necessary to allow measurement of the weak FIR absorption signal. Deviations from the parabolic confinement of electrons in quantum dots lead to the excitation of internal collective modes that can cause splitting of the upper plasmonic branch and make visible the classical Bernstein[@Bernstein58:10] modes.[@Gudmundsson95:17744; @Krahne01:195303] In the lower plasmonic branch they lead to weak oscillations caused by filling factor dependent screening properties.[@Bollweg96:2774; @Darnhofer96:591]
Resonant Raman scattering has been applied to quantum dots to analyze “single-electron” excitations and collective modes with monopole, dipole, or quadrupole symmetry ($\Delta M=0,\pm1,\pm2$).[@Steinebach99:10240; @Steinebach00:15600] As the monopole collective oscillations are excitations that can be exclusively described by internal relative coordinates one would expect them to be more influenced by the Coulomb interaction between the electrons than the dipole excitations that have to be described by relative and center-of-mass coordinates, or purely by the latter ones when the Kohn theorem holds.[@Kohn61:1242] The $\Delta M=0$ collective mode among others was measured by a very different method and calculated for a confined two-dimensional electron system in the classical regime on the surface of liquid Helium.[@PhysRevLett.54.1710]
In the far-infrared and the Raman measurements of arrays of dots the excitation has always been weak and some version of linear response has been an adequate approach to interpret the experimental results. All the same, curiosity has driven theoretical groups into questioning how the electron system in a quantum dot would respond as the linear regime is surpassed and a strong excitation would pump energy into the system.[@Puente01:235324; @Gudmundsson03:161301; @Gudmundsson03:0304571] These studies have been undertaken with some kind of a mean-field model to incorporate the Coulomb interaction between the electrons. Here, we will explore this nonlinear excitation regime with a model built on exact numerical diagonalization or configuration interaction (CI)[@Pfannkuche93:6] and compare the results with the predictions of three different mean-field approaches, and a time-dependent Hubbard model. Besides the question of what happens in the nonlinear regime, we want to see how close to the exact results the mean-field models can come for only two electrons in the dot, a regime that is indeed challenging for mean-field approaches which in general are more appropriate for a higher number of electrons. We will address issues of nonlinear behavior. What do we classify as nonlinear behavior? Can we see it emerging in an exact model? How, and when is it inherent in a mean-field approach?
Short excitation in the THz regime
==================================
In order to describe the response to an excitation of arbitrary strength we will follow the time-evolution of the system by methods that are appropriate to each model. At $t=0$ the quantum dot is radiated by a short THz pulse $$\begin{aligned}
W(t) &=& V_t r^{|N_p|}\cos{(N_p\phi)}\exp{(-sr^2-\Gamma t)}\nonumber\\
&{\ }&\sin{(\omega_1t)}\sin{(\omega t)}\theta (\pi -\omega_1t),
\label{Wt}\end{aligned}$$ where $\theta$ is the Heaviside step function. For the purpose of making the response strongly dependent on the Coulomb interaction between the electrons we select the monopole or the breathing mode with $N_p=0$. It should be kept in mind that this short excitation pulse perturbs the system in a wide frequency range.
The quantum dot will have a parabolic confinement potential $$V_{\mathrm{par}}(r)=\frac{1}{2}m^*\omega_0^2r^2,
\label{Vpar}$$ with $\hbar\omega_0 = 3.37$ meV. In addition, we will sometimes add a small potential hill in the center of the dot $$V_{\mathrm{c}}(r) = V_0\exp{(-\gamma r^2)},
\label{Vc}$$ with $V_0 = 3.0$ meV, and $a^2\gamma = 1.0$, where $a = \sqrt{\hbar/(m^*\omega_0)}$ is the characteristic length scale for the parabolic confinement. We will be assuming GaAs parameters here with $m^*=0.067m_e$ and a dielectric constant $\kappa = 12.4$. If we select $sa^2 = 0.8$, $\hbar\omega_1 = 0.658$ meV, $\hbar\omega = 2.63$ meV, and $\Gamma =2.0$ ps$^{-1}$, then the initial pulse of duration approximately $3$ ps represents a spatial circular Gaussian pulse rising from zero and vanishing after its amplitude gets negative. The system is perturbed by a radial compression followed by a slight radial expansion and then left to oscillate freely about the equilibrium point. The system will be kicked out of equilibrium and the time-evolution has to be described accordingly for each model.
The reason for adding the central hill (\[Vc\]) to the quantum dot is to avoid any special symmetry that could result from the parabolic confinement (\[Vpar\]).
Time-evolution of quantum dot Helium with a DFT interaction
===========================================================
The details of a density functional theoretical (DFT) approach to the model used to describe nonadiabatic excitation of electrons in a quantum ring or dots in an external magnetic field has been published earlier.[@Gudmundsson03:161301; @Gudmundsson03:0304571] Here, we will use the model for a vanishing external magnetic field and properly make clear the difference in the calculation of the time-evolution of this mean-field model to the CI model. To accomplish this we need to list few steps.
The “single-electron” energy spectrum of the model is presented in Fig. \[E-rof-dft\] at temperature $T=0.1$ K and for a small hill (\[Vc\]) placed in the center of the system.
![(Color online) The effective single-electron energy spectrum for the DFT-version of the model of two electrons in a parabolic quantum dot with a small central hill (\[Vc\]) as a function of the quantum number of angular momentum $M$. The chemical potential, $\mu$, needed to have two electrons in the dot is indicated by a solid green horizontal line. $V_0=3$ meV, $T=0.1$ K.[]{data-label="E-rof-dft"}](Fig01.pdf){width="42.00000%"}
The finite, but small temperature is used to stabilize the iteration process used to solve the DFT model. The chemical potential $\mu$ needed to have two electrons in the ground state of the system is indicated in the figure by a horizontal green line. The calculation is a “grid-free” approach utilizing the eigenstates of the noninteracting system as a functional basis $\{|nM\rangle\}$. The interacting states $|\alpha )$ can not be assigned a definite quantum number $n$ and $M$, but as the system is circularly symmetric here, by comparing the location in the energy spectrum and by checking the leading contribution to the interacting states we allow ourselves to assign, for educational purposes, the quantum numbers shown in Fig. \[E-rof-dft\]. The central hill (\[Vc\]) and the Coulomb interaction raise the energy of the states with high $M=0$ contribution.
To calculate the time-evolution of the system kicked out off equilibrium by the perturbing pulse (\[Wt\]) we use the Liouville-von Neumann equation for the density operator $$i\hbar \frac{d}{dt}{\rho}(t) = [H + W(t),\rho (t)],
\label{L-vN-dft}$$ represented in the noninteracting basis $\{|n,M\rangle\}$. The structure of this equation is inconvenient for numerical evaluation so we resort instead to the time-evolution operator $T$, defined by $\rho (t) = T(t)\rho_0T^+(t)$, which has the simpler equation of motion $$\begin{aligned}
i\hbar\dot T(t) &=& H(t)T(t)\nonumber\\
-i\hbar\dot T^+(t) &=& T^+(t)H(t).
\label{Teq} \end{aligned}$$ The single-electron basis is truncated after tests for convergence of the time-evolution with the parameters used here. We discretize time and use the Crank-Nicholson algorithm for the time-integration with the initial condition, $T(0)=1$.
The circular symmetry of the confinement potential (Eq.’s (\[Vpar\]) and (\[Vc\])) and the excitation pulse (\[Wt\]) suggest the mean value of the radius squared to be an ideal observable to be analyzed. In Fig. \[r2-dft-T01-hill\] we show $\langle r^2\rangle$ as function of time $t$ and the strength of the perturbing pulse $V_t$.
![(Color online) The time-evolution of the expectation value $\langle r^2\rangle$ as function of the strength of the initial perturbation pulse, $V_t$, for the DFT-version of the model of two electrons in a quantum dot. $V_0=3$ meV, $T=0.1$ K.[]{data-label="r2-dft-T01-hill"}](Fig02.pdf){width="42.00000%"}
We see already in Fig. \[r2-dft-T01-hill\] that the amplitude of the response to the initial perturbation (\[Wt\]) is nonlinear. To analyze this better we show the Fourier transform in Fig. \[FFT-r2-dft-T01-hill\](a), where we indeed see a local minimum around $V_t\approx 35-40$ meV.
![(Color online) The Fourier power spectrum for the time-evolution of $\langle r^2\rangle$ for the DFT-version of the model of two-electrons in a quantum dot. The lower panel is a side view to demonstrate the stability in frequency for different values of excitation $V_t$. $V_0=3$ meV, $T=0.1$ K.[]{data-label="FFT-r2-dft-T01-hill"}](Fig03a.pdf "fig:"){width="46.00000%"} ![(Color online) The Fourier power spectrum for the time-evolution of $\langle r^2\rangle$ for the DFT-version of the model of two-electrons in a quantum dot. The lower panel is a side view to demonstrate the stability in frequency for different values of excitation $V_t$. $V_0=3$ meV, $T=0.1$ K.[]{data-label="FFT-r2-dft-T01-hill"}](Fig03b.pdf "fig:"){width="46.00000%"}
Curiously enough, this local minimum can not be seen in the results if we turn off the exchange and the correlation functionals in the DFT model, i.e. if we use a Hartree approximation (HA) for the Coulomb interaction, see Fig. \[FFT-r2-hartree-T01\].
![(Color online) The Fourier power spectrum as a function of energy and perturbation strength $V_t$ for the Hartree Approximation. $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="FFT-r2-hartree-T01"}](Fig04a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum as a function of energy and perturbation strength $V_t$ for the Hartree Approximation. $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="FFT-r2-hartree-T01"}](Fig04b.pdf "fig:"){width="42.00000%"}
For a later discussion we note here that the time-dependent HA calculations for the present parameters are much more stable then the DFT version. We are thus able to go to higher values of $V_t$ and observe the time-evolution for longer time resulting in more accurate Fourier transforms. In the DFT or the HA model the part of the Hamiltonian describing the effective Coulomb interaction remains time-dependent at all times, even after the initial perturbing pulse has vanished, since the local effective potential depends on the electron density which is oscillating in time. It is thus of no surprise that in these mean-field models the occupation, the diagonal elements of the density matrix (\[L-vN-dft\]), remain time-dependent as can be seen in Fig. \[Occupation-T\].
![(Color online) Time-dependent occupation of effective single-electron states (of the noninteracting basis ${|nM\rangle}$) for the HA model with a central hill (\[Vc\]) for $V_t=10.0$ meV (left panel), and $V_t=200.0$ meV (right panel). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="Occupation-T"}](Fig05a.pdf "fig:"){width="23.00000%"} ![(Color online) Time-dependent occupation of effective single-electron states (of the noninteracting basis ${|nM\rangle}$) for the HA model with a central hill (\[Vc\]) for $V_t=10.0$ meV (left panel), and $V_t=200.0$ meV (right panel). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="Occupation-T"}](Fig05b.pdf "fig:"){width="23.00000%"}
This time-dependence of the occupation and the effective interaction will be in contrast to what happens in the CI calculation described below.
In a real system, an open system, the oscillations will be damped by phonon interactions[@PhysRevB.75.125324] or photons.[@PhysRevB.87.035314] In the far-infrared regime the radiation time scale is much longer than the 100 ps during which we follow the evolution of the system here.
It is possible to construct the time-dependent induced density, $\delta n(r,t)=n(r,t)-n(r,0)$, for the oscillations in the system in the hope to monitor the modes being occupied for different values of $V_t$. In Fig. \[dn-dft-050-350\] we see the induced density for the DFT model over approximately one oscillation for $V_t=5$ and $35$ meV. It is clear that for the higher value of excitation a second oscillation mode is superimposed on the fundamental mode visible for $V_t=5$ meV.
![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06a.pdf "fig:"){width="21.00000%"} ![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06b.pdf "fig:"){width="21.00000%"}\
![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06c.pdf "fig:"){width="21.00000%"} ![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06d.pdf "fig:"){width="21.00000%"}\
![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06e.pdf "fig:"){width="21.00000%"} ![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06f.pdf "fig:"){width="21.00000%"}\
![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06g.pdf "fig:"){width="21.00000%"} ![(Color online) The induced density $\delta n(r,t)=n(r,t)-n(r,0)$ within one period for the DFT model for $V_t=5$ meV (left), and $V_t=35$ meV (right). $V_0=3.0$ meV, $T=0.1$ K.[]{data-label="dn-dft-050-350"}](Fig06h.pdf "fig:"){width="21.00000%"}
For still higher excitation this becomes even more apparent. In Fig. \[E-rof-dft\] the main “single-electron” contribution to this collective oscillation is indicated by an arrow between $|00)$ and $|10)$. Higher excitation brings in a mixing from the $|10)$ to $|20)$ transition, and higher temperature would activate transitions from $|0-1)$ to $|1-1)$, and from $|01)$ to $|11)$.
Time-evolution of a quantum dot Helium described by a nonlinear Schr[ö]{}dinger-Poisson equation
================================================================================================
We will consider one more variant of a mean-field model for the two Coulomb interacting electrons in the quantum dot. This model could be considered a version of the HA for a special case, but we investigate it here for a different reason. It allows for the application of a nonlinear solution method to be described at the end of this section.[@Reinisch11:699; @Reinisch12:902]
Consider a $S=0$ electron pair located at $z_{1,2}=x_{1,2}+iy_{1,2}$ in the $x-y$ plane and confined by the 2D parabolic potential (\[Vpar\]). Since their spins are opposite, both electrons can stay, as fermions, in the same orbital state $\psi$. Moreover, they obey a pair orbital symmetry. Therefore the simplest two-electron wavefunction $\Psi_{\mathrm{pair}}(z_1,z_2,t)$ is $$\label{eq-orbital2el}
\Psi_{\mathrm{pair}}(z_1,z_2,t)=\psi(z_1,t) \psi(z_2,t),$$ where $|\Psi_{\mathrm{pair}}(z_1,z_2,t)|^2$ is the probability density to find at time $t$ either electron at $z_i$ while the other is at $z_j$ ($i\neq j = 1,\,2$). Therefore, the normalization condition reads $$\label{eq-norm}
\int d^2z_1 d^2z_2 |\Psi_{\mathrm{pair}}(z_1,z_2,t)|^2 =
\left[\int d^2z |\psi(z,t)|^2\right]^2=1.$$ We assume that $\psi(z,t)\equiv \psi(x,y,t)$ is a time-dependent nonlinear state defined by the following Schr[ö]{}dinger-Poisson (SP) differential system $$\begin{aligned}
\label{eq-Schroe_vraie}
i\hbar \frac{\partial}{\partial t} \psi &= H \psi,\\
\label{eq-Poisson_vraie}
\nabla^2 \Phi &= -2\pi{\cal N}\hbar\omega |\psi|^2,\end{aligned}$$ where ${\cal N}$ is a dimensionless order parameter of the SP system that defines the strength of the Coulomb repulsive interaction potential $\Phi$ between the particles in units of $\hbar\omega_0$ (in a loose sense, we call it the “norm”: see below Eq. (\[eq-normu\])). The 2D nonlinear Hamiltonian is defined by $$\label{eq-H_vrai}
H =-\frac{\hbar^2}{2m^*} \nabla^2 +\Phi(x,y,t)
+ \frac{1}{2} m^* \omega_0^2 (x^2+y^2).$$ Using the characteristic length $a$ of the parabolic confinement and its frequency $\omega_0$ we perform the following change of variables $$\label{eq-change_var}
X=\frac{x}{a};\enspace Y=\frac{y}{a};\enspace \tau=\omega t
;\enspace \psi =\sqrt{\frac{2m^*\omega_0}{\hbar{\cal N}}}u(X,Y,\tau).$$ Accordingly, Eq. (\[eq-norm\]) becomes $$\label{eq-normu}
\int |u(X,Y,\tau)|^2 dX dY ={\cal N},$$ while the SP time-space differential system (\[eq-Schroe\_vraie\]-\[eq-H\_vrai\]) yields $$\label{eq-Sdimless}
i\frac{\partial}{\partial \tau}u +\nabla_{X,Y}^2 u - V u =0,$$ $$\label{eq-Pdimless}
\nabla_{X,Y}^2 V +|u|^2-1 =0,$$ where $\nabla_{X,Y}$ operates on the new variables $X$ and $Y$. The (time-dependent) effective mean-field dimensionless potential experienced by the particles is $$\label{eq-POTdimless}
V= \frac{\frac{1}{2} m^* \omega^2 (x^2+y^2)+\Phi}{\hbar\omega_0}
=\frac{1}{4}(X^2+Y^2)+\frac{\Phi}{\hbar\omega_0}.$$ We wish to define the observable which allows comparison with the previous sections. Labelling $\bar z=\frac{1}{2}(z_1+z_2)$, $\bar x=\frac{1}{2}(x_1+x_2)$, and $\bar y=\frac{1}{2}(y_1+y_2)$, we have $\bar z \bar z^*={\bar x}^2+{\bar y}^2$ and therefore $$\label{eq-MeanVal}
\langle\langle \bar z \bar z^* \rangle\rangle=
\frac{1}{2}\Bigl[ \langle x^2 \rangle + \langle y^2 \rangle +
\langle x \rangle^2 + \langle y \rangle^2\Bigr],$$ where for any observable $A$ $$\label{eq-MeanPair}
\langle\langle A\rangle\rangle=\int d^2z_1 d^2z_2 A |\Psi_{\mathrm{pair}}|^2 ,$$ and $$\label{eq-MeanSP}
\langle A\rangle=\int dx dy A |\psi|^2,$$ (cf. Eq. (\[eq-norm\])). Obviously, $\sqrt{\langle\langle \bar z \bar z^* \rangle\rangle}$ is a sound measure of the time-dependent extension of the system. In the dimensionless variables (\[eq-change\_var\]), it reads $$\label{eq-MeanSPreduced}
R(\tau)= \frac{1}{\sqrt{2}}\Bigl[ \langle X^2 \rangle_u + \langle Y^2 \rangle_u +
\langle X \rangle_u ^2 + \langle Y \rangle_u ^2\Bigr]^{\frac{1}{2}},$$ where $$\label{eq-MeanSPred}
\langle A\rangle_u=\frac{1}{{\cal N}} \int dX dY A |u|^2 ,$$ (cf. Eq. (\[eq-normu\])).
The solution of system (\[eq-Schroe\_vraie\]-\[eq-Poisson\_vraie\]) demands the initial profile $\psi(x,y,0)$. For these means we use the radial symmetric ground state of the time-independent system. The Poisson equation (\[eq-Poisson\_vraie\]) is two-dimensional here and would thus produce a logarithmic Green function for homogeneous space instead of the $1/r$ three-dimensional that we should be using since the electric field can not be confined to 2D even though the electrons can be. But, we do accept this discrepancy for three reasons. First, the asymptotic behavior at $r\sim 0$ is not so dissimilar though the logarithm represents a bit softer repulsion, and second, the long range behavior will not carry much weight due to the parabolic confinement potential (\[Vpar\]). Third, and most important, the SP system (\[eq-Schroe\_vraie\]-\[eq-Poisson\_vraie\]) can be solved directly to obtain a nonlinear solution.[@Reinisch11:699; @Reinisch12:902] Generally, for physical mean-field models, which are of course nonlinear, the traditional method is to seek a solution by iteration. In case of the HA or the DFT model here, the effective interaction potential is calculated after an initial guess has been made for the wavefunctions. Then the new wavefunctions are sought by methods from linear algebra, and the iterations are continued until convergence is reached. The wavefunctions will be orthonormal. When the SP system is solved directly the wavefunctions are not in general orthonormal. Besides convenience, the reason for the iteration method is the connection of the Hartree and Hartree-approximations to higher order methods in many-body theory, that can only be established in case of orthonormal solutions. The hope is that the iteration method supplies the nonlinear solution in this sense or a solution very close to it. In fact, the nonlinear solutions are almost orthonormal with some small discrepancy of the order of $1-5\%$.
In Fig. \[FFT-Schr-Poisson-nohill\] we show the results for the time-evolution of the expectation value $\langle\langle \bar z \bar z^* \rangle\rangle$ and the corresponding Fourier transform for the SP model without a central hill in the quantum dot.
![(Color online) The time-dependent expectation value of $\langle r^2 \rangle$ and the corresponding Fourier power spectrum for the Schr[ö]{}dinger-Poisson model of the quantum dot without a central hill. $V_0=0$, $T=0$ K.[]{data-label="FFT-Schr-Poisson-nohill"}](Fig07a.pdf "fig:"){width="46.00000%"} ![(Color online) The time-dependent expectation value of $\langle r^2 \rangle$ and the corresponding Fourier power spectrum for the Schr[ö]{}dinger-Poisson model of the quantum dot without a central hill. $V_0=0$, $T=0$ K.[]{data-label="FFT-Schr-Poisson-nohill"}](Fig07b.pdf "fig:"){width="46.00000%"}
![(Color online) The time-dependent expectation value of $\langle r^2 \rangle$ and the corresponding Fourier power spectrum for the Schr[ö]{}dinger-Poisson model of the quantum dot with a central hill. $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-Schr-Poisson"}](Fig08a.pdf "fig:"){width="46.00000%"} ![(Color online) The time-dependent expectation value of $\langle r^2 \rangle$ and the corresponding Fourier power spectrum for the Schr[ö]{}dinger-Poisson model of the quantum dot with a central hill. $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-Schr-Poisson"}](Fig08b.pdf "fig:"){width="46.00000%"}
In Fig. \[FFT-Schr-Poisson\] we display the time-evolution of the expectation value $\langle\langle \bar z \bar z^* \rangle\rangle$ and the corresponding Fourier transform for the SP model with a central hill in the dot. Below, we will compare the location of the main peak or peaks for low $V_t$ for the different models, but here we notice that the main peak shows a local minimum around $V_t=40$ meV, a behavior not so different from the DFT model, but after $V_t=60$ meV the peak splits into a complex collection of smaller peaks. For the system without a central hill (Fig. \[FFT-Schr-Poisson-nohill\]) this disintegration of the main peaks happens earlier, and the resulting smaller peaks are fewer than in the system with a central hill.
The time-evolution in this essentially nonlinear model is very different from what is known for linear models. In order to appreciate this fact better we look at a linear model before we comment futher on the time-evolution of the SP model.
Exact time-evolution in a truncated Fock-space
==============================================
The CI-version of the model is capable to deliver the time-evolution of few Coulomb interacting electrons in a quantum dot in an external magnetic field. Here, we will use it for two electrons in the parabolic confinement introduced earlier (\[Vpar\]) with the option of the small central hill (\[Vc\]). The ground state for a vanishing external magnetic field is calculated in a truncated two-particle Fock-space. The truncation limits the two-electron Fock-space to the 16836 lowest states in energy. The Fock-space is constructed from the single-electron states of the parabolic confinement. The time evolution is again formally by the same Liouville-von Neuman equation (\[L-vN-dft\]) as was used for the mean-field version of the model, but now the density operator is a two-electron operator that is expressed in the Fock-space for the interacting two electrons. The main difference here is that the Hamiltonian of the system is only time-dependent as long as the initial perturbation (\[Wt\]) is switched on. The Coulomb part of the Hamiltonian is always time-independent and no iterations are necessary within each time step in order to attain convergence for the interaction like in the case of the DFT-model.
The penalty of this approach is instead the size of the matrices need for the calculation, but we have used two important technical items in order to attain the time-evolution to 100 ps. First, we tested for the present parameters how much we could reduce the Fock-space for the time-integration of the time-evolution operator (\[Teq\]). The states which contribute for $V_t=200$ meV to the density matrix with a contribution larger than $10^{-5}$ are less than 2415, so in the time-integration we further truncate the Fock-space to that size. We remind that these 2415 interacting two-electron states were initially calculated using 16836 noninteracting two-electron states. Still the matrices are considerably larger than in the DFT-case, so we then rewrote the time-integration to run on powerful GPU’s.[@Siro20121884] Furthermore, we tried two different methods for the time-integration, in one we refer the time-evolution operator to the initial time $t=0$, and in the other one we only refer it to the one earlier time step and accumulate the time-evolution in the density matrix. We selected a time-step small enough for the methods to give the same results.
After the initial perturbation pulse (\[Wt\]) dies out nothing is explicitly dependent on time in the Hamiltonian and therefore the diagonal elements of the density matrix, the occupation of the interacting two-electron states stays constant. In Figures \[FFT-exact\] and \[FFT-exact-occ\] we show the Fourier power spectrum for the collective oscillations of the model expressed in terms of the expectation value $\langle r^2\rangle$, together with the time-independent occupation of each interacting two-electron state participating in the collective oscillations. Here, we have a pure parabolic confinement without a central hill.
![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model without a central hill. The lower panel focuses in on the energy axis close to resonances. $V_0=0$, $T=0$ K.[]{data-label="FFT-exact"}](Fig09a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model without a central hill. The lower panel focuses in on the energy axis close to resonances. $V_0=0$, $T=0$ K.[]{data-label="FFT-exact"}](Fig09b.pdf "fig:"){width="42.00000%"}
![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model without a central hill (upper panel). The time-independent occupation of the interacting two-electron states $|\alpha )$ after the perturbation pulse has vanished (lower panel). $V_0=0$, $T=0$ K.[]{data-label="FFT-exact-occ"}](Fig10a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model without a central hill (upper panel). The time-independent occupation of the interacting two-electron states $|\alpha )$ after the perturbation pulse has vanished (lower panel). $V_0=0$, $T=0$ K.[]{data-label="FFT-exact-occ"}](Fig10b.pdf "fig:"){width="42.00000%"}
The logarithmic scale for the occupation in the lower panel of Fig. \[FFT-exact-occ\] hides the fact that for $V_t=200$ meV the occupation of the ground state has fallen to 77%. This is another measure of the strength of the excitation.
The results for the quantum dot with a central hill (\[Vc\]) added are shown in Figures \[FFT-exact-hill\] and \[FFT-exact-hill-occ\]
![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model with a central hill. The lower panel focuses in on the energy axis close to resonances. $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-exact-hill"}](Fig11a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model with a central hill. The lower panel focuses in on the energy axis close to resonances. $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-exact-hill"}](Fig11b.pdf "fig:"){width="42.00000%"}
![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model with a central hill (upper panel). The time-independent occupation of the interacting two-electron states $|\alpha )$ after the perturbation pulse has vanished (lower panel). $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-exact-hill-occ"}](Fig12a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum for the time-dependent expectation value of $\langle r^2\rangle$ for the CI model with a central hill (upper panel). The time-independent occupation of the interacting two-electron states $|\alpha )$ after the perturbation pulse has vanished (lower panel). $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-exact-hill-occ"}](Fig12b.pdf "fig:"){width="42.00000%"}
The main surprise for the exact results is that we do not find any local minimum for $V_t\approx 35-40$ meV. Indeed, the main peak found in the exact results shows behavior that is closer to the results of the HA if we consider only the height of the main peak found. There are more peaks visible in the exact results and that is reminiscent of the comparison in the linear response regime for the exact and the Hartree-Fock approach.[@Pfannkuche94:1221] One might of course worry about the possibility that the DFT-model could not predict the time-evolution properly or could not describe the excited states correctly, if it got stuck in some local minimum instead of a global minimum. We have tried to exclude this possibility by performing the DFT-calculation at higher temperatures, $T=1.0$ and $4.0$ K. In both cases a minimum around $V_t\approx 35-40$ meV is found. In addition, we have varied the minimum seeking, but in vain, the minimum always reappears.
The DFT-approach can be criticized by our use of a static functional instead of a more appropriate frequency dependent one, especially since we are using it to describe a collective oscillation in the system. We have no good excuse for this, but interestingly enough the DFT-model can reproduce the extended Kohn theorem valid for parabolic confinement for $|N_p|=1$ with ease. The same test has of course been used with success both for the exact CI-model and the Hartree-version of the DFT-model. Opposite to the CI-model the seeking of the ground state for the DFT model without a central hill is a very time-consuming and difficult affair. This behavior has to be related to the fact that the presence of the central hill (\[Vc\]) reduces the importance of the Coulomb interaction. In some sense this is also true for the nonphysical self-interaction in the Hartree-version of the DFT-model.
Corresponding reduction of the importance of the Coulomb interaction in the case of the CI-model can eventually be seen in the lower panel of Fig. \[FFT-exact-occ\] and Fig. \[FFT-exact-hill-occ\] for the occupation of the two-electron states caused by the initial perturbation. The energy spectra for the 100 lowest interacting two-electron states are compared in the upper panel of Fig. \[Exact-E\]. Besides the general behavior of the central hill (\[Vc\]) to increase the energy of each state we see a partial lifting of degeneracy.
![(Color online) The interacting two-electron spectra versus the state number $\mu$ (upper panel), and total energy versus the excitation strength $V_t$ (lower panel) compared for the exact model for the system with ($V_0=3.0$ meV) and without ($V_0=0$) a central hill, $T=0$ K.[]{data-label="Exact-E"}](Fig13a.pdf "fig:"){width="42.00000%"} ![(Color online) The interacting two-electron spectra versus the state number $\mu$ (upper panel), and total energy versus the excitation strength $V_t$ (lower panel) compared for the exact model for the system with ($V_0=3.0$ meV) and without ($V_0=0$) a central hill, $T=0$ K.[]{data-label="Exact-E"}](Fig13b.pdf "fig:"){width="42.00000%"}
We are here dealing with nonlinear response of a system as can be verified by looking at the expectation value for the total energy of the system described by the CI-model, after the excitation pulse has vanished, shown in Fig. \[Exact-E\]. The excitation pulse pumps a finite amount of energy into the system. This is important when interpreting the occupation of the interacting two-electron states in the system displayed in the lower panels of Fig. \[FFT-exact-occ\] and \[FFT-exact-hill-occ\]. If we look at the system without a central hill, Fig. \[FFT-exact-occ\], we see that the ground state $|1)$ is occupied with probability close to 1, and for low excitation, $V_t$, the next state is $|24)$ and for higher $V_t$ state $|26)$ competes with $|24)$. If we check the energy differences we find $E_{24}-E_1=6.139$ meV, and $E_{26}-E_1=6.746$ meV, which indeed fit with the main peak seen and a side peak appearing for higher $V_t$ in Fig. \[FFT-exact\].
For the case of a central hill in the system we find that again state $|24)$ has the next highest occupation, but now for the whole $V_t$ range. Next comes state $|33)$ for low values of $V_t$. Indeed, we get $E_{24}-E_1=5.698$ meV, and $E_{33}-E_1=6.472$ meV, which again fits very well with the location of the peaks in Fig. \[FFT-exact-hill\]. The graphs of the occupation of the interacting two-electron states $|\alpha )$ are thus indicating which states are being occupied as a result of the excitation of the system. We have verified that states $|24)$ and $|26)$ for the system without a central hill and states $|24)$ and $|33)$ for the system with one, all have a total angular momentum $\hbar{\cal M}=\hbar (M_1+M_2)=0$, where $M_i$ is the quantum number for angular momentum of electron $i$. As was noted earlier[@Pfannkuche93:2244] the CI-model allows for contributions to an ${\cal M}=0$ state two single electron states with angular momentum $\pm\hbar M$, a combination that is not possible in a HA with circular symmetry.[@Pfannkuche93:2244]
In Figure Fig. \[FFT-MaxMin\] we compare the Fourier power spectra for $V_t=10$ meV and $V_t=200$ meV in the case of the system with a central hill and without one, but here we have taken an extra long time-series, integrating the equations of motion for 1000 ps instead of the 100 ps we have used for the CI-model above.
![(Color online) The Fourier power spectra compared for $V_t=10$ meV and $V_t=200$ meV for the system without a central hill (a), and with a central hill (b). $T=0$ K.[]{data-label="FFT-MaxMin"}](Fig14.pdf){width="42.00000%"}
As could be expected for a linear model the peaks visible at low excitation are still present with unchanged frequency for strong excitation, but the strong excitation activates several more peaks. It is also clear that the presence of the central hill (Fig. \[FFT-MaxMin\](b)) shifts the frequency of the main peaks and allows for the excitation of many more. The central hill does break some special symmetry imposed by the parabolic confinement, that the Coulomb interaction alone does not break.
Time-evolution of a Hubbard model
=================================
Above, we have introduced mean-field theoretical models and a many-electron model that is solved exactly in a truncated Fock-space for two electrons to describe the strong radial excitation of electrons in a quantum dot. These models do all appear in different studies of linear response of quantum dots. The mean-field models tend, due to their nature, though to be used for dots with a higher number of electrons. The nonlinear SP-model can though be considered as an attempt to create a version of a mean-field approach fit for two electrons. For curiosity we like to add the last model, the Hubbard model, a many-electron model that has not often been applied to describe the electrons in a single parabolically confined quantum dot.
The Hamiltonian for the electrons in a quantum dot described by the Hubbard model is $$H = H_{\mathrm{int}} + H_{\mathrm{hop}} + H_{\mathrm{V}},
\label{Hub-H}$$ where the Coulomb interaction between the electrons is described by a spin dependent contact interaction $$H_{\mathrm{int}} = U \sum_{i=1}^N n_{i,\downarrow} n_{i,\uparrow},
\label{Hub-int}$$ and the hopping part has the form $$H_{\mathrm{hop}} = -t \sum_{\sigma=\downarrow,\uparrow} \sum_{\langle i,j\rangle}
c_{i,\sigma}^\dagger c_{j,\sigma} +h.c.,
\label{Hub-hop}$$ where $\langle i,j\rangle$ denotes a summation over the neighboring sites. The model is written in terms of the creation $c_{i,\sigma}^\dagger$, the destruction $c_{i,\sigma}$, and the number operator $n_{i,\sigma}$ for electrons with spin $\sigma$ on site $i$. The potential part $$H_\mathrm{V} = \sum_{\sigma=\downarrow,\uparrow} \sum_{i=1}^N V(r_i) n_{i,\sigma},
\label{Hub-Vpar}$$ includes the parabolic potential (\[Vpar\]) and possibly the central small hill (\[Vc\]).
We set the Hubbard model on a small square lattice with totally $N$ sites. We use the numbering of the states in the Fock-space suggested by Siro and Harju.[@Siro20121884] The height and the width of the lattice is fixed in terms of the characteristic length scale for the parabolic confinement to be $6a$. The lattice length is then $a_{\mathrm{latt}}=6a/(\sqrt{N}-1)$ and the hopping constant is $t=\hbar^2/(2m^*a_\mathrm{latt})$. The value for the strength of the Coulomb interaction is not so straightforward to find, but we fix the value of $U$ such that the energy of the ground state of the system is in accordance with the value found in the exact model. We keep in mind that there will always be a difference in the many-body energy spectrum of these two models, due to the different treatment of the Coulomb interaction and the finite square lattice that is bound to break the angular symmetry of the original model, but we want to see if we can identify some many-electron character in the excitation response.
The parabolic confinement of the electrons spreads out the energy spectrum of the Hubbard model that otherwise is extremely dense, and thus we can use the same approach as for the exact many-body model to solve it exactly within a truncated Fock-space. We performed this on GPU’s for a $5\times 5$ lattice. The results for the Fourier transform of the time-dependent oscillations in $\langle r^2\rangle$ are shown in Fig. \[FFT-hubbard\] for the pure parabolic confinement, and in Fig. \[FFT-hubbard-hill\] for the model with a small central hill (\[Vc\]).
![(Color online) The Fourier power spectrum of the expectation value of $\langle r^2\rangle$ (upper panel), and the occupation of the interacting two-electron states (lower panel) for the Hubbard model without a central hill. $V_0=0$, $T=0$ K.[]{data-label="FFT-hubbard"}](Fig15a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum of the expectation value of $\langle r^2\rangle$ (upper panel), and the occupation of the interacting two-electron states (lower panel) for the Hubbard model without a central hill. $V_0=0$, $T=0$ K.[]{data-label="FFT-hubbard"}](Fig15b.pdf "fig:"){width="42.00000%"}
The time-evolution of the system is calculated in the same way as was used for the CI model using the time-evolution operators presented above (\[Teq\]) in an interacting two-electron basis.
![(Color online) The Fourier power spectrum of the expectation value of $\langle r^2\rangle$ (upper panel), and the occupation of the interacting two-electron states (lower panel) for the Hubbard model with a central hill. $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-hubbard-hill"}](Fig16a.pdf "fig:"){width="42.00000%"} ![(Color online) The Fourier power spectrum of the expectation value of $\langle r^2\rangle$ (upper panel), and the occupation of the interacting two-electron states (lower panel) for the Hubbard model with a central hill. $V_0=3.0$ meV, $T=0$ K.[]{data-label="FFT-hubbard-hill"}](Fig16b.pdf "fig:"){width="42.00000%"}
The square symmetry of the lattice can be expected to produce deviations that should already be present in the excitation spectrum for low excitation.[@PhysRevB.60.16591] We have tested the Hubbard model for dipole active excitation modes, $N_p=\pm 1$, to verify this. The main peak (the lowest excitation) is indeed split for the Hubbard model in Figures \[FFT-hubbard\] and \[FFT-hubbard-hill\], and the modes at higher energy, only appear for a stronger excitation, i.e. a higher value of $V_t$. By looking at the lower panels in Figures \[FFT-hubbard\] and \[FFT-hubbard-hill\] we see again that in the system without a central hill (Fig. \[FFT-hubbard\]) more modes get active as the excitation grows. This is in accordance with our observation for the CI model. We have to admit that on this small lattice chosen the energy of the lowest mode is a bit higher than all the other models predict, even though we have chosen the interaction strength $U$ to give the similar energy for the ground state as the CI model does. We do not use the Hubbard model for higher excitation than $V_t=100$ meV to avoid artifacts created by the finite size of the model.
Comparison of model results and discussion
==========================================
For the quantum dot with no central hill present ($V_0=0$) at weak excitation, $V_t\sim 0$, all the models deliver one main peak that grows linearly with the excitation strength. As we have seen the Hubbard model due to the square symmetry imposed by the underlying lattice has two peaks,[@PhysRevB.60.16591] and in the case of the CI model we see a small side peak on the “blue” side, reminiscent of known results for the $N_p=\pm 1$ modes.[@Pfannkuche94:1221]
The location of the main peak in the case of the SP model is redshifted by an amount slightly surpassing 0.5 meV. This must be accredited to the slightly weaker repulsion of the electrons having a logarithmic singularity in the case of the SP model instead of the 3D Coulomb repulsion in the other mean-field approximations, and in the CI model. With the small central hill in the quantum dot the location of the main peak in the DFT model is blue shifted by 0.2 meV compared to the CI model, and the same analysis gives a blueshift of 0.3 meV for the HA. Calculations of the ground state for the CI, the SP, and the DFT model all give similar energy, but the HA gives results far off.[@Pfannkuche93:2244]
In the CI, the SP, and the Hubbard models the inclusion of a small central potential hill in the confinement potential of the quantum dot causes more collective modes to be activated with increasing excitation $V_t$. At the same time the lower panels of Figures \[FFT-exact-occ\], \[FFT-exact-hill-occ\], \[FFT-hubbard\], and \[FFT-hubbard-hill\] displaying occupation of interacting two-electron states indicates a slight simplification effects caused by the central hill, at least for some range of $V_t$. Amazingly, in the HA only one peak for the collective oscillations is seen for the whole range of excitation strength we try. This is probably caused by the artificial self-interaction that is specially large for two electrons described with the HA. On the other hand the dependence of the height of the main peak on $V_t$ for the Fourier transform of the expectation value $\langle r^2\rangle$ for the HA is very close to the results for the main peak for the CI model.
The finite occupation of higher energy states together with the increase of the mean total energy seen in the lower panel of Fig. \[Exact-E\] shows that we have left the linear response regime with increasing $V_t$. This fact is further demonstrated by the nonlinear growth of the height of the Fourier peak for the expectation value $\langle r^2\rangle$ for all models with increasing $V_t$ beyond the linear regime for low $V_t$. In addition, we notice that for low $V_t$ the electrons in the quantum dot oscillate with $\langle r^2\rangle$ very close to the ground state value. As the excitation is increased energy is pumped into the quantum dot and it increases in size.
We have identified nonlinear behavior in all the models when observing how the amplitude of the oscillations of $\langle r^2\rangle$ behave as a function of the excitation strength $V_t$ once we leave the linear response regime valid for very low excitation. The CI model is a purely linear model. All the possible excited states for the CI model are calculated before the time-integration of the system is started. This is clearly demonstrated in Fig. \[FFT-MaxMin\] where the excitations are compared for weak and strong excitation. The main peak at low $V_t$ is still visible in the excitation spectrum for large $V_t$, at exactly the same energy. Stronger excitation activates higher lying collective modes, and even in this simple system there very are many of them available.
The time-evolution for the mean-field models has to be viewed in different terms. In case of the DFT or the Hartree model information about the two-electron excitation spectrum does not exist before the time-integration is started. The effective potential changes in each time-step and the occupation of effective single-electron states becomes time-dependent, see Fig. \[Occupation-T\]. The effective potential (or equivalently, the density, or the density operator here) has to be found by iterations in each time-step in order to include the effects of the Coulomb interaction. Within each iteration the problem is treated as a linear one. In case of the SP model the nonlinear solution for the groundstate is sought directly without an iteration, and the same is true for the time-dependent solutions. The time-evolution of the SP model is thus nontrivial and could in principle bring forward phenomena that could be blocked by the linear solution requirement within each iteration step for the other mean-field models, especially in a long time series where small effects from this methodology gathered in each time-step might sum up.
Within the range for $V_t$ considered here the HA brings results that look very stable, one peak with no frequency shift as $V_t$ increases, but with a slight nonlinear behavior for the amplitude of the oscillations of $\langle r^2\rangle$. The SP and the DFT models bring similar results for $V_t\leq 60$ meV with a local minimum for the amplitude of the oscillations of $\langle r^2\rangle$. For larger values of $V_t$ the SP model brings a plethora of collective oscillations, and for the DFT model it becomes too difficult to stabilize a solution for a longer time interval. It should be kept in mind that also the CI model, especially for the case of no central hill, shows an increased number of active modes, but only for much stronger excitation and in a more “controlled” way. The different characteristics or the nuance of the nonlinear properties of the mean-field models may be influencing their response here to a strong excitation in a fundamentally different way than in the linear CI model.
It should be stated once more that extreme care has been taken in verifying and testing our numerical results by comparing different numerical methods, models, and variation of sizes and types of functional spaces and grids.
Conclusion
==========
The modeling of nonlinear response of confined quantum systems on the nanoscale is in its infancy, but may bring new insight into the systems as the measuring, processing, and growth techniques evolve opening up the field. For systems with many particles we most likely will have to rely on mean-field and DFT models, and only for systems of few particles can we expect to be able to rely on CI models. In anticipation of this we have studied here how some of these models fare describing the nonlinear response of a two electron model.
We have to expect the CI-model to deliver numerically exact results that we can compare the results of the other models to. The results of our implementation of a DFT-model do not compare well when leaving the linear response regime. This is not totally unexpected as we have not used any time-dependent functionals. In addition, the numerical time-integration of the DFT model is difficult to guarantee for strong excitation and long times. The Hartree model is easier to use and the overall qualitative nonlinear response of it is in accordance with the CI model, except for fine structure of side peaks visible in the CI model. Similar comparison has been seen in the linear response earlier [@Pfannkuche94:1221]. The results of the coarse lattice Hubbard model deviate quantitatively from the CI-results, but the qualitative behavior is similar, side peaks and occupation of higher modes with increased excitation.
Regarding the emergence of nonlinear effects the comparison to the SP model is valuable. In a mean field, or a local approximation to a DFT theory the results are usually obtained by iterations, and most often there is a condition that the underlying linear basis is orthogonal. In calculations of molecules this condition is sometimes relaxed, but most often it is used to guarantee a connection to higher order many-body methods. This is not done in the SP model. There the nonlinear solution is found directly and the resulting states are not orthogonal. Looking at our results we see that this essential nonlinearity does strongly affect the solution of the SP model beyond some excitation strength. These effects, emergence of many new excitation modes, splitting of modes, is not seen in any of the other models. So, even if the mean-field and the DFT models are nonlinear, then the iteration procedure in a linear functional space does protect them from this mode splitting and multiplication. As stated in the previous section the nonlinear behavior seen from the CI results is much more modest and probably only results from the “shape” of the many-body energy spectrum that can be reached with increasing excitation.
All these points in the end only stress how exciting and important experimental undertaking into this nonlinear regime will be.
This work was supported by the Research Fund of the University of Iceland, the Icelandic Research and Instruments Funds, and a Special Initiative for Students of the Directorate of Labour. Some of the calculations were performed on resources provided by the Nordic High Performance Computing (NHPC). C. Besse and G. Dujardin are partially supported by the French programme Labex CEMPI (ANR-11-LABX-0007-01).
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abstract: 'We prove stability estimates for the [Bakry-Émery ]{}bound on Poincaré and logarithmic Sobolev constants of uniformly log-concave measures. In particular, we improve the quantitative bound in a result of De Philippis and Figalli asserting that if a $1$-uniformly log-concave measure has almost the same Poincaré constant as the standard Gaussian measure, then it almost splits off a Gaussian factor, and establish similar new results for logarithmic Sobolev inequalities. As a consequence, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs are based on Stein’s method, optimal transport, and an approximate integration by parts identity relating measures and approximate optimizers in the associated functional inequality.'
author:
- |
Thomas A. Courtade$^*$ and Max Fathi$^{\dagger}$\
\
$^{*}$UC Berkeley, Department of Electrical Engineering and Computer Sciences\
$^{\dagger}$CNRS & Université Paul Sabatier, Institut de Mathématiques de Toulouse\
title: 'Stability of the [Bakry-Émery ]{}theorem on ${\mathbb{R}}^n$'
---
Introduction and main results
=============================
The purpose of this work is to establish stability estimates for the [Bakry-Émery ]{}theorem, which states that the sharp constant for various functional inequalities for uniformly log-concave measures must be better than the sharp constant for the standard Gaussian measure. We shall focus on two main inequalities: the Poincaré inequality and the logarithmic Sobolev inequality.
Poincaré inequality
-------------------
A probability measure on ${\mathbb{R}}^n$ is said to satisfy a Poincaré inequality with constant $C$ if for any smooth test function $f$, its variance satisfies the bound $$\operatorname{Var}_{\mu}(f) \leq C\int{|\nabla f|^2d\mu}.$$ The smallest possible constant in this inequality is called the Poincaré constant of $\mu$, which we shall denote by $C_P(\mu)$. Such inequalities play an important role in several areas of analysis, probability and statistics, such as concentration of measure, rates of convergence for stochastic dynamics and analysis of PDEs. This constant is also the inverse of the spectral gap of the Fokker-Planck (or overdamped Langevin) dynamic associated with $\mu$. A large class of probability measures satisfy such an inequality. In particular, if a probability measure is more log-concave than the standard Gaussian measure (that is, $\mu = e^{-V}dx$ with $\operatorname{Hess}V \geq \operatorname{I}_n$), then $C_P(\mu) \leq 1$. This result can be viewed as a consequence of the Brascamp-Lieb inequality [@BL76] or the [Bakry-Émery ]{}theory [@BE85].
More recently, Cheng and Zhou [@CZ] proved a rigidity property for the [Bakry-Émery ]{}theorem: if such a probability measure has its Poincaré constant equal to one, then it must be a product measure, with one of the factors being a Gaussian measure of unit variance. They also obtain a rigidity result in the more general setting of complete metric-measure spaces with positive Ricci curvature. See also [@HJS] for a weaker form of this rigidity in ${\mathbb{R}}^n$, and [@CL87] for rigidity in a different class of measures (and [@CFP18] for a corresponding stability estimate).
The convexity condition we shall assume here is a particular case of the [Bakry-Émery ]{}curvature-dimension condition, itself a generalization of Ricci curvature lower bounds. Splitting theorems for manifolds satisfying a curvature bound and a geometric condition have been the topic of some interest, going back to work of Cheeger and Gromoll [@CG71; @CC96]. More recently, rigidity and stability for a related (and stronger) isoperimetric inequality has been established [@CMM17] under the stronger curvature-dimension condition with finite dimension, using completely different techniques.
Poincaré inequalities can be viewed as estimates on the smallest eigenvalue of the diffusion operator $-\Delta + \nabla V \cdot \nabla$. Stability for other spectral problems have been considered, such as Poincaré inequalities on bounded domains [@BDPV1; @BDP] and a lower bound on the spectrum of Schrödinger operators [@CFL], respectively with applications in shape optimization and quantum mechanics.
The first main result of the present work is to improve the quantitative bounds in the following result of De Philippis and Figalli [@DPF], which establishes a strong form of quantitative stability for the Poincaré constant for uniformly log-concave measures.
\[thm\_dpf\] Let $\mu = e^{-V}dx$ be a probability measure with $\operatorname{Hess}V \geq \operatorname{I}_n$, and assume that there exists $k$ functions $u_i \in H^1(\mu)$, $k \leq n$, such that for any $i \in \{1,..,k\}$ we have $$\int{u_id\mu} = 0; \hspace{5mm} \int{u_i^2d\mu} = 1; \hspace{5mm} \int{\nabla u_i \cdot \nabla u_j d\mu} = 0, ~~~ \forall j\neq i$$ and $$\int{|\nabla u_i|^2d\mu} \leq 1 + \epsilon$$ for some $\epsilon \geq 0$. Then for any $\theta > 0$ there exists $C(n, \theta)$, a subspace $\mathcal{V}\subset {\mathbb{R}}^n$ with $\dim(\mathcal{V})=k$, and a vector $p \in \mathcal{V}$ such that $$W_1(\mu, \gamma_{p,\mathcal{V}} \otimes \bar{\mu}) \leq C(n,\theta)|\log \epsilon|^{-1/4 + \theta},$$ where $\gamma_{p,\mathcal{V}}$ is the standard Gaussian measure on $\mathcal{V}$ with barycenter $p$, and $\bar{\mu}$ is the marginal distribution of $\mu$ on $\mathcal{V}^{\bot}$, which enjoys the same convexity property as $\mu$.
In this statement, $W_1$ stands for the classical $L^1$ Kantorovitch-Wasserstein distance [@Vill03], and $H^1(\mu) := \{f; \int{(|f|^2 + |\nabla f|^2)d\mu} < \infty\}$ is a weighted Sobolev space with respect to $\mu$.
We shall obtain the following improvement:
\[thm\_improved\_dpf\] Under the same assumptions as in Theorem \[thm\_dpf\], we have $$W_1(\mu, \gamma_{p,\mathcal{V}}\otimes \bar{\mu}) \leq Ck^{3/2}\sqrt{\epsilon}.$$ In fact, we may take $C=18 \sqrt{2} < 26$.
Beyond the improved dependence on $\epsilon$, the fact that our bound depends on $k$ and not on $n$ is useful for high-dimensional situations.
The proof of [@DPF] is based on a stability version of Caffarelli’s contraction theorem [@Caf00], which is a regularity estimate on the nonlinear Monge-Ampère PDE. To obtain the improved bound, we shall rely instead on Stein’s method [@Ste72; @Ste86], which is a way of quantifying distances between probability measures using well-chosen integration by parts formulas. See [@Ros11] for an introduction to the topic. The main reason why this allows us to obtain better estimates is that this proof mostly remains at a linear level, instead of relying on nonlinear tools as in [@DPF]. The other main tool in the proof is that the test functions in the assumptions of the theorem can be viewed as approximate minimizers in a variational problem, which then give rise to an approximate Euler-Lagrange equation (up to reminder terms of order $\sqrt{\epsilon}$), which takes the form of an integration by parts formula. See Section \[strat\_proof\] for an overview of the strategy of proof.
When $k=n$, it is possible to improve the topology, and get an estimate of order $\sqrt{\epsilon}$ in the stronger $W_2$ distance, using results from [@LNP15; @CFP18]. We do not know how to get a $W_2$ estimate when $k < n$, due to regularity issues for the Poisson equation we shall make use of in the proof.
We do not know if the bound is optimal. Testing on Gaussian measures with variance $1-\epsilon$ shows that the optimal rate cannot be better than $\epsilon$ (instead of $\sqrt{\epsilon}$). See also Remark 1.4 in [@DPF] for computations in dimension one in a related problem which suggest the sharp rate could be $\epsilon$.
Our main result has the following immediate corollary, which can be viewed as a dimension-free improvement of the [Bakry-Émery ]{}theorem.
\[cor:1dimBE\] Let $\mu = e^{-V}dx$ be a probability measure with $\operatorname{Hess}V \geq \operatorname{I}_n$. There is a direction $\sigma \in \mathbb{S}^{n-1}$ and a vector $p \in \operatorname{Span}(\sigma)$ such that $$W_1(\mu, \gamma_{p,\sigma} \otimes \bar{\mu}) \leq C\sqrt{C_P(\mu)^{-1}-1},$$ where $\gamma_{p,\sigma}$ is the standard Gaussian measure on $\operatorname{Span}(\sigma)$ with barycenter $p$, and $\bar{\mu}$ is the marginal distribution of $\mu$ on $\operatorname{Span}(\sigma)^{\bot}$.
This corollary follows from Theorem \[thm\_improved\_dpf\] since there must be a function $u$ satisfying the assumptions of that theorem for any $\epsilon > C_P(\mu)^{-1}-1$ , by definition of the Poincaré constant. We make use here of the fact that the bound in Theorem \[thm\_improved\_dpf\] depends on $k$ and not $n$, unlike Theorem \[thm\_dpf\], to get a dimension-free estimate. A noteworthy consequence is the following refinement of the classical dimension-free concentration bound $\operatorname{Var}_{\mu}(F)\leq 1$ for 1-Lipschitz $F$.
Let the notation of Corollary \[cor:1dimBE\] prevail. For any $1$-Lipschitz $F:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$, there exists a direction $ \sigma(F) \in \mathbb{S}^{n-1}$ and a vector $ p(F)\in \operatorname{Span}(\sigma)$ such that $$W_1(\mu, \gamma_{p,\sigma} \otimes \bar{\mu}) \leq C \sqrt{ \operatorname{Var}_{\mu}(F)^{-1}-1}.$$
At this point, one might wonder if the convexity assumption is necessary. It cannot simply be dropped: if one looks at a general measure, its Poincaré constant may be equal to one, for example by rescaling an arbitrary (but nice) measure to enforce this, in which case there exists a function $u$ satisfying the assumptions, and in general there will not be a Gaussian factor. However, the convexity assumption will mainly be used to ensure the functions $u_i$ are close to coordinate functions, in a suitable basis of ${\mathbb{R}}^n$, and hence can be dropped if we assume extra knowledge on second moments. This leads to the following result, with improved dependence on $k$:
\[thm\_coord\] Assume $C_P(\mu) \leq 1$, and that there exists an orthonormal family $e_1,..,e_k$ of ${\mathbb{R}}^n$ such that $\operatorname{Var}_{\mu}(x \cdot e_i) \geq (1+\epsilon)^{-1}$, for each $i\leq k$. Then there exists $p\in \mathcal{V} = \operatorname{Span}(e_1, \dots, e_k)$ such that $$W_1(\mu, \gamma_{p,\mathcal{V}} \otimes \bar{\mu}) \leq k \sqrt{ \pi {\epsilon} },$$ where the measures $\gamma_{p,\mathcal{V}}$ and $\bar{\mu}$ are as defined in Theorem \[thm\_dpf\].
Logarithmic Sobolev inequality
------------------------------
According to the [Bakry-Émery ]{}theorem [@BE85], probability measures that are more log-concave than the standard Gaussian measure satisfy the logarithmic Sobolev inequality (LSI) $$\label{eq:LSImu}
\operatorname{Ent}_{\mu}(f^2) \leq 2C_{\mathrm{LSI}}(\mu)\int{|\nabla f|^2d\mu}; \hspace{5mm} C_{\mathrm{LSI}}(\mu) \leq 1$$ where $\operatorname{Ent}_{\mu}(f^2) = \int{f^2 \log f^2 d\mu} - \left(\int{f^2d\mu}\right) \log \int{f^2d\mu}$, and $C_{\mathrm{LSI}}(\mu)$ stands for the sharpest possible constant for $\mu$ in this inequality. This functional inequality, originally introduced by Gross [@Gro75], is strictly stronger than the Poincaré inequality, and the constant $1$ is sharp for the standard Gaussian measure. Moreover, Carlen [@Car91] showed that for the Gaussian measure equality holds in the LSI if and only if the function $f$ is of the form $f(x) = Ce^{p \cdot x}$ for some vector $p \in {\mathbb{R}}^n$. The LSI is used to derive Gaussian concentration inequalities, as well as estimates on the rate of convergence to equilibrium for certain stochastic processes. We refer to [@BGL14] for background on this inequality and its applications.
We study stability for the bound on the logarithmic Sobolev constant. Our second main result is the following estimate, showing that if $C_{\mathrm{LSI}}(\mu)$ is close to one, then $\mu$ still approximately splits off a Gaussian factor, provided the approximate optimizer satisfies regularity assumptions.
\[thm:QstableLSI\] Consider a probability measure $\mu= e^{-V}dx$ on ${\mathbb{R}}^n$ satisfying $\operatorname{Hess}V \geq \mathrm{I}_n$. Let $u:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$ be a nonnegative function such that $\log u$ is $\lambda$-Lipschitz and $\int{u^2d\mu} = 1$. There exists a constant $C(\lambda)$, depending only on $\lambda$, such that if $$\begin{aligned}
\operatorname{Ent}_{\mu}(u^2) \geq 2(1-\epsilon)\int{|\nabla u |^2d\mu}\label{eqn:fApproxExLSI}\end{aligned}$$ for some $\epsilon \geq 0$, then there is a direction $\sigma \in \mathbb{S}^{n-1}$ for which $$\begin{aligned}
W_1(\mu, \gamma_{b,\sigma} \otimes \bar{\mu}) \leq C(\lambda)\left(\int{|\nabla u |^2d\mu} \right)^{-1/2}\sqrt{\epsilon},\label{W1boundLSI}\end{aligned}$$ where $\gamma_{b,\sigma}$ denotes the standard Gaussian measure on $\operatorname{Span}(\sigma)$ with barycenter $b=\sigma \int x\cdot \sigma \, d\mu$, and $\bar{\mu}$ is the marginal distribution of $\mu$ on $\operatorname{Span}(\sigma)^{\bot}$, which enjoys the same convexity property as $\mu$.
The constant $C(\lambda)$ can, in principle, be made explicit. However, its expression would be quite complicated and our arguments make no attempt to optimize it, so we do not attempt to do so. Note that we should expect the bound to get worse if $\int{|\nabla u|^2d\mu}$ is small, since if $u$ was constant it would be a trivial minimizer of the LSI, no matter what $\mu$ would be, so the bound must rule out that situation in some way. Up to the regularity assumption that $\log u$ is Lipschitz, existence of such a function is a weaker assumption than the assumption of Theorem \[thm\_improved\_dpf\], since $C_{\mathrm{LSI}}(\mu) \geq C_P(\mu)$.
Like Theorem \[thm\_improved\_dpf\], it is possible to give a version of Theorem \[thm:QstableLSI\] with $k$ orthogonal minimizers, in the sense that $\mu$ approximately splits off a $k$-dimensional factor, provided $\int{\nabla \log u_i \cdot \nabla \log u_j d\mu} = 0$ for approximate minimizers $(u_i)_{i\leq k}$. The constant $C$ would depend on $k$, but not on the ambient dimension.
The stipulation that $\int{u^2d\mu} = 1$ is for made convenience, and comes without loss of generality. Indeed, the LSI is homogenous of degree 2, so rescaling $u \longrightarrow \alpha u$ for $\alpha \in {\mathbb{R}}$ affects neither $\epsilon$-optimality in the sense of , nor the $\lambda$-Lipschitz property assumed of $\log u$. Further, the assumed nonnegativity of $u$ is also for convenience, and comes without loss of generality since the log-lipschitz assumption already enforces it to have constant sign.
\[Rmk\_utdmu\] Theorem \[thm:QstableLSI\] can be strengthened to say that, for any $t\in {\mathbb{R}}$, the probability measure proportional to $u^t \mu$ satisfies . The only changes are (i) the barycenter $b$ becomes $b=Z^{-1} \sigma \cdot \int x\cdot \sigma u^t d\mu,$ where $Z:= \int u^t d\mu$ is a normalizing constant; and (ii) the constant $C$ will depend on both $\lambda$ and $t$. See Remark \[rmk:Measures\_ut\] for details.
An important consequence of the LSI is the classical concentration inequality for Lipschitz functions, established via Herbst’s argument: If $\mu$ satisfies and $F$ is $L$-Lipschitz, then $$\begin{aligned}
\int{e^{F }d\mu} \leq \exp\left( \int F d\mu + L^2/2 \right).\label{LipschitzConcentrationIneq}\end{aligned}$$ Equality is attained if $\mu$ splits off a standard Gaussian factor in a direction $\sigma \in \mathbb{S}^{n-1}$, in which case $F(x) = L \sigma \cdot x $ achieves equality. The following provides a quantitative stability estimate for this result, provided $\mu$ is uniformly log-concave.
\[thm:ThmExpConcentration\] Let $\mu= e^{-V}dx$ be a probability measure on ${\mathbb{R}}^n$ satisfying $\operatorname{Hess}V \geq \mathrm{I}_n$, and fix any $L>0$. There exists a constant $C(L)$ such that if $F: {\mathbb{R}}^n \longrightarrow {\mathbb{R}}$ satisfies $ \|F \|_{\mathrm{Lip}} \leq L$ and $$\int{e^{F }d\mu} \geq \exp\left( \int F d\mu + \frac{L^2}{2} (1 - \epsilon/2 ) \right)$$ for some $\epsilon \geq 0$, then there is a direction $\sigma \in \mathbb{S}^{n-1}$ for which $$\begin{aligned}
W_1(\mu, \gamma_{b,\sigma} \otimes \bar{\mu})\leq C(L) \sqrt{\epsilon}, \label{W1boundLipschitz}\end{aligned}$$ where $\gamma_{b,\sigma}$ and $\bar{\mu}$ are the same as in Theorem \[thm:QstableLSI\].
There have been some recent works on dimension-free stability for Gaussian concentration estimates [@BJ17; @CaMa17], which improve the bounds with reminder terms that compare the shape of level sets to hyperplane, and can be transferred to uniformly log-concave measures via the Caffarelli contraction theorem. It is unclear whether those results and ours can be compared.
Strategy of proof {#strat_proof}
-----------------
The proofs of Theorems \[thm\_improved\_dpf\] and \[thm:QstableLSI\] are based on the same broad strategy, with three main steps. To our knowledge, this way of implementing Stein’s method to study stability in variational problems is new.
The first step can be stated in a broad abstract framework. Consider a general minimization problem of the form $$\mu \longrightarrow \underset{f}{\inf} \hspace{1mm} \int{H(f, \nabla f)d\mu}$$ and assume the infimum over a class of probability measures $\mathcal{P}$ is known, say equal to zero, and that we can describe the subset of measures $\mu_0$ and associated functions $f_0$ such that $\int{H(f_0, \nabla f_0)d\mu_0} = 0$. Beyond the questions considered in this work, many relevant inequalities from analysis, geometry and probability can be cast in this form, such as sharp constants for Sobolev inequalities, variational problems in statistical physics, eigenvalue problems, and so on.
The Euler-Lagrange equation for problems of this form is $$\int{u\partial_1 H(f_0, \nabla f_0) + \nabla u \cdot \partial_2 H(f_0, \nabla f_0)d\mu} = 0 \hspace{2mm} \forall u.$$ So any minimization problem of this form gives rise to an integration by parts formula for measures that achieve the infimum. Now, if we consider a measure $\mu_1$ and a function $f_1$ such that $\int{H(f_1, \nabla f_1)d\mu_1} \leq \epsilon$, the problems we consider in this paper can be stated as trying to show that $\mu_1$ is close to the class of measures at which the infimum is reached. At a heuristic level, and maybe under extra assumptions on $f_1$, we expect an approximate Euler-Lagrange equation of the form $$\int{u\partial_1 H(f_1, \nabla f_1) + \nabla u \cdot \partial_2 H(f_1, \nabla f_1)d\mu_1} = o(1)\times F(||f||, ||u||')$$ to hold for some class of test functions, and norms $\|\cdot\|, \| \cdot\|'$ adapted to the problem. It is in this way that we obtain an approximate integration by parts identity, which is the basic setup required for Stein’s method.
The second step is to show that $f_1$ can be replaced up to small error by a function $f_0$ such that $\int{H(f_0, \nabla f_0)d\mu_0} = 0$ for some other probability measure. In the present paper, this is done by considering a transport map $T$ sending $\mu_0$ onto $\mu_1$, and proving that $f_1 \circ T$ approximately reaches the infimum when integrating with respect to $\mu_0$. If the minimization problem with fixed reference measure $\mu_0$ satisfies some quantitative stability property, this would mean $f_1 \circ T$ is close to some function $f_0$ for which the infimum is reached. We then deduce that $f_0$ is close to $f_1$ using specific regularity properties of the transport map, using the convexity assumptions in our problem. This part of the proof seems to be of less general scope than the other two steps. [As a tool, we use stability estimates for the sharp functional inequality with fixed reference measure.]{} The conclusion is that $\mu_1$ satisfies an approximate integration by parts formula $$\int{u\partial_1 H(f_0, \nabla f_0) + \nabla u \cdot \partial_2 H(f_0, \nabla f_0)d\mu_1} = o(1)\times F(||u||').$$
The third part of the proof is to compare $\mu_1$ to $\mu_0$ using Stein’s method [@Ste72; @Ste86] and the fact that they both satisfy the same integration by parts formula, up to small error. In our situation, $\mu_0$ is Gaussian in some direction and Stein’s method for such measures has been well-explored. We expect this type of argument to also apply to non-Gaussian situations, where Stein’s method has found some successes for other types of problems [@Ros11].
Stability of the Poincaré inequality
====================================
Proof of Theorem \[thm\_improved\_dpf\]
---------------------------------------
First, let us note that the assumptions constrain the value of the Poincaré constant
Under the assumptions of Theorem \[thm\_dpf\], we have $1 - \epsilon \leq C_P(\mu) \leq 1$.
The bound $C_P(\mu) \leq 1$ is true under the uniform convexity assumption of the potential. This is a classical result on Poincaré inequalities, which can be obtained for example via the [Bakry-Émery ]{}theory [@BGL14], or the Caffarelli contraction theorem [@Caf00]. The second bound comes from $$1 = \int{u_1^2d\mu} \leq C_P(\mu) \int{|\nabla u_1|^2d\mu} \leq C_P(\mu) (1+\epsilon)$$ so that $C_P(\mu) \geq (1+\epsilon)^{-1} \geq 1 - \epsilon$.
We have the following bounds on proximity between the $\nabla u_i$ and unit vectors, essentially proved in [@DPF]:
\[lem\_dpf\] Assume $\epsilon < (18 k)^{-2}$. Under the assumptions of Theorem \[thm\_dpf\], there exist unit vectors $\hat{w}_1,..,\hat{w}_k \subset {\mathbb{R}}^n$ such that $$\int{|\nabla u_i - \hat{w}_i|^2d\mu} \leq 9 \epsilon; \hspace{1cm} |\hat{w}_i \cdot \hat{w}_j| \leq 18 \sqrt{\epsilon}, ~~~i\neq j.$$ Moreover, $\operatorname{dim}(\operatorname{Span}(\hat{w}_1,..,\hat{w}_k))=k$.
[ In particular, this lemma implies that the functions $u_i$ are close to orthogonal affine functions.]{}
The proof follows the arguments of [@DPF], we include it for the sake of completeness.
First, let $T$ be the optimal transport (or Brenier map) [@Bre91] sending the standard Gaussian measure onto $\mu$, and define $v_i:= u_i \circ T$. According to the Caffarelli contraction theorem [@Caf00], $\nabla T$ is a symmetric, positive matrix satisfying $\| \nabla T\|_{op} \leq 1$. We then have $$\int{|\nabla v_i|^2d\gamma} \leq \int{|\nabla u_i|^2 \circ T d\gamma} = \int{|\nabla u_i|^2d\mu}$$ and $$\int{|\nabla u_i|^2d\mu} \leq (1+\epsilon)\int{u_i^2d\mu} = (1+\epsilon)\int{v_i^2d\gamma} \leq (1+\epsilon)\int{|\nabla v_i|^2d\gamma}.$$ Hence $$0 \leq \int{(|\nabla u_i|^2 \circ T -|\nabla v_i|^2)d\gamma} \leq \epsilon(1+\epsilon).$$ Additionally, since $(\operatorname{I} - \nabla T)^2 \leq \operatorname{I} - (\nabla T)^2$, we have $$\begin{aligned}
\int{|\nabla u_i \circ T - \nabla v_i|^2d\gamma} &= \int{|(\operatorname{I} - \nabla T) (\nabla u_i \circ T)|^2d\gamma} \notag \\
&\leq \int{|\nabla u_i \circ T|^2d\gamma} - \int{|\nabla T ( \nabla u_i \circ T ) |^2d\gamma} \notag \\
&= \int{(|\nabla u_i|^2 \circ T -|\nabla v_i|^2)d\gamma} \leq \epsilon(1+\epsilon).\end{aligned}$$
Note that $\int v_i d\gamma = \int u_i d\mu = 0$. Since the multivariate Hermite polynomials form an orthogonal basis for $L^2( \gamma)$, we may write $$v_i(x) = w_i \cdot x + z_i(x),$$ where $w_i\in \mathbb{R}^n$ and $z_i : {\mathbb{R}}^n \longrightarrow {\mathbb{R}}$, satisfying $\int z_i d\gamma = 0$ and $\int z_i x_j d\gamma =0$ for $j=1, \dots, n$. Using basic properties of Hermite polynomials, $$1 + \epsilon \geq \int{|\nabla v_i|^2d\gamma} = |w_i|^2 + \int |\nabla z_i|^2 d\gamma \geq 1+ \frac{1}{2}\int |\nabla z_i|^2 d\gamma.$$ The second inequality is a refinement of the Gaussian Poincaré inequality for functions orthogonal to the subspace spanned by constant and linear functions, combined with $ |w_i|^2 + \int z_i^2 d\gamma = \int v_i^2 d\gamma = \int u_i^2 d\mu = 1$. Hence, $$\int z_i^2 d\gamma\leq \frac{1}{2}\int |\nabla z_i|^2 d\gamma \leq \epsilon.$$ In particular, $\int{|\nabla v_i - w_i|^2 d\gamma} \leq 2\epsilon$ and $1-\epsilon\leq |w_i|^2\leq 1$. Together with the previous estimates, we have for $\hat w_i := w_i/|w_i|$, $$\int |\nabla u_i - \hat{w}_i|^2 d\mu \leq 3\left( \int |\nabla u_i\circ T - \nabla v_i|^2 d\gamma+ \int |\nabla v_i-w_i|^2 d\gamma + |w_i-\hat{w}_i|^2 \right)\leq 9 \epsilon.$$ As a consequence, for $j\neq i$, we have $$|\hat{w}_i \cdot \hat{w}_j| \leq 9\epsilon + 6 \sqrt{ \epsilon(1+\epsilon)} \leq 18 \sqrt{\epsilon}.$$ Finally, the matrix with coefficients $(\hat{w}_i \cdot \hat{w}_j)_{i,j \leq k}$ is strictly diagonally dominant when $\epsilon < (18 k)^{-2}$, and hence invertible. Thus, $\operatorname{dim}(\operatorname{Span}(\hat{w}_1,..,\hat{w}_k))=k$ as claimed.
The starting point to implement Stein’s method is the following approximate integration by parts formula for the measure $\mu$ and the approximate minimizers $u_i$:
\[lem\_approx\_euler\] Let $\mu$ be a probability measure satisfying a Poincaré inequality with constant $C_P \leq 1$. For any function $h \in H^1(\mu)$ and function $u$ satisfying $\int{ud\mu} = 0$, $\int{u^2d\mu} = 1$ and $\int{|\nabla u|^2d\mu} \leq 1 + \epsilon$, for some $\epsilon\geq 0$. We have $$\int{u h d\mu} - \int{\nabla u \cdot \nabla h d\mu} \leq \sqrt{\epsilon}\left(\int{|\nabla h|^2d\mu}\right)^{1/2}.$$
In particular, this applies for $u_i$ and $\mu$ satisfying the assumptions of Theorem \[thm\_dpf\].
The proof of the lemma is a variant of the argument used in [@CFP18; @Cou18] to establish integration by parts formula mimicking the Stein identity for measures satisfying a Poincaré inequality.
For any $h : {\mathbb{R}}\longrightarrow {\mathbb{R}}$ in the weighted Sobolev space $H^1(\mu)$, we have $$\left(\int{u h d\mu}\right)^2 \leq \operatorname{Var}_{\mu}(h )\int{u^2d\mu} \leq C_P\int{|\nabla h|^2d\mu}.$$ Hence the original term, viewed as a function of $h$, is a continuous linear form in $H^1(\mu)$, and as an application of the Lax-Milgram theorem there exists a function $g$ such that $$\int{uh d\mu} = \int{\nabla h \cdot \nabla gd\mu} \hspace{3mm} \forall h \in H^1(\mu); \hspace{5mm} \int{|\nabla h|^2d\mu} \leq C_P.$$ In particular, note that $\int{\nabla g \cdot \nabla u d\mu} = \int{u^2d\mu} = 1$.
Hence for any$h \in H^1(\mu)$, $$\int{(uh - \nabla u \cdot \nabla h) d\mu} = \int{\nabla h \cdot (\nabla g - \nabla u)d\mu} \leq \left(\int{|\nabla g - \nabla u|^2d\mu}\right)^{1/2}\left(\int{|\nabla h|^2d\mu}\right)^{1/2}.$$
Finally, we can expand the square and get $$\int{|\nabla g - \nabla u|^2d\mu} \leq C_P -2 + 1 + \epsilon \leq \epsilon$$ which concludes the proof.
We shall assume without loss of generality that $p = \int x d\mu = 0$.
Assume first that $\epsilon < 1/(18 k)^2$. Let $(\hat{w}_1,..,\hat{w}_k)$ be as in Lemma \[lem\_dpf\], and consider any orthonormal family $(e_1,..,e_k)$ such that $\operatorname{Span}(e_1,..,e_k)=\operatorname{Span}(\hat{w}_1,..,\hat{w}_k)$. Let $(\alpha_{ij})_{i,j\leq k}$ be real numbers such that $e_i = \sum_{j\leq k}\alpha_{ij} \hat{w}_j$. If $k=1$, then we may take $e_1 = \hat{w}_1$. On the other hand, if $k\geq 2$, we use $|\hat{w}_i\cdot\hat{w}_k|\leq 18\sqrt{\epsilon}$ for $i\neq j$ and recall $\epsilon < 1/(18 k)^2$ to conclude that $$\sum_{j\leq k}\alpha_{ij}^2 \leq (1-18\sqrt{\epsilon})^{-1} \leq 1+ \frac{18 k\sqrt{\epsilon}}{k-1} \leq 1+ \frac{1}{k-1}.$$ Hence, we always have $\sum_{j\leq k}\alpha_{ij}^2\leq 2$ for each $i\leq k$.
After suitable change of coordinates, we may assume without loss of generality that the vectors $(e_i)_{i\leq k}$ coincide with the first $k$ natural basis vectors of ${\mathbb{R}}^n$. Hence, from now on, we write $x = (y,z)$ where $y$ is the orthogonal projection of $x$ onto the vector space spanned by the $(e_i)_{i\leq k}$, with $y_i = x \cdot e_i$, and $z$ its projection onto $\operatorname{Span}(e_1,..,e_k)^{\bot}$. Let $\bar{\mu}$ be the distribution of $z$ when $x$ is distributed according to $\mu$, that is $\bar{\mu}(dz) = e^{-W(z)}dz$ with $W(z) = -\log \int_{\operatorname{Span}(e_1,..,e_k)}{e^{-V(y,z)}dy}$. As a consequence of the Prékopà-Leindler theorem, $W$ inherits uniform convexity from $V$, that is $\operatorname{Hess}W \geq \operatorname{I}_{n-k}$ (see for example [@BL76]).
Consider $1$-Lipschitz $f : {\mathbb{R}}^n \longrightarrow {\mathbb{R}}$; note this ensures $f$ is integrable with respect to both $\mu$ and $\gamma_k\otimes \bar{\mu}$, where $\gamma_k$ is the centered standard Gaussian measure on ${\mathbb{R}}^k$. For any $z$, there exists a function $h(\cdot, z) : {\mathbb{R}}^k \longrightarrow {\mathbb{R}}^k$ satisfying the Poisson equation $$\label{eq_poisson}
f(y,z) - \int{f(s,z)d\gamma_k(s)} = y \cdot h(y,z) - \operatorname{Tr}(\nabla_yh)(y,z).$$ In fact, as pointed out by Barbour [@Bar90], as a consequence of the representation of the Ornstein-Uhlenbeck semigroup via convolution with a Gaussian kernel, the function $h$ is given by $$\begin{aligned}
h_i(y,z) &= - \partial_{e_i}\int_0^1{\frac{1}{2t}\int{(f(\sqrt{t}y + \sqrt{1-t}w, z) - f(w,z))d\gamma_k(w)}dt}\\
&= -\int_0^1{\frac{1}{2\sqrt{t(1-t)}}\int{w_i f(\sqrt{t}y + \sqrt{1-t}w, z)d\gamma_k(w)}dt},\end{aligned}$$ where the second identity follows from the Gaussian integration by parts formula. Hence, by the Jensen and Cauchy-Schwarz inequalities, $$\begin{aligned}
|\nabla_y h_i(y,z) |^2 &= \left| \int_0^1{\frac{1}{2\sqrt{1-t}}\int{w_i \nabla_y f(\sqrt{t}y + \sqrt{1-t}w, z) d\gamma_k(w)}dt} \right|^2\\
&\leq \int_0^1 \frac{1}{2\sqrt{1-t} } \left| \int{w_i \nabla_y f(\sqrt{t}y + \sqrt{1-t}w, z) d\gamma_k(w)} \right|^2 dt\\
&\leq \int_0^1 \frac{1}{\sqrt{t(1-t)} } \int \left|\nabla_y f(\sqrt{t}y + \sqrt{1-t}w, z) \right|^2 d\gamma_k(w) dt.\end{aligned}$$ Similarly, $$\begin{aligned}
|\nabla_z h_i(y,z) |^2 &= \left| \int_0^1{\frac{1}{2\sqrt{t(1-t)}}\int{w_i \nabla_z f(\sqrt{t}y + \sqrt{1-t}w, z) d\gamma_k(w)}dt} \right|^2\\
&\leq \int_0^1 \frac{\pi }{4\sqrt{t(1-t)} } \left| \int{w_i \nabla_z f(\sqrt{t}y + \sqrt{1-t}w, z) d\gamma_k(w)} \right|^2 dt\\
&\leq \int_0^1 \frac{ 1 }{\sqrt{t(1-t)} } \int \left|\nabla_z f(\sqrt{t}y + \sqrt{1-t}w, z) \right|^2 d\gamma_k(w) dt.\end{aligned}$$ Combining the above, we have $$\begin{aligned}
|\nabla h_i(y,z) |^2 = |\nabla_y h_i(y,z) |^2+|\nabla_z h_i(y,z) |^2\leq \int_0^1 \frac{ 1 }{\sqrt{t(1-t)} } \| f\|^2_{\mathrm{Lip}} dt\leq \pi . \label{nablaHbound}\end{aligned}$$ The above computation follows the strategy of [@CM08; @Gau16]. Better regularity bounds on solutions of the Poisson equation have been derived in [@GMS; @FSX18], but for our purpose this bound will suffice. It follows that $h_i \in H^1(\mu)$, justifying the following manipulations: $$\begin{aligned}
\int{fd\mu} - \int{fd\gamma_k d\bar{\mu}} &= \int{ \Big( y \cdot h(y,z) - \operatorname{Tr}(\nabla_y h)(y,z)\Big) d\mu} \\
&= \sum_{i\leq k} \int \left(y_i h_i(y,z) - e_i \cdot \nabla h_i(y,z) \right) d\mu.\end{aligned}$$ Now, focusing on the $i$th term in the sum, we expand $$\begin{aligned}
\int &\left(y_i h_i(y,z) - e_i \cdot \nabla h_i(y,z) \right) d\mu = \sum_{j\leq k} \alpha_{ij} \int{(\nabla u_j-\hat{w}_j)\cdot \nabla h_i(y,z)d\mu} \\
&\hspace{1cm}+ \sum_{j\leq k} \alpha_{ij} \int{(\hat{w_j}\cdot x-u_j) h_i(y,z)d\mu} + \sum_{j\leq k} \alpha_{ij} \int{\left(u_j h_i(y,z) - \nabla u_j \cdot \nabla h_i(y,z) \right)d\mu}. \end{aligned}$$ We bound each of the three terms separately. By Cauchy-Schwarz, Lemma \[lem\_dpf\], and $$\begin{aligned}
&\sum_{j\leq k} \alpha_{ij} \int{(\nabla u_j-\hat{w}_j)\cdot \nabla h_i(y,z)d\mu}\\
&\leq \left( \sum_{j\leq k} \alpha_{ij}^2 \right)^{1/2} \left( \sum_{j\leq k } \left( \int{(\nabla u_j-\hat{w}_j)\cdot \nabla h_i(y,z)d\mu}\right)^2 \right)^{1/2} \leq \sqrt{2} \left( k \pi 9 \epsilon \right)^{1/2}.\end{aligned}$$ Similarly, with additional help from the Poincaré inequality for $\mu$ and the assumption that $\int x d\mu =\int u_i d\mu=0$, $$\begin{aligned}
\sum_{j\leq k} \alpha_{ij} &\int{(\hat{w_j}\cdot x-u_j) h_i(y,z)d\mu}
\leq \left( \sum_{j\leq k} \alpha_{ij}^2 \right)^{1/2} \left( \sum_{j\leq k } \left( \int{(\hat{w_j}\cdot x-u_j) h_i(y,z)d\mu}\right)^2 \right)^{1/2}\\
&\leq \sqrt{2} \left( \sum_{j\leq k } \left( \int|\hat{w_j}-\nabla u_j|^2 d\mu \right) \left( \int | \nabla h_i(y,z)|^2 d\mu\right) \right)^{1/2} \leq \sqrt{2} \left( k \pi 9 \epsilon \right)^{1/2}.\end{aligned}$$ Finally, by Lemma \[lem\_approx\_euler\] and , $$\begin{aligned}
&\sum_{j\leq k} \alpha_{ij} \int{\left(u_j h_i(y,z) - \nabla u_j \cdot \nabla h_i(y,z) \right)d\mu} \\
&\leq \left( \sum_{j\leq k} \alpha_{ij}^2 \right)^{1/2} \left( \sum_{j\leq k } \left( \int{\left(u_j h_i(y,z) - \nabla u_j \cdot \nabla h_i(y,z) \right)d\mu} \right)^2 \right)^{1/2} \leq \sqrt{2} \left( k \pi \epsilon \right)^{1/2}.\end{aligned}$$ Combining all of the above estimates with the Kantorovitch dual formulation of $W_1$ [@Vill03], we have $$W_1(\mu, \gamma_k \otimes \bar{\mu}) = \underset{f : \|f \|_{\mathrm{Lip}\leq 1}}{\sup} \hspace{1mm} \int{fd\mu} - \int{fd\gamma_k d\bar{\mu}}
\leq k^{3/2} 7 \sqrt{2 \pi \epsilon} < k^{3/2} 18 \sqrt{2 \epsilon} .$$
To finish the proof, we only need to consider $\epsilon \geq (18 k)^{-2}$. In this case, we bypass Lemma \[lem\_dpf\] and take $(e_1, \dots, e_k)$ to be any orthonormal family in ${\mathbb{R}}^n$, and define $\bar{\mu}$ in terms of this family, same as above. By the Poincaré inequality, $\operatorname{Var}_{\mu}(x\cdot e_i)\leq 1$ for each $i\leq k$, so it follows that $$W_1(\mu, \gamma_k \otimes \bar{\mu}) \leq W_2(\mu, \gamma_k \otimes \bar{\mu}) \leq \sqrt{2 k} \leq k^{3/2} 18 \sqrt{2 \epsilon} ,$$ where the last inequality holds under the assumption that $\epsilon \geq (18 k)^{-2}$.
Proof of Theorem \[thm\_coord\]
-------------------------------
The proof is essentially the same as for Theorem \[thm\_improved\_dpf\], except that our extra assumptions make Lemma \[lem\_dpf\] unnecessary, which allows us to drop the convexity assumption. Without loss of generality, we may assume $\mu$ has its barycenter at the origin. We then take $u_i = \frac{x \cdot e_i}{\sqrt{\operatorname{Var}_{\mu}(x \cdot e_i)}}$ in Lemma \[lem\_approx\_euler\] to get $$\int{x_i h(x) - \partial_i h(x)d\mu} \leq \sqrt{ {\epsilon} }\left(\int{|\nabla h|^2d\mu}\right)^{1/2}$$ for any real-valued smooth test function $h$. We can then introduce the same function $h$ associated to a $1$-Lipschitz function $f$ via the Poisson equation , and the proof continues in the same way as the proof of Theorem \[thm\_improved\_dpf\], but is simpler since we directly conclude: $$\begin{aligned}
\int{fd\mu} - \int{fd\gamma_k d\bar{\mu}} &= \int{ \Big( y \cdot h(y,z) - \operatorname{Tr}(\nabla_y h)(y,z)\Big) d\mu} \\
&= \underset{i \leq k}{\sum} \int{ (y_i h_i(y,z) - \partial_i h_i(y,z))d\mu} \leq k \sqrt{ \pi \epsilon}.\end{aligned}$$ Note that bypassing Lemma \[lem\_dpf\] gives improved dependence on $k$.
Stability for the logarithmic Sobolev inequality
================================================
Proof of Theorem \[thm:QstableLSI\]
-----------------------------------
The proof of Theorem \[thm:QstableLSI\] follows the strategy for that of Theorem \[thm\_improved\_dpf\], relying on an approximate integration by parts identity for extremizers of the LSI, combined with Stein’s method. However, the details are sufficiently different that the same argument can not be applied mutatis mutandis. The following sequence of lemmas provides the necessary ingredients for the proof.
The following approximate Euler-Lagrange equation for the LSI is the starting point of the proof. It is used as the counterpart of Lemma \[lem\_approx\_euler\].
\[lem:approxELeqn\] Assume $\mu$ satisfies the LSI , and let $u:{\mathbb{R}}^n \longrightarrow {\mathbb{R}}$ satisfy for some $\epsilon\geq 0$. For any smooth function $h$ we have $$\begin{aligned}
& \int{\nabla h \cdot \nabla u d\mu} - \frac{1}{2}\int{h u\log(u^2/\alpha)}d\mu \\
&\hspace{1cm}\leq \sqrt{\epsilon}\left(\int{|\nabla u|^2d\mu} \right)^{1/2} \left(\int{|\nabla h |^2d\mu} - \frac{1}{2}\int h^2 \log(u^2/\alpha) d\mu \right)^{1/2} ,\end{aligned}$$ where $\alpha := \int{u^2d\mu}$.
The quantity $\int{|\nabla h|^2d\mu} - \frac{1}{2}\int h^2 \log(u^2/\alpha) d\mu $ is nonnegative. Indeed, by the LSI for $\mu$, this quantity is at least $$\frac{1}{2}\operatorname{Ent}_{\mu}(h^2) - \frac{1}{2}\int h^2 \log(u^2/\alpha) d\mu = \frac{1}{2}\int h^2 \log \left( \frac{h^2 / \int h^2 d\mu }{ u^2 / \int u^2 d\mu} \right) d\mu,$$ which is proportional to a relative entropy, and therefore nonnegative.
We emphasize that Lemma \[lem:approxELeqn\] does not make any convexity assumptions on $\mu$, so may be of independent interest for other applications.
By convexity of the map $\varphi \longmapsto \operatorname{Ent}_{\mu}(\varphi)$ on nonnegative functions, for $t \geq 0$, it holds that $$\begin{aligned}
\operatorname{Ent}_{\mu}(\varphi + t \psi) \geq \operatorname{Ent}_{\mu}(\varphi ) + t \int \psi \log \left(\frac{\varphi}{\int \varphi d\mu } \right) d\mu\label{eq:EntLB}\end{aligned}$$ provided $\varphi \geq 0$ and $\varphi + t \psi \geq 0$.
Now, we observe $$\begin{aligned}
&2 \int |\nabla u|^2 d\mu + 4 t \int \nabla u \cdot \nabla h d\mu + 2 t \int |\nabla h |^2 d\mu \geq \operatorname{Ent}_{\mu}( (u+ t h)^2 ) \\
&\hspace{1cm} \geq \operatorname{Ent}_{\mu}( u^2 ) + t \int (2 u h + t h^2 ) \log \left(u^2/\alpha \right) d\mu\\
&\hspace{1cm}\geq 2(1-\epsilon)\int |\nabla u|^2 d\mu + t \int (2 uh + t h^2 ) \log \left(u^2/\alpha \right) d\mu,\end{aligned}$$ where the first inequality is the LSI for $\mu$ applied to the function $u + t h$, the second inequality is , and the third inequality is . Rearranging and dividing by $2 t$ gives $$\begin{aligned}
& \epsilon \, t^{-1} \int |\nabla u |^2 d\mu + t \left( \int |\nabla h |^2 d\mu -\frac{1}{2} \int h^2 \log \left(u^2/\alpha \right) d\mu \right) \\
&\hspace{1cm}\geq \int u h \log \left(u^2/\alpha \right) d\mu - 2 \int \nabla u \cdot \nabla h d\mu.\end{aligned}$$ Optimizing over $t>0$ gives $$\begin{aligned}
&\frac{1}{2} \int u h \log \left(u^2/\alpha \right) d\mu - \int \nabla u \cdot \nabla h d\mu \\
&\hspace{1cm}\leq
\sqrt{\epsilon}\left(\int{|\nabla u |^2d\mu} \right)^{1/2} \left(\int{|\nabla h|^2d\mu} - \frac{1}{2}\int h^2 \log(u^2/\alpha) d\mu \right)^{1/2}.\end{aligned}$$ We may now replace $h$ with $-h$ to obtain the desired inequality.
We now state the Aida-Shigekawa perturbation theorem for the LSI, which will be needed in the sequel. It will allow us to estimate certain terms that involve an extra weight $u^2$, using the fact that $\log u$ is Lipschitz. The following is a consequence of [@AS Theorem 3.4]:
\[thm:AS\] Let $\mu$ satisfy , and take $\mu_F$ to be the probability measure proportional to $e^F \mu$, where $F$ is $\lambda$-Lipschitz. There exists a $\tilde{\lambda}>0$, depending only on $\lambda$, for which $$\operatorname{Ent}_{\mu_F}(f^2) \leq 2\tilde{\lambda}\int |\nabla f|^2 d\mu_F.$$ In particular, $\mu_F$ also satisfies a Poincaré inequality with constant $C_P(\lambda)\leq \tilde{\lambda}$.
Together with [@FIL Theorem 1], this yields the following deficit estimate for the Gaussian LSI:
\[lem:FIL\] Let the notation of Theorem \[thm:AS\] prevail. If $\gamma$ is the standard Gaussian measure on ${\mathbb{R}}^n$, and $d\mu_F = v^2 d\gamma$, there is a constant $c(\lambda)<1$ for which $$\operatorname{Ent}_{\gamma}(v^2) \leq 2 c(\lambda) \int |\nabla v |^2 d\gamma.$$
The following specializes Lemma \[lem:approxELeqn\] under the hypothesis that $\log u$ is $\lambda$-Lipschitz.
\[lem:ELforg\] Let $u$, $\lambda$, $\epsilon$, and $\mu$ satisfy the assumptions of Theorem \[thm:QstableLSI\]. If $g:{\mathbb{R}}^n\to {\mathbb{R}}$ is Lipschitz, satisfying $\int g d\mu = 0$, then $$\int \nabla g \cdot \nabla \log(u) d\mu - \int g |\nabla \log u|^2 d\mu - \int g \log u \, d\mu \leq \|g\|_{\mathrm{Lip}} C(\lambda) \sqrt{\epsilon},$$ where $C(\lambda)$ is a constant depending only on $\lambda$.
We may assume without loss of generality that $\|g\|_{\mathrm{Lip}}\leq 1$.
Apply Lemma \[lem:approxELeqn\] to the test function $h=g/u$. This gives $$\begin{aligned}
&\int \nabla g \cdot \nabla \log(u) d\mu - \int g |\nabla \log u|^2 d\mu - \int g \log u \, d\mu \notag\\
&\leq \sqrt{\epsilon}\left(\int{|\nabla u|^2d\mu} \right)^{1/2} \left(2 \int{|\nabla g |^2u^{-2} d\mu}+ 2 \int{g^2 |\nabla \log u |^2 d\mu} - \frac{1}{2}\int (g/u)^2 \log(u^2) d\mu \right)^{1/2} \notag\\
&\leq \sqrt{\epsilon}\lambda \left(2 \int{ u^{-2} d\mu}+ 2 \lambda^2 \int{g^2 d\mu} - \frac{1}{2}\int (g/u)^2 \log(u^2) d\mu \right)^{1/2} \label{lastLine}\end{aligned}$$ Now, we claim that for any smooth enough $h$, we have $$\begin{aligned}
-\frac{1}{2}\int{h^2 \log u^2 d\mu}\leq \int{|\nabla h|^2d\mu} + 2\lambda^2\int{h^2d\mu}.\label{Hbound}\end{aligned}$$ From the classical entropy inequality, we have $$\begin{aligned}
-\int{h^2 \log u^2 d\mu} &\leq \operatorname{Ent}_{\mu}(h^2) + \int{h^2d\mu} \times \log \int{e^{-2\log u}d\mu}\\
&\leq 2\int{|\nabla h|^2d\mu} + \int{h^2d\mu} \times \log \int{e^{-2\log u}d\mu}.\end{aligned}$$ Now, we apply the concentration inequality . In particular, since $\log u$ is assumed to be $\lambda$-Lipschitz $$1 = \int{u^2d\mu} \leq \exp\left(2\int{\log u d\mu} + 2\lambda^2\right) ~~~\Longrightarrow ~~~ - \int{\log u \, d\mu}\leq \lambda^2.$$ On the other hand, using this together with gives, for any $t >0$, $$\begin{aligned}
\int u ^{-t}d\mu = \int{e^{-t \log u}d\mu} \leq \exp\left(-t\int{\log u \, d\mu} + (t\lambda)^2/2\right) \leq e^{t \lambda^2(1+t/2)},\label{eq:ExpLogf}\end{aligned}$$ which leads to by taking $t=2$.
Applying these estimates to gives $$\begin{aligned}
&\int \nabla g \cdot \nabla \log(u) d\mu - \int g |\nabla \log u|^2 d\mu - \int g \log u \, d\mu \\
&\leq \sqrt{\epsilon}\lambda \left(2 e^{4 \lambda^2} + 2 \lambda^2 \int{g^2 d\mu}
+\int{|\nabla (g/u)|^2d\mu} + 2\lambda^2\int{g^2u^{-2} d\mu}
\right)^{1/2}\\
&\leq \sqrt{\epsilon}\lambda \left(4 e^{4 \lambda^2} + 2 \lambda^2 + 4\lambda^2\int{g^2u^{-2} d\mu}
\right)^{1/2},\end{aligned}$$ where the last line made use of the Poincaré inequality $\int{g^2 d\mu} \leq \int{|\nabla g|^2 d\mu}\leq 1$. Now, since $\log u$ is $\lambda$-Lipschitz, the measure $u^{-2} \mu$ satisfies a Poincaré inequality with constant $C_P(\lambda)$. Hence, $$\begin{aligned}
\int g^2u^{-2} d\mu &\leq C_P(\lambda)\int |\nabla g|^2 u^{-2}d\mu + \left( \int g u^{-2} d\mu\right)^2 \left( \int u^{-2} d\mu \right)^{-1}\\
&\leq C_P(\lambda) e^{4\lambda^2} + \left( \int g u^{-2} d\mu\right)^2 .\end{aligned}$$ By Cauchy-Schwarz, the Poincaré inequality for $\mu$ and , we have $$\left( \int g u^{-2} d\mu\right)^2 \leq \int g^2 d\mu \times \int u^{-4} d\mu \leq e^{12 \lambda^2},$$ which completes the proof.
The next lemma quantifies the proximity between $\log u$ and an affine function. It is used as the counterpart to Lemma \[lem\_dpf\]. [ This step is more complicated than for the Poincaré inequality, since in this case stability for the Gaussian functional inequality is a much more difficult problem, as we cannot simply use a spectral decomposition of the function. ]{}
\[lem\_prox\_f\_linear\] Let $u$, $\lambda$, $\epsilon$, and $\mu$ be as in Theorem \[thm:QstableLSI\]. There exists $p\in {\mathbb{R}}^n$ and constants $C_1(\lambda)$ and $C_2(\lambda)$, depending only on $\lambda$, such that $$\begin{aligned}
\int{|\nabla \log u - p/2|^2u^2d\mu} &\leq C_1(\lambda) \epsilon\int{|\nabla u|^2d\mu};\label{L2nabla_f2mu} \\
\operatorname{Var}_{u^2\mu}(\log u - p\cdot x/2) &\leq C_2(\lambda) \epsilon\int{|\nabla u|^2d\mu}.\label{Var_f2mu}\end{aligned}$$
Let $T$ be the optimal transport map sending the standard Gaussian measure onto $\mu$ and define $$\begin{aligned}
p := \int{\xi u( T(\xi))^2d\gamma(\xi)} = 2\int u (T(\xi)) \nabla T(\xi) \nabla u (T(\xi)) d\gamma(\xi), \label{pDef}\end{aligned}$$ where the second identity follows from Gaussian integration by parts. The Caffarelli contraction theorem states that $T$ is $1$-Lipschitz. Define $v(x) = u(T(x+p) )e^{-p\cdot x/2 - |p|^2/4}$. We have $$\int{v^2d\gamma} = \int{u (T(x+p))^2e^{-|x+p|^2/2}(2\pi)^{-n/2}dx} = \int{u(T(\xi))^2d\gamma(\xi)} = \int{u^2d\mu} = 1;$$ $$\int{xv^2d\gamma} = \int{x u(T(x+p))^2e^{-|x+p|^2/2}(2\pi)^{-n/2}dx} = \int{(\xi-p)u(T(\xi))^2d\gamma(\xi)} = 0.$$ Hence $v^2d\gamma$ is a centered probability measure. Moreover, since $\log u(T(x+p))$ is $\lambda$-Lipschitz, the measure $v^2d\gamma$ satisfies a Poincaré inequality with a constant $C_P(\lambda)$ by Theorem \[thm:AS\].
We have $\operatorname{Ent}_{\gamma}(v^2) = \operatorname{Ent}_{\mu}(u^2) - |p|^2/2$ and, using the fact that $T$ is $1$-Lipschitz and the identity , $$\int{|\nabla v|^2d\gamma} = \int{|\nabla T(x+p) (\nabla \log u)(T(x+p)) - p/2|^2v^2d\gamma} \leq \int{|\nabla u|^2d\mu} - |p|^2/4.$$
Hence the deficit in the Gaussian LSI for the probability measure $v^2d\gamma$ is smaller than $2\epsilon\int{|\nabla u|^2d\mu}$. By Lemma \[lem:FIL\], this ensures that $$\begin{aligned}
\int{|\nabla (\log u \circ T) - p/2|^2(u^2\circ T)d\gamma} = \int{|\nabla v|^2d\gamma} \leq C_0(\lambda)\epsilon \int{|\nabla u|^2d\mu}.\label{boundFirstTerm}\end{aligned}$$ In a different direction, we use the Gaussian LSI to observe that $$\begin{aligned}
(1-\epsilon) \int |\nabla u|^2 d\mu &\leq \frac{1}{2 } \operatorname{Ent}_{\mu}(u^2) \notag\\
&= \frac{1}{2} \left( \operatorname{Ent}_{\gamma}(v^2) + |p|^2/2 \right)
\leq \left( \int |\nabla v|^2 d\gamma + |p|^2/4 \right).\label{revEst}\end{aligned}$$ Now, the proof continues along similar lines to that of Lemma \[lem\_dpf\]. First, we bound $$\begin{aligned}
&\frac{1}{2}\int{|\nabla \log u - p/2|^2u^2d\mu} \\
&\leq \int | \nabla \log(u\circ T)-p/2 |^2 (u^2\circ T)d\gamma + \int |(\nabla \log u)\circ T - \nabla \log(u\circ T)|^2 (u^2\circ T)d\gamma.\end{aligned}$$ The first term on the RHS is controlled by . The second term is bounded as $$\begin{aligned}
&\int |(\nabla \log u)\circ T - \nabla \log(u\circ T)|^2 (u^2\circ T)d\gamma\notag\\
&=\int |( I-\nabla T)(\nabla \log u)\circ T |^2 (u^2\circ T)d\gamma\notag\\
&\leq \int | (\nabla \log u)\circ T |^2 (u^2\circ T)d\gamma - \int |\nabla T(\nabla \log u)\circ T |^2 (u^2\circ T)d\gamma \label{psdIneq}\\
&= \int | \nabla u |^2 d\mu - \left(\int |\nabla v|^2 d\gamma + |p|^2/4 \right) \label{useVdefn} \\
&\leq \epsilon \int{|\nabla u|^2d\mu}, \label{useRevIneq}\end{aligned}$$ where follows since $(I-\nabla T)^2 \leq I - (\nabla T)^2$, follows by definition of $v$, and is due to . This establishes . Since $\log u$ is Lipschitz, the measure $u^2\mu$ satisfies a Poincaré inequality with constant depending only on $\lambda$ by Theorem \[thm:AS\], so that $\operatorname{Var}_{u^2\mu}(\log u - p\cdot x/2) \leq C_2(\lambda)\epsilon$ as desired.
Combining these estimates leads to the following approximate integration by parts formula, which is the crucial estimate we need:
\[lem:approxExpLinear\] Let $u$, $\lambda$, $\epsilon$, and $\mu$ satisfy the assumptions of Theorem \[thm:QstableLSI\], and let $p\in {\mathbb{R}}^n$ be as in Lemma \[lem\_prox\_f\_linear\]. For any Lipschitz function $g$, we have $$\begin{aligned}
\int{\left(g \times \left(x-\langle x\rangle_{\mu} \right) \cdot p - \nabla g \cdot p\right) d\mu} \leq \|g\|_{\mathrm{Lip}}C(\lambda) \sqrt{\epsilon},\label{eq:ApproxIntByParts}\end{aligned}$$ where $C(\lambda)$ is a constant depending only on $\lambda$, and $\langle x\rangle_{\mu} := \int x d\mu$.
\[rmk:Measures\_ut\] To modify Theorem \[thm:QstableLSI\] for measures $u^t \mu$ along the lines of Remark \[Rmk\_utdmu\], one should modify Lemma \[lem:ELforg\] by repeating the proof mutatis mutandis, except one should consider the test function $h=g u^{t-1}$, rather than $h=g/u$. The following proof can then be suitably modified to yield an approximate integration by parts formula for the measure $u^t \mu$. Lemma \[lem\_prox\_f\_linear\] does not need to be modified.
Since the statement to prove is invariant to adding a constant to $g$, we assume without loss of generality that $\int g d\mu =0$, and that $\|g\|_{\mathrm{Lip}}\leq 1$. Throughout, we let $C(\lambda)$ denote a constant depending only on $\lambda$ which may change line to line.
Letting $\beta = \int (\log u - x\cdot p/2)d\mu$, we have by Cauchy-Schwarz $$\begin{aligned}
\frac{1}{2}\int g x\cdot p d\mu - \int g \log u\, d\mu &= \int g (\beta - \log u + x\cdot p/2) d\mu\\
&\leq \left( \int g^2 u^{-2}d\mu \right)^{1/2} \operatorname{Var}_{u^2 \mu}\Big( \log u - x\cdot p/2 \Big)^{1/2}\\
&\leq C(\lambda) \sqrt{\epsilon},\end{aligned}$$ where the last line follows from , the fact that $\int |\nabla u|^2 d\mu = \int |\nabla \log u|^2 u^2 d\mu \leq \lambda^2$, and the estimate $\int g^2 u^{-2}d\mu\leq C(\lambda)$ established in the final steps of the proof of Lemma \[lem:ELforg\].
Next, we write $$\begin{aligned}
&\int \nabla g \cdot \nabla \log u \, du - \int g |\nabla \log u|^2 d\mu - \frac{1}{2}\int \nabla g \cdot p d\mu + \frac{1}{2} \int g\, p \cdot \nabla \log u d\mu\\
&= \int \nabla g \cdot (\nabla \log u - p/2) d\mu + \int g \nabla \log u \cdot (p/2 - \nabla \log u )d\mu \\
&\leq \left( \left( \int |\nabla g|^2u^{-2} d\mu \right)^{1/2} + \left( \int g^2 |\nabla u |u^{-2} d\mu \right)^{1/2}\right) \left( \int |\nabla \log u - p/2|u^2 d\mu \right)^{1/2} \\
&\leq \left( \left( \int u^{-2} d\mu \right)^{1/2} + \lambda \left( \int g^2 u^{-2} d\mu \right)^{1/2}\right) \left( \int |\nabla \log u - p/2|u^2 d\mu \right)^{1/2} \\
&\leq C(\lambda) \sqrt{\epsilon},\end{aligned}$$ where the final inequality follows similarly to before, except using .
Summing the estimates and applying Lemma \[lem:ELforg\], we have $$\begin{aligned}
\int \Big( g \times (x- \langle x\rangle_{\mu})\cdot p - \nabla g \cdot p \Big) d\mu + \int g\, p \cdot \nabla \log u d\mu \leq C(\lambda) \sqrt{\epsilon},\end{aligned}$$ where the $\langle x\rangle_{\mu}$ was inserted using the assumption that $\int g d\mu = 0$. Thus, it only remains to show that the error term is small. To this end, we again use $\int g d\mu = 0$ to write $$\begin{aligned}
\left| \int g\, p \cdot \nabla \log u d\mu \right| &= \left| \int g\, p \cdot (\nabla \log u -p/2) d\mu \right| \\
&\leq |p| \left(\int g^2 u^{-2}d\mu \right)^{1/2}\left( \int |\nabla \log u - p/2|u^2 d\mu \right)^{1/2} \\
&\leq C(\lambda)\sqrt{\epsilon},\end{aligned}$$ which follows from similar estimates as above, plus the fact that $|p|^2 \leq 4 \int |\nabla u|^2 d\mu = 4 \int |\nabla \log u|u^2 d\mu \leq 4 \lambda^2$, where the first inequality was observed in the proof of Lemma \[lem\_prox\_f\_linear\].
Combining this last lemma and Stein’s method, we now prove Theorem \[thm:QstableLSI\].
Since the statement to prove is translation invariant, we assume $\int x d\mu=0$. We assume first that $\epsilon \leq 1/(4 C_1(\lambda))$, where $C_1(\lambda)$ is as defined in Lemma \[lem\_prox\_f\_linear\]. The same lemma ensures existence of $p\in {\mathbb{R}}^n$ such that $$\int{|\nabla \log u - p/2|^2u^2d\mu} \leq C_1(\lambda) \epsilon\int{|\nabla u|^2d\mu}.$$ Thus, using the assumption that $\epsilon \leq 1/(4 C_1(\lambda))$, we apply the elementary inequality $|A-B|^2\geq \frac{1}{2}|A|^2-|B|^2$ to the above to conclude $$|p|^2 \geq (2 - \epsilon\, 4 C_1(\lambda))\int{|\nabla u|^2d\mu}\geq \int{|\nabla u|^2d\mu}.$$ Henceforth, we let $C(\lambda)$ denote a constant depending on $\lambda$, which may change from line to line. The vector $p$ above is the same as in Lemma \[lem:approxExpLinear\], so we apply it and combine with the above estimate on $|p|$ to find for $e := p/|p|$, $$\int{(g \, x\cdot e - \nabla g \cdot e)d\mu} \leq \| g\|_{\mathrm{Lip}}C(\lambda) \sqrt{\epsilon}\left(\int{|\nabla u|^2d\mu}\right)^{-1/2} ,$$ holding for any Lipschitz $g: {\mathbb{R}}^n \longrightarrow {\mathbb{R}}$.
Now, we implement Stein’s method following the proof of Theorem \[thm\_improved\_dpf\]. In particular, we begin by writing $x = (y,z)$ where $y$ is the orthogonal projection of $x$ onto $e$, and $z$ its projection onto $e^{\bot}$. Consider $1$-Lipschitz $f : {\mathbb{R}}^n \longrightarrow {\mathbb{R}}$. For any $z\in {\mathbb{R}}^{n-1}$, there exists a function $g(\cdot, z) : {\mathbb{R}}\longrightarrow {\mathbb{R}}$ satisfying $$f(y,z) - \int_{\operatorname{Span}(e)}{f(s,z)d\gamma_{0,e}(s)} = y g(y,z) - \partial_y g (y,z),$$ where $\gamma_{0,e}$ is the centered standard Gaussian measure on $\operatorname{Span}(e)$.
The function $g$ is measurable and satisfies $\|g \|_{\mathrm{Lip}}\leq \sqrt{\pi}$, as already shown in . Hence, we integrate with respect to $\mu$ to conclude $$\begin{aligned}
\int{f d\mu} - \int{ f d\gamma_{0,e} d\bar{\mu}} &= \int{ \Big( y g(y,z) - \partial_y g(y,z)\Big) d\mu} \\
&= \int \left( g\times (x\cdot e) - e \cdot \nabla g \right) d\mu \leq C(\lambda) \sqrt{\epsilon} \left(\int{|\nabla u |^2d\mu} \right)^{-1/2} .\end{aligned}$$ Since $f$ was an arbitrary 1-Lipschitz function, the theorem follows from the Kantorovich dual formulation of $W_1$, provided $\epsilon \leq 1/(4 C_1(\lambda))$.
Now, by the triangle inequality for $W_1$ and simple variance bounds, it is easy to see that $W_1(\mu, \gamma_{b,\sigma}\otimes \bar{\mu})\leq 2$ for any $\sigma \in \mathbb{S}^{n-1}$ and $b = \sigma \int x\cdot \sigma d\mu$. Hence, the $W_1$ estimate can not become active until $ {\epsilon} \leq 4 \left( \int |\nabla u|^2 d\mu \right) /C(\lambda)^2 \leq 4 \lambda^2 / C(\lambda)^2$. By suitable modification of $C(\lambda)$, we may assume $C(\lambda)^2 \geq 16 \lambda^2 C_1(\lambda)$, so that the claim of the theorem is automatically satisfied whenever $\epsilon > 1/(4 C_1(\lambda))$. This completes the proof.
Proof of Theorem \[thm:ThmExpConcentration\]
--------------------------------------------
By suitable modification, we can assume without loss of generality that $\int F d\mu =0$ and $C(L)\geq 2\sqrt{2}$, so that we may restrict attention to the case where $\epsilon\leq 1/2$ (in the complementary case, the $W_1$ estimate will be automatically satisfied for similar reasons as argued in the final steps of the proof of Theorem \[thm:QstableLSI\]). The Herbst argument establishes by considering the function $$H(\lambda) = \log\left( \int e^{\lambda F} d\mu \right) ,$$ and using the LSI to establish the differential inequality $$\frac{d}{d\lambda}\left( \frac{H(\lambda)}{\lambda}\right) = \frac{H'(\lambda)}{\lambda} - \frac{H(\lambda)}{\lambda^2}\leq \frac{L^2}{2}.$$ This is then integrated with respect to $\lambda$ on $(0,1)$ to establish the inequality . So, by Markov’s inequality, $$\begin{aligned}
\left|\left\{s\in (0,1) : \frac{L^2}{2} - \left. \frac{d}{d\lambda}\left(\frac{H(\lambda)}{\lambda} \right)\right|_{\lambda =s} \geq \frac{\epsilon L^2}{2} \right\}\right| &\leq \frac{2}{\epsilon L^2} \left( \frac{L^2}{2} - \log\left( \int e^{ F} d\mu \right) \right) \leq \frac{1}{2}, \end{aligned}$$ where the last inequality follows by our hypothesis on $F$. Therefore, there exists $\lambda_0 \in [1/2,1]$ for which $$\frac{\int e^{\lambda_0 F}\log(e^{\lambda_0 F} ) d\mu }{\int e^{\lambda_0 F} d\mu } - \log\left( \int e^{\lambda_0 F} d\mu \right)
= \lambda_0 H'(\lambda_0) - H(\lambda_0) \geq (1-\epsilon) \lambda_0^2 \frac{L^2}{2} .$$ Multiplying through by $\int e^{\lambda_0 F} d\mu $, we have $$\begin{aligned}
\operatorname{Ent}_{\mu}(e^{\lambda_0 F} ) \geq (1-\epsilon) \lambda_0^2 \frac{L^2}{2} \int e^{\lambda_0 F} d\mu \geq 2(1-\epsilon) \int \left| \nabla e^{\lambda_0 F/2}\right|^2 d\mu.\label{almostLSI}\end{aligned}$$ Since $\lambda_0\leq 1$, we have that $\log e^{\lambda_0 F/2}$ is $L/2$-Lipschitz. As a consequence, Theorem \[thm:QstableLSI\] applies to yield the estimate $$\begin{aligned}
W_1(\mu, \gamma_{b,\sigma} \otimes \bar{\mu} )\leq C(L) \left( \int \left|\nabla e^{\lambda_0 F/2}\right|^2 d\mu \right)^{-1/2}\sqrt{\epsilon},\label{fromThmW1}\end{aligned}$$ for probability measures $\gamma_{b,\sigma}, \bar{\mu}$ as defined in the statement of the theorem.
By the LSI for $\mu$ together with , we have $$\int \left| \nabla e^{\lambda_0 F/2}\right|^2 d\mu \geq (1-\epsilon) \lambda_0^2 \frac{L^2}{4} \int e^{\lambda_0 F} d\mu \geq (1-\epsilon) \frac{L^2}{16} \int e^{\lambda_0 F} d\mu .$$ Using the fact that $\frac{d}{d\lambda}\left(\lambda \frac{L^2}{2} - \frac{H(\lambda)}{\lambda}\right) \geq 0$, we have $\frac{L^2}{2} - H(1) \geq \lambda_0 \frac{L^2}{2} - \frac{H(\lambda_0)}{\lambda_0}$. Rearranging yields, for $\epsilon < 1$, $$\int e^{\lambda_0 F} d\mu \geq e^{-\lambda_0(1-\lambda_0)L^2/2 }\left( \int e^{ F}d\mu\right)^{\lambda_0} \geq
e^{-\lambda_0(1-\lambda_0)L^2/2 }\left( e^{L^2/2(1-\epsilon/2)} \right)^{\lambda_0}\geq e^{L^2/8}.$$ Therefore, $\int \left| \nabla e^{\lambda_0 F/2}\right|^2 d\mu \geq \frac{L^2}{32} e^{L^2/8}$, so that this term can be absorbed into the constant $C(L)$ in , completing the proof.
We conclude with a stability estimate for another formulation of Gaussian concentration. The Markov inequality argument applied to shows that any 1-Lipschitz $F$ satisfies the Gaussian concentration inequality $$\mu\left(\left\{ F \geq t + \int F d\mu \right\} \right) \leq e^{-t^2/2}, ~~~~ t\geq 0.$$ Unlike , the inequality here is actually strict, and this form of concentration inequality is actually strictly weaker. A simple corollary of Theorem \[thm:ThmExpConcentration\] is the following stability version of this result.
Let $\mu$ and $C$ be as in Theorem \[thm:QstableLSI\], and consider 1-Lipschitz $F$. If $$\mu\left(\left\{ F \geq t + \int F d\mu \right\} \right) \geq \exp\left( -(1+\epsilon/2)\frac{t^2}{2}\right)$$ for some $t>0$ and $\epsilon \geq 0$, then $\mu$ satisfies for $L=t$.
However, the classical concentration bound $\mu\left(\left\{ F \geq t + \int F d\mu \right\} \right) \leq e^{-t^2/2}$ can be sharpened into a bound of the form $\mu\left(\left\{ F \geq t + \int F d\mu \right\} \right) \leq Ce^{-t^2/2}/t$, using for example the Bakry-Ledoux isoperimetric inequality [@BL96] or the Caffarelli contraction theorem and refined concentration bounds for the Gaussian measure. Because the dependence of $C$ on $t$ is not explicit, it may be that the above Corollary is vacuous, in that taking $\epsilon$ small enough relative to $C(t)$ (to activate the $W_1$ estimate ) always makes the above lower bound greater than the improved upper bound. As such, it is not clear if this statement is of any interest, but we include it because the question of stability for this way of encoding Gaussian concentration for uniformly log-concave measures seemed like a natural question the reader may wonder about after reading this work.
We may assume that $\int F d\mu = 0$. By the hypothesis and the Markov inequality, we have $$\exp\left( -(1+\epsilon/2)\frac{t^2}{2}\right)\leq \mu\left(\{ F \geq t \} \right) \leq e^{-t^2} \int e^{t F}d\mu .$$ Multiplying through by $\exp(t^2)$ gives $$\int e^{t F}d\mu \geq \exp\left( \frac{t^2}{2}(1-\epsilon/2)\right).$$ Hence, Theorem \[thm:ThmExpConcentration\] applies to the $t$-Lipschitz function $t F$.
**** This work benefited from support from the France-Berkeley fund, and ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. M.F. was partly supported by Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR) T.C. was partly supported by NSF grants CCF-1750430 and CCF-1704967. We thank Lorenzo Brasco and Michel Ledoux for their helpful remarks on a preliminary version of this work.
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---
author:
- 'Qiang Du[^1] andXiaobo Yin[^2]'
bibliography:
- 'references.bib'
title: A conforming DG method for linear nonlocal models with integrable kernels
---
> [**Abstract.**]{} Numerical solution of nonlocal constrained value problems with integrable kernels are considered. These nonlocal problems arise in nonlocal mechanics and nonlocal diffusion. The structure of the true solution to the problem is analyzed first. The analysis leads naturally to a new kind of discontinuous Galerkin method that efficiently solve the problem numerically. This method is shown to be asymptotically compatible. Moreover, it has optimal convergence rate for one dimensional case under very weak assumptions, and almost optimal convergence rate for two dimensional case under mild assumptions.
>
> [**Keywords.**]{} nonlocal diffusion; peridynamic model; nonlocal model; integrable kernel; discontinuous Galerkin; finite element; convergence analysis; condition number
>
> [**2010 Mathematics Subject Classification**]{} 82C21, 65R20, 74S05, 46N20, 45A05
Introduction
============
Nonlocal models have generated much interests in recent years [@du2018icm]. For example, the peridynamic (PD) model proposed by Silling [@silling2000reformulation] is an integral-type nonlocal theory of continuum mechanics which provides an alternative setup to classical continuum mechanics based on PDEs. Since PD avoids the explicit use of spatial derivatives, it is especially effective for modeling problems involving discontinuities or other singularities in the deformation [@askari2008peridynamics; @du18je; @kilic2010coupling; @oterkus2012peridynamic; @silling2010crack; @silling2010peridynamic]. The nonlocal diffusion (ND) model, described in [@du2012analysis] is another example of integro-differential equations. More recently, mathematical analysis of nonlocal models is also paid more attention, which could be found in [@aksoylu2010results; @aksoylu2011variational; @andreu2010nonlocal; @burch2011classical; @du2019cbms; @du2013nonlocal; @du2011mathematical; @emmrich2007well]. Meanwhile, to simulate nonlocal models, various numerical approximations have been proposed and studied, including finite difference, finite element, meshfree method, quadrature and particle-based methods [@bobaru2009convergence; @chen2011continuous; @du2013posteriori; @kilic2010coupling; @macek2007peridynamics; @seleson2009peridynamics; @silling2005meshfree; @tian2013analysis; @tian2014asymptotically; @wang2012fast; @zhou2010mathematical].
Let $\Omega\subset\mathbb{R}^{d}$ denote a bounded, open and convex domain with Lipschitz continuous boundary. For $u({\bf x}): \Omega\rightarrow \mathbb{R}$, the nonlocal operator $\mathcal{L}$ on $u({\bf x})$ is defined as $$\label{Nolocal_Operator}
\mathcal{L}u({\bf x})=\int_{\mathbb{R}^{d}}{\big (}u({\bf y})-u({\bf x}){\big )}\gamma({\bf x},{\bf y})d{\bf y}\quad \forall {\bf x}\in\Omega,$$ where the nonnegative symmetric mapping $\gamma({\bf x},{\bf y}): \mathbb{R}^{d}\times\mathbb{R}^{d} \rightarrow \mathbb{R}$ is called a kernel. The operator $\mathcal{L}$ is regarded nonlocal since the value of $\mathcal{L}u$ at a point $\bf x$ involves information about $u$ at points ${\bf y}\neq{\bf x}$. In this article, we consider the following nonlocal Dirichlet volume-constrained diffusion problem $$\label{nonlocal_diffusion}
\left \{
\begin{array}{rl}
-\mathcal{L}u({\bf x})&=b({\bf x})\: \mbox{on}\: \Omega, \\
u({\bf x})&=g({\bf x}) \: \mbox{on} \:\Omega_I,
\end{array}
\right .$$ where $\Omega_I=\{{\bf y}\in \mathbb{R}^d\setminus\Omega:\, \mbox{dist}({\bf y},\partial \Omega)<\delta\}$ with $\delta$ a constant called horizon parameter, $b({\bf x})\in L^2(\Omega)$ and $g({\bf x})\in L^2(\Omega_I)$ are given functions. Volume constraints are natural extensions, to the nonlocal case, of boundary conditions for differential equations. Nonlocal versions of Neumann and Robin boundary conditions are also naturally defined [@du2012analysis].
We assume that the nonnegative symmetric kernel $\gamma({\bf x},{\bf y})$ satisfies, there exists a positive constant $\gamma_0$, for all ${\bf x}\in\Omega\cup\Omega_I$, $$\label{kernel_basic}
\gamma({\bf x},{\bf y})\geq \gamma_0\quad\forall {\bf y}\in B_{\delta/2}({\bf x}),\quad
\gamma({\bf x},{\bf y})=0\quad \forall {\bf y}\in(\Omega\cup\Omega_I)\setminus B_{\delta}({\bf x}),$$ where $B_{\delta}({\bf x}) := \{{\bf y}\in \Omega\cup\Omega_I: |{\bf y}-{\bf x}|\leq\delta\}$. Obviously, (\[kernel\_basic\]) implies that although interactions are nonlocal, they are limited to a ball of radius $\delta$. A few important classes of kernels are considered in [@du2012analysis]. Of particular interests here is a special choice of $\gamma$ being a radial function of ${\bf x}-{\bf y}$ (which also makes the kernel translation invariant). Such a case has also been studied earlier in [@andreu2010nonlocal] where $$\label{int_kernel}
\gamma({\bf x},{\bf y})=\tilde{\gamma}(|{\bf y}-{\bf x}|)\geq 0, \quad
\int_{\mathbb{R}^{d}}\tilde{\gamma}(|{\bf z}|) d{\bf z} = 1.$$ This condition on $\gamma$ implies that $\mathcal{L}$ is a bounded mapping from $L^2(\mathbb{R}^{d})$ to $L^2(\mathbb{R}^{d})$. As we will discuss here, even though for smooth enough $b({\bf x})$ in $\Omega$, unlike the classical local PDE boundary value problems, the solution $u({\bf x})$ is possibly discontinuous across $\partial\Omega$ which makes the numerical solution to the corresponding nonlocal problem challenging.
An intuitive idea to overcome the possible loss of continuity is to use discontinuous Galerkin (DG) methods. The latter are in fact conforming, which is in stark contrast to DG methods for the discretization of second order elliptic partial differential equations for which they are nonconforming [@arnold2002unified]. While nonconforming DG has also been studied recently for nonlocal models [@du18dg], if the structure of the solution could be studied carefully, a well designed conforming DG method could be a more competitive option as it could lessen the cost of computation. In this work, we propose a new kind of conforming DG method to approximate the nonlocal problem (\[nonlocal\_diffusion\]) with kernels satisfying (\[kernel\_basic\]) and (\[int\_kernel\]) where the key is to adopt a hybrid version of continuous elements with DG at the boundary.
The paper is organized as follows. In Section \[section:structure\], the structure of the solution for the given problem is analyzed, which is a generalization of the results in [@silling2003deformation]. We also convert the original inhomogeneous problem (\[nonlocal\_diffusion\]) into the homogeneous problem (\[modified\_problem\]) whose solution is smoother, so we just need to discuss a smoother homogeneous problem (\[nonlocal\_diff\_homo\]). Based on the structure of the solution, in Section \[section:DG\_method\] we propose a new DG method which solves the problem (\[nonlocal\_diff\_homo\]) efficiently. Convergence analysis and condition number estimates along with asymptotic compatibility for the method are given in Section \[section:Theoretical considerations\]. Results of numerical experiments are reported in Section \[section:numerical\_experiment\].
The structure of the solution {#section:structure}
=============================
To design more efficient numerical discretization, we first present some regularity studies on the nonlocal constrained value problem. We recall first some one dimensional results presented in [@silling2003deformation]: when using peridynamic theory to model the elasticity on $\mathbb{R}=(-\infty,\infty)$, the displacement field $u$ has the same smoothness as the body force field $b$. In addition, any discontinuity in the kernel $\gamma$ (or in one of its derivatives) has some additional effect on the smoothness of $u$. For a peridynamic bar, suppose that $b$ has a discontinuity in its $N$th derivative at some $x=x_b$, and $\gamma$ has a discontinuity in its $L$th derivative at some $x=x_c$, then $u$ has a discontinuity in its $(N + nL + n)$th derivative at $x=x_b+nx_c, n=1,2,\cdots$. This propagation of discontinuities is illustrated schematically in [@silling2003deformation Figure 3]. Roughly speaking, the smoothness of $u$ increases as one moves away from the location where the solution is discontinuous due to the discontinuity of the body force field $b$. These types of step-wise improved regularity associated with a finite horizon parameter have also been observed for nonlocal initial value problems of nonlocal-in-time dynamic systems in [@du17dcdsb].
The structure of the solution for general dimensional cases on bounded domains
------------------------------------------------------------------------------
Recall for the 1D case in an unbounded domain, the regularity of the solution for a nonlocal problem is affected by both the right hand side function and the kernel function, assuming good behavior of the solution at infinities. In this subsection we consider the effect of these two sources on the regularity of the solution for general multidimensional constrained value problem on a bounded domain.
First, let us present a result to reduce the problem (\[nonlocal\_diffusion\]) which we are concerned with to be a problem with a homogeneous nonlocal constraint. Denoted by $$\label{btrans}
\overline{b}({\bf x})=\left \{
\begin{array}{ll}
b({\bf x}), & {\bf x}\in \Omega,\\
g({\bf x}), & {\bf x}\in \Omega_I,
\end{array}
\right .$$ and $$\label{utrans}
\overline{u}({\bf x})=u({\bf x})-\overline{b}({\bf x}).$$ Then the nonlocal operator $-\mathcal{L}$ on $\overline{u}({\bf x})$ is $$\begin{aligned}
-\mathcal{L}\overline{u}({\bf x})&=f({\bf x})=\int_{B_{\delta}({\bf x})\cap\Omega}\overline{b}({\bf y})\gamma({\bf y}-{\bf x})d{\bf y}\\
&=\int_{B_{\delta}({\bf x})\cap\Omega}b({\bf y})\gamma({\bf y}-{\bf x})d{\bf y}+\int_{B_{\delta}({\bf x})\cap\Omega_I}g({\bf y})\gamma({\bf y}-{\bf x})d{\bf y}, \quad \forall {\bf x} \in \Omega.\end{aligned}$$ Thus, $\overline{u}({\bf x})$ is the solution of the following homogeneous nonlocal problem $$\label{modified_problem}
\left \{
\begin{array}{rl}
-\mathcal{L}\overline{u}({\bf x})&=f({\bf x}),\:\: \mbox{on}\: \Omega, \\
\overline{u}({\bf x})&=0,\qquad \mbox{on} \:\Omega_I.
\end{array}
\right .$$ Due to $\overline{b}({\bf y})\in L^2(\Omega\cup\Omega_I)$, $\gamma({\bf s})\in L^2(\mathbb{R}^d)$, and the fact that convolution of functions in dual $L^p(\mathbb{R}^d)$-spaces is continuous, we know that $f({\bf x})\in C(\Omega)$.
The problem (\[nonlocal\_diffusion\]) is equivalent to the problem (\[modified\_problem\]) which has a homogeneous nonlocal constraint. That is, we just need to study the following homogeneous nonlocal problem: $$\label{nonlocal_diff_homo}
\left \{
\begin{array}{rl}
-\mathcal{L}u({\bf x})&=b({\bf x}),\:\: \mbox{on}\: \Omega, \\
u({\bf x})&=0,\quad\:\:\: \mbox{on} \:\Omega_I,
\end{array}
\right .$$ with $b({\bf x})\in C(\Omega)$, $\gamma({\bf x},{\bf y})$ satisfying (\[kernel\_basic\]) and (\[int\_kernel\]). We will show that if $\gamma({\bf x},{\bf y})$ satisfies some mild assumptions, the results recalled earlier for the one dimensional case can be generalized to multidimensional case on a bounded domain.
Discontinuities due to the right hand side function
---------------------------------------------------
\[Thm:Continuity\_Zeroboundary\] If $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]), and $b({\bf x})\in C(\Omega)$, the solution of (\[nonlocal\_diff\_homo\]) is continuous in $\Omega$, i.e., $u({\bf x})\in C(\Omega)$.
Since $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]), we easily see that $$\begin{aligned}
u({\bf x})\in L^2(\Omega\cup\Omega_I).\end{aligned}$$ From (\[nonlocal\_diff\_homo\]), we have $$\begin{aligned}
\label{u_relation}
u({\bf x})=b({\bf x})+\int_{B_{\delta}({\bf x})}u({\bf y})\gamma({\bf y}-{\bf x})d{\bf y}, \quad \forall {\bf x} \in \Omega.\end{aligned}$$ Since $u({\bf y})\in L^2(\Omega\cup\Omega_I)$ and $\gamma({\bf s})\in L^2(\mathbb{R}^d)$, we have that $$\int_{B_{\delta}({\bf x})}u({\bf y})\gamma({\bf y}-{\bf x})d{\bf y}$$ is continuous for ${\bf x}$ in $\Omega$. Together with the condition $b({\bf x})\in C(\Omega)$, we complete the proof.
It is obvious from (\[u\_relation\]) that the discontinuity at a point $\bf x$ of the right hand side $b$ will lead to the discontinuity at the same point of the solution $u$. With an additional assumption on the kernel, we can bootstrap a higher order regularity result as follows.
\[Thm:H1:smooth\] Suppose that $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]). If $\tilde{\gamma}(r)\in C^1(0,\delta)$, $\tilde{\gamma}(\delta)=0$ and $b({\bf x})\in C^1(\Omega)$, then $u({\bf x})\in C^1(\Omega)$.
From (\[u\_relation\]), we know $$\begin{aligned}
\label{u_relation_s}
u({\bf x})=b({\bf x})+\int_{B_{\delta}({\bf 0})}u({\bf x}+{\bf s})\gamma({\bf s})d{\bf s}.\end{aligned}$$ For any unit vector ${\bf t}$, we take the corresponding directional derivative for (\[u\_relation\_s\]), so $$\begin{aligned}
\frac{\partial u({\bf x})}{\partial{\bf t}}=\frac{\partial b({\bf x})}{\partial{\bf t}}+\int_{B_{\delta}({\bf 0})}\frac{\partial u({\bf x}+{\bf s})}{\partial{\bf t}}\gamma({\bf s})d{\bf s}.\end{aligned}$$ Since $\tilde{\gamma}(r)\in C^1(0,\delta)$ and $\tilde{\gamma}(\delta)=0$, we have $$\begin{aligned}
&\int_{B_{\delta}({\bf 0})}\frac{\partial u({\bf x}+{\bf s})}{\partial{\bf t}}\gamma({\bf s})d{\bf s}\\
&=\int_{\partial B_{\delta}({\bf 0})}u({\bf x}+{\bf s})\gamma({\bf s}){\bf t}\cdot{\bf n}_{\bf s}d{\bf S}_{\bf s}-\int_{B_{\delta}({\bf 0})}u({\bf x}+{\bf s}){\bf t}\cdot\nabla\gamma({\bf s})d{\bf s}\\
&=\tilde{\gamma}(\delta)\int_{\partial B_{\delta}({\bf 0})}u({\bf x}+{\bf s}){\bf t}\cdot{\bf n}_{\bf s}d{\bf S}_{\bf s}-\int_{B_{\delta}({\bf 0})}u({\bf x}+{\bf s})\tilde{\gamma}'(|{\bf s}|){\bf t}\cdot{\bf n}_{\bf s}d{\bf s}\\
&=-\int_{B_{\delta}({\bf 0})}u({\bf x}+{\bf s})\tilde{\gamma}'(|{\bf s}|){\bf t}\cdot{\bf n}_{\bf s}d{\bf s}.\end{aligned}$$ Thus, $$\begin{aligned}
\label{u_de_intbypart}
\frac{\partial u({\bf x})}{\partial{\bf t}}=\frac{\partial b({\bf x})}{\partial{\bf t}}-\int_{B_{\delta}({\bf 0})}u({\bf x}+{\bf s})\tilde{\gamma}'(|{\bf s}|){\bf t}\cdot{\bf n}_{\bf s}d{\bf s}.\end{aligned}$$ Since the convolution of functions in dual $L^p(\mathbb{R}^{d})$-spaces is continuous, the second term in the right hand side of (\[u\_de\_intbypart\]) is continuous with respect to ${\bf x}$. Together with the condition $b({\bf x})\in C^1(\Omega)$, we complete the proof.
From (\[u\_de\_intbypart\]) we get that under the assumptions of Theorem \[Thm:H1:smooth\], if the first derivative of the right hand side $b$ is discontinuous at some point $\bf x$, the first derivative of the solution $u$ will be discontinuous at the same point. However, if we may have both the condition $\tilde{\gamma}(\delta)>0$ and the condition $b({\bf x})\notin C^1(\Omega)$, the conclusion $u({\bf x})\in C^1(\Omega)$ may still hold (see Example 1, where in fact $u({\bf x})\in C^{\infty}(\Omega)$). This is not contradicting as *the sum of two discontinuous function could be continuous, and even infinitely differentiable*.
Propagation of discontinuities due to the kernel
------------------------------------------------
For the convenience of discussion, let us denote by $\Omega_{1}=\{{\bf x}\in \Omega: \mbox{dist}(x,\partial\Omega)>\delta\}$ and $\Omega_{2}=\Omega\setminus\overline{\Omega_{1}}$. In fact, the significance of the result Theorem \[Thm:Continuity\_Zeroboundary\] is twofold. First, it indicates *the smoothness of $u({\bf x})$ is the same as $b({\bf x})$ for general multidimensional case on a bounded domain*. Second, it reveals the possible propagation of discontinuities due to the kernel. Although $b({\bf x})\in C(\Omega)$, it might be discontinuous across $\partial\Omega$. So does $u({\bf x})$, and this discontinuity will be propagated to those points on $\partial\Omega_{1}$, which are $\delta$ distance from $\partial\Omega$ onto one order higher derivatives by Theorem \[Thm:Continuity\_Zeroboundary\], which is consistent with the conclusion for 1D case on $\mathbb{R}$ with $N=0$ and $L=0$. Similar argument can be given for Theorem \[Thm:H1:smooth\] which is consistent with the conclusion for 1D case on $\mathbb{R}$ with $N=0$ and $L=1$. This bootstrap procedure could be repeated again and again, and the corresponding results for general $N$ and $L$ then follow.
Let us now emphasize on the importance and necessity for the smoothness of the kernel function. For instance, in Theorem \[Thm:H1:smooth\] $\tilde{\gamma}(\delta)=0$ is required such that $\tilde{\gamma}$ has a discontinuity in its first (but not zeroth) derivative at $x=\delta$. If $\tilde{\gamma}(\delta)>0$, then from the proof of the Theorem \[Thm:H1:smooth\], we see that $u({\bf x})\in C^1(\Omega)$ may not hold. In fact, for all ${\bf x}_0\in\partial\Omega_{1}$ and any unit vector ${\bf t}$, since $$\begin{aligned}
\frac{\partial u({\bf x}_0)}{\partial{\bf t}}&=\frac{\partial b({\bf x}_0)}{\partial{\bf t}}+\tilde{\gamma}(\delta)\int_{\partial B_{\delta}({\bf 0})}u({\bf x}_0+{\bf s}){\bf t}\cdot{\bf n}_{\bf s}d{\bf S}_{\bf s}\\
&-\int_{B_{\delta}({\bf 0})}u({\bf x}_0+{\bf s})\tilde{\gamma}'(|{\bf s}|){\bf t}\cdot{\bf n}_{\bf s}d{\bf s},\end{aligned}$$ if $\tilde{\gamma}(\delta)>0$ (that is $\tilde{\gamma}$ has a discontinuity in its zeroth derivative) and $b$ (thus $u$) is discontinuous across $\partial\Omega$, then the term $$\int_{\partial B_{\delta}({\bf 0})}u({\bf x}+{\bf s}){\bf t}\cdot{\bf n}_{\bf s}d{\bf S}_{\bf s}$$ is likely to be discontinuous across ${\bf x}_0$. If so, $\frac{\partial u}{\partial{\bf t}}$ would be discontinuous at ${\bf x}_0$ ($u$ has a discontinuity in its first derivative). This situation might happen, as illustrated in Example 2. Using a similar argument we could prove the following theorem.
\[Theorem:C2\] Suppose that $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]). If the following two conditions hold:
\(i) $b({\bf x})\in C^1(\Omega)$, $b\in C^{2}(\Omega_{1})$, $b\in C^2(\Omega_{2})$,
\(ii) $\tilde{\gamma}(r)\in C^1(0,\delta)$ and $\tilde{\gamma}(\delta)=0$.\
Then $u\in C^1(\Omega)$, $u\in C^2(\Omega_{1})$ and $u\in C^2(\Omega_{2})$.
The conditions of Theorem \[Thm:H1:smooth\] hold due to the conditions (i) and (ii), so $u\in C^{1}(\Omega)$. Furthermore, for any two unit vectors ${\bf t}_1$ and ${\bf t}_2$, take a directional derivative for (\[u\_de\_intbypart\]), $$\begin{aligned}
\frac{\partial^2 u({\bf x})}{\partial{\bf t}_1\partial{\bf t}_2}&=\frac{\partial^2 b({\bf x})}{\partial{\bf t}_1\partial{\bf t}_2}
-\int_{B_{\delta}({\bf 0})}\frac{\partial u({\bf x}+{\bf s})}{\partial {\bf t}_2}\tilde{\gamma}'(|{\bf s}|){\bf t}_1\cdot{\bf n}_{\bf s}d{\bf s}.\end{aligned}$$ Applying this equality in $\Omega_{1}$ and $\Omega_{2}$ will lead to the conclusion $u\in C^{2}(\Omega_{1})$ and $u\in C^{2}(\Omega_{2})$, respectively.
To get the optimal convergence order, we always need the regularity $u\in H^2(\Omega)$, the following corollary give a sufficient condition to guarantee this property.
\[Corollary:H2\] Suppose that $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]). If the following two conditions hold:
\(i) $b({\bf x})\in C^1(\Omega)$, $b\in H^{2}(\Omega_{1})$, $b\in H^2(\Omega_{2})$,
\(ii) $\tilde{\gamma}(r)\in C^1(0,\delta)$ and $\tilde{\gamma}(\delta)=0$.\
Then $u\in H^2(\Omega)$.
Using the density of $C^2(\Omega_{i})$ in $H^2(\Omega_{i})$ ($i=1,2$) and Theorem \[Theorem:C2\] we could get the result $u\in H^{2}(\Omega_{i})$. Since $u\in C^{1}(\Omega)$ is proven, the result $u\in H^{2}(\Omega)$ holds.
A new DG method for nonlocal models with integrable kernels {#section:DG_method}
===========================================================
Here and after, for a function $\phi({\bf x})$ we denote $\lim\limits_{h\rightarrow 0-}\phi({\bf x}+h{\bf n_x})$ by $\phi({\bf x}-)$ in the case of no ambiguity. Under the condition of Theorem \[Thm:Continuity\_Zeroboundary\], we know that for given ${\bf x}\in \partial \Omega$, $$\lim\limits_{h\rightarrow 0+}u({\bf x}+h{\bf t})=u({\bf x}-), \forall {\bf t}\cdot{\bf n}_{\bf x}<0.$$ However $u({\bf x}-)$ does not need to be zero, that is $u({\bf x})$ is possibly discontinuous across $\partial\Omega$. Thus, it might be inefficient to use continuous FEM on the whole domain $\Omega\cup\Omega_I$. Moreover, since we do not specify the value of the right hand side function on $\Omega_I$ a priori, we have no control on the amount of the jump across $\partial\Omega$ where the solution is likely to be discontinuous. Thus, we propose a suitable DG method by adopting a hybrid version of DG (across the domain boundary) and continuous elements (in the interior domain).
As in [@du2012analysis] the nonlocal energy inner product, the nonlocal energy norm, nonlocal energy space, and nonlocal volume constrained energy space are defined by $$(u,v)_{\||} :={\Big (}\int_{\Omega\cup\Omega_I}\int_{\Omega\cup\Omega_I}{\big(}u({\bf y})-u({\bf x}){\big)}{\big(}v({\bf y})-v({\bf x}){\big)}\gamma({\bf x},{\bf y})d{\bf y}d{\bf x}{\Big )},$$ $$\|| u\|| :=(u,u)^{1/2}_{\||},$$ $$V(\Omega\cup\Omega_I):=\{u\in L^2(\Omega\cup\Omega_I): \|| u\||<\infty\},$$ $$V_{c,0}(\Omega\cup\Omega_I):=\{u\in V(\Omega\cup\Omega_I): u({\bf x})=0\:\mbox{on} \:\Omega_I\},$$ respectively. Similar to the definition $V_{c,0}(\Omega\cup\Omega_I)$, the subspace of $L^{2}(\Omega\cup\Omega_I)$ is defined as follows: $$L^{2}_{c,0}(\Omega\cup\Omega_I):=\{u\in L^{2}(\Omega\cup\Omega_I): u({\bf x})=0\:\mbox{on} \:\Omega_I\}.$$ The authors in [@du2012analysis] prove that if the kernel function $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]), then $V_{c,0}(\Omega\cup\Omega_I)$ is equivalent to $L^{2}_{c,0}(\Omega\cup\Omega_I)$. Denote by $$V_{c,0}^{pc}(\Omega\cup\Omega_I)=\{u\in V_{c,0}(\Omega\cup\Omega_I): u|_{\Omega} \in C(\Omega)\},$$ where the superscripts $pc$ means *partly continuous* (continuous in $\Omega$).
For a given triangulation $\mathcal{T}_h$ of $\Omega\cup\Omega_I$ that simultaneously triangulates $\Omega$, let $\Omega_h=\mathcal{T}_h\cap\overline{\Omega}$. Next, let $V^{pc,h}_{c,0}$ consist of those functions in $V_{c,0}^{pc}(\Omega\cup\Omega_I)$ that are piecewise linear. Since $\Omega$ is convex, this *conforming* property is satisfied, that is, $$\label{Set_conf}
V_{c,0}^{pc,h}\subset V_{c,0}^{pc}(\Omega\cup\Omega_I).$$
We assume that $\mathcal{T}_h$ is shape-regular and quasi-uniform [@brenner2007mathematical] as $h\rightarrow 0$, where $h$ denotes the diameter of the largest element in $\mathcal{T}_h$. For a fixed $\mathcal{T}_h$, the set of inner nodes of $\Omega_h$, i.e., all nodes in $\Omega_h\setminus\partial\Omega$, is denoted by $N\!I=\{{\bf x}_j: j=1,2,\cdots, m\}$, with piecewise linear basis functions defined on $\mathcal{T}_h$ being $\phi_j({\bf x}),\:j=1,2,\cdots, m$. The set of all nodes in $\Omega_h\cap\partial\Omega$ is denoted by $N\!B=\{{\bf x}_{m+j}: j=1,2,\cdots, n\}$ with piecewise linear basis functions defined on $\mathcal{T}_h$ being $\phi_{m+j}({\bf x}),\:j=1,2,\cdots, n$. The basis functions for the space $V_{c,0}^{pc,h}$ are as follows: for $j=1,2,\cdots,m+n$, $$\widehat{\phi}_{j}({\bf x})=\left \{
\begin{array}{ll}
\phi_{j}({\bf x})|_{\Omega_h}, & {\bf x}\in \Omega_h,\\
0, & {\bf x}\in (\Omega\cup\Omega_I)\setminus\Omega_{h}.
\end{array}
\right .$$ Throughout the paper, the generic constant $C$ is always independent of the finite element mesh parameter $h$.
Since we know the structure of the true solution and the space it belongs to, we could design a DG method for its approximation. First, variational form in $V_{c,0}^{pc}(\Omega\cup\Omega_I)$ finds $u({\bf x})\in V_{c,0}^{pc}(\Omega\cup\Omega_I)$, such that $$\label{modi_prob_var}
-\int_{\Omega}\mathcal{L}u({\bf x}) w({\bf x})d{\bf x}=\int_{\Omega}b({\bf x})w({\bf x})d{\bf x},\: \forall w({\bf x}) \in V_{c,0}^{pc}(\Omega\cup\Omega_I).$$ The finite dimensional approximation for (\[modi\_prob\_var\]) finds $u_h({\bf x})\in V^{pc,h}_{c,0}$, such that $$\label{modi_disc_var}
-\int_{\Omega_{h}}\mathcal{L}u_h({\bf x}) w_h({\bf x})d{\bf x}=\int_{\Omega_{h}}b({\bf x})w_h({\bf x})d{\bf x}, \: \forall w_h({\bf x}) \in V_{c,0}^{pc,h}.$$ Set $u_h({\bf x})=\sum\limits_{j=1}^{m+n}u_j\widehat{\phi}_{j}({\bf x})$, ${\bf u}=(u_{1}, u_{2},\cdots,u_{m+n})^T$. Denote by $$\begin{aligned}
{\bf d}=(d_1, d_2, \cdots, d_{m+n})^T,\end{aligned}$$ and $$\begin{aligned}
A_{II}=(a_{i,j})_{m\times m},\: A_{IB}=(a_{i,{m+j}})_{m\times n},\: A_{BB}=(a_{m+i,{m+j}})_{n\times n},\end{aligned}$$ with $$\begin{aligned}
d_i=\int_{\Omega_{h}}b({\bf x}) \widehat{\phi}_i({\bf x})d{\bf x},\:
a_{i,j}=-\int_{\Omega_{h}}\mathcal{L}\widehat{\phi}_j({\bf x}) \widehat{\phi}_i({\bf x})d{\bf x}.\end{aligned}$$ Set $w_h=\widehat{\phi}_i$, $i=1,2,\cdots,m+n$, the algebraic system of (\[modi\_disc\_var\]) is $$\begin{aligned}
A{\bf u}={\bf d},\end{aligned}$$ with $$\label{Stiff_matrix}
A=\left(\begin{array}{cc}
A_{II} & A_{IB} \\
A_{IB}^T& A_{BB}
\end{array}\right).$$
In the process to solve for $u_h({\bf x})$ we use the finite element space $V^{pc,h}_{c,0}$ which is continuous in $\Omega_{h}$, however, discontinuous across $\partial\Omega_{h}$, thus we regard it a conforming but hybrid version of DG and continuous FEM. This method possesses some advantages as follows:
\(i) The method leads to a linear algebraic system with the coefficient matrix $A$ in (\[Stiff\_matrix\]) that is symmetric and positive definite, just as in the case using either the conforming DG or continuous FEM, thus many efficient solvers suitable to such linear systems could still be used.
\(ii) The method is asymptotically compatible: as shown in [@tian2014asymptotically], as long as the finite element space contains continuous piecewise linear functions (which is the case for our hybrid algorithm), the Galerkin finite element approximation is always asymptotically compatible, and thus offers robust numerical discretizations to problems involving nonlocal interactions.
\(iii) The method has optimal convergence rate provided that the solution is smooth on $\Omega$, that is $O(h^2)$ ($O(h)$) for error in $L^2$ ($H^1$) norm provided that the true solution $u\in H^2(\Omega)$. This result is in sharp contrast to the assumption given that in [@du2012analysis] where to insure the optimal convergence rate the true solution is required to be in $H^2(\Omega\cup\Omega_I)$ which generally not holds for the problem (\[nonlocal\_diff\_homo\]). Furthermore, it has optimal convergence rate for 1D case under very weak assumptions, and nearly optimal convergence rate for two dimensional (2D) case under some mild assumptions, as shown in the next section.
\(iv) The method, in comparison with the direct use of DG in all discrete elements, uses a smaller degree of freedoms. For example, the degree of freedoms is $n+1$ versus $2n$ for a mesh with $n+1$ nodes in 1D case, and $(n+1)^2$ versus $6n^2$ for a uniform triangulation with $n^2$ nodes in 2D case.
Theoretical Analysis {#section:Theoretical considerations}
====================
We now provide further theoretical analysis on the new DG approximations. Given what has already been discussed in (ii) of the above section, the asymptotic compatibility is assured and we thus focus on the case where the problems remain strictly nonlocal.
Convergence
-----------
The following convergence result describes the best approximation property of the finite-dimensional Ritz-Galerkin solution.
\[Thm:Convergence\] If $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]), $b({\bf x})\in C(\Omega)$, $u({\bf x})$ is the solution of (\[nonlocal\_diff\_homo\]), $u_{h}({\bf x})$ is the solution of (\[modi\_disc\_var\]). We define $$\tilde{u}({\bf x},\Omega_h)=\left \{
\begin{array}{ll}
u({\bf x}), & {\bf x}\in \Omega_h,\\
0, & {\bf x}\in (\Omega\cup\Omega_I)\setminus\Omega_{h}.
\end{array}
\right .$$ Then we have $$\label{cea0}
\|| \tilde{u}-u_{h}\||\leq \inf_{w_h \in V_{c,0}^{pc,h}}
\|| \tilde{u}-w_h\||\, .$$ Consequently, $$\label{cea}
\|u-u_{h}\|_{\Omega_{h}}\leq C \min_{w_{h}\in V^{pc,h}_{c,0}}\|u-w_{h}\|_{\Omega_{h}}\rightarrow 0 \quad as \: h\rightarrow 0.$$
Since $V^{pc,h}_{c,0}\subset V^{pc}_{c,0}(\Omega\cup\Omega_I)$ as in (\[Set\_conf\]), then for all $w_h \in V_{c,0}^{pc,h}$, $$-\int_{\Omega_{h}}\mathcal{L}\tilde{u}({\bf x},\Omega_h) w_h({\bf x})d{\bf x}=\int_{\Omega_{h}}b({\bf x})w_h({\bf x})d{\bf x},$$ together with (\[modi\_disc\_var\]), we have $$-\int_{\Omega_{h}}\mathcal{L}{\big(}\tilde{u}({\bf x},\Omega_h)-u_{h}({\bf x}){\big)} w_h({\bf x})d{\bf x}=0, \: \forall w_h \in V_{c,0}^{pc,h}.$$ Using the nonlocal Green’s first identity [@du2013nonlocal], we have $$(\tilde{u}-u_{h},w_h)_{\||}=0,\:\forall w_h \in V_{c,0}^{pc,h}.$$ Then we get the following estimate $$\begin{aligned}
\|| \tilde{u}-u_{h}\||^2 &=(\tilde{u}-u_{h},\tilde{u}-u_{h})_{\||}=(\tilde{u}-u_{h},\tilde{u}-w_h)_{\||}\\
&\leq \|| \tilde{u}-u_{h}\|| \|| \tilde{u}-w_h\||, \: \forall w_h({\bf x}) \in V_{c,0}^{pc,h},\end{aligned}$$ and then $$\begin{aligned}
\|| \tilde{u}-u_{h}\||\leq\|| \tilde{u}-w_h\||, \: \forall w_h({\bf x}) \in V_{c,0}^{pc,h}.\end{aligned}$$ By the equivalence between $V_{c,0}(\Omega\cup\Omega_I)$ and $L^{2}_{c,0}(\Omega\cup\Omega_I)$, we complete the proof.
Let us note that due to the use of norm equivalence in the above proof, generally speaking, the constant $C$ in the lemma could depend on the nonlocal space and thus the nonlocal kernel. One may not infer that this constant remains uniformly bounded when we consider the local limit of the nonlocal problem. Fortunately, as alluded to earlier, with the asymptotic compatibility already established in [@tian2014asymptotically], we hereby only focus on the strict nonlocal case.
We now combine the theory of the interpolation error estimate and (\[cea\]) to give the convergence rate estimate.
\[Thm:Convergence\_rate\] If $\gamma({\bf x},{\bf y})$ satisfies (\[kernel\_basic\]) and (\[int\_kernel\]), $b({\bf x})\in C(\Omega)$, $u({\bf x})$ is the solution of (\[nonlocal\_diff\_homo\]), $u_{h}({\bf x})$ is the solution of (\[modi\_disc\_var\]). Suppose that $u\in H^t(\Omega)$ holds, there exists a constant $C$ such that, for sufficiently small $h$, $$\label{Error_u}
\|u-u_{h}\|_{\Omega_{h}}\leq C h^s\|u\|_{s,\Omega},$$ with $s=\min(t,2)>d/2$. If $s>1$ $$\label{Error_du}
\|\nabla(u-u_{h})\|_{\Omega_{h}}\leq C h^{s-1}\|u\|_{s,\Omega}.$$ Moreover, if the following two conditions hold:
\(i) $b({\bf x})\in C^1(\Omega)$, $b\in H^{2}(\Omega_{1})$, $b\in H^2(\Omega_{2})$;
\(ii) $\gamma({\bf x},{\bf y})$ is a radial function such that $\tilde{\gamma}(r)\in C^1(0,\delta)$ and $\tilde{\gamma}(\delta)=0$.\
Then $u\in H^2(\Omega)$, thus $s=2$.
Denote by $\mathcal{I}_hu$ the Lagrange interpolant from $C(\Omega_h)$ to $V_{c,0}^{pc,h}|_{\Omega_h}$$$w_{h}({\bf x})=\left \{
\begin{array}{ll}
\mathcal{I}_hu({\bf x}), & {\bf x}\in \Omega_h,\\
0, & {\bf x}\in (\Omega\cup\Omega_I)\setminus\Omega_{h},
\end{array}
\right .$$ then $w_{h}\in V^{pc,h}_{c,0}$, and $$\label{Interp_err}
\|u-w_{h}\|_{\Omega_{h}}=\|u-\mathcal{I}_hu\|_{\Omega_{h}}\leq C h^{s}\|u\|_{s,\Omega},$$ with $s=\min(t,2)$. Combination of (\[cea\]) and (\[Interp\_err\]) leads to (\[Error\_u\]).
Using the inverse estimate for finite element space, we have $$\begin{aligned}
\|\nabla(u-u_{h})\|_{\Omega_{h}}
&\leq\|\nabla(\mathcal{I}_hu-u_{h})\|_{\Omega_{h}}+\|\nabla(u-\mathcal{I}_hu)\|_{\Omega_{h}}\\
&\leq Ch^{-1}\|\mathcal{I}_hu-u_{h}\|_{\Omega_{h}}+ C h^{s-1}\|u\|_{s,\Omega}\leq C h^{s-1}\|u\|_{s,\Omega}\end{aligned}$$ This is the desired result (\[Error\_du\]).
The conditions (i) and (ii) lead to $u\in H^{2}(\Omega)$ due to Corollary \[Corollary:H2\].
We recall by Theorem 6.2 in [@du2012analysis] that, when continuous FEM is used to approximate the nonlocal problem (\[nonlocal\_diff\_homo\]), the approximation $u_{n}$ has an error estimate of the form $\|u-u_{n}\|_{\Omega\cup\Omega_I}\leq C h^s\|u\|_{s,\Omega\cup\Omega_I}$. Since the solution of nonlocal problem (\[nonlocal\_diff\_homo\]) could be discontinuous across $\partial\Omega$, we see that $u\in H^{s}(\Omega\cup\Omega_I)$ does not hold for $s\geq 1/2$, let alone for $s=2$. For 1D case, the best to expect is $s=1/2-\epsilon$ for arbitrary small positive $\epsilon$ if continuous FEM is used. Theorem \[Thm:Convergence\_rate\] improves the convergence rate from $1/2-\epsilon$ to $3/2-\epsilon$ for this case since we have the regularity of $H^{3/2-\epsilon}(\Omega)$. This convergence rate is still not optimal. If the points on $\partial\Omega_{1}$ are selected as the mesh grids, the optimal convergence rate could be obtained.
For the 1D case, assume $u\in C(0,1)$, $u\in H^2(0,\delta)$, $u\in H^2(\delta,1-\delta)$, $u\in H^2(1-\delta,1)$. If $\delta$ and $1-\delta$ are all selected as the mesh grids, then there exists a constant $C$ such that, for sufficiently small $h$, $$\label{Error_1D}
\|u-u_{h}\|_{(0,1)}+h\|u'-u'_{h}\|_{(0,1)}\leq C h^2{\big(}\|u\|_{2,(0,\delta)}+\|u\|_{2,(\delta,1-\delta)}+\|u\|_{2,(1-\delta,1)}{\big)}.$$
Using the interpolation error estimate in three intervals $(0,\delta)$, $(\delta,1-\delta)$ and $(1-\delta,1)$, respectively and add them together, we get $$\|u-\mathcal{I}_hu\|_{(0,1)}\leq C h^2{\big(}\|u\|_{2,(0,\delta)}+\|u\|_{2,(\delta,1-\delta)}+\|u\|_{2,(1-\delta,1)}{\big)}.$$ Together with (\[cea\]) and the inverse estimate we get (\[Error\_1D\]).
For 2D case, under mild assumptions on the smoothness of the boundary, the regularity of the solution, and the conformity between the mesh and the boundary, we have the almost optimal convergence rate, that is optimal up to a factor $|\log h|^{1/2}$.
For 2D case, assume $u\in C(\Omega)$, $u\in H^{2}(\Omega_{1})$, $u\in H^2(\Omega_{2})$. If $\partial\Omega$ is of class $C^2$, and $N\!B\subset\partial \Omega_{1}$, then there exists a constant $C$ such that, for sufficiently small $h$, $$\|u-u_{h}\|_{\Omega_{h}}+h\|\nabla(u-u_{h})\|_{\Omega_{h}}\leq C h^2|\log h|^{1/2}{\big(}\|u\|_{2,\Omega_{1}}+\|u\|_{2,\Omega_{2}}{\big)}.$$
We cite from [@chen1998finite] the results for the linear interpolation error estimate for the interface problem, that is $$\|u-\mathcal{I}_hu\|_{\Omega_{h}}\leq C h^2|\log h|^{1/2}{\big(}\|u\|_{2,\Omega_{1}}+\|u\|_{2,\Omega_{2}}{\big)}.$$ Together with (\[cea\]) and the inverse estimate we complete the proof.
\[Remark:kernel\] Due to the structure of the solution for the problem (\[nonlocal\_diff\_homo\]) we have discussed, the solution $u$ is often discontinuous across $\partial\Omega$. It may cause the discontinuity of the first derivative across $\partial\Omega_{1}$ if $\tilde{\gamma}(r)$ has a discontinuity at $r=\delta$ (that is $\tilde{\gamma}(\delta)>0$), or the discontinuity of the second derivative across $\partial\Omega_{1}$ if $\tilde{\gamma}(r)$ is continuous at $r=\delta$ (that is $\tilde{\gamma}(\delta)=0$) but $\tilde{\gamma}'(r)$ has a discontinuity at $r=\delta$ (that is $\tilde{\gamma}'(\delta-)\neq 0$). In the next section, we will discuss two kinds of kernels representing the above two cases, respectively.
Condition number estimate
-------------------------
The condition number of the stiffness matrix is an indicator of the sensitivity of the discrete solution with respect to the data and the performance of iterative solvers such as the conjugate-gradient method. For the DG method we propose in Section \[section:DG\_method\], consider the $(m+n)\times(m+n)$ stiffness matrix $A$ defined in (\[Stiff\_matrix\]). We have the following condition number estimate whose proof is standard and is given for completeness. Similar discussions can be found in earlier studies[@AU14; @du10sinum].
For the stiffness matrix $A$ defined in (\[Stiff\_matrix\]) associated with the kernel $\gamma({\bf x},{\bf y})$ that satisfies (\[kernel\_basic\]) and (\[int\_kernel\]), there exists a constant $C$ such that $${\rm cond}_{2}(A)\leq C.$$
For the given finite element nodal basis, there exist two generic constants $c_{2}\geq c_{1}>0$ such that $$c_{1}h^{d}|{\bf w}|^{2}\leq \|w_{h}\|^{2}\leq c_{2}h^{d}|{\bf w}|^{2}\,,\quad \forall w_{h}=\sum_{j=1}^{m+n}w_{j}\widehat{\phi}_j\in V_{c,0}^{pc,h},$$ where $\{w_{j}\}$, $j=1,2,\cdots, m+n$, are components of the vector ${\bf w}$. Since the space $V_{c,0}(\Omega\cup\Omega_I)$ is equivalent to the space $L^{2}_{c,0}(\Omega\cup\Omega_I)$, we get the theorem immediately.
We note again that the constant $C$ may depend on the kernel, as demonstrated in [@du10sinum], hence the result is only meaningful for nonlocal problems with a fixed kernel that satisfies the assumptions (\[kernel\_basic\]) and (\[int\_kernel\]).
Numerical results {#section:numerical_experiment}
=================
We now report results of numerical experiments which substantiate the theoretical analysis in Section \[section:Theoretical considerations\]. For 1D case, the problem (\[nonlocal\_diff\_homo\]) becomes the following form $$\label{1D_prob}
\left \{
\begin{array}{rl}
-\int_{-\delta}^{\delta}{\big(}u(x+s)-u(x){\big)}\gamma(s)ds&=b(x)\quad \mbox{on}\: (0,1), \\
u(x)&=0 \: \qquad \mbox{on} \:(-\delta,0]\cup[1,1+\delta).
\end{array}
\right .$$ We solve the nonlocal problem first on uniform meshes and take $\delta$ to be a constant multiple of $h$ and reduce $h$ to check convergence and condition number properties of the proposed DG method, and then solve the problem on non-uniform meshes which are obtained by random disturbance to uniform meshes. Here we choose two popular examples of kernel functions representing two cases as discussed in Remark \[Remark:kernel\].
Constant kernel function
------------------------
We first consider the following kernel function $$\label{Con_ker}
\gamma(s)=\left \{
\begin{array}{ll}
(2\delta)^{-1}, & |s|\leq \delta,\\
0, & |s|>\delta.
\end{array}
\right .$$ Obviously $\gamma$ defined as (\[Con\_ker\]) is discontinuous at points $\pm\delta$, if $b$ is discontinuous at $x=0$ or $x=1$, the solution of (\[1D\_prob\]) will probably be (however, not necessarily) discontinuous in its first derivative at $x=\delta$ or $x=1-\delta$. In fact, in Example 1 the smoothness pick-up is beyond first order, that is, although $b$ is discontinuous at $x=1$, $u$ is infinitely continuously differentiable at $x=1-\delta$. While in Example 2 the smoothness pick-up is only first order and could not be improved, that is, since $b$ is discontinuous at $x=0$ and $x=1$, the first derivative of $u$ is is discontinuous at $x=\delta$ and $x=1-\delta$.
[**Example 1.**]{} In order to get simpler benchmark solutions, we calculate the right-hand side of (\[1D\_prob\]) based on an exact solution $u(x)=x^2,\: x\in \Omega=(0,1)$ and $u(x)=0,\: x\in \Omega_I=(-\delta,0)\cup(1,1+\delta)$, with kernel function (\[Con\_ker\]). This naturally leads to a $\delta$-dependent right-hand side $b(x)=b_{\delta}(x)$, see Figure \[fig:1\] for the plots of $u(x)$ and $b(x)$. The DG method we proposed in Section \[section:DG\_method\] is used to discretize it with $\delta=0.4$.
We first use the proposed DG method on uniform meshes and conclude from Table \[table:ex1:uni\] that convergence rates for errors in $L^2$ and $H^1$ norms are all optimal. The spectral condition number of the stiffness matrix is almost constant when the mesh size decreases, indicating the insensitivity of the discrete solution regardless how small $h$ is.
[lllllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64 & 128\
$\|u-u_{h}\|$ & 7.45e-4 & 1.86e-4 & 4.66e-5 & 1.16e-5 & 2.91e-6 & 7.28e-7\
Rate & – & 2.0000 & 2.0000 & 2.0000 & 2.0000 & 2.0000\
$\|u'-u'_{h}\|$ & 5.77e-2 & 2.89e-2 & 1.44e-2 & 7.22e-3 & 3.61e-3 & 1.80e-3\
Rate & – & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000\
Cond & 5.8875 & 6.7743 & 6.9926 & 7.0294 & 7.0282 & 7.0225\
We then use a kind of non-uniform meshes obtained by random disturbance to uniform meshes. To be specific, for a fixed $m$, let $h=\delta/m$, the non-uniform mesh is obtained by adding a random vector ${\bf \varepsilon}\in\mathbb{R}^{m-1}$ (which obeys the uniform distribution on $[-0.1h,0.1h]$) to $x_i$ to reach $x_i+\varepsilon_i,\:i=1,2,\cdots,m-1$. Together with $x_0$ and $x_m$ we get the new mesh grids $$\label{nonuni_grid}
x_i^n=x_i+\varepsilon_i,\:i=1,2,\cdots,m-1,\: x_0^n=x_0,\: x_m^n=x_m.$$ We have done over twenty tests, and the convergence rates and the spectral condition numbers are all similar. Thus, instead of listing all of them, we just select one test to verify our theoretical analysis. Similar actions and presentations are made also in later examples. It is seen from Table \[table:ex1:non-uni\] that the errors in $L^2$ and $H^1$ norms and convergence rates are comparable with that in uniform meshes case. This is consistent with the theoretical result in Theorem \[Thm:Convergence\_rate\] since $u\in C^{\infty}(\Omega)$, thus $s=2$. The spectral condition numbers of the stiffness matrices behave similar as in the uniform meshes case, too.
[llllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64\
$\|u-u_{h}\|$ & 8.04e-4 & 2.50e-4 & 6.00e-5 & 1.24e-5 & 3.14e-6\
Rate & – & 1.6827 & 2.0629 & 2.2710 & 1.9846\
$\|u'-u'_{h}\|$ & 5.84e-2 & 2.95e-2 & 1.48e-2 & 7.30e-3 & 3.65e-3\
Rate & – & 0.9848 & 0.9965 & 1.0179 & 0.9983\
Cond & 5.9830 & 6.7507 & 7.0829 & 7.0786 & 7.0646\
[**Example 2.**]{} We consider (\[1D\_prob\]) with kernel function (\[Con\_ker\]) and $b(x)=e^{x}$. The DG method we proposed in Section \[section:DG\_method\] is used to discretize it with $\delta=0.4$.
Since the exact solution $u(x)$ is not known for this problem we compute errors using the solution on finer meshes as approximation of the true solution. We first use the proposed DG method on uniform meshes. The right hand side function $b(x)$ and the approximation $u_{h}$ with $h=0.00625$ are plotted in Figure \[fig:2\]. Although $b(x)$ is in $C^{\infty}(0,1)$, it has two discontinuous points $x=0$ and $x=1$, which causes the discontinuity in the first derivative of $u$ at $x=\delta=0.4$ and $x=1-\delta=0.6$. From Table \[table:ex2:uni\] it is seen that the method has optimal convergence rates for errors in $L^2$ and $H^1$ norms. The spectral condition numbers of the stiffness matrices for the method behave similarly as Example 1. Since the results for the spectral condition numbers of the stiffness matrices are all similar in the rest of the numerical tests, we no longer list them to avoid repetitions.
[llllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64\
$\|u-u_{h}\|$ & 7.54e-3 & 1.79e-3 & 4.38e-4 & 1.08e-4 & 2.69e-5\
Rate & – & 2.0717 & 2.0349 & 2.0170 & 2.0084\
$\|u'-u'_{h}\|$ & 4.90e-1 & 2.41e-1 & 1.19e-1 & 5.94e-2 & 2.97e-2\
Rate & – & 1.0259 & 1.0114 & 1.0053 & 1.0026\
Cond & 5.8875 & 6.7743 & 6.9926 & 7.0294 & 7.0282\
Next, we use the non-uniform meshes (\[nonuni\_grid\]) to solve the problem. In this example the true solution $u(x)\in H^{1.5-\epsilon}(\Omega)$ for arbitrary small positive $\epsilon$. So for general non-uniform meshes, we expect the convergence rates for errors in $L^2$ ($H^1$) norm to be at most $1.5$ ($0.5$). This fact is verified in Table \[table:ex2:non-uni\]. However, since we know the two discontinuous points of $u'(x)$, if they are selected as mesh grids, optimal convergence rates could be recovered. To be specific, random disturbances are added to the original mesh grids except $\delta$ and $1-\delta$. The corresponding results are shown in Table \[table:ex2:non-uni-grid\]. That is the convergence rates for errors in $L^2$ and $H^1$ norms are $2$ and $1$, respectively, which is consistent with the theoretical result (\[Error\_1D\]). We then re-examine the computation on the meshes used in Table \[table:ex2:non-uni\]. The convergence rates, however, are not necessarily similar, which is different with Example 1. To be specific, if the perturbation for the points $\delta$ and $1-\delta$ is large in absolute value, such as $0.1h$, the convergence rates remain similar to Table \[table:ex2:non-uni\]. On the other hand, if the perturbation for the two points is small or close to zero, the convergence rates of the test are similar to Table \[table:ex2:non-uni-grid\].
[llllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64\
$\|u-u_{h}\|$ & 9.34e-3 & 3.41e-3 & 1.24e-3 & 4.26e-4 & 1.59e-4\
Rate & – & 1.4519 & 1.4611 & 1.5411 & 1.4222\
$\|u'-u'_{h}\|$ & 3.95e-1 & 2.66e-1 & 2.13e-1 & 1.42e-1 & 1.21e-1\
Rate & – & 0.5712 & 0.3204 & 0.5815 & 0.3432\
[llllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64\
$\|u-u_{h}\|$ & 6.98e-3 & 1.72e-3 & 4.10e-4 & 1.01e-4 & 2.49e-5\
Rate & – & 2.0229 & 2.0661 & 2.0243 & 2.0180\
$\|u'-u'_{h}\|$ & 4.17e-1 & 2.14e-1 & 1.01e-1 & 4.74e-2 & 2.42e-2\
Rate & – & 0.9646 & 1.0781 & 1.0941 & 0.9671\
Non-constant kernel function
----------------------------
We then consider the kernel function $$\label{Lin_ker}
\gamma(s)=\left \{
\begin{array}{ll}
(1-|s|/\delta)/\delta, & |s|\leq \delta,\\
0, & |s|>\delta.
\end{array}
\right .$$ Obviously the first derivative of $\gamma$ is discontinuous at points $\pm\delta$, if $b$ is discontinuous at $x=0$ or $x=1$, thus $u$ will likely be discontinuous in its second derivative at $x=\delta$ or $x=1-\delta$. This is the case in Example 3 where the regularity pick-up is only second order and could not be further improved. That is, since $b$ is discontinuous at $x=0$ and $x=1$, the second derivative of $u$ is is discontinuous at $x=\delta$ and $x=1-\delta$.
[**Example 3.**]{} We consider (\[1D\_prob\]) with kernel function (\[Lin\_ker\]) and $b(x)=0.01e^{6x}$. The proposed DG method in Section \[section:DG\_method\] is used to discretize it with $\delta=0.4$.
As in Example 2, since we do not know the exact solution $u(x)$, errors are computed using the solution on finer meshes as approximation of the true solution. We first implement the proposed DG method on uniform meshes. The approximation $u_{h}(x)$ with $h=0.00625$ are plotted in Figure \[fig:3\]. To see the discontinuity of the second derivative, the first derivative of $u_{h}(x)$ is also plotted in Figure \[fig:3\]. Although $b(x)\in C^{\infty}(0,1)$, however, $b(x)$ has two discontinuous points $x=0$ and $x=1$, which causes the discontinuity for the second derivative of $u$ at $x=\delta=0.4$ and $x=1-\delta=0.6$. From Table \[table:ex3:uni\] it is seen that optimal convergence rates for errors in $L^2$ and $H^1$ norms are achieved.
[llllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64\
$\|u-u_{h}\|$ & 1.18e-2 & 2.74e-3 & 6.61e-4 & 1.62e-4 & 4.03e-5\
Rate & – & 2.1019 & 2.0525 & 2.0246 & 2.0116\
$\|u'-u'_{h}\|$ & 7.20e-1 & 3.60e-1 & 1.79e-1 & 8.90e-2 & 4.44e-2\
Rate & – & 1.0018 & 1.0089 & 1.0056 & 1.0030\
Next, we use non-uniform meshes (\[nonuni\_grid\]) to solve the problem. In this example the true solution satisfies $u(x)\in H^{2.5-\epsilon}(\Omega)$ for arbitrary small positive $\epsilon$. So for general non-uniform meshes the theoretical convergence rates for errors in $L^2$ and $H^1$ norms are all optimal. This is indeed verified in Table \[table:ex3:non-uni\].
[llllll]{} $\delta/h$ & 4 & 8 & 16 & 32 & 64\
$\|u-u_{h}\|$ & 1.11e-2 & 2.61e-3 & 6.08e-4 & 1.51e-4 & 3.76e-5\
Rate & – & 2.0891 & 2.0993 & 2.0143 & 1.9985\
$\|u'-u'_{h}\|$ & 6.70e-1 & 3.32e-1 & 1.51e-1 & 7.47e-2 & 3.67e-2\
Rate & – & 1.0128 & 1.1331 & 1.0189 & 1.0261\
We have considered the homogeneous problem (\[nonlocal\_diff\_homo\]) with a right hand side function $b({\bf x})\in C(\Omega)$. The inhomogeneous problem (\[nonlocal\_diffusion\]) could be converted to a homogeneous one like (\[nonlocal\_diff\_homo\]) via the transforms (\[btrans\]) and (\[utrans\]).
Conclusion
==========
We propose a new kind of DG method in this paper to numerically solve the nonlocal models with integrable kernels. The existed references tell us that if the right hand side function and the volume constraint function are in $L^{2}(\Omega)$ and $L^{2}(\Omega_{I})$, respectively, then the true solution of that nonlocal model also belongs to $L^{2}(\Omega\cup\Omega_{I})$. Such a general result makes the numerical approximation difficult to operate, or easy to operate but not so efficiently. To make the approximation easier and more efficient simultaneously, we first convert the original nonhomogeneous problem with right hand side function in $L^{2}(\Omega)$ to a homogeneous problem with right hand side function continuous in $\Omega$. Then we analyze the structure of true solution of the homogeneous problem, especially for higher dimensional cases. The main result is, this kind of problem often encounters the discontinuity across the boundary $\partial\Omega$, thus possibly causes the discontinuity of first or second derivative (perhaps higher order derivatives, depending on the smoothness of the kernels) across $\partial\Omega_{1}$. Based on this observation, an appropriate DG method is proposed which has some good properties, such as, the matrix of the algebraic system is symmetrical positive definite and has almost constant spectral condition number independent of the mesh size, the method is asymptotically compatible and uses less degrees of freedom compared with direct use of DG method. Moreover, it has optimal convergence rate for 1D case under very weak assumption, and the almost optimal convergence rate for 2D case under mild assumption. This is the essential improvement over the existed theory for standard approximation like continuous FEM.
The error and condition number estimates for the method are proven in Section \[section:Theoretical considerations\] for any dimensional case. However, the numerical experiments are implemented just for 1D case in Section \[section:numerical\_experiment\]. This is because the implementation of a FEM for a nonlocal problem involves calculation of double integral which is rather complicated for higher dimensional cases. To be more specific, since the kernels we discussed are supported on a ball of radius $\delta$, when the inner integral of the double integral is written by a sum of some components (also integrals) over the elements which overlap with the support of the kernel, the integral region for some components is strictly contained within the support of the kernel. For such special integrals, some specifically designed quadrature rule should be used to obtain good accuracy. Authors in [@xu2015multiscale; @xu2016multiscale] discuss this issue in details and use the quadrature rule for multiscale implementation for PD models. Interested readers are referred to them and references cited therein.
[^1]: Department of Applied Physics and Applied Mathematics, Columbia University, NY 10027, USA (email: qd2125@columbia.edu)
[^2]: Corresponding author, School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China (email: yinxb@mail.ccnu.edu.cn)
|
---
abstract: 'Gapless superconductivity can arise when pairing occurs between fermion species with different Fermi surface sizes, provided there is a sufficiently large mismatch between Fermi surfaces and/or at sufficiently large coupling constant. In gapless states, secondary Fermi surfaces appear where quasiparticle excitation energy vanishes. This work focuses on homogeneous and isotropic superfluids in the $s$-wave channel, with either zero (conventional superconductor), one, or two spherical Fermi surfaces. The stability conditions for these candidate phases are analyzed. It is found that gapless states with one Fermi surface are stable in the BEC region, while gapless states with two Fermi surfaces are unstable in all parameter space. The results can be applied to ultracold fermionic atom systems.'
address: |
Center for Theoretical Physics, Massachusetts Institute of Technology,\
Cambridge, MA 02139, USA\
E-mail: elena1@mit.edu
author:
- 'E. GUBANKOVA'
title: STABILITY CONDITIONS IN GAPLESS SUPERCONDUCTORS
---
Introduction
============
This work focuses on stability conditions and their possible relations to different Fermi surface topologies in a superconductor with unequal number densities of fermions (or unequal chemical potentials)[@1]. The physics of paired fermion systems with unequal densities of the two fermion species is of interest in the study of two physical systems: (1) Ultracold atomic fermionic gases, where one can freely choose populations in two hyperfine states of the fermionic atom. Experimental work is currently being conducted with these systems. (2) Quark matter in the interior of neutron stars, which is believed to be a color superconductor. There, the mismatch in quark Fermi surfaces is driven by differences in quark masses and electric and color neutrality conditions. In both cases, the ground state of such a system is the subject of current debate. The study of fermionic superfluids with imbalanced populations is a novel subject, because until recently, experimental evidence for this situation was elusive. In a superconductor, an imbalance in population between spin-up and spin-down electrons can be created by a magnetic field. However, due to the Meissner effect, the magnetic field is either completely shielded from the superconductor bulk or enters in the form of quantized flux lines or vortices.
Cold fermionic atomic gases provide new possibilities in the experimental exploration of fermionic superfluids with unequal mixtures of fermions. In cold atoms, Feshbach resonance permits tuning of the interaction from (1) attraction and a resulting superfluid of loosely bound pairs (the BCS regime at $\zeta>n^{-1/3}$); to (2) repulsion and a resulting Bose-Einstein condensate of tightly bound molecules (the BEC regime at $\zeta<n^{-1/3}$), where $\zeta$ is the size of a pair and $n^{-1/3}$ is the interparticle separation (or, more precisely, the mean free path). With equal mixtures, the BEC–BCS crossover is smooth, with no phase transition. With asymmetric densities, one or more phase transitions are expected, and a more complex phase diagram may result.
In the weak coupling regime, a BCS superfluid remains stable as long as the difference in chemical potential is small compared to the pairing gap, $\delta\mu<\Delta$; the gap prevents the excess unpaired atoms from entering the superfluid state. By either increasing the mismatch or reducing the binding energy (and hence decreasing the gap), a quantum phase transition from the superfluid to normal state takes place, and superflidity ceases. The point at which the phase transition occurs is known as the Clogston limit, which can be estimated as $\delta\mu\sim\Delta\sim\mu\exp(-1/g)\ll\mu$, where $\mu$ is the Fermi energy. Thus only an exponentially small population imbalance is allowed in weak coupling. In strong coupling, as one approaches Feshbach resonance, the situation is quite different. On the BEC side, superfluidity with imbalanced population is robust over a wide range of parameter space. Surprisingly, on the BCS side, macroscopic imbalance is also possible in a superfluid state, possibly due to the formation of a gapless superfluid at strong coupling. This gapless superfluid incorporates large numbers of unpaired fermions which reside at the secondary Fermi surfaces. Here, we consider stability conditions for gapless states with different Fermi surface topologies.
Definitions for the screening masses and susceptibility.
========================================================
We consider two species of nonrelativistic fermions, $\psi=(\psi_1,\psi_2)$ with the same mass but with different chemical potentials, $\mu={\rm diag}(\mu_1,\mu_2)$. There is an attractive interaction only between different species of fermions, $g\left(\psi^{\dagger}\sigma_2 \psi^*\right)\left(\psi^T\sigma_2\psi\right)$. The order parameter is $\Phi^{\dagger}=\Delta\sigma_2$ where $\Delta=2g<\psi^T\sigma_2\psi>$, it defines pairing in the singlet channel, spin $0$, $[2]\times [2]=[1]+[3]$. The order parameter breaks the original group associated with conservation of particle numbers of each species down to the diagonal subgroup associated with conservation of difference in particle numbers, $U(1)_{\alpha_1}\times U(1)_{\alpha_2}\rightarrow U(1)_{\alpha_1-\alpha_2}$ which is invariant under simultaneous rotation $\alpha_1=-\alpha_2$. Total number of particles is not conserved. Formally, one can express this pattern of symmetry breaking by gauging the theory. We introduce two external gauge fields, and couple each species of fermions to its gauge field, $\psi_1$ couples to $A_1$ with $g_1$ and $\psi_2$ couples to $A_2$ with $g_2$, that is reflected by the generators of the gauge group, $T_1={\rm diag}(1,0)$ and $T_2={\rm diag}(0,1)$. According to the symmetry breaking pattern, gauge fields mix in a superconductor, rotated sum of fields $\tilde A_1=A_1+A_2$ is screened with non-zero Meissner mass via Andersen-Higgs mechanism, $m_M^2\neq 0$ corresponding to $U(1)_{\alpha_1+\alpha_2}$, and rotated difference of fields $\tilde A_2=A_1-A_2$ propagates in a superconductor unscreened, Meissner mass is zero $m_M^2=0$ corresponding to $U(1)_{\alpha_1-\alpha_2}$. There is a different mixing of gauge fields in case of Debye masses. In QCD, with color and electromagnetic gauge groups, $SU(3)_c\times U(1)_{EM}$, mixing of gauge fields is more complicated.
We treat four-fermion interaction on a mean field level. We integrate over the fermions, obtain fermion determinant, $${\cal Z} = \int {\cal D}A\,\exp\left[S_A +
\frac{|\Delta|^2}{4g}-\frac{1}{2}{\rm Tr}\ln({\cal S}^{-1} + {\cal A})\right]\,.
\label{part}$$ We use the Nambu-Gorkov formalism with particle-hole basis, $\Psi=(\psi,\psi^*)$. Inverse fermion propagator and gauge field are $4\times 4$ matrices in this basis, including the Nambu-Gorkov and two fermion species indices, $${\cal S}^{-1} \equiv \left(\begin{array}{cc} [G_0^+]^{-1} & \Phi^- \\ \Phi^+ &
[G_0^-]^{-1}
\end{array}\right) \, ,$$ where the inverse free fermion propagators are $[G_0^\pm]^{-1} = i\partial_t \pm \frac{\nabla^2}{2m} \pm \mu$, and we have abbreviated for the gauge fields ${\cal A}={\rm diag}(A^+, A^-)$ with $A^\pm = \pm \Gamma_a A_a^0 \mp\frac{\Gamma_a^2}{2m}\,{\bf A}_a^2
-\, \frac{i\Gamma_a}{2m}(\nabla\cdot{\bf A}_a+
{\bf A}_a\cdot\nabla)$ and $\Gamma_a=g_a T_a$. Performing derivative expansion and collecting terms quadratic in the gauge field, we produce fermion loops, $\Pi_{ab}^{00}$, $\Pi_{ab}^{i0}$, $\Pi_{ab}^{0i}$, $\Pi_{ab}^{ij}$. Debye mass in one-loop is defined by the temporal component of polarization operator, Meissner mass is given by the spatial component, $m_{D,ab}^2\equiv -\lim_{{\bf p}\to 0}\Pi_{ab}^{00}(0,{\bf p})$ and $m_{M,ab}^2\equiv \frac{1}{2}\lim_{{\bf p}\to
0}(\delta_{ij}-\hat{p}_i\hat{p}_j)\Pi_{ab}^{ij}(0,{\bf p})$, where $\hat{p}_i\equiv p_i/p$. Screening masses are given[@1] $$\begin{aligned}
m_{D,ab}^2&=& -\lim_{P\to 0}\frac{1}{2}\frac{T}{V}\sum_K {\rm Tr}[S(K)\Gamma_a^-S(K-P)\Gamma_b^-] \, ,
\nonumber\\
m_{M,ab}^2&=& \frac{1}{2m}\lim_{P\to 0}\frac{T}{V}\sum_K \left(\phantom{\frac{k^2}{m}}\hspace{-0.5cm}
\delta_{ab}{\rm Tr}[S(K)\bar{\Gamma}_a^2]\right.
\label{screening} \\
&+&\left.\frac{k^2}{2m}\,[1-(\hat{p}\cdot\hat{k})^2]\,{\rm Tr}[S(K)\Gamma_a^+S(K-P)\Gamma_b^+]\right)\, ,
\nonumber
%\label{screening}\end{aligned}$$ where $S$ is the fermion propagator and we have introduced the following matrices in Nambu-Gorkov space, $\Gamma_a^{\pm}= {\rm diag}(\Gamma_a,\pm \Gamma_a)$ and $\bar{\Gamma}_a^2= {\rm diag}(\Gamma_a^2, - \Gamma_a^2)$. Physically, Debye mass to all loops, including ladder diagrams, is equivalent to compressibility of a system, and Meissner mass can be associated with the density of superconducting fermions. These $2\times 2$ matrices in two-fermion space shall be evaluated in the following section in order to obtain stability conditions for gapless superconductors.
One may derive the pressure from the partition function in (\[part\]) using the Cornwall-Jackiw-Tomboulis formalism. The pressure is the negative of the effective potential at its stationary point (i.e., with the propagators determined to extremize the effective potential). The fermionic part of the pressure is $p = \frac{1}{2}\frac{T}{V}{\rm Tr}\ln{\cal S}^{-1} + \frac{1}{2}\frac{T}{V}{\rm
Tr}[{\cal S}_0^{-1}{\cal S} -1]
+\Gamma_2[{\cal S}]$, where ${\cal S}_0 = {\rm diag}(G_0^+,G_0^-)$ is the tree-level fermion propagator in Nambu-Gorkov space and $\Gamma_2[{\cal S}]$ is the sum of all two-particle irreducible diagrams. The number densities is defined $n_a = \partial p/\partial \mu_a$. The number susceptibility $\chi$ is defined as the derivative of the number density with respect to the chemical potential (at constant volume and temperature). Using the expression for the pressure, we obtain[@1] $$\begin{aligned}
\label{susc}
\chi_{ab}=\frac{\partial n_a}{\partial \mu_b}&=& -\frac{1}{2g_ag_b}\frac{T}{V}\sum_K{\rm Tr}[\Gamma_a^-{\cal S}(K)\Gamma_b^-{\cal S}(K)]
\nonumber\\
&-&\,\frac{1}{2g_a}\frac{T}{V}\sum_K{\rm Tr}\left[\Gamma_a^-{\cal S}(K)\frac{\partial \Sigma(K)}
{\partial\mu_b}{\cal S}(K)\right] \, ,\end{aligned}$$ where $\Sigma$ is the fermion self-energy, ${\cal S}^{-1} = {\cal S}_0^{-1} + \Sigma$. The first term on the right-hand side of this equation is given by the one-loop result for the electric screening mass, cf. Eq. (\[screening\]). For the second term, we assume that the self-energy $\Sigma$ depends on $\mu$ only through the gap, then we obtain[@1] $$\label{chidef}
\chi_{ab} = \frac{m_{D,ab}^2}{g_ag_b} +
\frac{\partial n_a}{\partial \Delta}\frac{\partial \Delta}{\partial \mu_b} \, .$$ In general, the self-energy $\Sigma$ contains terms of any number of fermion loops. Consequently, the number susceptibility contains terms of arbitrary many fermion loops too, corresponding to the exact Debye mass including all possible perturbative insertions. Remarkably, the free fermion result for $\chi$, i.e. $\Sigma=0$, gives the one-loop result for $m_D^2$. Susceptibility is equivalent to compressibility of a system. We shall use Eq. (\[chidef\]) in the following section to compute the number susceptibility. As this equation shows, it goes beyond the one-loop result for the electric screening mass.
Results
=======
We consider three cases distinguished by how many zeros quasiparticle dispersion has, $\varepsilon_k^-=\sqrt{(k^2/2m-\bar{\mu})^2+\Delta^2}-\delta\mu$, where $\mu$ is the average chemical potential and $\delta\mu$ is the difference between potentials, Fig. \[figoccupation\]. In this case, number of zeros is equivalent to number of the Fermi surfaces. No zeros corresponds to the fully gapped state. In case of one Fermi surface, momenta outside the Fermi surface contribute to the pairing, while the excess of fermions resides inside the Fermi ball. In case of two Fermi surfaces, the excess of fermions resides between the two Fermi surfaces in momentum space. As was noticed by Son, apart from number of zeros, dispersion can have two different characteristic behaviors, distinguished by the position of the minimum. Minimum is located at nonzero momentum for positive $\bar{\mu}$, and corresponds to BCS, and minimum shifts to $p=0$ for negative $\mu$, and corresponds to BEC. This behavior manifests itself in stability conditions. We depict different topologies which are distinguished by the number of Fermi surfaces on the phase diagram in dimensionless average chemical potential $\bar{\mu}/\Delta$ and difference in chemical potentials $\delta\mu/\Delta$, the gap is the energy scale, in Fig. \[figglobal\] regions between the solid lines. We have fully gapped $F_0$, and gapless states with one $F_1$ and two $F_2$ Fermi surfaces. At small mismatch, there is a fully gapped state $F_0$, which is at positive $\bar{\mu}$ and $\delta\mu<\Delta$, BCS, and at small and negative $\bar{\mu}$, $\bar{\mu}<-\sqrt{\delta\mu^2-\Delta^2}$, BEC. Increasing mismatch, when mismatch is at least larger than the gap, there is a gapless state with one Fermi surface $F_1$ when $\mu$ is around the Feshbach resonance, $-\sqrt{\delta\mu^2-\Delta^2}<\bar{\mu}<\sqrt{\delta\mu^2-\Delta^2}$, and gapless state with two Fermi surfaces $F_2$ for positive $\mu$ restricted from below, $\sqrt{\delta\mu^2-\Delta^2}<\bar{\mu}$. We therefore expect that $F_1$ exists at strong coupling, while $F_2$ probably exists only at weak coupling.
\[ht\]
The question we are solving here is, “What is the ground state in a degenerate Fermi system with asymmetric number densities of fermions?”. We avoid solving for the ground state explicitly. Instead, we check stability criteria, positive definite eigenvalues of screening masses and number susceptibility. If stability conditions are satisfied, homogeneous superconducting gapless state is indeed the ground state in this parameter range. If stability conditions are not satisfied, the alternative state may be realized as the ground state. These include LOFF, spatially separated mixtures, normal (not superconducting) states.
We analyze stability conditions, Eq. (\[screening\]) and Eq. (\[chidef\]), in all parameter space $(\bar{\mu},\delta\mu)$, and depict stable/unstable regions together with topology regions, Fig. \[figglobal\] left panel. Debye mass e.v. are positive in all parameter space, hence Debye mass does not impose a constraint. All entries for the Meissner mass matrix are the same, it is trivial to diagonalize. We obtain[@1] $m_M^2=0$ corresponding to the unbroken sector $U(1)_{\alpha_1-\alpha_2}$, and $m_M^2=2L$ corresponding to the broken group $U(1)_{\alpha_1+\alpha_2}$, where $L=\tilde{I}-\frac{\rho_+^{3/2}+\rho_-^{3/2}}{2\eta\sqrt{\eta^2-1}}$, which defines the dashed-dotted (blue online) curve in Fig. \[figglobal\] left panel. It renders all states between the dashed-dotted curve and the solid vertical line unstable. There is a strip left in gapless superconductor state with two Fermi surfaces $F_2$ which is stable, gray area. $F_0$ and $F_1$ are stable with respect to $m_M^2$ everywhere. In magnetic sector, mixing does not depend on chemical potentials and it is defined by the pattern of symmetry breaking. In electric sector, mixing depends on chemical potentials, e.g. in QCD mixing depends on $\delta\mu$. Mixing in electric and magnetic sectors is the same only for the fully gapped case.
\[ht\]
Analyzing the number susceptibility matrix, Eq. (\[chidef\]), we obtain[@1] the expression defining the sign of e.v., which is very similar in structure to that for the Meissner mass, see expression for $L$, $R=I-\frac{\rho_+^{1/2}+\rho_-^{1/2}}{2\eta\sqrt{\eta^2-1}}$. Here, we defined $I\equiv I_{\rho}(0,\infty)-I_{\rho}(\sqrt{\rho_-},\sqrt{\rho_+})$ and $\tilde{I}\equiv \tilde{I}_{\rho}(0,\infty)-
\tilde{I}_{\rho}(\sqrt{\rho_-},\sqrt{\rho_+ } )$ with elliptic integrals $I_{\rho}(a,b)\equiv \int_a^b dx\,x^2/[(x^2-\rho)^2+1]^{3/2}$ and $\tilde{I}_{\rho}(a,b)\equiv \int_a^b dx\,x^4/[(x^2-\rho)^2+1]^{3/2}$, and $\rho_{\pm}=\rho\pm\sqrt{\eta^2-1}$ are zeros of $\varepsilon_k^{-}=0$ with $\rho=\bar{\mu}/\Delta$, $\eta=\delta\mu/\Delta$. In expression for $R$ one should put $\rho_{-}=0$ when applied to $F_1$ region. It defines the dashed (red online) line, and it renders all states between dashed and solid vertical lines unstable. Thus all $F_2$ states are unstable,and there is a strip in $F_1$ which is stable, dark gray area. $F_0$ is stable everywhere. Stable states with respect to $\chi$ correspond to local maximum of pressure.
We obtained stable regions which are local maxima of pressure, Fig. \[figglobal\] left panel. Now we consider global maxima, Fig. \[figglobal\] right panel. For this we compare pressure of the superconducting and normal states. Superconducting states which pressure is higher than that of the normal state are stable, $\Delta p=p_s-p_n>0$. The dashed-dotted (blue on line) line is $\Delta p=0$, it renders all states above and to the right of it unstable. All unstable regions with respect to the screening masses and number susceptibility are subset of unstable region with respect to the pressure. In a weak coupling, the vertical dashed-dotted line reproduces the known Clogston limit, $\delta\mu=\Delta/\sqrt{2}$, above which BCS is unstable. The global stability line cuts through the stable strip of $F_1$ state, below is a stable superconducting state, above is a metastable state. Both lines coincide at large mismatches.
Currently, experiments are being performed with unequal mixtures of fermions to map superfluid regions as a function of population imbalance, interaction strength and temperature. The experimental signature of superfluidity is the existence of vortices, which prove phase coherence in a sample.
[9]{} E. Gubankova, A. Schmitt, F. Wilczek, cond-mat/0603603; accepted for publication in Phys. Rev. B; and references therein.
|
---
abstract: 'We will investigate practical aspects for a recently introduced blind (noncoherent) communication scheme, called modulation on conjugate-reciprocal zeros (MOCZ), which enables reliable transmission of sporadic and short-packets at ultra-low latency in unknown wireless multipath channels, which are static over the receive duration of one packet. Here the information is modulated on the zeros of the transmitted discrete-time baseband signal’s $z-$transform. Due to ubiquitous impairments between transmitter and receiver clocks a carrier frequency offset (CFO) will occur after a down-conversion to the baseband, which results in a common rotation of the zeros. To identify fractional rotations of the base angle in the zero-pattern, we propose an oversampled direct zero testing decoder to identify the most likely one. Integer rotations correspond to cyclic shifts of the binary message, which we compensate by a cyclically permutable code (CPC). Additionally, the embedding of CPCs into cyclic codes, allows to exploit additive error correction which reduces the bit-error-rate tremendously. Furthermore, we use the trident structure in the signals autocorrelation to estimate timing offsets and the channels effective delay spread. We finally demonstrate how these impairment estimations can be largely improved by receive antenna diversity, which enables extreme bursty reliable communication at low latency and SNR.'
author:
-
bibliography:
- 'jabref\_philipp\_utf2.bib'
- 'peter.bib'
title: 'MOCZ for Blind Short-Packet Communication: Some Practical Aspects'
---
Introduction
============
We introduced in [@WJH18a; @WJH18b] a novel blind (noncoherent) communication scheme for the physical layer, called modulation on conjugate-reciprocal zeros (MOCZ), to reliably transmit sporadic short-packets of fixed size over unknown wireless multipath channels with bandwidth $W$ at an incredible low-latency. Here the information of the packet is modulated on the zeros of the transmitted discrete-time baseband signal’s $z-$transform. We will call the discrete-time baseband signal a *MOCZ symbol*, similar to an orthogonal frequency division multiplexing (OFDM) symbol, which is a finite length sequence of complex-valued coefficients. These coefficients will then modulate a continuous-time pulse shape at a sample period of $T=1/W$ to generate the continuous-time baseband waveform. Since the MOCZ symbols (sequences) are neither orthogonal in time nor frequency domain, the MOCZ design can be seen as a non-orthogonal multiplexing scheme. After up-converting to the desired carrier frequency, the transmitted passband signal will propagate in space such that, due to reflections, diffractions, and scattering, different delays of the attenuated signal will interfere at the receiver. Hence, multipath propagation causes a time-dispersion which results in a frequency-selective fading channel [@TV05]. Due to ubiquitous impairments between transmitter and receiver clocks a *carrier frequency offset* (CFO) will be present after a down conversion to the baseband. Doppler shifts due to relative velocity causes additional frequency dispersion which can be also approximated in first order by a CFO. This is a known weakness in many multi-carrier modulation schemes, such as OFDM [@TV05; @Moo94; @ZGX10; @LLTC04], and various approaches have been developed to estimate or eliminate the CFO effect. A common approach for OFDM systems is to learn the CFO in a training phase or from blind estimation algorithms, such as MUSIC [@LT98] or ESPRIT [@TLZ00]. Furthermore, due to the unknown distance and asynchronous transmission, a *timing offset* (TO) of the received symbol has to be determined as well, which will otherwise destroy the orthogonality of the OFDM symbols [@CKYK10 5.1],[@PKPKKH06]. By “sandwiching” the data symbol between two training symbols a timing and frequency offset can be estimated [@SC97],[@SC96]. By using antenna arrays at the receiver, antenna diversity of a single-input-multiple output (SIMO) system can be exploited to improve the performance [@ZGX10].
Whereas OFDM is typically used in long frames, consisting of many successive OFDM symbols and hence much longer signal lengths, we consider here only one single symbol transmission with a very short signal length. This places high demands on such a bursty signaling scheme, since timing and carrier frequency offsets have to be addressed from only one received symbol. Here our MOCZ scheme will be a promising solution. Since any communication will be scheduled and timed on the MAC layer by a certain bus, running with a known bus clock-rate, timing-offsets of the symbols can be assumed as fractions of the bus clock-rate. We will introduce here an improved receiver design for a coded binary MOCZ (BMOCZ) scheme and demonstrate by bit-error-rate (BER) simulations the robustness against these impairments.
In the MOCZ design, a CFO will result in an unknown common rotation of all received zeros. Since the angular zero spacing in a BMOCZ symbol of length $K+1$ is given by a base angle of $2\pi/K$, a fractional rotation can be easily obtained at the receiver by an oversampling during the post-processing to identify the most likely transmitted zeros (zero-pattern). Rotations, which are integer multiples of the base angle, correspond to cyclic shifts of the binary message word. By using a *cyclically permutable code* (CPC) for the binary message, the BMOCZ symbol becomes invariant against any cyclic shift and hence against any CFO. This prevents any further symbol transmissions for estimating the CFO, which will reduce overhead, latency, and complexity. As a byproduct, this has the appealing feature of providing a CFO estimation from the decoding process of a single BMOCZ symbol. Furthermore, due to the embedding into a cyclic code, such as BCH codes, we can use their error correction capabilities to improve the BER and moreover the *block error-rate* (BLER) performance tremendously. By measuring the energy of the expected symbol length with a sliding window in the received signal, we can identify arbitrary TOs at the receiver. We will show the robustness of the TO estimation analytically, which reveals another strong property of the MOCZ design.
At last, we will combine CFO and TO with error correction over multiple receive antennas and demonstrate antenna diversity of the SIMO system. By simulating BER over the received SNR for various average power delay profiles, with constant and exponential decay as well as random sparsity constraints, we will demonstrate the performance in various indoor and outdoor scenarios by using the simulation framework Quadriga [@JRBT14].
Notation
--------
We will use small letters for complex numbers in $\C$. Capital Latin letters denote natural numbers $\N$ and refer to fixed dimensions, where small letters are used as indices. Boldface small letters denote row vectors and capitalized letters refer to matrices. Upright capital letters denote complex-valued polynomials in $\C[z]$. We will denote the first $N$ natural numbers in $\N$ as $[N]:=\{0,1,\dots,N-1\}$. For $K\in\N$ we denote by $K+[N]=\{K,K+1,\dots K+N-1\}$ the $K-$shift of the set $[N]$. The Kronecker-delta symbol is given by $\del_{nm}$ and is $1$ if $n=m$ and $0$ else. For a complex number $x=a+\im b$, given by its real part $\Re(x)=a\in\R$ and imaginary part $\Im(x)=b\in\R$ with imaginary unit $\im=\sqrt{-1}$, its complex-conjugation is given by $\cc{x}=a-\im b$ and its absolute value by $|x|=\sqrt{x\cc{x}}$. For a vector $\vx\in\C^N$ we denote by $\cc{\vx^-}$ its complex-conjugated time-reversal or *conjugate-reciprocal*, given as $\cc{x_k^-} = \cc{x_{N-k}}$ for $k\in[N]$. We use $\vA^*=\cc{\vA}^T$ for the complex-conjugated transpose of the matrix $\vA$. For the $N\times N$ identity matrix we write $\id_N$ and for a $N\times M$ matrix with all elements zero we write $\vzero_{N,M}$. By $\vD_{\vx}$ we refer to the diagonal matrix generated by $\vx\in\C^N$. The $N\times N$ unitary Fourier matrix $\Fmatrix=\Fmatrix_N$ is given entry-wise by $f_{l,k}=e^{-\im 2\pi lk/N}/\sqrt{N}$ for $l,k\in[N]$. By $\vT\in\R^{N\times M}$ denote the elementary Toeplitz matrix given element-wise as $\del_{n-1 m}$. The all one and all zero vectors in dimension $N$ will be denoted by $\eins_N$ and $\zero_N$, respectively. The $\ell_p$-norm of a vector $\vx=(x_0,\dots,x_{N-1})\in\C^N$ is given by $\Norm{\vx}_p=(\sum_{k=0}^{N-1}|x_k|^p)^{1/p}$ for $p\geq 1$. If $p=\infty$ we write $\Norm{\vx}_{\infty}=\max_k |x_k|$ and for $p=2$ we set $\Norm{\vx}_2=\Norm{\vx}$. The expectation of a random variable $x$ is denoted by $\Expect{x}$.
System Model and Requirements
=============================
We are interested in a blind and asynchronous transmission of a short [**single MOCZ symbol**]{} at a designated bandwidth $W$. In this “one-shot” communication we assume no synchronization and no packet scheduling between transmitter and receiver. Such extreme sporadic, asynchronous, and ultra short-packet transmissions are required, for example, in critical control applications, exchange of channel state information (CSI), signaling protocols, secret keys, authentication, commands in wireless industry applications, or initiation, synchronization and channel probing packets to prepare for longer or future transmission phases. By choosing the carrier frequency, transmit sequence length, and bandwidth accordingly, a receive duration in the order of the channel delay spread can be obtained, which pushes the latency at the receiver to the lowest possible. Since the next generation of mobile wireless networks aims for large bandwidths with carrier frequencies beyond $10$Ghz, in the so called *mmWave* band, the transmitted signal duration will be in the order of nano seconds. Hence, even at moderate mobility, the wireless channel in an indoor or outdoor scenario can be considered as approximately time-invariant over such a short time duration. On the other hand, wideband channels are highly frequency selective, which is due to the superposition of different delayed versions (echos) of the transmitted signal at the receiver. This makes equalizing in time-domain very challenging and is commonly simplified by using OFDM instead. But conventional OFDM requires an additional cyclic prefix to convert the frequency-selective channel to parallel scalar channels and in coherent mode it requires additional pilots (training) to learn the channel coefficients. This will increase the latency for short messages dramatically.
For a communication in mmWave band massive antenna arrays are exploited to overcome the large attenuation, which increases the complexity and energy consumption in estimating the huge amount of channel parameters and becomes the bottleneck in mmWave MIMO systems, especially for mobile scenarios. However, in a sporadic communication only one symbol will be transmitted and a next symbol may follow at an unknown time later. In a random access channel (RACH), a different user may transmit the next symbol from a different location, which will therefore experience an independent channel realization. Hence, the receiver can barely use any channel information learned from past communications. OFDM systems approach this by transmitting many successive OFDM symbols as a long frame, to estimate the channel impairments, which will cause a considerable overhead and latency if only a few data-bits need to be communicated. Furthermore, to achieve orthogonal subcarriers in OFDM, the cyclic prefix has to be at least as long as the channel impulse response (CIR) length, resulting in signal lengths at least twice as the CIR length during which the channel also needs to be static. Using OFDM signal lengths much longer than the coherence time might be not feasible for fast time-varying block-fading channels. Furthermore, the maximal CIR length needs to be known at the transmitter and if underestimated will lead to a serious performance loss. This is in high contrast to our MOCZ design, where the signal length can be chosen for a single MOCZ symbol independently from the CIR length. The goal in this work is to address the ubiquitous impairments of the MOCZ design under such ad-hoc communication assumptions and signal lengths in the order of the CIR length. After up-converting the MOCZ symbol, which is a discrete-time complex-valued baseband signal $\vx=(x_0,x_1,\dots,x_K)\in\C^{K+1}$ of two-sided bandwidth $W$, to the desired carrier frequency $f_c$, the transmitted passband signal will propagate in space. Regardless of directional or omnidirectional antennas, the signal will be reflected and diffracted at point-scatters, resulting in different delays of the attenuated signal which interfere at the receiver if the maximal delay spread $T_d$ of the channel is larger than the sample period $T=1/W$. Hence, the multipath propagation causes time dispersion resulting in a frequency-selective fading channel. Due to ubiquitous impairments between transmitter and receiver clocks an unknown *frequency offset* $\Deltaf$ will be present after the down-conversion to the received continuous-time baseband signal $$\begin{aligned}
\tilr(t)=r(t)e^{\im 2\pi t\Deltaf}.\end{aligned}$$ By sampling $\tilr_n=\tilr(nT)$ at the sample period $T$, the received discrete-time baseband signal can be represented by a *tapped delay line* (TDL) model. Here the channel action is given as the convolution of the MOCZ symbol $\vx$ with a finite impulse response $\{h_\ell\}$, where the $\ell$th complex-valued channel tap $h_\ell$ describes the $\ell$th averaged path over the bin $[\ell T,(\ell+1)T)$, which we model by a circularly symmetric Gaussian random variable in $\CN(0, s_{\ell}p^{\ell})$ for $l\in[L]$ and zero elsewhere. The average *power delay profile* (PDP) of the channel can be sparse and exponentially decaying, where $s_{\ell}\in\{0,1\}$ defines the sparsity pattern of $S=|\supp{\vh}|=\sum_{\ell=0}^{L-1} s_{\ell}=\Norm{\vs}_1$ non-zero coefficients and $\pdp\leq 1$ the exponential decay rate. To obtain equal average transmit and average receive power we will eliminate in our analysis the overall channel gain by normalizing the CIR realization $\vh=(h_0,\dots,h_{L-1})$ by its average energy $\sum_{l=0}^{L-1} s_lp^l$ (for a given sparsity pattern), such that $\Expect{\Norm{\vh}^2}=1$. The convolution output is then additively distorted by Gaussian noise $w_n\in\CN(0,N_0)$ of zero mean and variance (average power density) $N_0$ for $n\in\N$ as $$\begin{aligned}
\begin{split}
\tilr_{n} &= e^{\im n\phi} \sum_{k=0}^{K} x_{k} h_{n-\toff-k} \!+\! e^{\im n\phi}
w_n =\sum_{k=0}^K e^{\im k\phi}x_k e^{\im (n-k)\phi}h_{n-\toff-k} \!+\! \tw_n =\sum_{k=0}^K \tx_k\tilh_{n-\toff-k} \!+\! \tw_n.
\end{split}\label{eq:receivedsamples}\end{aligned}$$ Here $\phi=2\pi\Deltaf/W \mod 2\pi\in[0,2\pi)$ denotes the *carrier frequency offset* (CFO) and $\tau_0\in\N$ the *timing offset* (TO), which marks the delay of the first symbol coefficient $x_0$ via the first channel path $h_0$, measured in integer multiples of the sample time $T$. The modulated MOCZ symbol $\vtx\in\C^K$ will have rotated coefficients $\tx_k=e^{jk\phi}x_k$ as well as the channel $\tilh_{\ell}=e^{\im (\ell+\tau_0)\phi}h_{\ell}$, which will be also effected by a *global phase* $\tau_0\phi$. Since the channel taps have a uniform independent phase the distribution does not change. By the same argument, the Gaussian noise distribution is not alternated by the phase, hence we have $\tilde{w}_n\in\CN(0,N_0)$ for any $n$ and $\phi$.
In [@WJH18b; @WJH18a] a good signal-codebook is given for Binary MOCZ (BMOCZ) for the set of normalized *Huffman sequences* $\Code(R,K)=\set{\vx\in\C^{K+1}}{\va(R,K)=\vx*\cc{\vx^-},x_0>0}$, i.e., by all $\vx\in\C^{K+1}$ with positive first coefficientand “impulsive-like“ autocorrelation [@HUf62], given by $$\begin{aligned}
\va=\va(R,K)=\vx*\cc{\vx^-} =(-\eta,\zero_{K},1,\zero_K,-\eta)
\quad \text{with}\quad\eta=1/(R^K+R^{-K})
\label{eq:huffmanauto}\end{aligned}$$ for some $R>1$. The absolute value of forms a *trident* with one main peak at the center, given by the energy $\Norm{\vx}^2=1$, and two equal side-peaks of $\eta\in[0,1/2)$, see . From an analytical and empirical investigation [@WJH18a], the BMOCZ symbols are most robust against noise if $$\begin{aligned}
R=R(K)=\sqrt{1+\sin(\pi/K)}>1 \quad,\quad K\geq 2.
\label{eq:optimalR}\end{aligned}$$ Hence, the BMOCZ codebook (constellation set) $\Code=\Code(K)$ is only determined by the number $K$. Each BMOCZ symbol (constellation, Huffman sequence) defines the coefficients of a polynomial of degree $K$, where the $K$ zeros are uniformly placed on a circle of radius $R$ or $R^{-1}$, selected by the message bits $\vm=(m_1,\dots,m_K)\in\{0,1\}^K$ as $$\begin{aligned}
\uX(z)=\sum_{k=0}^{K} x_k z^k=x_K \Pro_{k=1}^K (z -R^{2m_k-1} e^{\im 2\pi (k-1)/K})=x_K \Pro_{k=1}^K
(z-\alp_k^{(m_k)})\label{eq:Xzeros}\end{aligned}$$ see also . Hence, the BMOCZ encoder is defined iteratively for $q=2,\dots,K$ by its *zero codeword* $\valp(\vm)=(\alp_1^{(m_1)},\dots,\alp_K^{(m_K)})\in\Zero\subset\C^K$ as $$\begin{aligned}
\vx_q = (0, \vx_{q-1}) - (R^{2m_q-1} e^{\im \frac{2 \pi q}{K} } \vx_{q-1})
\quad\text{with}\quad \vx_1=(-R^{2m_1-1}e^{\im \frac{2\pi}{K}},1),\end{aligned}$$ where we normalize after the last iteration step $\vx=\vx_K/\Norm{\vx_K}$. From the received $N=L+K$ noisy signal samples (no CFO and TO) $$\begin{aligned}
y_n=\sum_{k=0}^K x_k h_{n-k} + w_n=(\vx*\vh)_n+w_n,\label{eq:convolutionoutput}\end{aligned}$$ the decoder is given as a *Direct Zero Testing* (DiZeT) of the received polynomial $\uY(z)=\sum_{n=0}^{N-1}y_n z^n$ at the $2K$ possible zero positions as $$\begin{aligned}
\hm_k = \begin{cases}
1,& |\uY(R e^{\im 2\pi \frac{k-1}{K}})| <R^{N-1}|\uY(R^{-1} e^{\im 2\pi \frac{k-1}{K}})|\\
0,& \text{else}
\end{cases}\label{eq:dizetpoly} \quad,\quad k=1,\dots,K,\end{aligned}$$ see [@WJH18a; @WJH18b]. A global phase in $\uY(z)$ will have no affect to the DiZeT decoder and to the received zeros. But the CFO $\phi$ modulates the BMOCZ symbol in and causes a rotation[^1] of its zeros by $-\phi$ in , which will destroy the hypothesis test of the DiZeT decoder. Hence, one needs to either estimate $\phi$ or use an outer code for BMOCZ to be invariant against an arbitrary rotation of the entire zero codebook $\Zero$, which we will introduce in . However, before we can apply the DiZeT decoder, we have to identify the timing offset of the symbol which yields to the convolution output in .
Conclusion
==========
We proposed a timing-offset and carrier frequency offset estimation for the novel BMOCZ modulation scheme in wideband frequency-selective fading channels. The CFO robustness is realized by a cyclically permutable code, which allows to identify the integer CFO. An oversampled DiZeT decoder allows to estimate the fractional CFO. The CPC code construction with cyclic BCH codes allow to correct additional bit errors which enhances the performance of the BMOCZ design for moderate SNRs. Furthermore, we used a novel simulation software Quadriga version 2.0, to generate random CIR at a bandwidth of $150$Mhz. Due to the low-latency of BMOCZ the CFO and TO estimation from one single BMOCZ symbol, this blind scheme is ideal for control-channel applications, where few critical and control data need to be exchanged while at the same time, channel and impairments information need to be communicated and estimated. Coded BMOCZ with ACPC is therefore a promising scheme to enable low-latency and ultra-reliable short-packet communications over unknown wideband channels.
Acknowledgements
================
The authors would like to thank Richard Kueng, Anatoly Khina, and Urbashi Mitra for many helpful discussions. Peter Jung is supported by DFG grant JU 2795/3.
References {#references .unnumbered}
==========
[^1]: The CFO would rotate the zeros in any scheme of modulation on zeros, but we will consider here for simplicity only the BMOCZ scheme.
|
---
abstract: 'Let $E$ be a row-finite directed graph. We prove that there exists a $C^*$-algebra ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ with the following co-universal property: given any $C^*$-algebra $B$ generated by a Toeplitz-Cuntz-Krieger $E$-family in which all the vertex projections are nonzero, there is a canonical homomorphism from $B$ onto ${\ensuremath{C^*_{\operatorname{min}}(E)}}$. We also identify when a homomorphism from $B$ to ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ obtained from the co-universal property is injective. When every loop in $E$ has an entrance, ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ coincides with the graph $C^*$-algebra $C^*(E)$, but in general, ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ is a quotient of $C^*(E)$. We investigate the properties of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ with emphasis on the utility of co-universality as the defining property of the algebra.'
address: |
School of Mathematics and Applied Statistics\
Austin Keane Building (15)\
University of Wollongong\
NSW 2522\
AUSTRALIA
author:
- Aidan Sims
title: 'The co-universal C\*-algebra of a row-finite graph'
---
Introduction
============
The aim of this paper is to initiate a study of $C^*$-algebras defined by what we refer to as co-universal properties, and to demonstrate the utility of such a property in investigating the structure of the resulting $C^*$-algebra. We do this by considering the specific example of co-universal $C^*$-algebras associated to row-finite directed graphs.
A directed graph $E$ consists of a countable set $E^0$ of vertices, and a countable set $E^1$ of directed edges. The edge-directions are encoded by maps $r, s : E^1 \to E^0$: an edge $e$ points from the vertex $s(e)$ to the vertex $r(e)$. In this paper, we follow the edge-direction conventions of [@Raeburn2005]; that is, a path in $E$ is a finite sequence $e_1e_2...e_n$ of edges such that $s(e_i) = r(e_{i+1})$ for $1
\le i < n$.
Let $E$ be a directed graph. A *Toeplitz-Cuntz-Krieger $E$-family* in a $C^*$-algebra $B$ consists of sets $\{p_v : v
\in E^0\}$ and $\{s_e : s \in E^1\}$ of elements of $B$ such that
1. \[it:TR1\] the $p_v$ are mutually orthogonal projections;
2. \[it:TR2\] $s^*_e s_e = p_{s(e)}$ for all $e \in
E^1$; and
3. \[it:TR3\] $p_v \ge \sum_{e \in F} s_e s^*_e$ for all $v \in E^0$ and all finite $F \subset r^{-1}(v)$.
A Toeplitz-Cuntz-Krieger $E$-family $\{p_v : v \in E^0\}$, $\{s_e : s \in E^1\}$ is called a *Cuntz-Krieger $E$-family* if it satisfies
1. \[it:CK\] $p_v = \sum_{r(e) = v} s_e s^*_e$ whenever $0 < |r^{-1}(v)| < \infty$.
The graph $C^*$-algebra $C^*(E)$ is the universal $C^*$-algebra generated by a Cuntz-Krieger $E$-family.
To see where (\[it:TR1\])–(\[it:TR3\]) come from, let $E^*$ denote the path category of $E$. That is, $E^*$ consists of all directed paths $\alpha = \alpha_1 \alpha_2 \dots
\alpha_m$ endowed with the partially defined associative multiplication given by concatenation. There is a natural notion of a “left-regular" representation $\lambda$ of $E^*$ on $\ell^2(E^*)$: for a path $\alpha \in E^*$, $\lambda(\alpha)$ is the operator on $\ell^2(E^*)$ such that $$\label{eq:lrr}
\lambda(\alpha) \xi_\beta =
\begin{cases}
\xi_{\alpha\beta} &\text{ if $s(\alpha) = r(\beta)$} \\
0 &\text{ otherwise.}
\end{cases}$$ It is not hard to verify that the elements $P_v := \lambda(v)$ and $S_e := \lambda(e)$ satisfy (\[it:TR1\])–(\[it:TR3\]). Indeed, it turns out that the $C^*$-algebra generated by these $P_v$ and $S_e$ is universal for Toeplitz-Cuntz-Krieger $E$-families.
The final relation (\[it:CK\]) arises if we replace the space $E^*$ of paths in $E$ with its boundary $E^{\le\infty}$ (this boundary consists of all the infinite paths in $E$ together with those finite paths that originate at a vertex which receives no edges). A formula more or less identical to defines a Cuntz-Krieger $E$-family $\{P^\infty_v : v \in E^0\}$, $\{S^\infty_e : e \in E^1\}$ in ${\mathcal B}(\ell^2(E^{\le\infty}))$. The Cuntz-Krieger uniqueness theorem [@BPRS2000 Theorem 3.1] implies that when every loop in $E$ has an entrance, the $C^*$-algebra generated by this Cuntz-Krieger family is universal for Cuntz-Krieger $E$-families. When $E$ contains loops without entrances however, universality fails. For example, if $E$ has just one vertex and one edge, then a Cuntz-Krieger $E$-family consists of a pair $P,S$ where $P$ is a projection and $S$ satisfies $S^*S = P = SS^*$. Thus the universal $C^*$-algebra $C^*(E)$ is isomorphic to $C^*({{\mathbb{Z}}}) = C({{\mathbb{T}}})$. However, $E^{\le\infty}$ consists of a single point, so $C^*(\{P^\infty_v, S^\infty_e\})
\cong {{\mathbb{C}}}$.
The definition of $C^*(E)$ is justified, when $E$ contains loops with no entrance, by the gauge-invariant uniqueness theorem (originally due to an Huef and Raeburn; see [@HR1997 Theorem 2.3]), which says that $C^*(E)$ is the unique $C^*$-algebra generated by a Cuntz-Krieger $E$-family in which each $p_v$ is nonzero and such that there is a *gauge action* $\gamma$ of ${{\mathbb{T}}}$ on $C^*(E)$ satisfying $\gamma_z(p_v) = p_v$ and $\gamma_z(s_e) = zs_e$ for all $v \in
E^0$, $e \in E^1$ and $z \in {{\mathbb{T}}}$.
Recently, Katsura developed a very natural description of this gauge-invariant uniqueness property in terms of what we call here a *co-universal property*. In the context of graph $C^*$-algebras, Proposition 7.14 of [@Katsura2007] says that $C^*(E)$ is co-universal for gauge-equivariant Toeplitz-Cuntz-Krieger $E$-families in which each vertex projection is nonzero. That is, $C^*(E)$ is the unique $C^*$-algebra such that
- $C^*(E)$ is generated by a Toeplitz-Cuntz-Krieger $E$-family $\{p_v, s_e\}$ such that each $p_v$ is nonzero, and $C^*(E)$ carries a gauge action; and
- For every Toeplitz-Cuntz-Krieger $E$-family $\{q_v,
t_e\}$ such that each $q_v$ is nonzero and such that there is a strongly continuous action $\beta$ of ${{\mathbb{T}}}$ on $C^*(\{q_v, t_e\})$ satisfying $\beta_z(q_v) = q_v$ and $\beta_z(t_e) = zt_e$ for all $v \in E^0$ and $e
\in E^1$, there is a homomorphism $\psi_{q,t} :
C^*(\{q_v, t_e\}) \to C^*(E)$ satisfying $\psi_{q,t}(q_v) = p_v$ and $\psi_{q,t}(t_e) = s_e$ for all $v \in E^0$ and $e \in E^1$.
The question which we address in this paper is whether there exists a co-universal $C^*$-algebra for (not necessarily gauge-equivariant) Toeplitz-Cuntz-Krieger $E$-families in which each vertex projection is nonzero. Our first main theorem, Theorem \[thm:Cr(E)-Existence\] shows that there does indeed exist such a $C^*$-algebra ${\ensuremath{C^*_{\operatorname{min}}(E)}}$, and identifies exactly when a homomorphism $B \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ obtained from the co-universal property of the latter is injective. The bulk of Section \[sec:existence\] is devoted to proving this theorem. Our key tool is Hong and Szymański’s powerful description of the primitive ideal space of the $C^*$-algebra of a directed graph. We realise ${\mathcal T}C^*(E)$ as the universal $C^*$-algebra of a modified graph ${\widetilde{E}}$ to apply Hong and Szymański’s results to the Toeplitz algebra.
Our second main theorem, Theorem \[thm:Cr(E)-Properties\] is a uniqueness theorem for the co-universal $C^*$-algebra. We then proceed in the remainder of Section \[sec:properties\] to demonstrate the power and utility of the defining co-universal property of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ and of our uniqueness theorem by obtaining the following as fairly straightforward corollaries:
- a characterisation of simplicity of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$;
- a characterisation of injectivity of representations of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$;
- a description of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ in terms of a universal property, and a uniqueness theorem of Cuntz-Krieger type;
- a realisation of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ as the Cuntz-Krieger algebra $C^*(F)$ of a modified graph $F$;
- an isomorphism of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ with the $C^*$-subalgebra of ${\mathcal B}(\ell^2(E^{\le \infty}))$ generated by the Cuntz-Krieger $E$-family $\{P^{\le\infty}_v,
S^{\le\infty}_e\}$ described earlier; and
- a faithful representation of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ on a Hilbert space ${\mathcal H}$ such that the canonical faithful conditional expectation of ${\mathcal B}({\mathcal H})$ onto its diagonal subalgebra implements an expectation from ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ onto the commutative $C^*$-subalgebra generated by the range projections $\{{s^m_{\alpha}} {{({s^m_{\alpha}})^{*}}} : \alpha \in
E^*\}$.
Our results deal only with row-finite graphs to simplify the exposition. However, it seems likely that a similar analysis applies to arbitrary graphs. Certainly Hong and Szymański’s characterisation of the primitive ideal space of a graph $C^*$-algebra is available for arbitrary graphs. In principle one can argue along exactly the same lines as we do in Section 3 to obtain a co-universal $C^*$-algebra for an arbitrary directed graph. Alternatively, the results of this paper could be bootstrapped to the non-row-finite situation using Drinen and Tomforde’s desingularisation process [@DT2005].
Acknowledgements. {#acknowledgements. .unnumbered}
-----------------
The author thanks Iain Raeburn for lending a generous ear, and for helpful conversations. The author also thanks Toke Carlsen for illuminating discussions, and for his suggestions after a careful reading of a preliminary draft.
Preliminaries
=============
We use the conventions and notation for directed graphs established in [@Raeburn2005]; in particular our edge-direction convention is consistent with [@Raeburn2005] rather than with, for example, [@BHRS2002; @BPRS2000; @HS2004].
A *path* in a directed graph $E$ is a concatenation $\lambda = \lambda_1\lambda_2 \ldots \lambda_n$ of edges $\lambda_i \in E^1$ such that $s(\lambda_i) = r(\lambda_{i+1})$ for $i < n$; we write $r(\lambda)$ for $r(\lambda_1)$ and $s(\lambda)$ for $s(\lambda_n)$. We denote by $E^*$ the collection of all paths in $E$. For $v \in E^0$ we write $vE^1$ for $\{e \in E^1 : r(e) = v\}$; similarly $E^1v = \{e \in E^1 :
s(e) = v\}$.
A *cycle* in $E$ is a path $\mu = \mu_1 \dots \mu_{|\mu|}$ such that $r(\mu) = s(\mu)$ and such that $s(\mu_i) \not=
s(\mu_j)$ for $1 \le i < j \le |\mu|$. Given a cycle $\mu$ in $E$, we write $[\mu]$ for the set $$[\mu] = \{\mu,\; \mu_2\mu_3\cdots\mu_{|\mu|}\mu_1,\; \dots,\; \mu_{|\mu|}
\mu_1\cdots\mu_{n-1}\}$$ of cyclic permutations of $\mu$. We write $[\mu]^0$ for the set $\{s(\mu_i) : 1 \le i \le |\mu|\} \subset E^0$, and $[\mu]^1$ for the set $\{\mu_i : 1 \le i \le |\mu|\} \subset E^1$. Given a cycle $\mu$ in $E$ and a subset $M$ of $E^0$ containing $[\mu]^0$, we say that $\mu$ has *no entrance in M* if $r(e) = r(\mu_i)$ and $s(e) \in M$ implies $e = \mu_i$ for all $1 \le i \le |\mu|$. We denote by $C(E)$ the set $\{[\mu] :
\mu$ is a cycle with no entrance in $E^0\}$. By $C(E)^1$ we mean $\bigcup_{C \in C(E)} C^1$, and by $C(E)^0$ we mean $\bigcup_{C \in C(E)} C^0$.
A *cutting set* for a directed graph $E$ is a subset $X$ of $C(E)^1$ such that for each $C \in C(E)$, $X \cap C^1$ is a singleton. Given a cutting set $X$ for $E$, for each $x \in X$, we write $\mu(x)$ for the unique cycle in $E$ such that $r(\mu)
= r(x)$, and let $\lambda(x) = \mu(x)_2\mu(x)_3 \dots
\mu(x)_{|\mu(x)|}$; so $\mu(x) = x\lambda(x)$ for all $x \in
X$, and $C(E) = \{[\mu(x)] : x \in X\}$.
Existence of the co-universal C\*-algebra {#sec:existence}
=========================================
Our main theorem asserts that every row-finite directed graph admits a co-universal $C^*$-algebra and identifies when a homomorphism obtained from the co-universal property is injective.
\[thm:Cr(E)-Existence\] Let $E$ be a row-finite directed graph.
1. \[it:existence\] There exists a $C^*$-algebra ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ which is co-universal for Toeplitz-Cuntz-Krieger $E$-families of nonzero partial isometries in the sense that ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ is generated by a Toeplitz-Cuntz-Krieger $E$-family $\{P_v : v \in E^0\},
\{S_e : e \in E^1\}$ with the following two properties.
1. \[it:generators\] The vertex projections $\{P_v : v \in E^0\}$ are all nonzero.
2. \[it:co-universal\] Given any Toeplitz-Cuntz-Krieger $E$-family $\{q_v : v
\in E^0\}, \{t_e : e \in E^1\}$ such that each $q_v \not= 0$ and given any cutting set $X$ for $E$, there is a function $\kappa : X \to {{\mathbb{T}}}$ and a homomorphism $\psi_{q,t} : C^*(\{q_v, t_e
: v \in E^0, e \in E^1\}) \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ satisfying $\psi_{q,t}(q_v) = P_v$ for all $v
\in E^0$, $\psi_{q,t}(t_e) = S_e$ for all $e
\in E^1 \setminus X$, and $\psi_{q,t}(t_x) =
\kappa(x)S_x$ for all $x \in X$.
2. \[it:injectivity\] Given a Toeplitz-Cuntz-Krieger $E$-family $\{q_v : v \in E^0\}$, $\{t_e : e \in E^1\}$ with each $q_v$ nonzero, the homomorphism $\psi_{q,t} :
B \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ obtained from (\[it:co-universal\]) is an isomorphism if and only if for each cycle $\mu$ with no entrance in $E$, the partial isometry $t_\mu$ is a scalar multiple of $q_{r(\mu)}$.
It is convenient in practise to work with cutting sets $X$ and functions from $X$ to ${{\mathbb{T}}}$ as in Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]). However, property (\[it:co-universal\]) can also be reformulated without respect to cutting sets. Indeed:
1. The asymmetry arising from the choice of a cutting set $X$ in Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]) can be avoided. The property could be reformulated equivalently as follows: *given a Toeplitz-Cuntz-Krieger $E$-family $\{q_v : v \in E^0\},
\{t_e : e \in E^1\}$ such that each $q_v \not= 0$, there is a function $\rho : C(E)^1 \to {{\mathbb{T}}}$ and a homomorphism $\psi_{q,t} : C^*(\{q_v, t_e : v \in E^0,
e \in E^1\}) \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ satisfying $\psi_{q,t}(q_v) =
P_v$ for all $v \in E^0$, $\psi_{q,t}(t_e) = S_e$ for all $e \in E^1 \setminus C(E)^1$, and $\psi_{q,t}(t_e)
= \rho(e) S_e$ for all $e \in C(E)^1$.* One can prove that an algebra satisfying this modified condition (\[it:co-universal\]) exists using exactly the same argument as for the current theorem after making the appropriate modification to Lemma \[lem:Ikappas same\]. That the resulting algebra coincides with ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ follows from applications of the co-universal properties of the two algebras.
2. Fix a row-finite graph $E$ with no sources and a function $\kappa : C(E) \to {{\mathbb{T}}}$. Let $\{q_v : v \in
E^0\}, \{t_e : e \in E^1\}$ be a Toeplitz-Cuntz-Krieger $E$-family such that each $q_v \not= 0$. Then there is a homomorphism as in Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]) with respect to the fixed function $\kappa$ for some cutting set $X$ if and only if there is such a homomorphism for every cutting set $X$. One can see this by following the argument of Lemma \[lem:what’s the big ideal\] below to see that $\kappa$ does not depend on $X$.
3. Given a Toeplitz-Cuntz-Krieger $E$-family $\{q_v : v
\in E^0\}, \{t_e : e \in E^1\}$ such that each $q_v
\not= 0$ and a cutting set $X$, the functions $\kappa :
X \to {{\mathbb{T}}}$ which can arise in Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]) are precisely those for which $\kappa([\mu])$ belongs to the spectrum ${\operatorname{sp}}_{q_v C^*(\{q_v, t_e : v \in E^0,
e \in E^1\}) q_v}(t_\mu)$ for each cycle $\mu$ without an entrance in $E$. To see this, one uses Hong and Szymański’s theorems to show that in the first paragraph of the proof of Lemma \[lem:what’s the big ideal\], the complex numbers $z$ which can arise are precisely the elements of the spectrum of the unitary $s_{\alpha(\mu)} + I$ in the corner $(p_{\alpha(r(\mu))} + I) \big(C^*({\widetilde{E}})/I\big)
(p_{\alpha(r(\mu))} + I)$.
Let $E$ be a row-finite directed graph in which every cycle has an entrance. Then $C^*(E) \cong {\ensuremath{C^*_{\operatorname{min}}(E)}}$. In particular, if $\{q_v : v \in E^0\}$, $\{t_e : e \in E^1\}$ is a Toeplitz-Cuntz-Krieger $E$-family in a $C^*$-algebra $B$ such that each $q_v$ is nonzero, then there is a homomorphism $\psi_{q,t} : C^*(\{q_v, t_e : v\in E^0, e \in E^1\}) \to
C^*(E)$ such that $\psi_{q,t}(q_v) = p_v$ for all $v \in E^0$ and $\psi_{q,t}(t_e) = s_e$ for all $e \in E^1$.
For the first statement, observe that the co-universal property of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ induces a surjective homomorphism $\psi_{p,s} :
C^*(E) \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$. Since every cycle in $E$ has an entrance, the condition in Theorem \[thm:Cr(E)-Existence\](\[it:injectivity\]) is trivially satisfied, and it follows that $\psi_{p,s}$ is an isomorphism.
Since every cycle in $E$ has an entrance, a cutting set for $E$ has no elements. Hence the second statement is just a re-statement of Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]).
The remainder of this section will be devoted to proving Theorem \[thm:Cr(E)-Existence\]. Our key technical tool in proving Theorem \[thm:Cr(E)-Existence\] will be Hong and Szymański’s description of the primitive ideal space of a graph $C^*$-algebra. To do this, we first realise the Toeplitz algebra of $C^*(E)$ as a graph algebra in its own right. This construction is known, but we have found it difficult to pin down in the literature.
\[ntn:TE\] Let $E$ be a directed graph. Define a directed graph ${\widetilde{E}}$ as follows: $$\begin{gathered}
{\widetilde{E}}^0 = \{\alpha(v) : v \in E^0\} \sqcup \{\beta(v) : v \in
E^0, 0 < |vE^1| < \infty\} \\
{\widetilde{E}}^1 = \{\alpha(e) : e \in E^1\} \sqcup \{\beta(e) : e \in
E^1, 0 < |s(e) E^1| < \infty\} \\
r(\alpha(e)) = r(\beta(e)) = \alpha(r(e)),\\
s(\alpha(e)) = \alpha(s(e)),\text{ and } s(\beta(e)) =
\beta(s(e)).\end{gathered}$$ For $\lambda \in E^*$ with $|\lambda| \ge 2$, we define $\alpha(\lambda) :=
\alpha(\lambda_1)\dots\alpha(\lambda_{|\lambda|})$. Since $E$ is row-finite, ${\widetilde{E}}$ is also row-finite.
\[lem:TE alg\] Let $E$ be a directed graph and let ${\widetilde{E}}$ be as in Notation \[ntn:TE\]. For $v \in E^0$ and $e \in E^1$, let $$\begin{aligned}
q_v &:=
\begin{cases}
p_{\alpha(v)} + p_{\beta(v)} & \text{ if $0 < |vE^1| < \infty$} \\
p_{\alpha(v)} & \text{ otherwise,}
\end{cases} \\
\intertext{and}
t_e &:=
\begin{cases}
s_{\alpha(e)} + s_{\beta(e)} & \text{ if $0 < |s(e)E^1| < \infty$}\\
s_{\alpha(e)} & \text{ otherwise.}
\end{cases}\end{aligned}$$ Then there is an isomorphism $\phi : {\mathcal T}C^*(E) \to C^*({\widetilde{E}})$ satisfying $\phi({p^{\mathcal T}_{v}} = q_v$ and $\phi({s^{\mathcal T}_{e}}) = t_e$ for all $v \in E^0$ and $e \in E^1$.
Routine calculations show that $\{q_v : v \in E^0\}$, $\{t_e :
e \in E^1\}$ is a Toeplitz-Cuntz-Krieger $E$-family in $C^*({\widetilde{E}})$. The universal property of ${\mathcal T}C^*(E)$ therefore implies that there is a homomorphism $\phi : {\mathcal T}C^*(E) \to
C^*({\widetilde{E}})$ satisfying $\phi({p^{\mathcal T}_{v}}) = q_v$ for all $v \in E^0$ and $\phi({s^{\mathcal T}_{e}}) = t_e$ for all $e \in E^1$.
To see that $\phi$ is surjective, fix $v \in {\widetilde{E}}^0$ and $E \in
{\widetilde{E}}^1$. To see that $p_v\in {\operatorname{range}}(\phi)$, we consider three cases: (a) $v = \alpha(w)$ for some $w$ with $wE^1$ either empty or infinite; (b) $v = \alpha(w)$ for some $w$ with $0 <
|wE^1| < \infty$; or (c) $v = \beta(w)$ for some $w$ with $0 <
|wE^1| < \infty$. In case (a), we have $p_v = p_{\alpha(w)} =
\phi({p^{\mathcal T}_{w}})$ by definition. In case (b), the set $v{\widetilde{E}}^1 =
\{\alpha(e), \beta(e) : e \in wE^1\}$ is nonempty and finite. Hence the Cuntz-Krieger relation in $C^*({\widetilde{E}})$ ensures that $$\label{eq:CK for Toeplitz}
p_v = p_{\alpha(w)}
= \sum_{e \in vE^1} s_{\alpha(e)} s^*_{\alpha(e)} + s_{\beta(e)} s^*_{\beta(e)}
= \sum_{e \in vE^1} t_e t^*_e \in {\operatorname{range}}(\phi).$$ In case (c), we have $p_v = q_v - p_{\alpha(w)} \in
{\operatorname{range}}(\phi)$ by case (b). Now to see that $s_e \in
{\operatorname{range}}(\phi)$, observe that if $e = \alpha(f)$ for some $f \in
E^1$, then $s_e = s_{\alpha(f)} = \phi({s^{\mathcal T}_{f}}) p_{\alpha(s(f))}
\in {\operatorname{range}}(\phi)$, and if $e = \beta(f)$, then $s_e =
s_{\beta(f)} = \phi({s^{\mathcal T}_{f}}) p_{\beta(s(f))} \in {\operatorname{range}}(\phi)$ also.
To finish the proof, observe that if $0 < |vE^1| < \infty$, then $q_v - \sum_{r(e) = v} t_e t^*_e = p_{\beta(v)} \not= 0$. Since the $t_e$ are all nonzero and have mutually orthogonal ranges, it follows that for each $v \in E^0$ and each finite subset $F$ of $vE^1$, we have $q_v - \sum_{e \in F} t_e t^*_e
\not= 0$. Thus the uniqueness theorem [@FR1999 Theorem 4.1] for ${\mathcal T}C^*(E)$ implies that $\phi$ is injective.
\[ntn:Ikappa\] Let $E$ be a directed graph.
1. For $v \in E^0$ such that $0 < |vE^1| < \infty$, we define $\Delta_v := {p^{\mathcal T}_{v}} - \sum_{r(e) = v}
{s^{\mathcal T}_{e}}{{({s^{\mathcal T}_{e}})^{*}}} \in {\mathcal T}C^*(E)$.
2. Given a function $\kappa : C(E) \to {{\mathbb{T}}}$, we denote by $I^\kappa$ the ideal of ${\mathcal T}C^*(E)$ generated by $\{\Delta_v : v \in E^0\} \cup \{\kappa(C) {p^{\mathcal T}_{r(\mu)}}
- {s^{\mathcal T}_{\alpha(\mu)}} : C \in C(E), \mu \in C\}$.
\[lem:Ikappas same\] Let $E$ be a directed graph. Let $\kappa$ be a function from $C(E) \to {{\mathbb{T}}}$, and let $1 : C(E) \to {{\mathbb{T}}}$ denote the constant function $1(C) = 1$ for all $C \in C(E)$. Fix a cutting set $X$ for $E$, and for each $x \in X$, let $C(x)$ be the unique element of $C(E)$ such that $x \in C(x)^1$. Then there is an isomorphism $\widetilde{\tau_{\overline{\kappa}}} : {\mathcal T}C^*(E)
/ I^1 \to {\mathcal T}C^*(E) / I^\kappa$ satisfying $$\begin{array}{rcll}
\widetilde{\tau_{\overline{\kappa}}}({p^{\mathcal T}_{v}}+ I^1) &=& p_v + I^\kappa &\qquad\text{for all $v \in E^0$} \\
\widetilde{\tau_{\overline{\kappa}}}({s^{\mathcal T}_{e}} + I^1) &=& s_e + I^\kappa &\qquad\text{for all $e \in E^1 \setminus X$, and} \\
\widetilde{\tau_{\overline{\kappa}}}({s^{\mathcal T}_{x}} + I^1) &=& \overline{\kappa(C(x))}s_x + I^\kappa &\qquad\text{for all $x \in X$.}
\end{array}$$
By the universal property of ${\mathcal T}C^*(E)$, there is an action $\tau$ of ${{\mathbb{T}}}^{C(E)}$ on ${\mathcal T}C^*(E)$ such that for $\rho \in
{{\mathbb{T}}}^{C(E)}$, we have $$\begin{array}{rcll}
{\tau_\rho}({p^{\mathcal T}_{v}}) &=& {p^{\mathcal T}_{v}} &\qquad\text{for all $v \in E^0$} \\
{\tau_\rho}({s^{\mathcal T}_{e}}) &=& {s^{\mathcal T}_{e}} &\qquad\text{for all $e \in E^1 \setminus X$, and} \\
{\tau_\rho}({s^{\mathcal T}_{x}}) &=& \rho(C(x)) {s^{\mathcal T}_{x}} &\qquad\text{for all $x \in X$.}
\end{array}$$ By definition of $I^1$ and $I^\kappa$ and of the action $\tau$, we have $\tau_{\overline{\kappa}}(I^1) = I^\kappa$. Hence $\tau_{\overline{\kappa}}$ determines an isomorphism $$\widetilde{\tau_{\overline{\kappa}}} : {\mathcal T}C^*(E)/I^1 \to \tau_{\overline{\kappa}}({\mathcal T}C^*(E)) / I^\kappa = {\mathcal T}C^*(E) / I^\kappa$$ satisfying $\widetilde{\tau_{\overline{\kappa}}}(a + I^1) =
\tau_{\overline{\kappa}}(a) + I^\kappa$ for all $a \in {\mathcal T}C^*(E)$.
\[lem:Ikappa vert proj free\] Let $E$ be a directed graph. Fix a function $\kappa : C(E) \to
{{\mathbb{T}}}$. Then ${s^{\mathcal T}_{v}} \not\in I^\kappa$ for all $v \in E^0$.
To prove this lemma, we need a little notation.
\[ntn:bdryCKfam\] Given a directed graph $E$, we denote by $E^{\le\infty}$ the collection $E^\infty \cup \{\alpha \in E^* : s(\alpha)E^1 =
\emptyset\}$. There is a Cuntz-Krieger $E$-family in ${\mathcal B}(\ell^2(E^{\le\infty}))$ determined by $$P^\infty_v \xi_x
= \begin{cases}
\xi_x &\text{ if $r(x) = v$} \\
0 &\text{ otherwise,}
\end{cases}$$ and $$S^\infty_e \xi_x
= \begin{cases}
\xi_{ex} &\text{ if $r(x) = s(e)$} \\
0 &\text{ otherwise.}
\end{cases}$$
If $\mu$ is a cycle with no entrance in $E$, then $r(\mu)E^{\le
\infty} = \{\mu^\infty\}$, so $S^\infty_{\mu} =
P^\infty_{r(\mu)}$.
By Lemma \[lem:Ikappas same\], it suffices to show that $I^1$ contains no vertex projections. Let $\pi_{P^\infty, S^\infty} :
{\mathcal T}C^*(E) \to {\mathcal B}(\ell^2(E^{\le\infty}))$ be the representation obtained from then universal property of ${\mathcal T}C^*(E)$ applied to the Cuntz-Krieger family of Notation \[ntn:bdryCKfam\]. Then $\ker(\pi_{P^\infty,
S^\infty})$ contains all the generators of $I^1$, so $I^1
\subset \ker(\pi_{P^\infty, S^\infty})$. Moreover, $\ker(\pi_{P^\infty, S^\infty})$ contains no vertex projections because each vertex of $E$ is the range of at least one $x \in
E^{\le \infty}$.
From this point onward we make the simplifying assumption that our graphs are row-finite. Though there is no obvious obstruction to our analysis without this restriction, the added generality would complicate the details of our arguments. In any case, if the added generality should prove useful, it should not be difficult to bootstrap our results to the non-row-finite setting by means of the Drinen-Tomforde desingularisation process applied to the graph ${\widetilde{E}}$ of Notation \[ntn:TE\].
\[prop:Toeplitz big ideal\] Let $I$ be an ideal of ${\mathcal T}C^*(E)$ such that ${p^{\mathcal T}_{v}} \not\in
I$ for all $v \in E^0$. There is a function $\kappa : C(E) \to
{{\mathbb{T}}}$ such that $I \subset I^\kappa$.
To prove the proposition, we first establish our key technical lemma. This lemma is implicit in Hong and Szymański’s description [@HS2004] of the primitive ideal space of $C^*({\widetilde{E}})$, but it takes a little work to tease a proof of the statement out of their two main theorems.
\[lem:what’s the big ideal\] Let ${\widetilde{E}}$ be the directed graph of Notation \[ntn:TE\]. Let $I$ be an ideal of $C^*({\widetilde{E}})$ such that $p_{\alpha(v)} \not\in
I$ for all $v \in E^0$. There is a function $\kappa : C(E) \to
{{\mathbb{T}}}$ such that $I$ is contained in the ideal $J^\kappa$ of $C^*({\widetilde{E}})$ generated by $\{p_{\beta(v)} : v \in E^0\}$ and $\{\kappa(C) p_{\alpha(r(\mu))} - s_{\alpha(\mu)} : C \in C(E),
\mu \in C\}$.
Before proving the lemma, we summarise some notation and results of [@HS2004] as they apply to the row-finite directed graph ${\widetilde{E}}$ in the situation of Lemma \[lem:what’s the big ideal\]. A *maximal tail* of ${\widetilde{E}}$ is a subset $M$ of ${\widetilde{E}}^0$ such that
- $w \in M$ and $v {\widetilde{E}}^* w \not= \emptyset$ imply $v \in M$;
- if $v \in M$ and $v{\widetilde{E}}^1 \not= \emptyset$, then there exists $e \in v{\widetilde{E}}^1$ such that $s(e) \in
M$; and
- if $u, v \in M$, then there exists $w \in M$ such that $u {\widetilde{E}}^* w \not= \emptyset$ and $v {\widetilde{E}}^* w
\not= \emptyset$.
We denote by ${\mathcal M}_\gamma({\widetilde{E}})$ the collection of maximal tails $M$ of ${\widetilde{E}}$ such that every cycle $\mu$ satisfying $[\mu]^0
\subset M$ has an entrance in $M$. We denote by ${\mathcal M}_\tau({\widetilde{E}})$ the collection of maximal tails $M$ of ${\widetilde{E}}$ such that there is a cycle $\mu$ in ${\widetilde{E}}$ for which $[\mu]^0 \subset M$ but $\mu$ has no entrance in $M$. Since each $\beta(v)$ is a source in ${\widetilde{E}}$, the cycles in ${\widetilde{E}}$ are the paths of the form $\alpha(\mu)$ where $\mu$ is a cycle in $E$. Moreover $\alpha(\mu) \in C({\widetilde{E}})$ if and only if $\mu \in C(E)$. Thus if $M \in M_\tau({\widetilde{E}})$, then there is a unique $C_M \in C(E)$ such that $\alpha(C_M^0) \subset M$ and $\alpha(\mu)$ has no entrance in $M$ for each $\mu \in C_M$. We may recover $M$ from $C_M$: $$M = \{\alpha(v) : v \in E^0, v E^* C_M^0 \not=
\emptyset\}.$$
The gauge-invariant primitive ideals of $C^*({\widetilde{E}})$ are indexed by ${\mathcal M}_\gamma({\widetilde{E}})$; specifically, $M \in {\mathcal M}_\gamma({\widetilde{E}})$ corresponds to the primitive ideal ${\operatorname{PI}}^\gamma_M$ generated by $\{p_w : w \in {\widetilde{E}}^0 \setminus M\}$. The non-gauge-invariant primitive ideals of $C^*({\widetilde{E}})$ are indexed by ${\mathcal M}_\tau({\widetilde{E}})
\times {{\mathbb{T}}}$; specifically, the pair $(M,z)$ corresponds to the primitive ideal ${\operatorname{PI}}^\tau_{M,z}$ generated by $\{p_w : w \in
E^0 \setminus M\} \cup \{zp_{r(\mu)} - s_\mu\}$ for any $\mu
\in C_M$ (the ideal does not depend on the choice of $\mu \in
C_M$). Corollary 3.5 of [@HS2004] describes the closed subsets of the primitive ideal space of $C^*({\widetilde{E}})$ in terms of subsets of ${\mathcal M}_\gamma({\widetilde{E}}) \sqcup {\mathcal M}_\tau({\widetilde{E}}) \times {{\mathbb{T}}}$.
We begin by constructing the function $\kappa$. Fix $C \in
C(E)$. Since $p_{\alpha(v)} \not\in I$ for each $v \in C^0$, we may fix a primitive ideal $J^{C}$ of $C^*({\widetilde{E}})$ such that $I
\subset J^{C}$ and $p_{\alpha(v)} \not\in J^{C}$ for $v \in
C^0$. By [@HS2004 Corollary 2.12], we have either $J^{C} =
{\operatorname{PI}}^\gamma_M$ for some $M \in {\mathcal M}_\gamma({\widetilde{E}})$, or $J^{C} =
{\operatorname{PI}}^\tau_{M,z}$ for some $M \in {\mathcal M}_\tau({\widetilde{E}})$ and $z \in {{\mathbb{T}}}$. Since $p_{\alpha(v)} \not\in J^{C}$ for $v \in C^0$, we must have $\alpha(C^0) \subset M$, and then the maximal tail condition forces $$M = M_C := \{\alpha(v) : v \in E^0, v E^* C^0
\not= \emptyset\} \in M_\tau({\widetilde{E}}).$$ Hence $J^{C} = {\operatorname{PI}}^\tau_{M,z}$ for some $z \in {{\mathbb{T}}}$; we set $\kappa(C) := z$.
For $C \in C(E)$ and $v \in C^0$, let $J^v :=
{\operatorname{PI}}^\tau_{M_C,\kappa(C)}$. Since $\beta(v) \not \in M_C$ for all $v \in E^0$, [@HS2004 Lemma 2.8] implies that $p_{\beta(v)} \in J^v$ for all $v \in E^0$, and our definition of $J^v$ ensures that $p_{\alpha(v)} \not \in J^v$.
We claim that the assignment $v \mapsto J^v$ of the preceding paragraph extends to a function $v \mapsto J^v$ from $E^0$ to the primitive ideal space of $C^*({\widetilde{E}})$ such that for every $v
\in E^0$,
1. $I \subset J^v$;
2. \[it:no p(alpha)s\] $p_{\alpha(v)} \not\in J^v$;
3. $p_{\beta(w)} \in J^v$ for all $w \in E^0$; and
4. \[it:kappa(C) or gi\] either $J^v = {\operatorname{PI}}^\tau_{M_C,
\kappa(C)}$ for some $C \in C(E)$, or $J^v$ is gauge-invariant.
To prove the claim, fix $v \in E^0$. If $v E^* C^0
\not=\emptyset$ for some $C \in C(E)$, then $J^v :=
{\operatorname{PI}}^\tau_{M_C, \kappa(C)}$ has the desired properties. So suppose that $v E^* C^0 = \emptyset$ for all $C \in C(E)$. Then there exists $x^v \in vE^{\le\infty}$ such that $x^v$ does not have the form $\lambda\mu^\infty$ for any $\lambda \in E^*$ and cycle $\mu \in E$. The set $$M^v := \{\alpha(w) : w \in E^0, w E^* x^v(n) \not=\emptyset\text{ for some }n \in {{\mathbb{N}}}\}$$ is a maximal tail of ${\widetilde{E}}$ which contains $\alpha(v)$ and does not contain $\beta(w)$ for any $w \in E^0$. By construction of $x^v$, every cycle in $M^v$ has an entrance in $M^v$, and so $J^v := {\operatorname{PI}}^\gamma_{M^v}$ satisfies (\[it:no p(alpha)s\])–(\[it:kappa(C) or gi\]). It therefore suffices to show that $I \subset {\operatorname{PI}}^\gamma_{M^v}$. For each $n \in
{{\mathbb{N}}}$, we have $p_{\alpha(x^v(n))} \not\in I$, so there is a primitive ideal $J$ of $C^*({\widetilde{E}})$ containing $I$ and not containing $p_{\alpha(x^v(n))}$. The set $M_n = \{w \in {\widetilde{E}}^0 :
p_w \not\in J\}$ is a maximal tail of ${\widetilde{E}}$, so either $M_n \in
{\mathcal M}_\gamma({\widetilde{E}})$ and $J = {\operatorname{PI}}^\gamma_{M_n}$, or $M_n \in
{\mathcal M}_\tau({\widetilde{E}})$ and $J = {\operatorname{PI}}^\tau_{M_n, z}$ for some $z \in
{{\mathbb{T}}}$. By definition, $x^v(n) \in M_n$, and then (MT1) forces $\alpha(w) \in M_n$ for all $w \in E^0$ such that $w E^* x^v(m)
\not= \emptyset$ for some $m \le n$. Hence $M^v \subset \cup_{n
\in {{\mathbb{N}}}} M_n$. Hence parts (1) and (3) of [@HS2004 Corollary 3.5] imply that $J^v$ belongs to the closure of the set of primitive ideals of $C^*({\widetilde{E}})$ which contain $I$, and hence contains $I$ itself. This proves the claim.
Observe that $I \subset \bigcap_{v \in E^0} J^v$. To prove the result, it therefore suffices to show that $\bigcap_{v \in E^0}
J^v$ is generated by $\{p_{\beta(v)} : v \in E^0\}$ and $\{\kappa(C) p_{\alpha(r(\mu))} - s_{\alpha(\mu)} : C \in C(E),
\mu \in C\}$.
For this, first observe that for $v \in E^0$, we have $p_{\beta(v)} \in \bigcap_{v \in E^0} J^v$ because $p_{\beta(v)}$ belongs to each $J^v$. Fix $C \in C(E)$ and $\mu
\in C$. Since the cycle without an entrance belonging to a given maximal tail of $E$ is unique, for each $v \in E^0$, we have either $J^v = {\operatorname{PI}}^\tau_{M_C, \kappa(C)}$, or $p_{r(\mu)}
\in J^v$. In particular, $\kappa(C) p_{\alpha(r(\mu))} -
s_{\alpha(\mu)} \in J^v$ for each $v \in E^0$, so $\kappa(C)
p_{\alpha(r(\mu))} - s_{\alpha(\mu)} \in \bigcap_{v \in E^0}
J^v$. Hence $J^\kappa \subset \bigcap_{v \in E^0} J^v$, and it remains to establish the reverse inclusion.
Fix a primitive ideal $J$ of $C^*({\widetilde{E}})$ which contains all the generators of $J^\kappa$. It suffices to show that $\bigcap_{v
\in E^0} J^v \subset J$. Under the bijection between primitive ideals of $C^*({\widetilde{E}})$ and elements of ${\mathcal M}_\gamma({\widetilde{E}}) \sqcup
{\mathcal M}_\tau({\widetilde{E}}) \times {{\mathbb{T}}}$, the collection $\{J_v : v \in E^0\}$ is sent to $\{M^v : v E^* C^0 = \emptyset\text{ for all }C \in
C(E)\} \sqcup \{(M_C, \kappa(C)) : C \in C(E)\}$. Since each $J^v$ trivially contains $\bigcap_{v \in E^0} J^v$, it therefore suffices to show that the element $N_J$ of ${\mathcal M}_\gamma({\widetilde{E}}) \sqcup {\mathcal M}_{\tau}({\widetilde{E}}) \times {{\mathbb{T}}}$ corresponding to $J$ satisfies $$\label{eq:MJ in closure}
N_J \in \overline{\{M^v : v E^*
C^0 = \emptyset\text{ for all }C \in C(E)\}} \sqcup \overline{\{(M_C,
\kappa(C)) : C \in C(E)\}}.$$ Let $M_J := \{v \in {\widetilde{E}}^0 : p_v \not\in J\}$. Then either $J$ is gauge-invariant and $N_J = M_J$, or $J$ is not gauge-invariant, and $N_J = (M_J, z)$ for some $z \in {{\mathbb{T}}}$. Observe that $$\label{eq:MJ in alphas}
\begin{split}
M_J &\subset \{\alpha(v) : v \in E^0\} \\
&= \Big(\bigcup\big\{M^v : v E^* C^0 = \emptyset\text{ for all }C \in C(E)\big\}\Big) \\
&\hskip4cm\cup \Big(\bigcup\big\{(M_C, \kappa(C)) : C \in C(E)\big\}\Big).
\end{split}$$ We now consider three cases.
Case 1: $J$ is gauge-invariant. Then $M_J \in M_\gamma({\widetilde{E}})$, and $N_J = M_J$. In this case, together with parts (1) and (3) of [@HS2004 Corollary 3.5] give .
Case 2: $J$ is not gauge-invariant, and $M_J \not= M_C$ for all $C \in C(E)$. We have $N_J = (M_J, z)$ for some $z \in {{\mathbb{T}}}$, and since $M_J \not= M_C$ for all $C \in C(E)$, it follows that $N_J$ does not belong to the subset $\{(M_C, \kappa(C)) : C \in
C(E)\}_{\rm min}$ of $\{(M_C, \kappa(C)) : C \in C(E)\}$ defined on page 58 of [@HS2004]. Hence parts (2) and (4ii) of [@HS2004 Corollary 3.5] imply .
Case 3: $M_J = M_C$ for some $C \in C(E)$. Fix $\mu \in C$. Since $J$ contains $\kappa(C)p_{\alpha(r(\mu))} -
s_{\alpha(\mu)}$, we have $N_J = (M_C, \kappa(C))$ and then part (4iii) of [@HS2004 Corollary 3.5] implies . This completes the proof.
Let $\phi : {\mathcal T}C^*(E) \to C^*({\widetilde{E}})$ be the isomorphism of Lemma \[lem:TE alg\]. Observe that by , we have $\phi(\Delta_v) = p_{\beta(v)}$ for all $v
\in E^0$ such that $vE^1 \not=\emptyset$. We claim that $p_{\alpha(v)} \not\in \phi(I)$ for all $v \in E^0$. To see this, first suppose that $vE^1 = \emptyset$. Then $p_{\alpha(v)} = \phi({p^{\mathcal T}_{v}}) \not\in \phi(I)$ by assumption. Now suppose that $vE^1 \not= \emptyset$, say $r(e) = v$. Then $$s_{\alpha(e)}^* p_{\alpha(v)} s_{\alpha(e)} + s_{\beta(e)}^* p_{\alpha(v)} s_{\beta(e)}
= p_{\alpha(s(e))} + p_{\beta(s(e))} = \phi({p^{\mathcal T}_{s(e)}}) \not\in \phi(I)$$ by assumption. This forces $p_{\alpha(v)} \not\in \phi(I)$.
Lemma \[lem:what’s the big ideal\] therefore applies to the ideal $\phi(I)$ of $C^*({\widetilde{E}})$. Let $\kappa : C(E) \to {{\mathbb{T}}}$ and $J^\kappa \lhd C^*({\widetilde{E}})$ be the resulting function and ideal. Then $I^\kappa := \phi^{-1}(J^\kappa)$ is generated by $\{\Delta_v : v \in E^0\}$ and $\{\kappa(C) {p^{\mathcal T}_{r(\mu)}} -
{s^{\mathcal T}_{\alpha(\mu)}} : C \in C(E), \mu \in C\}$ by definition of $\phi$, and contains $I$ by construction.
We are now ready to prove our main theorem.
With $I^1 \lhd {\mathcal T}C^*(E)$ defined as in Notation 3.6 and Lemma 3.7, define ${\ensuremath{C^*_{\operatorname{min}}(E)}} := {\mathcal T}C^*(E) / I^1$. For $v \in E^0$ and $e \in E^1$, let $P_v := {p^{\mathcal T}_{v}} + I^1$ and $S_e := {s^{\mathcal T}_{v}} +
I^1$. Then $\{P_v : v \in E^0\}$, $\{S_e : e \in E^1\}$ is a Cuntz-Krieger $E$-family which generates ${\ensuremath{C^*_{\operatorname{min}}(E)}}$. The $P_v$ are all nonzero by Lemma \[lem:Ikappa vert proj free\].
Now let $\{q_v : v \in E^0\}, \{t_e : e \in E^1\}$ be a Toeplitz-Cuntz-Krieger $E$-family such that $q_v \not= 0$ for all $v$, and let $B := C^*(\{q_v, t_e : v \in E^0, e \in
E^1\})$. The universal property of ${\mathcal T}C^*(E)$ implies that there is a homomorphism $\pi_{q,t} : {\mathcal T}C^*(E) \to B$ satisfying $\pi_{q,t}({p^{\mathcal T}_{v}}) = q_v$ for all $v \in E^0$ and $\pi_{q,t}({s^{\mathcal T}_{e}}) = t_e$ for all $e \in E^1$. Since each $q_v$ is nonzero, $I = \ker(\pi_{q,t})$ is an ideal of ${\mathcal T}C^*(E)$ such that ${p^{\mathcal T}_{v}} \not\in I$ for all $v \in E^0$. Let $\kappa :
C(E) \to {{\mathbb{T}}}$ and $I^\kappa$ be as in Corollary \[prop:Toeplitz big ideal\]. Since $I \subset
I^\kappa$, there is a well-defined homomorphism $\psi_0 : B \to
{\mathcal T}C^*(E)/I^\kappa$ satisfying $\psi_0(q_v) = {p^{\mathcal T}_{v}} +
I^\kappa$ for all $v \in E^0$ and $\psi_0(t_e) = {s^{\mathcal T}_{e}} +
I^\kappa$ for all $e \in E^1$. Let $\widetilde{\tau_{\overline{\kappa}}} : {\mathcal T}C^*(E)/I^1 \to {\mathcal T}C^*(E)/I^\kappa$ be as in \[lem:Ikappas same\]. Then $\psi_{q,t} := \widetilde{\tau_{\overline{\kappa}}}^{-1} \circ
\psi_0$ has the desired property. This proves statement \[it:existence\].
For statement \[it:injectivity\], suppose first that $\psi_{q,t}$ is injective. For each $\mu \in C(E)$, we have $S_\mu = P_{r(\mu)}$ by definition of $I^1$. Let $x(\mu)$ be the unique element of the cutting set $X$ which belongs to $[\mu]^1$. With $\psi_{q,t}$ and $\kappa$ are as in (\[it:existence\]), we have $\psi_{q,t}(t_\mu) =
\kappa(x(\mu)) \psi_{q,t}(q_{r(\mu)})$. Since $\psi_{q,t}$ is injective, we must have $t_\mu = \kappa(x(\mu)) q_{r(\mu)}$ for all $\mu \in C(E)$. Now suppose that there is a function $\kappa : C(E) \to {{\mathbb{T}}}$ such that $t_\mu = \kappa([\mu])
q_{r(\mu)}$ for every cycle $\mu$ with no entrance in $E$. Then the kernel $I_{q,t}$ of the canonical homomorphism $\pi_{q,t} :
{\mathcal T}C^*(E) \to B$ contains the generators of $I^\kappa$, and hence contains $I^\kappa$. Since the $q_v$ are all nonzero, Corollary \[prop:Toeplitz big ideal\] implies that we also have $I_{q,t} \subset I^{\lambda}$ for some $\lambda : C(E) \to
{{\mathbb{T}}}$. We claim that $\kappa = \lambda$; for if not, then there exists $C \in C(E)$ such that $k := \kappa(C)$ is distinct from $l = \lambda(C)$. For $\mu \in C$, we then have $k{p^{\mathcal T}_{v}} -
{s^{\mathcal T}_{\mu}}, l{p^{\mathcal T}_{v}} - {s^{\mathcal T}_{\mu}} \in I^\lambda$. But then $(k - l)
{p^{\mathcal T}_{v}} \in I^\lambda$, which is impossible by Lemma \[lem:Ikappa vert proj free\]. Hence $I^\lambda =
I_{q,t} = I^\kappa$, and $\psi_{q,t}$ is injective.
Properties of the co-universal C\*-algebra {#sec:properties}
==========================================
In this section we prove a uniqueness theorem for ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ in terms of its co-universal property. We go on to explore the structure and properties of the co-universal algebra. Throughout this section we have preferred proofs which emphasise the utility of the co-universal property over other techniques.
Let $E$ be a row-finite directed graph. We say that a Cuntz-Krieger $E$-family $\{p_v : v \in E^0\}$, $\{s_e : e \in
E^1\}$ is a *reduced Cuntz-Krieger $E$-family* if
- for every cycle $\mu$ without an entrance in $E^0$, there is a scalar $\kappa(\mu) \in {{\mathbb{T}}}$ such that $s_\mu = \kappa(\mu)p_{r(\mu)}$.
We say that $\{p_v : v \in E^0\}$, $\{s_e : e \in E^1\}$ is a *normalised reduced Cuntz-Krieger $E$-family* if $s_\mu =
p_{r(\mu)}$ for each cycle $\mu$ without an entrance in $E^0$.
\[thm:Cr(E)-Properties\] Let $E$ be a row-finite directed graph.
1. \[it:generation\] There is a normalised reduced Cuntz-Krieger $E$-family $\{{p^m_{v}} : v \in E^0\} \cup
\{{s^m_{e}} : E \in E^1\}$ which generates ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ and satisfies Theorem \[thm:Cr(E)-Existence\](\[it:generators\]) and (\[it:co-universal\]). In particular, given any cutting set $X$ for $E$, ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ is generated by $\{{p^m_{v}} : v \in E^0\} \cup
\{{s^m_{e}} : E \in E^1 \setminus X\}$.
2. \[it:uniqueness\] Any other $C^*$-algebra generated by a Toeplitz-Cuntz-Krieger $E$-family satisfying Theorem \[thm:Cr(E)-Existence\](\[it:generators\]) and (\[it:co-universal\]) is isomorphic to ${\ensuremath{C^*_{\operatorname{min}}(E)}}$.
For (\[it:generation\]) let $\{P^\infty_v : v \in E^0\}$, $\{S^\infty_e : e \in E^1\}$ be the Cuntz-Krieger $E$-family of Notation \[ntn:bdryCKfam\]. Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]) ensures that there is a function $\kappa : X \to {{\mathbb{T}}}$ and a homomorphism $\psi_{P^\infty, S^\infty}$ from $C^*(\{P^\infty_v, S^\infty_e : v \in E^0, e \in E^1\})$ onto ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ such that $\psi_{P^\infty, S^\infty}(P^\infty_v) =
P_v$ for all $v \in E^0$, $\psi_{P^\infty,
S^\infty}(S^\infty_e) = S_e$ for all $e \in E^1 \setminus X$, and $\psi_{P^\infty, S^\infty}(S^\infty_x) = \kappa(x)S_x$ for all $x \in X$. Hence ${p^m_{v}} := \psi_{P^\infty,
S^\infty}(P^\infty_v)$ and ${s^m_{e}} := \psi_{P^\infty,
S^\infty}(S^\infty_e)$ generate ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ and satisfy Theorem \[thm:Cr(E)-Existence\](\[it:generators\]) and (\[it:co-universal\]). To prove the last assertion of (\[it:generation\]), fix $x
\in X$ and calculate: $$\begin{aligned}
S^\infty_x
&= S^\infty_x S^\infty_{\lambda(x)} (S^\infty_{\lambda(x)})^* \\
&= S^\infty_{\mu(x)} (S^\infty_{\lambda(x)})^* \\
&= (S^\infty_{\lambda(x)})^* \\
&\in C^*(\{P^\infty_v, S^\infty_e : v \in E^0, E \in E^1 \setminus X\}.\end{aligned}$$
For (\[it:uniqueness\]), let $A$ be another $C^*$-algebra generated by a Toeplitz-Cuntz-Krieger family $\{p^A_v : v \in
E^0\}$, $\{s^A_e : e \in E^1\}$ with each $p^A_v$ nonzero, and suppose that $A$ has the same two properties as ${\ensuremath{C^*_{\operatorname{min}}(E)}}$. Applying the co-universal properties, we see that there are surjective homomorphisms $\phi : {\ensuremath{C^*_{\operatorname{min}}(E)}} \to A$ and $\psi : A
\to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ such that $\phi({p^m_{v}}) = p^A_v$, $\phi({s^m_{e}}) =
s^A_e$, $\psi(p^A_v) = {p^m_{v}}$, and $\psi(s^A_e) = {s^m_{e}}$ for all $v \in E^0$ and $e \in E^1 \setminus X$. In particular, $\phi$ and $\psi$ are inverse to each other, and hence are isomorphisms.
Of course statement (\[it:generation\]) of Theorem \[thm:Cr(E)-Properties\] follows from the definition of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ (embedded in the proof of Theorem \[thm:Cr(E)-Existence\]). However the argument given highlights how it follows from the co-universal property.
\[cor:injectivity criterion\] Let $E$ be a row-finite directed graph. Let $\phi : {\ensuremath{C^*_{\operatorname{min}}(E)}} \to
B$ be a homomorphism. Then $\phi$ is injective if and only if $\phi({p^m_{v}}) \not= 0$ for all $v \in E^0$.
Suppose that $\phi$ is injective. Then that each ${p^m_{v}} \not=
0$ implies that each $\phi({p^m_{v}}) \not= 0$ also.
Now suppose that $\phi({p^m_{v}}) \not= 0$ for all $v \in E^0$. Then the co-universal property of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ ensures that for any cutting set $X$ for $E$, there is a homomorphism $\psi :
\phi({\ensuremath{C^*_{\operatorname{min}}(E)}}) \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ satisfying $\psi(\phi({p^m_{v}})) =
{p^m_{v}}$ for all $v \in E^0$ and $\psi(\phi({s^m_{e}})) = {s^m_{e}}$ for all $e \in E^1 \setminus X$. Theorem \[thm:Cr(E)-Properties\](\[it:generation\]) implies that $\psi$ is surjective and an inverse for $\phi$.
\[cor:simplicity\] Let $E$ be a row-finite directed graph. The $C^*$-algebra ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ is simple if and only if $E$ is cofinal.
First suppose that $E$ is cofinal. Fix a homomorphism $\phi :
{\ensuremath{C^*_{\operatorname{min}}(E)}} \to B$. We must show that $\phi$ is either trivial or injective. The argument of [@Raeburn2005 Proposition 4.2] shows that $\phi({p^m_{v}}) = 0$ for any $v \in E^0$, then $\phi({p^m_{w}}) = 0$ for all $v \in E^0$, which forces $\phi =
0$. On the other hand, if $\phi({p^m_{w}}) \not = 0$ for all $w
\in E^0$, then Corollary \[cor:injectivity criterion\] implies that $\phi$ is injective.
Now suppose that $E$ is not cofinal. Fix $v \in E^0$ and $x \in
E^{\le \infty}$ such that $v E^* x(n) = \emptyset$ for all $n
\in {{\mathbb{N}}}$. Standard calculations show that $$I_x := {\overline{{\mathop{\operatorname{span}}\nolimits}}}\{{s^m_{\alpha}} {{({s^m_{\beta}})^{*}}} : s(\alpha) = s(\beta) = x(n)\text{ for some } n \le |x|\}$$ is an ideal of ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ which is nontrivial because it contains ${p^m_{x(0)}}$. To see that ${p^m_{v}} \not\in I_x$, fix $n \le |x|$ and $\alpha, \beta \in E^* x(n)$. It suffices to show that ${p^m_{v}} {s^m_{\alpha}} {{({s^m_{\beta}})^{*}}} = 0$. Let $l := |\alpha|$. By Theorem \[thm:Cr(E)-Existence\](\[it:injectivity\]), $\{{p^m_{v}} : v \in E^0\}$, $\{{s^m_{e}}, e \in E^1\}$ is a reduced Cuntz-Krieger $E$-family, and in particular a standard inductive argument based on relation (CK) shows that $${p^m_{v}} = \sum_{\lambda \in v E^{\le l}} {s^m_{\lambda}}{{({s^m_{\lambda}})^{*}}}.$$ Fix $\lambda \in v E^{\le l}$. Since $v E^* x(n) = \emptyset$ for all $n \in {{\mathbb{N}}}$, we have $\alpha \not= \lambda\lambda'$ for all $\lambda' \in E^*$. Since $|\alpha| = l \ge |\lambda|$, it follows that ${{({s^m_{\lambda}})^{*}}} {s^m_{\alpha}} = 0$. Hence ${p^m_{v}}
{s^m_{\alpha}} {{({s^m_{\beta}})^{*}}} = 0$ as claimed.
\[cor:universal property\] Let $E$ be a row-finite directed graph. The $C^*$-algebra ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ is the universal $C^*$-algebra generated by a normalised reduced Cuntz-Krieger $E$-family. That is, if $\{q_v
: v \in E^0\}$, $\{t_e : e \in E^1\}$ is another normalised reduced Cuntz-Krieger family in a $C^*$-algebra $B$, then there is a homomorphism $\pi_{q,t} : {\ensuremath{C^*_{\operatorname{min}}(E)}} \to B$ such that $\pi_{q,t}({p^m_{v}}) = q_v$ for all $v \in E^0$ and $\pi_{q,t}({s^m_{e}}) = t_e$ for all $e \in E^1$.
The universal property of ${\mathcal T}C^*(E)$ implies that there is a homomorphism $\pi^{\mathcal T}_{q,t} : {\mathcal T}C^*(E) \to B$ such that $\pi^{\mathcal T}_{q,t}({p^{\mathcal T}_{v}}) = q_v$ and $\pi^{\mathcal T}_{q,t}({s^{\mathcal T}_{e}}) = t_e$ for all $v \in E^0$ and $e \in E^1$. Let $I_{q,t} :=
\ker(\pi^{\mathcal T}_{q,t})$, and let $I_{{p^m_{}},{s^m_{}}}$ be the kernel of the canonical homomorphism $\pi^{\mathcal T}_{{p^m_{}}, {s^m_{}}} : {\mathcal T}C^*(E) \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$. Let $K := I_{q,t} \cap I_{{p^m_{}},{s^m_{}}}$. Define $p^K_v := {p^{\mathcal T}_{v}} + K$ and $s^K_e := {s^{\mathcal T}_{e}} + K$ for all $v \in E^0$ and $e \in E^1$. Since both $\{{p^m_{v}}, {s^m_{e}}\}$ and $\{q_v, t_e\}$ are normalised reduced Cuntz-Krieger families, $\{p^K_v, s^K_e\}$ is also. Since no ${p^{\mathcal T}_{v}}$ belongs to $I_{{p^m_{}}, {s^m_{}}}$, each $p^K_v$ is nonzero. Hence Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]) and (\[it:injectivity\]) imply that there is an isomorphism $\psi_{p^K, s^K} : {\mathcal T}C^*(E)/K \to {\ensuremath{C^*_{\operatorname{min}}(E)}}$ such that $\psi_{p^K, s^K}(p^K_v) =
{p^m_{v}}$ and $\psi_{p^K, s^K}(s^K_e) = {s^m_{e}}$ for all $v, e$. By definition of $K$, the homomorphism $\pi^{\mathcal T}_{q,t} : {\mathcal T}C^*(E) \to B$ descends to a homomorphism $\widetilde{\pi^{\mathcal T}_{q,t}} : {\mathcal T}C^*(E)/K \to B$, and then $\pi_{q,t} := \pi^{\mathcal T}_{q,t} \circ (\psi_{p^K, s^K})^{-1}$ is the desired homomorphism.
\[lem:EX-fam -> E-fam\] Let $E$ be a row-finite directed graph. Fix a cutting set $X$ for $E$. Define a directed graph $F$ as follows: $$\begin{gathered}
F^0 = \{\zeta(v) : v \in E^0\} \\
F^1 = \{\zeta(e) : e \in E^1 \setminus X\} \\
s(\zeta(e)) = \zeta(s(e))\quad\text{and}\quad r(\zeta(e)) =
\zeta(r(e)).\end{gathered}$$ There is an isomorphism $\phi$ from $C^*(F)$ to ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ such that $\phi(p_{\zeta(v)}) = {p^m_{v}}$ for all $v \in E^0$ and $\phi(s_{\zeta(e)}) = {s^m_{e}}$ for all $e \in F^1$.
Let $\{p_{\zeta(v)} : v \in E^0\}$, $\{s_{\zeta(e)} : e \in E^1
\setminus X\}$ denote the universal generating Cuntz-Krieger $F$-family in $C^*(F)$. Recall that for $x \in X$, we write $\mu(x)$ for the unique cycle with no entrance in $E$ such that $\mu(x)_1 = x$, and we define $\lambda(x)$ to be the path such that $\mu(x) = x\lambda(x)$. For $\nu \in E^*$ with $|\nu| \ge
2$ and $\nu_i \not\in X$ for all $i$, we write $\zeta(\nu)$ for the path $\zeta(\nu_1)\cdots\zeta(\nu_{|\nu|}) \in F$. Define $$\begin{aligned}
q_v &:= p_{\zeta(v)}\text{ for all }v \in E^0,\\
t_e &:= s_{\zeta(e)}\text{ for all }e \in E^1 \setminus X,\text{ and}\\
t_x &:= s_{\zeta(\lambda(x))}^*\text{ for all }x \in X.\end{aligned}$$ It suffices to show that the $q_v$ and $t_e$ form a normalised reduced Cuntz-Krieger $E$-family; the result will then follow from Theorem \[thm:Cr(E)-Existence\](\[it:co-universal\]) and (\[it:injectivity\]).
The $q_v$ are mutually orthogonal projections because the $p_{\zeta(v)}$ are. This establishes (\[it:TR1\]).
For $e \in F^1$ we have $t^*_e t_e = s^*_{\zeta(e)}
s_{\zeta(e)} = p_{s(\zeta(e))} = q_{s(e)}$. For each $x \in X$, since $\mu(x)$ has no entrance in $E$, we have $r(x)(F)^{|\lambda(x)|} = \{\zeta(\lambda(x))\}$, so the Cuntz-Krieger relation forces $s_{\zeta(\lambda(x))}
s_{\zeta(\lambda(x))}^* = p_{\zeta(s(x))}$. Hence $$t^*_x t_x
= s_{\zeta(\lambda(x))} s_{\zeta(\lambda(x))}^*
= p_{\zeta(s(x))} = q_{s(x)}.$$ This establishes (\[it:TR2\]).
Fix $v \in F^0$ such that $vE^1 \not= \emptyset$. If $v = r(x)$ for some $x \in X$, then $r_E^{-1}(v) = \{x\}$, and we have $$q_v = p_{\zeta(v)}
= s^*_{\zeta(\lambda(x))} s_{\zeta(\lambda(x))}
= t_x t^*_x
= \sum_{e \in r_E^{-1}(v)} t_e t^*_e.$$ If $v \not= r(x)$ for all $x \in X$, then $vF^1 = \{\zeta(e) :
e \in vE^1\}$, and so $$q_v = p_{\zeta(v)}
= \sum_{f \in vF^1} s_f s^*_f
= \sum_{e \in vE^1} t_e t^*_e.$$ This establishes both (\[it:TR3\]) and (CK).
\[cor:bdry rep\] Let $E$ be a row-finite directed graph. There is an isomorphism $$\psi_{P^\infty, S^\infty} : {\ensuremath{C^*_{\operatorname{min}}(E)}} \to C^*(\{P^\infty_v, S^\infty_e : v \in E^0, e \in E^1\})$$ satisfying $\psi_{P^\infty, S^\infty}({p^m_{v}}) = P^\infty_v$ for all $v \in E^0$ and $\psi_{P^\infty, S^\infty}({s^m_{e}}) =
S^\infty_e$ for all $e \in E^1$.
As observed above, $\{P^\infty_v : v \in E^0\}$, $\{S^\infty_e
: e \in E^1\}$ is a normalised reduced Cuntz-Krieger $E$-family with each $P^\infty_v$ nonzero. The result therefore follows from Corollaries \[cor:injectivity criterion\] and \[cor:universal property\].
We now identify a subspace of $\ell^2(E^{\le \infty})$ which is invariant under the Cuntz-Krieger family of Notation \[ntn:bdryCKfam\]. We use the resulting Cuntz-Krieger family to construct a faithful conditional expectation from ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ onto its diagonal subalgebra.
Let ${\Omega}$ denote the collection $$\begin{aligned}
{\Omega}= \{&\alpha \in E^* : s(\alpha)E^1 = \emptyset\} \\
&\cup \{\alpha\mu^\infty : \alpha \in E^*, \mu\text{ is a cycle with no entrance }\} \\
&\cup \{x \in E^\infty : x \not= \alpha\rho^\infty\text{ for any }\alpha,\rho \in E^*\text{ such that } s(\alpha) = r(\rho) = s(\rho)\}.\end{aligned}$$ So $x \in E^{\le \infty}$ belongs to ${\Omega}$ if and only if either $x$ is aperiodic, or $x$ has the form $\alpha\mu^\infty$ for some cycle $\mu$ with no entrance in $E$. Observe that $$\label{eq:STE invariance}
\text{if $x \in {\Omega}$ and if $y \in E^{\le\infty}$ and $m,n \in
{{\mathbb{N}}}$ satisfy $\sigma^m(x) = \sigma^n(y)$, then $y \in
{\Omega}$.}$$
We regard $\ell^2({\Omega})$ as a subspace of $\ell^2(E^{\le\infty})$. The condition implies that $\ell^2(E^{\le\infty})$ is invariant for the Cuntz-Krieger $E$-family of Notation \[ntn:bdryCKfam\]. We may therefore define a Cuntz-Krieger $E$-family $\{{P^{\Omega}_{v}} : v \in E^0\}$, $\{{S^{\Omega}_{e}}
: e \in E^1\}$ in ${\mathcal B}(\ell^2({\Omega}))$ by $${P^{\Omega}_{v}} = P^\infty_v|_{\ell^2({\Omega})}
\qquad\text{ and }\qquad
{S^{\Omega}_{e}} = S^\infty_e|_{\ell^2({\Omega})}$$ for all $v \in E^0$ and $e \in E^1$. Since every vertex of $E$ is the range of at least one element of ${\Omega}$, we have ${P^{\Omega}_{v}} \not= 0$ for all $v \in E^0$.
\[lem:abCK rep\] Let $E$ be a row-finite directed graph. There is an isomorphism $\psi_{{P^{\Omega}_{}}, {S^{\Omega}_{}}} : {\ensuremath{C^*_{\operatorname{min}}(E)}} \to C^*(\{{P^{\Omega}_{v}}, {S^{\Omega}_{e}} : v
\in E^0, e \in E^1\})$ satisfying $\psi_{{P^{\Omega}_{}},
{S^{\Omega}_{}}}({p^m_{v}}) = {P^{\Omega}_{v}}$ for all $v \in E^0$ and $\psi_{{P^{\Omega}_{}},{S^{\Omega}_{}}}({s^m_{e}}) = {S^{\Omega}_{e}}$ for all $e \in E^1$.
The proof is identical to that of Corollary \[cor:bdry rep\].
For the next proposition, let ${W}$ denote the collection of paths $\alpha \in E^*$ such that $\alpha \not= \beta\mu$ for any $\beta \in E^*$ and any cycle $\mu$ with no entrance in $E$.
Let $E$ be a row-finite directed graph.
1. \[it:Cr spanning\] The $C^*$-algebra ${\ensuremath{C^*_{\operatorname{min}}(E)}}$ satisfies $${\ensuremath{C^*_{\operatorname{min}}(E)}} = {\overline{{\mathop{\operatorname{span}}\nolimits}}}\{{s^m_{\alpha}} {{({s^m_{\beta}})^{*}}} : \alpha,\beta \in {W}, s(\alpha) = s(\beta)\}.$$
2. \[it:FCE\] Let $D :=
{\overline{{\mathop{\operatorname{span}}\nolimits}}}\{{s^m_{\alpha}}{{({s^m_{\alpha}})^{*}}}
: \alpha \in E^*\}$. There is a faithful conditional expectation $\Psi : {\ensuremath{C^*_{\operatorname{min}}(E)}} \to D$ such that $$\Psi({s^m_{\alpha}} {{({s^m_{\beta}})^{*}}}) =
\begin{cases}
{s^m_{\alpha}}{{({s^m_{\alpha}})^{*}}} &\text{ if $\alpha = \beta$} \\
0 &\text{ otherwise}
\end{cases}$$ for all $\alpha,\beta \in {W}$ with $s(\alpha) =
s(\beta)$.
By Lemma \[lem:abCK rep\] it suffices to prove the corresponding statements for the $C^*$-algebra $B :=
C^*(\{{P^{\Omega}_{v}}, {S^{\Omega}_{e}} : v \in E^0, e \in E^1\}$.
(\[it:Cr spanning\]) We have $B =
{\overline{{\mathop{\operatorname{span}}\nolimits}}}\{{S^{\Omega}_{\alpha}}{{({S^{\Omega}_{\beta}})^{*}}} : \alpha,\beta \in E^*\}$ because the same is true of ${\mathcal T}C^*(E)$. If $\alpha \in E^*
\setminus {W}$, then $\alpha = \alpha'\mu^n$ for some $\alpha' \in {W}$, some cycle $\mu$ with no entrance in $E$ and some $n \in {{\mathbb{N}}}$. Since $\{{P^{\Omega}_{v}} : v \in E^0\}$, $\{{S^{\Omega}_{e}} : e \in E^1\}$ is a normalised reduced Cuntz-Krieger $E$-family, ${({S^{\Omega}_{\mu}})^{n}} = {P^{\Omega}_{r(\mu)}}$, so ${S^{\Omega}_{\alpha}} =
{S^{\Omega}_{\alpha'}}$.
(\[it:FCE\]) Let $\{\xi_x : x \in {\Omega}\}$ denote the standard orthonormal basis for $\ell^2({\Omega})$. For each $x
\in {\Omega}$, let $\theta_{x,x} \in {\mathcal B}(\ell^2({\Omega}))$ denote the rank-one projection onto ${{\mathbb{C}}}\xi_x$. Let $\Psi$ denote the faithful conditional expectation on ${\mathcal B}(\ell^2({\Omega}))$ determined by $\Psi(T) = \sum_{x \in
{\Omega}} \theta_{x,x} T \theta_{x,x}$, where the convergence is in the strong operator topology. It suffices to show that $$\label{eq:Psi eqn}
\Psi({S^{\Omega}_{\alpha}} {{({S^{\Omega}_{\beta}})^{*}}}) =
\begin{cases}
{S^{\Omega}_{\alpha}}{({S^{\Omega}_{\alpha}})^{*}} &\text{ if $\alpha = \beta$} \\
0 &\text{ otherwise}
\end{cases}$$ for all $\alpha,\beta \in {W}$ with $s(\alpha) = s(\beta)$.
Fix $\alpha,\beta \in {W}$ with $s(\alpha) = s(\beta)$. If $\alpha = \beta$, then $${S^{\Omega}_{\alpha}}{({S^{\Omega}_{\alpha}})^{*}} = \sum_{y \in
s(\alpha){\Omega}} p_{\alpha x},$$ and is immediate. So suppose that $\alpha
\not= \beta$. For $x \in {\Omega}$, we have $$\theta_{x,x} {S^{\Omega}_{\alpha}}{({S^{\Omega}_{\beta}})^{*}} \theta_{x,x} =
\begin{cases}
\theta_{x,x} &\text{ if $x = \alpha y = \beta y$} \\
0 &\text{ otherwise.}
\end{cases}$$ Hence we must show that $\alpha y \not= \beta y$ for all $y \in
s(\alpha) {\Omega}$. Fix $y \in s(\alpha) {\Omega}$. First observe that if $|\alpha| = |\beta| = l$, then $(\alpha y)(0,
l) = \alpha \not= \beta = (\beta y)(0,l)$. Now suppose that $|\alpha| \not= |\beta|$; we may assume without loss of generality that $|\alpha| < |\beta|$. We suppose that $\alpha y
= \beta y$ and seek a contradiction. That $\alpha y = \beta y$ implies that $\beta = \alpha\beta'$ and $y = \beta' y$. Hence $r(\beta') = s(\beta')$ and $y = (\beta')^\infty$. Since $y \in
{\Omega}$, it follows that $\beta' = \mu^n$ for some cycle $\mu$ with no entrance and some $n \in {{\mathbb{N}}}$, contradicting $\beta \in {W}$.
[10]{} T. Bates, J. H. Hong, I. Raeburn, and W. Szyma[ń]{}ski, *The ideal structure of the [${C}\sp*$]{}-algebras of infinite graphs*, Illinois J. Math. **46** (2002), no. 4, 1159–1176.
T. Bates, D. Pask, I. Raeburn, and W. Szyma[ń]{}ski, *The [${C}\sp*$]{}-algebras of row-finite graphs*, New York J. Math. **6** (2000), 307–324 (electronic).
D. Drinen and M. Tomforde, *The [$C\sp
*$]{}-algebras of arbitrary graphs*, Rocky Mountain J. Math. **35** (2005), no. 1, 105–135.
N. J. Fowler and I. Raeburn, *The [T]{}oeplitz algebra of a [H]{}ilbert bimodule*, Indiana Univ. Math. J. **48** (1999), no. 1, 155–181.
J. H. Hong and W. Szyma[ń]{}ski, *The primitive ideal space of the [${C}\sp\ast$]{}-algebras of infinite graphs*, J. Math. Soc. Japan **56** (2004), no. 1, 45–64.
A. Huef and I. Raeburn, *The ideal structure of [C]{}untz-[K]{}rieger algebras*, Ergodic Theory Dynam. Systems **17** (1997), no. 3, 611–624.
T. Katsura, *Ideal structure of ${C}^*$-algebras associated with ${C}^*$-correspondences*, Pacific J. Math. **230** (2007), no. 1, 107–146.
A. Kumjian, D. Pask, and I. Raeburn, *Cuntz-[K]{}rieger algebras of directed graphs*, Pacific J. Math. **184** (1998), no. 1, 161–174.
I. Raeburn, Graph algebras, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2005, vi+113.
|
---
author:
- |
Siavosh R. Behbahani $^{1,2}$, Martin Jankowiak$^{1,2}$, Tomas Rube$^{2}$, Jay G. Wacker$^{1,2}$\
$^1$Theory Group, SLAC National Accelerator Laboratory, Menlo Park, CA 94025\
$^2$Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305
title: 15 pt Nearly Supersymmetric Dark Atoms
---
Introduction {#Sec: Introduction}
============
The nature of dark matter is unknown and its relation to the Standard Model (SM) is an open question. The recent spate of anomalies in direct detection experiments [@DAMA] and cosmic ray signatures [@anomalies] has motivated re-examining the standard assumptions about the identity of dark matter. Most models of dark matter assume that dark matter is an elementary particle with no relevant or long range interactions. If supersymmetry is present in these models, the supersymmetric mass splittings are so large that the supersymmetric structure of dark matter is unimportant. This article provides a framework to illustrate the exact opposite case: dark matter is composite with long range interactions and supersymmetry breaking effects are small.
Recent anomalies have several common features that motivate considering dark sectors that support bound states. Bound states naturally enjoy a hierarchy of different scales. Inelastic Dark Matter explanations of DAMA, [*e.g.*]{} [@iDM; @StatusiDM; @TheoriesOfiDM], require several scales to reconcile the anomalies with the null results of other direct detection experiments. A hierarchy of scales is also employed in the Exciting Dark Matter scenario [@Finkbeiner:2007kk] to explain the 511 keV signal from INTEGRAL/SPI. Additional structure in the dark sector is also motivated by positron excesses in cosmic ray data, which might be a result of cascade decays in the dark sector. Examining the Standard Model, one finds a variety of different bound state systems: mesons and baryons, nuclei, atoms, and molecules. Given the prevalence of bound states in Standard Model systems, it is natural to explore the possibility [@CompositeDarkMatter; @Alves:2009nf; @Kaplan:2009de] that dark matter is composed of bound states in a separate sector.
Fermions with gauge interactions are a ubiquitous ingredient in theories beyond the Standard Model. It is plausible that there are additional gauge sectors that SM fermions are not charged under. If there are no SM particles directly charged under the new gauge interaction, then experimental limits on decoupled gauge sectors are extremely weak. If supersymmetry breaking is only weakly mediated to the dark sector, perhaps through dark matter’s interactions with the Standard Model, then the magnitude of supersymmetry breaking effects can be extremely small. This allows for the possibility that dark matter is nearly supersymmetric. If there are any bound states in the dark sector, the spectrum will exhibit near Bose-Fermi degeneracy. Such weakly coupled hidden sectors also naturally sit near the GeV scale, which makes for interesting dark matter phenomenology [@GevScale] and experimental signatures [@ExperSignatures].
Investigating nearly supersymmetric bound states arising from perturbative Coulombic interactions is a relatively intricate process and the standard techniques from quantum mechanics involve computing first and second order $\TT$-matrix elements and then diagonalizing the Hamiltonian. At each step the calculation is not supersymmetric, although the final answer is supersymmetric. Ultimately, the states have organized themselves into supersymmetric multiplets and the admixtures of different supersymmetric particles that each composite state consists of is known. For instance, a spin zero fermion-fermion bound state will mix with a spin zero scalar-scalar bound state. Since phenomenological applications depend on these admixtures, it would be convenient to understand their structure and how they generalize to other bound state systems. Similarly, phenomenological studies would be made easier by understanding how bound state interactions are constrained by supersymmetry. This article develops a simple formalism to do this using off shell superfields.
The organization of the paper is as follows. Sec.\[Sec: NR\] reviews non-relativistic supersymmetric bound states, focusing on how supersymmetry organizes the spectrum and superspin wavefunctions of the states. The free effective action is also introduced, which will form the basis for computing the supersymmetric interactions of the bound states. Sec.\[Sec: SSB Effects\] incorporates the effects of supersymmetry breaking into the spectrum for the case where the dominant source of supersymmetry breaking is the soft masses of the scalar constituents. Sec.\[Sec: Interactions\] computes the interactions of the bound states when interacting with weakly coupled external gauge interactions. Sec.\[Sec: Nearly Susy\] constructs a realistic model of nearly supersymmetric atomic dark matter. Sec.\[Sec: Discussion\] discusses possible directions of future research for models along these lines, including recombination and the formation of supersymmetric molecules. Sec.\[Sec: Conc\] makes some concluding remarks.
Non-relativistic Supersymmetric Bound States {#Sec: NR}
============================================
This section studies how non-relativistic supersymmetric bound states organize themselves into supermultiplets. Sec. \[Sec: Wavefunctions Fields\] outlines a general procedure for determining the composition of non-relativistic bound states formed from massive superfields. When applicable, this procedure has the advantage of sidestepping a detailed perturbative calculation in favor of some superfield algebra. This procedure is illustrated in the particular case of bound states formed from two chiral multiplets. Sec. \[Sec: Effective Action\] continues the study of this particular example by introducing an effective field theory description of the ground state. This will provide the basis for Sec. \[Sec: Interactions\], in which bound state interactions are discussed.
Wavefunctions From Superspace {#Sec: Wavefunctions Fields}
-----------------------------
Non-relativistic bound states have a structure that is simple to understand because they benefit from a good expansion parameter: the velocity $v$. This is especially the case for two-body systems, where an expansion in powers of $v$ not only helps to organize calculations but also determines the relevant scales of the problem. The gross structure of the spectrum can be organized into principle excitations split by energies of order $$\begin{aligned}
m_{\text{prin}} \propto \mu\, v^2,\end{aligned}$$ where $\mu$ is the reduced mass. Fine structure effects are the next order correction in the non-relativistic expansion, appearing as $$\begin{aligned}
m_{\text{FS}} \propto \mu\, v^4 .\end{aligned}$$ Recent papers [@Rube:2009yc; @Herzog:2009fw] have computed the fine structure of supersymmetric hydrogen through explicit calculation. This section rederives these results by considering how supersymmetry acts on the bound states in the non-relativistic limit. The organization of the spectrum into supermultiplets does not depend on the details of the binding dynamics except for specific quantities, such as energy splittings. Most notably, the superspin wavefunctions are completely determined by supersymmetry alone if there are no accidental degeneracies in the spectrum. This method of using supersymmetry to fix the superspin wavefunctions is applicable to a wider class of non-relativistic bound states than Coulombic bound states and more cleanly delineates which quantities depend upon dynamics versus the structure of supersymmetry.
For simplicity, assume that the bound state is supported by a central potential that is spin-independent at $\OO(v^2)$. This is true for a wide range of composite states, including those bound together by the exchange of light vector or chiral multiplets. The ground state then has a non-degenerate radial wavefunction with $l=0$ [@Downs:1963] and factorizes as $$\begin{aligned}
|\Psi\rangle = |\psi(r)\rangle \otimes | \SS\rangle
\label{eqn:StateProd}\end{aligned}$$ to leading order, where $| \SS\rangle$ is the superspin part of the wavefunction. At leading order in $v$ the supercharges act only on $|\SS\rangle$, leaving $|\psi(r)\rangle$ intact, because gradients of non-relativistic wavefunctions are suppressed, $\partial_i \psi \sim \OO(v)$. Since $\psi(r)$ has trivial angular dependence, decomposing $| \SS\rangle $ into irreducible representations decomposes $|\Psi\rangle$ into irreducible supersymmetry representations, $\Omega_j$, were $j$ refers to the spin of the Clifford vacuum (e.g.$\Omega_0$ is the chiral multiplet). At $\OO(v^4)$ the Hamiltonian is typically spin-dependent and any degeneracy among the $\Omega_j$’s will generically be lifted in the absence of any special symmetries. For $\Omega_j$’s that are accidentally degenerate at $\OO(v^2)$, there can be large mixing that depends on the details of the dynamics, though in many cases the appropriate mass eigenstates are determined by the action of addtional symmetries on the supermultiplets.
As an example that illustrates this decomposition, consider the model bound state system that will form the main subject of this article. It consists of four massive chiral superfields ($E$, $E^c$, $P$, $P^c$) with Dirac masses $m_e$ and $m_p$ satisfying $m_e \le m_p$. The binding dynamics respect parity, under which the coordinates and superfields transform as $$\begin{aligned}
x^\mu\leftrightarrow(-1)^\mu x^\mu\qquad \theta^\alpha\leftrightarrow\bar\theta_{\dot\alpha}\qquad P\leftrightarrow P^{c\dagger} \qquad E\leftrightarrow
E^{c\dagger}\end{aligned}$$ The dynamics also respect a $U(1)_R$-symmetry and a $U(1)_e \times U(1)_p$ flavor symmetry. The charges of the component fields are taken to be $$\begin{aligned}
\begin{array}{|c|cccccc|}
\hline
&\;\;\;\tilde p\;\;\;&\left(\begin{array}{c}p^\alpha\cr \bar p_{\dot\alpha}^c\end{array}\right)&\;\;\;\tilde p^{c\dagger}\;\;\;
&\;\;\;\tilde e\;\;\;&\left(\begin{array}{c}e^\alpha\cr \bar e^c_{\dot\alpha}\end{array}\right)&\;\;\;\tilde e^{c\dagger}\;\;\;
\cr
\hline
U(1)_R&1&0&-1&1&0&-1\cr
U(1)_{e+p}&1&1&1&1&1&1\\
\hline
\end{array}
\label{eqn:charges}\end{aligned}$$ Significantly, the $U(1)_R$ symmetry and $\mathbb{Z}_{2}$ parity do [*not*]{} commute and combine into an $O(2)_R$ symmetry. This can be seen by considering the “selectrons" $\tilde{e}$ and $ \tilde{e}^{c}$. Parity, $\PP$, acts upon the selectrons as $$\begin{aligned}
\mathcal{P} \tilde{e} = \tilde{e}^{c\dagger}\end{aligned}$$ while under a $U(1)_R$ transformation, $R(\alpha)$, the selectrons transform as $$\begin{aligned}
R(\alpha) \tilde{e} = e^{i \alpha}\tilde{e} \qquad \text{and} \qquad R(\alpha) \tilde{e}^{c\dagger} = e^{-i \alpha}\tilde{e}^{c\dagger}
\end{aligned}$$ so that $[\mathcal{P}, R(\alpha)] \tilde{e} \ne 0$. Thus $U(1)_R$ and $\mathbb{Z}_{2}$ are not a direct product and instead combine as the semi-direct product $U(1)_R \rtimes \mathbb{Z}_{2} \cong O(2)_R$. This is important because $O(2)_R$ has two-dimensional irreducible representations that are realized in the bound state spectrum. In particular any state that transforms non-trivially under $U(1)_R$ must sit in an $O(2)_R$ doublet.
For this system, the superspin wavefunction $| \SS\rangle $ in Eq. \[eqn:StateProd\] decomposes as two chiral multiplets and one vector multiplet ($\VVV$), as can be verified by counting degrees of freedom. As a consequence of the $O(2)_R$ symmetry, however, the two chiral multiplets combine into a hypermultiplet $\HHH$ so that the decomposition of $| \SS\rangle $ reads $$\begin{aligned}
| \SS\rangle = 2\Omega_0 \oplus \Omega_{\frac{1}{2}} = \HHH \oplus \VVV
\label{eqn:chiralxchiral}\end{aligned}$$ Both $\HHH$ and $\VVV$ are charged under the global $U(1)_{e+p}$ flavor symmetry of the theory. The superspin wavefunctions of the ground state are fixed by (super)symmetry at leading order because $\HHH$ and $\VVV$ are irreducible under the full symmetry group and therefore insensitive to mixing.
Supersymmetry organizes non-relativistic pairs of *free* particles into supermultiplets, determining the bound state wavefunctions at leading order in $v$ in terms of the constituent particles. The organization of pairs of free particles into supermultiplets is found by putting $E$, $E^c$, $P$ and $P^c$ on shell and constructing all possible superfield bilinears. The resulting bilinears will have spins ranging from 0 to 1. For example, the superfields $P$ and $E^{c\dagger}$ yield the bilinears $PE^{c\dagger}$, $\DD^\alpha PE^{c\dagger}$, $P \bar \DD_{\dot\alpha}E^{c\dagger}$ and $\DD^\alpha P\bar \DD_{\dot \alpha}E^{c\dagger}$. These bilinears can then be decomposed into irreducible supersymmetry representations with the help of projection operators, which in the case of spin zero superfields are given by $$\begin{aligned}
\PP_1=\frac{\DD^2\bar\DD^2}{16\Box}
\qquad \PP_2=\frac{\bar\DD^2\DD^2}{16\Box}
\qquad \text{and}\qquad
\PP_T=-\frac{\DD\bar\DD^2\DD}{8\Box}
\label{eqn:projectionoperators}\end{aligned}$$ where $\PP_1+\PP_2+\PP_T=1$ [@Wess:1992cp].
The decomposition is simplified by noting that the same state can appear in many different bilinears. In fact the bilinears $$\begin{aligned}
\label{eqn:bilinears}
\PP_2PE=PE \qquad \PP_1 P^{c\dagger} E^{c\dagger}=P^{c\dagger} E^{c\dagger} \qquad\text{and}\qquad \PP_TPE^{c\dagger}\end{aligned}$$ contain all the states as can be verified by counting degrees of freedom. Expanding the first two bilinears in Eq. \[eqn:bilinears\] using the non-relativistic fields[^1] $$\begin{aligned}
\tilde p=\frac{e^{im_p t}}{\sqrt{2m_p}}\phi_p\quad\text{ and }\quad
\Psi_p^D=e^{im_pt}\left(\!\begin{array}{c}\psi_p\cr\frac{i\vec{\sigma}\cdot\vec{\nabla}}{2m_p}\psi_p\end{array}\!\right)
\label{eqn:non-relfields}\end{aligned}$$ gives the superfields (cf. [@DiVecchia:1985xm]) $$\begin{aligned}
\label{eqn:pe}
PE\propto &&\phi_p\phi_e+\sqrt{2}\Theta^a\left(c_\theta\psi_p^a\phi_e+s_\theta\phi_p\psi_e^a\right)\\
\nonumber
&&-\Theta^2\left(s_\theta^2\phi_p\phi_{e^c}^\dagger+c_\theta^2\phi_{p^c}^\dagger\phi_e-s_{2\theta}(\psi_p\psi_e)_0\right)\end{aligned}$$ and $$\begin{aligned}
\label{eqn:pcec}
P^{c\dagger} E^{c\dagger}\propto &&\phi_{p^c}^\dagger\phi_{e^c}^\dagger+\sqrt{2}\bar\Theta_{a}\left(c_\theta\psi_{p}^{a}\phi_{e^c}^\dagger
+s_\theta\phi_{p^c}^\dagger\psi_{e}^{a}\right)\\
\nonumber
&&-\bar\Theta^2\left(c_\theta^2\phi_p\phi_{e^c}^\dagger+s_\theta^2\phi_{p^c}^\dagger\phi_{e}+s_{2\theta}(\psi_p\psi_e)_0\right)\end{aligned}$$ where the dimensionless $\Theta^\alpha=\sqrt{m_p+m_e}\theta^\alpha$ has been introduced, and the mixing angle $\theta$ is defined by $$\begin{aligned}
\tan^2 \theta = \frac{ m_e}{m_p} \end{aligned}$$ These two superfields have $U(1)_R$-charges of $\pm2$ and transform into each other under parity; they correspond to the two $\Omega_0$’s in $\HHH$. The $\VVV$ wavefunctions are found by decomposing the bilinear $\PP_T P E^{c\dagger}$, which gives a complex vector (curl) superfield with components $$\begin{aligned}
D&\propto c_{2\theta}(\psi_p\psi_e)_0+s_{2\theta}(\phi_{p^c}^\dagger\phi_e-\phi_p\phi_{e^c}^\dagger)/{\sqrt{2}}\cr
\bar\lambda_1&\propto s_\theta\psi_p\phi_{e^c}^\dagger-c_\theta\phi_{p^c}^\dagger\psi_e\cr
\lambda_2&\propto s_\theta\psi_p\phi_e-c_\theta\phi_p\psi_e\cr
v^\mu&\propto\psi_p\vec{\sigma}\psi_e\end{aligned}$$ Going to the parity eigenbasis and introducing notation for the various states gives $$\begin{aligned}
\nonumber
\VVV &=& \begin{cases}
v_\mu \;\;\;\;\;\;\;\; & {|\vec{v}\rangle} = {|(\psi_p\psi_e)_1 \rangle}\ \ \\
\chi_+, \bar{\chi}^c_+&{|\psi_{\chi_+}\rangle}=c_\theta {|\phi_{p+} \psi_e\rangle} - s_\theta{| \psi_{p} \phi_{e+}\rangle}\\
\chi_-, \bar{\chi}^c_- &{|\psi_{\chi_-}\rangle}= c_\theta{|\phi_{p-}\psi_e\rangle}-s_\theta{|\psi_p \phi_{e-} \rangle}\\
\varsigma_- &{|\varsigma_-\rangle}= c_{2\theta} {|(\psi_p \psi_e)_0\rangle} + \tfrac{s_{2\theta}}{\sqrt{2}} ({|\phi_{p+}\phi_{e-}\rangle} - {|\phi_{p-}\phi_{e+}\rangle})
\end{cases} \\
\ \HHH &=&\begin{cases}
\omega_+ \;\;\;\;\;\;\; & {|\omega_+\rangle} = \tfrac{1}{\sqrt{2}}({|\phi_{p+} \phi_{e+}\rangle} - {|\phi_{p-}\phi_{e-}\rangle}) \\
\omega_-&{| \omega_-\rangle}= \tfrac{c_{2\theta}}{\sqrt{2}} ({|\phi_{p-}\phi_{e^+}\rangle} - {|\phi_{p+}\phi_{e-}\rangle})+s_{2\theta} {|(\psi_p \psi_e)_0 \rangle}\\
\xi_+, \bar{\xi}_+^c & {|\psi_{\xi_+}\rangle}=c_\theta {|\psi_p \phi_{e+}\rangle} + s_\theta {|\phi_{p+} \psi_e\rangle}\\
\xi_-, \bar{\xi}_-^c&{|\psi_{\xi_-}\rangle}= c_\theta {|\psi_p \phi_{e-}\rangle} + s_\theta {|\phi_{p-} \psi_e\rangle}\\
\varpi_+&{| \varpi_+\rangle}= \tfrac{1}{\sqrt{2}}({|\phi_{p+} \phi_{e+}\rangle} +{| \phi_{p-}\phi_{e-}\rangle})\\
\varpi_-& {|\varpi_-\rangle}= \tfrac{1}{\sqrt{2}}({|\phi_{p+} \phi_{e-}\rangle} + {|\phi_{p-}\phi_{e+}\rangle})
\end{cases}
\label{eqn:wavefunctions}\end{aligned}$$ where $(\psi_p\psi_e)_0=\tfrac{1}{\sqrt{2}}\epsilon^{ab}\psi_p^a\psi_e^b$, $\phi_{p/e\pm}=\tfrac{1}{\sqrt{2}}(\phi_{p/e}\pm\phi_{p^c/e^c}^\dagger)$, and $c_\theta, s_\theta$ are $\cos \theta$ and $\sin\theta$, respectively. These are the same wavefunctions found in [@Rube:2009yc; @Herzog:2009fw] by means of a detailed perturbative calculation in the particular case of supersymmetric hydrogen.
Although the $\VVV$ and $\HHH$ wavefunctions have been determined here without specifying the binding dynamics, the mass splitting between $\VVV$ and $\HHH$ can only be determined by doing a dynamical calculation. In the absence of any special symmetries, however, it is expected that $m_{\text{FS}} \equiv m_{\VVV}-m_{\HHH}$ will be at the fine structure scale, $m_{\text{FS}} \sim \OO(v^4 \mu)$, and in the case of supersymmetric hydrogen one finds $$\begin{aligned}
m_{\text{FS}} = {{\frac{1}{2} }}\alpha_\vv^4 \mu.\end{aligned}$$
The states in Eq. \[eqn:wavefunctions\] are organized according to their $O(2)_R$ representations, with simple transformation properties under parity, because the breaking of $O(2)_R$ plays an important role in lifting degeneracies in the spectrum once supersymmetry is broken. The states $\chi_\pm$, $\xi_\pm$ and $\varpi_\pm$ transform in two-dimensional representations of $O(2)_R$ with $U(1)_R$ charges of 1, 1, and 2, respectively. For example, the doublet $$\begin{aligned}
\varpi = \begin{pmatrix}{\varpi_+}\\ {i \varpi_- } \end{pmatrix}\end{aligned}$$ transforms irreducibly as $$\begin{aligned}
\mathcal{P}:\varpi \to \sigma_3 \varpi \qquad \qquad R(\alpha): \varpi \to e^{2 i \alpha \sigma_2} \varpi\end{aligned}$$ The states $v_\mu$, $\varsigma_-$ and $\omega_\pm$ are invariant under $R(\alpha)$ and thus transform as $O(2)_R$ singlets.
To illustrate the action of supersymmetry on the ground states, consider the heavy proton limit, $\theta \rightarrow 0$. In this limit, supersymmetry clocks the states of the heavier constituent, leaving the valence particle intact. In particular, the $\VVV$ states consist of a light electron orbiting a heavy proton multiplet and the $\HHH$ states consist of a light selectron orbiting a heavy proton multiplet.
This method of calculating superspin wavefunctions through decomposing products of superfields is general and can be applied to a wide class of non-relativistic supersymmetric bound state problems. For example, the superspin wavefunctions of non-relativistic $SU(3)$ baryons can be found by studying the decomposition of superfield trilinears. In this case, acting with the projection operators in Eq. \[eqn:projectionoperators\] on spin zero trilinears does not give all of the wavefunctions, and a spin ${{\frac{1}{2} }}$ trilinear is necessary. Similarly, the study of the bound states of a massive chiral and a massive vector superfield requires higher spin projections.
### Excited state wavefunctions {#excited-state-wavefunctions .unnumbered}
This prescription for finding the superspin wavefunctions can also be applied to the excited states. For a given spatial wavefunction $|nl\rangle$, the various excited states can be built by acting with supersymmetry on the Clifford vacua defined by the particle content, $$\begin{aligned}
\left\{|nl\rangle\otimes|\Omega_s\rangle,\quad
|nl\rangle\otimes\left(a^\dagger \otimes|\Omega_s\rangle\right),\quad
|nl\rangle\otimes\left(a^{\dagger2}|\Omega_s\rangle\right)\right\}
\label{eqn:ls-basis}\end{aligned}$$ where $|\Omega_s\rangle$, $a^\dagger|\Omega_s\rangle$ and $a^{\dagger2}|\Omega_s\rangle$ are the superspin wavefunctions derived in the previous section. For example in the case considered above $|\Omega_s\rangle$ is either $|\Omega_0\rangle$ or $|\Omega_{{{\frac{1}{2} }}}\rangle$, and the raising operators fill out the various states in $\VVV$ and $\HHH$. Decomposing Eq. \[eqn:ls-basis\] into supermultiplets is equivalent to switching to the basis $$\begin{aligned}
\left\{|nl\rangle\otimes|\Omega_s\rangle,\quad
a^\dagger\otimes\left(|nl\rangle\otimes|\Omega_s\rangle\right),\quad
a^{\dagger2}\left(|nl\rangle\otimes|\Omega_s\rangle\right)\right\},
\label{eqn:j-basis}\end{aligned}$$ since irreducible representations of supersymmetry are obtained by acting with the raising operator on Clifford vacua that are irreducible representations of the rotation group. This basis switch is just a matter of Clebsch-Gordon algebra and results in the decomposition $$\begin{aligned}
l\otimes\Omega_s=\Omega_{|l-s|}\oplus...\oplus\Omega_{|l+s|}.\end{aligned}$$ For example, in the case considered above, where the bound state is formed from two chiral multiplets, the decomposition gives $$\begin{aligned}
\label{eqn:decompexample}
l\otimes(\Omega_0\otimes\Omega_0)= l \otimes( \Omega_0 \oplus \Omega_0 \oplus \Omega_{{\frac{1}{2} }}) =\Omega_{l-1/2}\oplus \Omega_l\oplus \Omega_l\oplus\Omega_{l+1/2}\end{aligned}$$ with the two $\Omega_l$ related to one another by parity.
Thus provided that a given $\Omega_j$ does not undergo large mixing, the excited state angular/superspin wavefunctions can be found just as for the ground state. One does a (single) superfield calculation as before to determine $\Omega_s$ and then transforms from the basis of Eq. \[eqn:ls-basis\] to that in Eq. \[eqn:j-basis\] using Clebsch-Gordan coefficients.
Effective Action for the Ground State {#Sec: Effective Action}
-------------------------------------
Once the ground state spectrum is known, it is important to determine how the various states interact with one another as well as with the SM. There are a variety of interactions, many of which are related through supersymmetric Ward identities. Superfields thus offer a convenient method for packaging all these interactions into manifestly supersymmetric forms. This section uses the standard off shell superfield formalism to formulate an effective free action for the ground state, postponing until Sec. \[Sec: Interactions\] a discussion of ground state interactions.
$\HHH$ is described by two chiral superfields that satisfy the following relations on shell $$\begin{aligned}
\HH_1\propto PE \qquad \text{and} \qquad \HH_2^\dagger\propto P^{c\dagger}E^{c\dagger} .\end{aligned}$$ A second set of chiral superfields, $\HH_1^{c\dagger}$ and $\HH_2^c$, is introduced to give the $F$-terms of $\HH_1$ and $\HH_2^\dagger$ dynamics. The free Lagrangian for $\HHH$ is given by $$\begin{aligned}
\label{eqn:haction}
\LL_{\HHH} = \int\!\!d^4\theta\; \delta^{ij}\!\! \left( \HH^\dagger_i \HH_j + \HH^c{}^\dagger_i \HH_j^c\right)
+\int\!\! d^2\theta \; \delta^{ij} m_{\HH} \HH_i \HH^c_j +\!\!{\text{ h.c. }}\label{eqn:Haction}\end{aligned}$$ The equations of motion which follow from Eq. \[eqn:haction\] then result in the identification $$\begin{aligned}
\HH_1^{c\dagger} \propto \mathcal{P}_1 P^{c\dagger} E \qquad \text{and} \qquad \HH_2^c\propto \mathcal{P}_2 P^{c\dagger} E\end{aligned}$$ With $\HH_i$ and $\HH_i^c$ identified as above the appropriate $U(1)_R$ and $U(1)_{e+p}$ charges are given by $$\begin{aligned}
\begin{array}{|c|cccc|}
\hline
& \HH_1 & \HH_2^\dagger & \HH_1^{c\dagger} & \HH_2^c \\
\hline
U(1)_R& 2 & -2 & 0 & 0 \\
U(1)_{e+p}& 2 & 2 & 2 & 2 \\
\hline
\end{array}\end{aligned}$$ so that the Lagrangian is properly invariant under $U(1)_R$ and $U(1)_{e+p}$. Parity acts on the composite superfields as $$\begin{aligned}
\HH_1\leftrightarrow \HH_2^{\dagger}\;\;\qquad\;\;\HH_1^c\leftrightarrow \HH_2^{c\dagger} .\end{aligned}$$ so that the Lagrangian is also invariant under parity.
$\VVV$ is described by an off shell field, $\VV$, and an action consistent with the on shell constraint $\VV\propto \PP_T PE^{c\dagger}$. $\VV$ is a charged vector superfield – a general superfield with no Lorentz index $$\begin{aligned}
\VV(x, \theta,\bar{\theta}) \ne \VV^\dagger(x,\theta,\bar{\theta}).\end{aligned}$$ The action is written with the help of the supersymmetric field strengths $$\begin{aligned}
\WW_{1\,\alpha} = -\frac{1}{4}\, \bar{\DD}^2 \DD_\alpha \VV \qquad \text{and} \qquad \WW_{2\,\alpha} = -\frac{1}{4}\,\bar{\DD}^2 \DD_\alpha \VV^\dagger \end{aligned}$$ which have $U(1)_R$ charges of $+1$. Under parity $\VV$ and $\WW_i$ transform as $$\begin{aligned}
\VV\leftrightarrow -\VV \qquad\text{and}\qquad \WW_1^\alpha \leftrightarrow -\bar{\WW}_{2\dot\alpha} \end{aligned}$$ The free Lagrangian, which is properly invariant under $U(1)_R$ and parity, is given by $$\begin{aligned}
\LL_{\VVV} = \int\!\!d^4\theta\; 2\,m_{\VV}^2 \VV^\dagger \VV + \int\!\!d^2\theta\; \frac{1}{2} \WW_1^\alpha \WW_{2\,\alpha} +\!{\text{ h.c. }}\end{aligned}$$ Varying the action yields the equation of motion $\DD_\alpha\WW_1^\alpha=2\,m_\vv^2\VV$, implying that $\PP_T \VV=\VV$ on shell.
Supersymmetry Breaking in the Ground State {#Sec: SSB Effects}
==========================================
The previous section calculated the composition of non-relativistic supersymmetric bound states using supersymmetric group theory, focusing on the particular example of bound states formed from two chiral superfields. This section builds on Sec. \[Sec: NR\] by incorporating the effects of weak supersymmetry breaking on the ground state spectrum. The exact changes to the spectrum resulting from supersymmetry breaking depend on the details of the binding dynamics. In many theories, however, supersymmetry breaking level splittings induced by the binding dynamics are accompanied by powers of the velocity, $v$. Consequently in the non-relativistic limit supersymmetry breaking in the bound state spectrum will be dominated by the differences in the rest energies of the constituent fermions and bosons. For such theories the resulting spectrum is insensitive to the details of the binding dynamics.
Constituent Mass Effects {#Sec: CME}
------------------------
The leading supersymmetry breaking effects can be calculated by folding in the perturbed rest energies of the constituents with the ground state superspin wavefunctions calculated in Sec. \[Sec: NR\]. This leading order effect is straightforward to compute if the effective scale of supersymmetry breaking in the bound states spectrum, $m_{\text{soft}}$, is smaller than the scale of principle excitations $$\begin{aligned}
m_{\text{soft}} \ll m_{\text{prin}} \simeq \OO(\mu v^2) .\end{aligned}$$ In this case mixing with excited states is unimportant and the incorporation of supersymmetry breaking into the bound state spectrum reduces to a finite dimensional quantum mechanical perturbation theory problem.
The bound state spectrum has two effective mass scales for supersymmetry breaking effects. The first scale is set by the $U(1)_R$-preserving soft masses, $m_{R-{\text{pres}}}$, while the second is set by the $U(1)_R$-violating $B$-term masses, $m_{R-{\text{viol}}}$. The breaking of the $U(1)_R$ symmetry induces splittings between states that are doublets under the $O(2)_R$ symmetry. In many implementations of dark sector supersymmetry breaking, $U(1)_R$-violating soft terms will be suppressed relative to the $U(1)_R$-preserving soft terms and for simplicity the relative ordering of the scales is taken to be $$\begin{aligned}
\label{assumption}
m_{R-\text{pres}}, m_{\text{FS}}\gg m_{R-\text{viol}}
\end{aligned}$$ throughout, where $m_{\text{FS}} = \OO(\mu v^4)$.
The soft supersymmetry breaking Lagrangian for the chiral-chiral bound state system introduced in Sec. \[Sec: NR\] contains a $U(1)_R$-preserving piece, $$\begin{aligned}
\label{eqn:lpreserve}
-\LL_{R-\text{pres}} \supset \Delta^2_{\tilde{e}} (|\tilde{e}|^2 + |\tilde{e}^c|^2) + \Delta^2_{\tilde{p}} (|\tilde{p}|^2 + |\tilde{p}^c|^2)\end{aligned}$$ and additional supersymmetry breaking terms that break the $U(1)_R$ symmetry: $$\begin{aligned}
\label{eqn:lbreak}
-\LL_{R-\text{viol}} \supset B_e m_e \tilde{e} \tilde{e}^c + B_p m_p \tilde{p}\tilde{p}^c +\!\!{\text{ h.c. }}\end{aligned}$$ For simplicity the soft parameters are assumed to obey the relations $$\begin{aligned}
\Delta^2_{\tilde{e}} \simeq \Delta^2_{\tilde{p}} \qquad\text{ and }\qquad B_p\simeq B_e \equiv B\end{aligned}$$ In the presence of $\Delta^2_{\tilde{e}}$ and $B_e$ the selectron mass eigenstates become $$\begin{aligned}
\tilde{e}_\pm = \frac{1}{\sqrt 2}(\tilde{e}\pm\tilde{e}^{c\dagger})\end{aligned}$$ with masses $$\begin{aligned}
m_{\tilde{e}\pm} = m_e + \delta m_{\tilde{e}_\pm} \equiv m_e + {{\frac{1}{2} }}\frac{ \Delta^2_{\tilde{e}}}{m_{\text{e}}} \pm {{\frac{1}{2} }}B \end{aligned}$$ Analogous expressions hold for the mass eigenstates $\tilde{p}_\pm$. See Sec. \[Sec: SSB\] for details on a particular implementation of supersymmetry breaking in the dark sector that satisfies the above assumptions.
The leading supersymmetry breaking perturbation on the ground state spectrum is encapsulated in the perturbing Hamiltonian $$\begin{aligned}
H_{\text{soft}} = \delta m_{\tilde{p}\pm} |\phi_{p\pm}\rangle \langle\phi_{p\pm} | + \delta m_{\tilde{e}\pm } |\phi_{e\pm}\rangle\langle\phi_{e\pm} |\end{aligned}$$ The $U(1)_R$-preserving contributions of $H_{\text{soft}}$ will appear in the combination $$\begin{aligned}
m_{\text{soft}} \equiv {{\frac{1}{2} }}(\delta m_{\tilde{p}+}+\delta m_{\tilde{p}-} + \delta m_{\tilde{e}+}+\delta m_{\tilde{e}-})\end{aligned}$$ The rest energy perturbations can now be read off directly from the supersymmetric wavefunctions in Eq. \[eqn:wavefunctions\]. For example consider the state $$\begin{aligned}
{|\varsigma_-\rangle}= c_{2\theta} {|(\psi_p \psi_e)_0\rangle} + s_{2\theta} ({|\phi_{p+}\phi_{e-}\rangle} - {|\phi_{p-}\phi_{e+}\rangle})/\sqrt{2}\end{aligned}$$ The fermion-fermion component is insensitive to $H_{\text{soft}}$, but the scalar-scalar component results in a perturbation $$\begin{aligned}
\Delta m_{\varsigma_-} \simeq \langle \varsigma_- | H_{\text{soft}} |\varsigma_-\rangle =
{{\frac{1}{2} }}s_{2\theta}^2 (\delta m_{\tilde{p}+}+\delta m_{\tilde{e}-} + \delta m_{\tilde{p}-}+\delta m_{\tilde{e}+}) = s^2_{2 \theta} \; m_{\text{soft}}\end{aligned}$$ which is the leading supersymmetry breaking contribution to the mass of $\varsigma_-$ in the limit that $m_{\text{FS}} \gg m_{\text{soft}}$. For many physical applications, such as decays or scattering off of SM nuclei, knowing only the leading breaking is sufficient. Using the superspace approach for finding the wavefunctions, as in Sec. \[Sec: Wavefunctions Fields\], the leading supersymmetry breaking can thus be found for a broad range of perturbative bound states.
Subdominant Effects {#Sec: Subdominant}
-------------------
Supersymmetry breaking effects begin to grow in complexity beyond the rest mass perturbation. The next most important term in the non-relativistic expansion is the kinetic energy perturbation $$\begin{aligned}
H_{v^2} \simeq -\frac{p^2}{2\mu} \frac{\delta \mu}{\mu }\end{aligned}$$ This changes the principle structure of the bound state and leads to a $\OO(v^2)$ perturbing Hamiltonian $$\begin{aligned}
\label{Eq: KineticPert}
H_{v^2} = -\left \langle \frac{p^2}{2\mu^2} \right\rangle
\left( \cos^4\!\theta \; \delta m_{\tilde{e}\pm} |\phi_{e\pm}\rangle\langle \phi_{e\pm}| + \sin^4\! \theta \; \delta m_{\tilde{p}\pm} |\phi_{p\pm}\rangle\langle \phi_{p\pm}|
\right) .\end{aligned}$$ At the level of fine structure many new effects arise. These include additional kinematic effects from $\OO(p^4)$ terms and, in the case of supersymmetric hydrogen, gaugino mass effects and $D$-term contributions. Incorporating all these effects requires using the $\TT$ matrix and computing all tree-level Feynman diagrams contributing to $ep \rightarrow ep$ matrix elements. The $\TT$ matrix is proportional to an effective non-relativistic Hamiltonian that can be used to do perturbation theory, as in the calculation of the fine structure of supersymmetric hydrogen [@Buchmuller:1981bp; @Rube:2009yc; @Herzog:2009fw].
Eigenstates
-----------
In this section the ground state spectrum with weakly broken supersymmetry is presented by diagonalizing the perturbation $H_{\text{soft}}$.
### Scalars {#scalars .unnumbered}
In the absence of supersymmetry breaking the hypermultiplet contains the degenerate pair of positive parity scalar bound states $\varpi_+$ and $\omega_+$. In the presence of $H_{\text{soft}}$ these states mix maximally: $$\begin{aligned}
\left(\begin{array}{ccc}
\varpi_+ \\ \omega_+ \end{array}\right)^{\!\!\! \dagger}
\left(\begin{array}{ccc}
m_{\text{soft}} & B \\
B & \;\;\; m_{\text{soft}}
\end{array}\right)
\left(\begin{array}{ccc}
\varpi_+ \\ \omega_+ \end{array}\right)\end{aligned}$$ $B$ characterizes the size of $O(2)_R$ breaking and mixes states of different $R$-charge. The mass eigenstates are $$\begin{aligned}
\nonumber
\label{eqn:omegaoneplus}
\omega_{1+} &\equiv& \phi_{p+}\phi_{e+} = \frac{1}{\sqrt 2}(\varpi_+ + \omega_+)
\qquad\quad m_{\omega_{1+}} = \delta m_{\tilde{e}+}+\delta m_{\tilde{p}+}=m_{\text{soft}} + B \\
\omega_{2+} &\equiv& \phi_{p-}\phi_{e-} = \frac{1}{\sqrt 2}(\varpi_+ - \omega_+)
\qquad\quad m_{\omega_{2+}} = \delta m_{\tilde{e}-}+\delta m_{\tilde{p}-}= m_{\text{soft}} -B .\end{aligned}$$
In the supersymmetric limit the ground state contains three parity odd scalars, one of which, $\varsigma_-$, is in the vector multiplet and two of which, $\omega_-$ and $\varpi_-$, are in the hypermultiplet. In the presence of $H_{\text{soft}}$ and $H_{v^2}$ all three states mix: $$\begin{aligned}
\label{3x3matrix}
\left(\begin{array}{ccc}
\varsigma_- \! \\ \omega_-\! \\ \varpi_-\! \end{array}\right)^{\!\!\! \dagger}\!\!
\left( \begin{array}{ccc}
m_{\text{FS}} + s_{2\theta}^2 m_{\text{soft}} & -{{\frac{1}{2} }}s_{4\theta} m_{\text{soft}} &{{\frac{1}{2} }}s_{4\theta} B^{\prime} \\
-{{\frac{1}{2} }}s_{4\theta} m_{\text{soft}} & c_{2\theta}^2 m_{\text{soft}} &-c_{2\theta}^2 B^{\prime} \\
{{\frac{1}{2} }}s_{4\theta} B^{\prime} & -c_{2\theta}^2 B^{\prime} &m_{\text{soft}}
\end{array} \right)
\left(\begin{array}{ccc}
\varsigma_- \! \\ \omega_- \! \\ \varpi_-\! \end{array}\right)
\end{aligned}$$ Here $B^{\prime}$ characterizes the $U(1)_R$-breaking in this sector and comes about through $H_{v^2}$ in Eq. \[Eq: KineticPert\] or from the difference in the $B$-term masses between the $\tilde{e}$ and $\tilde{p}$ and is of the order $$\begin{aligned}
\label{negparity1}
B' \sim \OO(Bv^2), \OO(B_e - B_p).\end{aligned}$$ By specializing to the regime where $$\begin{aligned}
B^{\prime} \ll m_{\text{FS}}, \;\tan^2 2\theta \,m_{\text{soft}} \qquad\end{aligned}$$ so that mixing between the three states occurs primarily between $\varsigma_-$ and $\omega_-$, one obtains simple formulae for the approximate energy levels: $$\begin{aligned}
\label{eqn:varsigma-}
m_{\varsigma_-}& =&
\frac{m_{\text{FS}} + m_{\text{soft}}}{2} + \left (\frac{m_{\text{FS}} - c_{4\theta}m_{\text{soft}}}{2} \right)
\sqrt{1+\frac{ s_{4\theta}^2 \; m_{\text{soft}}^2 }{ (m_{\text{FS}} - c_{4\theta}m_{\text{soft}})^2}}\\
\label{eqn:omega-}
m_{\omega_-}& =&\frac{m_{\text{FS}} + m_{\text{soft}}}{2} - \left (\frac{m_{\text{FS}} - c_{4\theta} m_{\text{soft}}}{2} \right)
\sqrt{1+\frac{ s_{4\theta}^2 \; m_{\text{soft}}^2}{ ( m_{\text{FS}} - c_{4\theta}m_{\text{soft}})^2}} \\
m_{ \varpi_-}& =& m_{\text{soft}} \end{aligned}$$ Non-zero $B$-terms split the $O(2)_R$ doublet containing $ \varpi_+$ and $ \varpi_-$.
In the limit $m_{\text{soft}}/m_{\text{FS}}\rightarrow \infty$ with $0 \le \theta < \tfrac{\pi}{8}$ (respectively $\tfrac{\pi}{8} < \theta \le \tfrac{\pi}{4}$), the state $\varsigma_-$ (respectively $\omega_-$) becomes the $(\psi_e \psi_p)_0$ bound state. Naively, for $\theta^2 \simeq \frac{m_e}{m_p} \simeq \tfrac{1}{1836}$ and $m_{\text{FS}}={{\frac{1}{2} }}\alpha_\vv^4 \mu$ the splitting between this state and the vector $(\psi_e \psi_p)_1$ should give the hyperfine splitting in regular hydrogen; however, Eq. \[eqn:varsigma-\] yields instead: $$\begin{aligned}
\label{Eq: HyperfineEmergence}
m_{(\psi_e \psi_p)_1} - m_{(\psi_e \psi_p)_0} = m_{v_\mu} - m_{\varsigma_-} \rightarrow s_{2\theta}^2 m_{\text{FS}} \rightarrow \frac{2\alpha_\vv^4 m_e^2}{ m_p} \end{aligned}$$ This is not the correct hyperfine splitting of regular hydrogen which is $$\begin{aligned}
m_{\text{HFS}} = \frac{8}{3} \frac{ \alpha^4 m_e^2}{m_p}\end{aligned}$$ for a point-like proton. This difference arises because $m_{\text{soft}}/m_{\text{FS}}\rightarrow \infty$ is [*not*]{} the full decoupling limit. In particular, the ground state of supersymmetric hydrogen contains admixtures of higher principle excitations arising from gaugino exchange at second order in perturbation theory. These effects contribute to the hyperfine splitting in Eq. \[Eq: HyperfineEmergence\] but disappear in the full decoupling limit where the gaugino mass goes to infinity, $m_{\tilde{\vv}}\rightarrow \infty$.
### Fermions {#fermions .unnumbered}
In the absence of supersymmetry breaking the vector multiplet (hypermultiplet) contains the degenerate pair of $j={{\frac{1}{2} }}$ bound states $\xi_\pm$ ($\chi_\pm$). In the presence of $H_{\text{soft}}$ the states of equal parity mix with one another: $$\begin{aligned}
\left(\begin{array}{ccc}
\bar{\chi}_\pm \\ \bar{\xi}_\pm \end{array}\right)^{\!\!\! T}
\left(\begin{array}{ccc}
m_{\text{FS}} +{{\frac{1}{2} }}s_{2\theta}^2 m_{\text{soft}} \pm{{\frac{1}{2} }}B & -\tfrac{1}{4} s_{4\theta} m_{\text{soft}} \\
-\tfrac{1}{4} s_{4\theta} m_{\text{soft}} & (s_{\theta}^4+c_{\theta}^4) m_{\text{soft}} \pm{{\frac{1}{2} }}B
\end{array}\right)
\left(\begin{array}{ccc}
\chi_\pm \\ \xi_\pm \end{array}\right)
\label{eqn:fermionmatrix}\end{aligned}$$ The spectrum is given by $$\begin{aligned}
m_{\chi_\pm}& =&
\frac{m_{\text{FS}} + m_{\text{soft}}}{2} + \left (\frac{m_{\text{FS}} - c_{2\theta}^2 m_{\text{soft}}}{2} \right)
\sqrt{1+\frac{\tfrac{1}{4} s_{4\theta}^2 m_{\text{soft}}^2 }{ (m_{\text{FS}} - c_{2\theta}^2 m_{\text{soft}})^2}}\pm {{\frac{1}{2} }}B \\
m_{\xi_\pm}& =&\frac{m_{\text{FS}} + m_{\text{soft}}}{2} - \left (\frac{m_{\text{FS}} - c_{2\theta}^2 m_{\text{soft}}}{2} \right)
\sqrt{1+\frac{\tfrac{1}{4} s_{4\theta}^2 m_{\text{soft}}^2 }{ (m_{\text{FS}} - c_{2\theta}^2 m_{\text{soft}})^2}}\pm {{\frac{1}{2} }}B \end{aligned}$$ For non-zero $B$ the $O(2)_R$ symmetry that ensured the degeneracy of the pair of states $\chi_{\pm}$ as well as the pair of states $\xi_{\pm}$ is broken and the fermionic spectrum splits completely.
### Vector {#vector .unnumbered}
The vector state ${|\vec{v}\rangle} = {|(\psi_p\psi_e)_1 \rangle}$ is insensitive to $H_{\text{soft}}$ and, as a consequence, does not feel supersymmetry breaking at leading order.
Interactions {#Sec: Interactions}
============
Composite systems have a wide range of interactions that are controlled by selection rules and form factors that results in these systems having a much richer phenomenology than elementary particles. This section uses the effective field theory of Sec. \[Sec: Effective Action\] to study the interactions that arise when composite states inherit gauge interactions from their constituents (cf. [@Bagnasco]).
Sec. \[Sec: U1v\] considers the case where the constituents are charged under an unbroken vectorial gauge symmetry $U(1)_\vv$ such that the composite state is neutral with the following charge and parity assignments $$\begin{aligned}
\begin{array}{|c|cccc|}
\hline
& E& E^c& P & P^c\\
\hline
U(1)_\vv& -1& +1 & + 1& -1\\
\hline
\end{array}
\qquad
\text{ and }\qquad \vv \leftrightarrow -\vv\end{aligned}$$ $\vv$ does not need to be responsible for binding the chiral multiplets together; [*e.g.*]{}, the binding could arise from a Yukawa force. The $U(1)_\vv$ gauge interactions of the constituents induce a number of effective operators, including charge radius, Rayleigh scattering, and spin flip operators. Specializing to the case where the hypermultiplet $\HH$ is lighter than the vector multiplet $\VV$ and $m_{\text{FS}} \gg m_{\text{soft}}$, decays within the ground state are discussed in detail. It is found that the states of the vector multiplet $\VV$ decay relatively quickly down to $\HH$, while the decays within $\HH$ are much slower.
Sec. \[Sec: U1a\] briefly considers the case where the constituents are charged under a broken axial gauge symmetry $U(1)_\aa$ with charge and parity assignments $$\begin{aligned}
\label{Eq: Axial Charges}
\begin{array}{|c|cccc|}
\hline
& E& E^c& P & P^c\\
\hline
U(1)_\aa& +1 & +1 & -1 & -1\\
\hline
\end{array} \qquad\text{ and }\qquad \aa \leftrightarrow \aa\end{aligned}$$ In models such as that of Sec. \[Sec: Nearly Susy\] where $\aa$ undergoes kinetic mixing with the SSM, these interactions mediate the dominant coupling of dark atoms to the Standard Model. Sec. \[Sec: U1a\] discusses the allowed scattering channels and finds the leading supersymmetric axial interactions.
$U(1)_\vv$ interactions {#Sec: U1v}
-----------------------
The interactions of neutral bound states with an external vector superfield, $\vv$, are characterized by two scales corresponding to the charge radius, $R_e$, and magnetic radius, $R_m$. Physically $R_e$ corresponds to the size of the bound state, $R_e \sim \sqrt{\langle r^2 \rangle}$. In the case of Coulombic bound states $R_e$ is given by the Bohr radius, $R_e^{-1} = \alpha_\vv \mu$. $R_m$ is just the Compton wavelength, $R_m^{-1} = \mu$. For convenience, this section will restrict its discussion to supersymmetric hydrogen, although it is generally applicable to chiral-chiral bound states.
Before considering the supersymmetric case, it is instructive to review the leading interactions of the photon with the spin-singlet ground state of regular hydrogen. The leading elastic interaction comes from the charge radius operator $$\begin{aligned}
g_\vv c_{2\theta} R_e^2(\psi_p\psi_e)_0^\dagger\partial_\mu(\psi_p\psi_e)_0 \partial_\nu \vv^{\mu\nu}
\label{eqn:charge-radius-ff}\end{aligned}$$ which is fully determined by the charge distribution of the bound state. The leading inelastic interaction comes from the magnetic spin-flip operator which is determined by the fermion content: $$\begin{aligned}
g_\vv R_m\partial_\mu(\psi_p\psi_e)_{1,\nu}(\psi_p\psi_e)_0\tilde \vv^{\mu\nu}
\label{eqn:magnetic1}\end{aligned}$$ Finally there is the Rayleigh scattering operator $$\begin{aligned}
g_\vv R_e^3m_H(\psi_p\psi_e)_0^\dagger(\psi_p\psi_e)_0 \vv_{\mu\nu}\vv^{\mu\nu}
\label{eqn:rayleigh}\end{aligned}$$ which makes the sky blue. All other operators are higher order in either $g_\vv$ or $\mu^{-1}$.
The next step is to find the set of operators necessary to satisfy the supersymmetric Ward identities. $\VV\leftrightarrow \VV$ interactions can be important for scattering processes if the states of $\VV$ are long-lived but have a subdominant effect on the lifetimes of the states in $\VV$. Because the leading $\VV \leftrightarrow \HH$ decay is relatively fast $\VV$ tends to be short-lived and therefore $\VV\leftrightarrow \VV$ interactions are ignored here. The $\HH \leftrightarrow \HH$ interactions are found in the following, since they determine the relaxation timescale of the $\HH$ supermultiplet.
The charge radius operator in Eq. \[eqn:charge-radius-ff\] only depends on the charge distribution, and therefore the scalar-scalar bound states must share identical (diagonal) interactions: $$\begin{aligned}
g_\vv c_{2\theta} R_e^2(\phi_{p\pm}\phi_{e\pm})^\dagger \partial_{\mu} (\phi_{p\pm}\phi_{e\pm}) \partial_\nu \vv^{\mu\nu}
\label{eqn:charge-radius-ss}\end{aligned}$$ Rewriting Eq. \[eqn:charge-radius-ff\] and \[eqn:charge-radius-ss\] in terms of the wavefunctions in Eq. \[eqn:wavefunctions\], the charge radius interactions become $$\begin{aligned}
g_\vv c_{2\theta} R_e^2\left(\omega^\dagger_\pm\partial_\mu\omega_\pm+\varpi^\dagger_\pm\partial_\mu\varpi_\pm+
\varsigma^\dagger_-\partial_\mu\varsigma_-\right)\! \partial_\nu \vv^{\mu\nu}
\label{eqn:charge-radius-SUSY1}\end{aligned}$$ Similarly, the spin-flip and Rayleigh scattering operators become $$\begin{aligned}
g_\vv R_m\partial_\mu v_\nu (c_{2\theta}\varsigma_-+s_{2\theta}\omega_-)\tilde \vv^{\mu\nu}
\label{eqn:spin-flip-SUSY1}\end{aligned}$$ and $$\begin{aligned}
g_\vv R_e^3(m_e+m_p)\left(\omega^\dagger_\pm\omega_\pm+\varpi^\dagger_\pm\varpi_\pm+
\varsigma^\dagger_-\varsigma_-\right)\!\vv_{\mu\nu}\vv^{\mu\nu},
\label{eqn:rayleigh-SUSY}\end{aligned}$$ respectively. The operators in Eq. \[eqn:charge-radius-SUSY1\] to \[eqn:rayleigh-SUSY\] represent the leading single photon and two-photon interactions for the scalar states in $\VV$ and $\HH$. Several interactions remain to be found, [*e.g.*]{}, the leading single photino interactions as well as the interactions for the fermionic states. The coefficients of the remaining interactions are found by forming operators from the effective fields of Sec. \[Sec: Effective Action\]. The matching coefficients are determined by expanding the supersymmetric operators in terms of their components and identifying the corresponding interactions from Eq. \[eqn:charge-radius-SUSY1\] to \[eqn:rayleigh-SUSY\]. This procedure allows for the various supersymmetric interactions to be systematically enumerated by building upon the known interactions of regular hydrogen.
### Interactions of the Hypermultiplet with Higher States {#interactions-of-the-hypermultiplet-with-higher-states .unnumbered}
A variety of processes cause the decay of the excited states to the ground state. For example, supersymmetric hydrogen inherits the (fast) electric dipole and magnetic dipole transitions of regular hydrogen. Decays from $\VV$ to $\HH$, however, are not as fast and merit further discussion.
The states in $\VV$ are connected to $\HH$ through two one-photon operators of dimension five: $$\begin{aligned}
\label{Eq: HV Int1}
\LL_{\VV \HH \vv} &=&c_M s_{2\theta}\,g_\vv R_m\! \int\!\! d^2\theta\; (\HH_1^c\WW_1+\HH_2^c\WW_2)\WW_{\vv}+{\text{ h.c. }}\\
\label{Eq: HV Int2}
&&+ c_{M}^{\prime}(g_\vv, \theta) g_\vv R_m\! \int\!\! d^4\theta\; (\HH_1^c+\bar\HH_2^c)\VV \DD\WW_\vv+{\text{ h.c. }}\end{aligned}$$ These two operators are the most general forms for $\HH\leftrightarrow\VV$ interactions mediated by $U(1)_\vv$. Higher dimensional operators can be reduced to these two forms with additional factors of $\partial^2$ acting on $\WW_\vv$ by using the matter field equations of motion.
Only Eq. \[Eq: HV Int1\] contains the magnetic spin-flip interaction, $\omega_-\partial_\mu v_\nu \tilde F^{\mu\nu}_\vv$. The factor of $s_{2\theta}$ is fixed by comparison with Eq. \[eqn:magnetic1\]. In supersymmetric hydrogen, some of the component interactions contained in Eq. \[Eq: HV Int1\] arise from $\OO(\alpha_\vv)$ mixing between the ground state and higher principle excitations. For example, excited $\omega_+$ states ($2p$, $3p$, etc.) mix with $v^\mu$, allowing for $v^\mu$ to decay to $\omega_+$ through electric dipole transitions. This mixing with excited states is the origin of the “electric” interaction $\omega_+\partial_\mu v_\nu F^{\mu\nu}_\vv$ contained in the operator of Eq. \[Eq: HV Int1\]. In this sense, the operator of Eq. \[Eq: HV Int1\] is neither purely magnetic or electric. The role that excited state mixing plays in ensuring this supersymmetric result is familiar from the calculation of the supersymmetric spectrum in [@Buchmuller:1981bp; @Rube:2009yc; @Herzog:2009fw], where second order perturbation theory is needed to determine the spectrum to $\OO(\alpha_\vv^4)$.
The operator in Eq. \[Eq: HV Int2\] does not mediate decays in the supersymmetric limit. This can be seen by using the equations of motion to replace $\DD\WW_\vv$ with the current $\JJ_\vv$. Decays through this operator are kinematically forbidden because the mass splitting between $\VV$ and $\HH$ is much smaller than the mass of any particle charged under $U(1)_\vv$. For this reason we leave the coefficient of this operator undetermined, noting however that it can only come in at higher order than $g_\vv R_e$, since it contains off-diagonal scalar-scalar transitions, which do not arise from charge radius scattering.
The various decay channels induced by the interactions in Eq. \[Eq: HV Int1\] cause each state in $\VV$ to have the same inclusive decay width to the states of $\HH$ in the supersymmetric limit—otherwise the component propagators of $\VV$ would have different poles. Therefore, the decay width can be calculated by considering the state with the simplest decay modes, in this case $\varsigma_-$: $$\begin{aligned}
\LL_{\VV \HH \vv}
\supset
c_M g_\vv R_m m_\vv \; \frac{i s_{2\theta} }{2\sqrt{2}}\left(i\bar\xi_+\gamma_5+\bar\xi_-\right) \Lambda_\vv\,\varsigma_-\end{aligned}$$ Here $\Lambda_\vv$ is the four-component Majorana gaugino of $U(1)_\vv$ and $m_{\tilde{\vv}}$ is its mass. This gives the decay rate $$\begin{aligned}
\Gamma_{\VV\rightarrow\HH \vv} \simeq |c_M g_\vv s_{2\theta} R_m m_\vv|^2 \frac{ m_{\text{FS}}^2}{4\pi m_\vv }
= |c_M|^2 \alpha_\vv^9 \mu\;
\label{eqn:spinflipdecayrate}
\end{aligned}$$ This is a factor of $\frac{\mu}{m_{\text{FS}}}$ faster than the corresponding spin-flip transition in regular hydrogen, which scales as $\alpha m_{\text{FS}}^3$. This is because the decays are dominated by $\Lambda_\vv$ emission rather than $\vv^\mu$ emission, for which the amplitude carries an additional factor factor of $E^{{\frac{1}{2} }}$, where $E$ is the energy of the emitted gauge particle.
### Interactions within the Hypermultiplet {#interactions-within-the-hypermultiplet .unnumbered}
Supersymmetry restricts the form of possible interactions significantly, and these restrictions are particularly severe for interactions connecting two chiral superfields. For instance, the only allowed single photon operator, up to possible additional factors of $ \partial^2$, is $$\begin{aligned}
\int\! d^4\theta\; \Phi_1\Phi_2^\dagger \DD\WW_\vv+ \! \!{\text{ h.c. }}\label{eqn:PhPhDW}\end{aligned}$$ In the case of $\HH \leftrightarrow \HH$ interactions, the only operators of this form allowed by the $O(2)_R$ and $U(1)_{\text{e+p}}$ symmetries of the theory are $$\begin{aligned}
\int\!d^4\theta(\HH_1\HH_1^\dagger-\HH_2\HH_2^\dagger)\DD\WW_\vv \qquad \text{and} \qquad
\int\!d^4\theta(\HH_1^c\HH_1^{c\dagger}-\HH_2^c\HH_2^{c\dagger})\DD\WW_\vv\end{aligned}$$ These operators contain terms like $\varpi_+^\dagger\partial_\mu\varpi_+ \partial_\nu F^{\mu\nu}_\vv$ and $\omega_+^\dagger\partial_\mu\omega_+ \partial_\nu F^{\mu\nu}_\vv$, respectively, and thus correspond to charge radius interactions. Matching to Eq. \[eqn:charge-radius-SUSY1\] then gives the supersymmetric completion of the charge radius interactions: $$\begin{aligned}
\LL_{\HH \HH \vv} = c_E g_\vv c_{2\theta} R_e^2 \int\!\!d^4 \theta (\HH^\dagger_1 \HH_1 - \HH^{\dagger}_2 \HH_2 -\HH^{c}_1\HH_1^{c\dagger} + \HH^{c\dagger }_2 \HH^c_2)\DD \WW_\vv
\label{eqn:HHinteraction}\end{aligned}$$ Replacing $\DD\WW_V$ with the current $\JJ_\vv$ gives atom-ion scattering. Similarly matching onto the Rayleigh scattering operator in Eq. \[eqn:rayleigh-SUSY\] yields $$\begin{aligned}
\LL_{\HH\HH\vv\vv}= c_{E}^{\prime} g_\vv R_e^3 &\int\!d^4\theta\; (\HH_1\HH_1^c+\HH_2\HH_2^c)^\dagger \WW_\vv\WW_\vv +\!\!{\text{ h.c. }}\label{eqn:HHVVinteraction}\end{aligned}$$ Just like the operator in Eq. \[eqn:HHinteraction\], this operator will mediate decays within the hypermultiplet once supersymmetry is broken.
The restriction to operators of the form in Eq. \[Eq: HV Int2\] is a supersymmetric analog of the statement that any interaction involving two scalars and a field strength can be written as “$(\textup{derivatives})\times \phi\partial_\mu\phi'\partial_\nu F^{\mu\nu}$,” which implies that transitions between scalar states cannot proceed via single photon emission. Thus, for example, direct single photon/photino decays from the $2s$ hypermultiplet to the ground state hypermultiplet are forbidden. The decay will instead proceed through either two photon/photino transitions or a cascade decay via magnetic operators of the form in Eq. \[Eq: HV Int1\].
### Hypermultiplet Decays {#hypermultiplet-decays .unnumbered}
In the supersymmetric limit, the hypermultiplet is exactly stable. Once supersymmetry is broken and decays within the hypermultiplet become kinematically allowed, it is interesting to ask what decay channels determine the relaxation timescale. This question is complicated by the fact that supersymmetry breaking enters the physics of decays in a number of ways. On the one hand, supersymmetry breaking perturbs eigenvalues and eigenstates; this opens up phase space, changes the equations of motion, and induces decay channels through mixing. On the other hand, supersymmetry breaking perturbs the effective interactions of the non-relativistic constituents. The rest of this section considers these possibilities in more detail, with the conclusion that eigenstate mixing in the magnetic spin-flip operator, Eq. \[Eq: HV Int1\], induces the largest decay rates.
In the presence of soft masses, the supersymmetric operators in Eq. \[eqn:HHinteraction\] and \[eqn:HHVVinteraction\] can mediate decays within the hypermultiplet. In the case of the three-body decays mediated by the Rayleigh scattering operator in Eq. \[eqn:HHVVinteraction\], these soft masses appear in the eight powers of phase space: $$\begin{aligned}
\Gamma_{\{ \omega_{1+},\; \omega_{2+},\;\varpi_-\}\rightarrow\xi_\pm \Lambda_\vv \vv} \simeq \Gamma_{\xi_\pm\rightarrow\omega_- \Lambda_\vv \vv}
&\simeq& \nonumber
|c^{\prime}_E g_\vv R_e^3|^2 \frac{(m_{\xi_\pm}-m_{\omega_-})^8}{ 64 \pi^3 m_\HH} \\ &=&|c^{\prime}_E|^2 \frac{\alpha_\vv^{27}}{16 \pi^2} \left(\frac{s_{2\theta}}{2}\right)^{18} \left(\frac{m_{\text{soft}}}{m_{\text{FS}}}\right)^8 \mu\end{aligned}$$ In the case of the two-body decays mediated by the charge radius operator in Eq. \[eqn:HHinteraction\], these soft masses appear in phase space as well as in an overall factor of $m_{\tilde{\vv}}^2$. This latter factor arises from the modified equations of motion for $\Lambda_\vv$, which imply that $\DD\WW_V \supset \bar\theta \slashed{\partial} \Lambda_\vv \propto m_{\tilde{\vv}} \bar\theta \Lambda_\vv$. The resulting decay rate is $$\begin{aligned}
\nonumber
\Gamma_{\{ \omega_{1+},\; \omega_{2+},\;\varpi_-\}\rightarrow\xi_\pm \Lambda_\vv } \simeq
\Gamma_{\xi_\pm\rightarrow\omega_- \Lambda_\vv } &\simeq& |c_E g_\vv c_{2\theta} R_e^2|^2 \;|m_{\tilde{\vv}}(m_\HH-m_{\xi_\pm})|^2 \frac{(m_{\xi_{\pm}}-m_{\omega_-})^2}{4\pi m_\HH}\\
&\simeq& |c_E|^2 c_{2\theta}^2 \frac{\alpha_\vv^{21}}{16} \left(\frac{s_{2\theta}}{2}\right)^{6} \left(\frac{m_{\text{soft}}}{m_{\text{FS}}}\right)^4 \left(\frac{m_{\tilde{\vv}}}{m_{\text{FS}}}\right)^2 \mu.
\label{eqn:HHV-rate}\end{aligned}$$ Here two powers of $m_{\text{soft}}$ arise from cancellations between the terms involving $\HH$ and $\HH^c$ in Eq. \[eqn:HHinteraction\]. Higher order operators may not have this cancellation.
Next consider how the magnetic spin-flip operator in Eq. \[Eq: HV Int1\] induces decays in the presence of supersymmetry breaking. Mixing between, e.g., the fermionic states $\chi_\pm$ and $\xi_\pm$ allows all the states in $\HH$ to decay down to $\omega_-$ through Eq. \[Eq: HV Int1\], which contains interactions of the form $$\begin{aligned}
c_M g_\vv R_m s_{2\theta} m_{\tilde{\vv}} \bar\Lambda_\vv \begin{pmatrix}{1}\\ { \gamma_5 } \end{pmatrix} \chi \begin{pmatrix}{\varpi}\\ { \omega } \end{pmatrix}^{\!\dagger}
\label{eqn:magneticchi}\end{aligned}$$ Comparison with Eq. \[eqn:fermionmatrix\] shows that, in Eq. \[eqn:magneticchi\], this fermionic mixing is accounted for by making a replacement of the form $$\begin{aligned}
\chi \rightarrow \chi +\frac{s_{4\theta}}{4} \frac{m_{\text{soft}}}{m_{\text{FS}}} \xi\end{aligned}$$ which leads to the decay rate $$\begin{aligned}
\Gamma_{\{ \omega_{1+},\; \omega_{2+},\;\varpi_-\}\rightarrow\xi_\pm \lambda_\vv} &\simeq& \Gamma_{\xi_\pm\rightarrow\omega_- \lambda_\vv} \nonumber
\simeq |c_M g_\vv s_{2\theta} R_m m_\vv|^2 \left( \frac{s_{4\theta}}{4} \frac{m_{\text{soft}}}{m_{\text{FS}}} \right)^{\!2} \frac{ (\tfrac{1}{2} s_{2\theta}^2 m_{\text{soft}})^2}{4\pi m_\HH } \\
&=& |c_M|^2 \alpha_\vv^9 \left( \frac{s_{2\theta} s_{4\theta}}{8} \right)^{\!2} \left( \frac{m_{\text{soft}}}{m_{\text{FS}}} \right)^{\!4} \mu\;
\label{eqn:gammamixie}\end{aligned}$$ This decay rate also receives contributions from supersymmetry breaking in the effective Yukawa operators of the non-relativistic theory, since the coefficients carry factors of $m_e^{-1/2}$ and $m_p^{-1/2}$ from the non-relativistic normalization of the scalar constituents in Eq.\[eqn:non-relfields\]. These contributions, however, are parametrically smaller by an amount $\OO(m_{\text{FS}}^2/m_e^2)$. Hence the decay rate Eq.\[eqn:gammamixie\], which is suppressed by only four powers of the largest supersymmetry breaking spurion, $m_{\text{soft}}$, characterizes the relaxation timescale of the hypermultiplet.
$U(1)_\aa$ interactions {#Sec: U1a}
------------------------
This section outlines the dominant interactions between dark atoms and an axial $U(1)$ with charges given in Eq.\[Eq: Axial Charges\] and mediated by a vector superfield $\aa$. Axial gauge symmetry forbids mass terms for fermions and therefore the gauge symmetry must be broken if non-relativistic bound states exist. As in the previous section, there are several allowed supersymmetric operators and the interactions of the vector boson are sufficient to fix the coefficients of the operators.
The leading $\aa_\mu$ interactions are determined by the axial charges of the constituents. The scalars $\varpi_\pm$ in Eq. \[eqn:wavefunctions\] have zero axial charge, but the combinations $$\label{eqn:AxialInteraction0}
\frac{1}{\sqrt{2}}\left(\omega_+ \mp c_{2\theta}\omega_-\pm s_{2\theta}\varsigma_-\right)$$ have charges of $\pm 2$ respectively. This leads to the inelastic interactions $$2ig_\aa\left(c_{2\theta}\omega_+^\dagger\overleftrightarrow{\partial_\mu}\omega_- -s_{2\theta}\omega_+^\dagger\overleftrightarrow{\partial_\mu}\varsigma_-\right)\aa^\mu.
\label{eqn:AxialInteraction1}$$ Similarly, the fermion-fermion bound states are charged with interactions given by$$\begin{aligned}
g_\aa m_\vv \left(c_{2\theta}\varsigma_-^\dagger+s_{2\theta}\omega_-^\dagger\right)v_\mu\aa^\mu.
\label{eqn:AxialInteraction2}\end{aligned}$$ The interactions of Eq.\[eqn:AxialInteraction0\]-\[eqn:AxialInteraction2\] can be embedded in the following superspace operators: $$\begin{aligned}
&\LL_{\HH\HH\aa}\propto& g_\aa c_{2\theta}\int\!\!d^4 \theta\; \left(\HH_1^{c\dagger} \HH_1^c + \HH_2^{c\dagger}\HH_2^c\right)\aa
\label{eqn:AxialHyper}\\
&\LL_{\HH\VV\aa} \propto& g_\aa s_{2\theta}m_\vv\int\!\!d^4 \theta\; (\HH_1^c-\HH_2^{c\dagger})\VV \aa+{\text{ h.c. }}\\
&\LL_{\VV\VV\aa}\propto& g_\aa m_\vv^2 c_{2\theta}\int\!\!d^4 \theta\; \VV^\dagger\VV\aa.\end{aligned}$$ The interactions of the linear superfield eaten by $\aa$ can be obtained by going out of unitary gauge $$\begin{aligned}
\aa \rightarrow \aa + \frac{ \pi_\aa + \pi^\dagger_\aa}{\sqrt{2} m_\aa} .\end{aligned}$$ In models like that of Sec.\[Sec: Nearly Susy\], where $\aa_\mu$ undergoes kinetic mixing with Standard Model hypercharge, $\aa_\mu$ interactions mediate the dominant coupling of dark atoms to the Standard Model. This setup can be used to realize inelastic dark matter because the elastic interaction of the ground state, $\omega_-$, with $\aa_\mu$ is forbidden due to parity. After supersymmetry breaking, $\tfrac{1}{\sqrt 2}(\omega_+\pm\varpi_+)$ become mass eigenstates. The interactions in Eq.\[eqn:AxialInteraction1\]-\[eqn:AxialInteraction2\] then allow $\omega_-$ to upscatter to $\omega_{1+}$ and $\omega_{2+}$, which are heavier by an amount $\sim \OO(m_{\text{soft}})$, and to $v_\mu$, which is heavier by an amount $\sim \OO(m_{\text{FS}})$.
Higher dimension operators also contribute to the interactions with the standard model. For example, the axial spin flip operator $$\begin{aligned}
\LL_{\HH\VV\aa}^{D=5}\propto g_\aa s_{2\theta}R_m \int\!\!d^2 \theta\; \left(\HH_1^c\WW_1-\HH_2^c\WW_2\right)\WW_\aa+\textup{h.c.}\end{aligned}$$ leads to scattering which can be important in certain regions of parameter space.
Two body decays mediated by $\aa_\mu$ are either kinematically forbidden or severely suppressed, since $m_\aa$ can only be made smaller than $m_{FS}$ by choosing $g_\aa \lesssim \OO(\alpha_\vv^4)$. Similarly, three body decays mediated by an off-shell $\aa_\mu$ are subdominant.
Kinetically Mixed Supersymmetric Hydrogen {#Sec: Nearly Susy}
=========================================
This section constructs a minimal model for a nearly supersymmetric dark sector that supports Coulombic bound states. Sec. \[Sec: Kinetic Mixing\] introduces a minimal Higgs sector and discusses how kinetic mixing of the dark $U(1)_\aa$ with hypercharge in the supersymmetric Standard Model (SSM) drives gauge symmetry breaking in the hidden sector. Sec. \[Sec: Charged Matter\] adds matter fields that are charged under a second Abelian gauge symmetry, $U(1)_\vv$, that introduces hydrogen-like bound states into the spectrum of the theory. In the low energy limit this theory reduces to supersymmetric QED with two massive flavors. Sec. \[Sec: SSB\] discusses how supersymmetry breaking is communicated to the dark sector from the SSM. Finally, Sec. \[Sec: Benchmarks\] illustrates the scales of the resulting model by calculating three benchmark points.
Kinetic Mixing {#Sec: Kinetic Mixing}
--------------
Abelian field strengths are gauge invariant and therefore no symmetry principle forbids mixed field strength terms [@Holdom:1985ag]. Kinetic mixing occurs in extensions of the Standard Model with additional $U(1)$ gauge factors if there are fields that are charged under both the new $U(1)$ and hypercharge. In supersymmetric theories, the entire gauge supermultiplet undergoes gauge kinetic mixing, leading to both gaugino kinetic mixing and $D$-term mixing [@SusyKineticMixing; @Cheung:2009]. If there are light fields charged under the new $U(1)$, then kinetic mixing drives gauge symmetry breaking.
Consider a minimal example where a dark $U(1)_\aa$ couples to a pair of chiral superfields $\Phi$ and $\Phi^c$ with charges $\pm 2$ (chosen for later convenience). The Lagrangian is given by $$\begin{aligned}
\label{Eq: Simple L}
\!\!\!\!\!\!\LL_{\text{Hidden}} = \int\!\!d^4 \theta \; (\Phi^\dagger e^{4 g_\aa \aa} \Phi + \Phi^c{}^\dagger e^{-4 g_\aa \aa} \Phi^c)
+\!\! \int\!\!d^2\theta\;(\tfrac{1}{4} \WW_\aa^2 -\tfrac{\epsilon}{2} \WW_\aa \WW_Y + W_0)+\!\!{\text{ h.c. }}\end{aligned}$$ where $\aa$ is the supersymmetric gauge potential of the hidden $U(1)_\aa$, $\WW_\aa$ is the supersymmetric gauge field strength of $\aa$, and $\WW_Y$ is the supersymmetric gauge field strength of SSM hypercharge. For $\epsilon \ll 1$ the hidden sector is only a small perturbation to the SSM so that all SSM fields will have their normal vacuum expectation values; in particular the SSM Higgs fields will acquire vevs along a non $D$-flat direction: $$\begin{aligned}
D_Y = \frac{g_Y v^2}{4} \cos 2\beta \end{aligned}$$ This SSM vev now acts as a source term for $\WW_\aa$ in Eq. \[Eq: Simple L\] and forces $\phi$, the lowest component of $\Phi$, to acquire a vev, since $D_Y$ acts an effective Fayet-Illiopoulos term for $U(1)_\aa$. The resulting effective Lagrangian is $$\begin{aligned}
\LL_{D} = -{{\frac{1}{2} }}D_\aa^2 + D_\aa( \epsilon D_Y -2g_\aa( |\phi|^2 -|\phi^c|^2) ) \;\;\; \Rightarrow \;\;\;
|\phi|^2 = |\phi^c|^2 + \frac{ \epsilon D_Y}{ 2g_\aa} \ne 0\end{aligned}$$ This $D$-term potential has a residual flat direction, which can be lifted by $W_0$. This section uses a superpotential $$\begin{aligned}
W_0=\lambda S (\Phi \Phi^c -\mu_0^2)
\label{eqn:W0}\end{aligned}$$ where $S$ is a new singlet chiral superfield. With the addition of $W_0$ the vevs of all fields are fixed and there are no massless fermions. It is convenient to let the superfields acquire vevs and to expand around the new field origin $$\begin{aligned}
\langle \Phi \rangle = v_\aa \cos \beta_\aa \qquad \langle \Phi^c\rangle = v_\aa \sin \beta_\aa \qquad \langle \Phi \Phi^c \rangle = \mu_0^2
\label{eqn:vevs}\end{aligned}$$ where the last expression is enforced by the $F$-term for $S$. Solving for $v_\aa$ and $\tan\beta_\aa$ gives $$\begin{aligned}
v^4_\aa= \left( \frac{\epsilon D_Y}{2g_\aa}\right)^2 + 4 \mu_0^4\qquad
\tan 2\beta_\aa = \frac{ 4 g_\aa \mu_0^2}{\epsilon D_Y}.
\label{eqn:va}\end{aligned}$$ In the limit $\mu_0^2\ll \epsilon D_Y/g_\aa$, $\tan\beta_\aa \rightarrow 0$ and in the opposite limit, $\tan \beta_\aa \rightarrow 1$. Fluctuations around the vacuum in Eq. \[eqn:vevs\] can be diagonalized using the field definitions $$\begin{aligned}
\Phi = ( v_\aa + \pi)\cos\beta_\aa + \sin\beta_\aa S^c \qquad \Phi^c =( v_\aa - \pi)\sin\beta_\aa + \cos\beta_\aa S^c\end{aligned}$$ so that the superpotential becomes $$\begin{aligned}
W_0 = \lambda S( \Phi \Phi^c -\mu_0^2) = \lambda v_\aa S S^c + \cdots \end{aligned}$$ clearly showing that $S$ picks up a Dirac mass $m_S= \lambda v_\aa$. In the Kähler term, the super-Higgs mechanism takes place: $$\begin{aligned}
\nonumber
K &=& \Phi^\dagger e^{4 g_\aa \aa}\Phi + \Phi^c{}^\dagger e^{-4g_\aa \aa} \Phi^c + S^\dagger S\\
&=&
S^c{}^\dagger S^c + S^\dagger S+ m_{\aa}^2\left( \aa + \frac{ \pi + \pi^\dagger}{\sqrt{2} m_\aa}\right)^2 +\cdots\end{aligned}$$ The superfield $\pi$ is clearly identified as the eaten linear superfield, and the vector field has picked up a mass $$\begin{aligned}
m_\aa = 2\sqrt{2} g_\aa v_\aa. \end{aligned}$$
In addition to driving $U(1)_\aa$ gauge symmetry breaking, kinetic mixing also leads to $U(1)_R$-breaking mass effects in the gaugino sector of the theory. This is because for $\epsilon \ne0$ gaugino kinetic mixing between $U(1)_Y$ and $U(1)_\aa$ entangles the $U(1)_R$-preserving Dirac mass $m_\aa$ with the $U(1)_R$-breaking bino mass $M_1$. The effective Lagrangian for the gauginos is $$\begin{aligned}
\!\!\LL_{ \lambda} =
\bar{\lambda}_Y i {\partial\hspace{-0.155in}\not\hspace{0.075in}}\lambda_Y
+ \bar{\lambda}_\aa i {\partial\hspace{-0.155in}\not\hspace{0.075in}}\lambda_\aa
+\bar{\chi}_\pi i {\partial\hspace{-0.155in}\not\hspace{0.075in}}\chi_\pi
-( M_1 \lambda_Y \lambda_Y
+\epsilon \bar{\lambda}_\aa {\partial\hspace{-0.155in}\not\hspace{0.075in}}\lambda_Y + m_\aa \chi_\pi \lambda_\aa +\!\!{\text{ h.c. }}\!\!)\end{aligned}$$ where $\chi_\pi$ is the fermion component of the linear superfield $\pi$. The three eigenvalues to $\OO(\epsilon^2)$ are $$\begin{aligned}
m = m_\aa \left( 1 - \frac{ \epsilon^2 m_\aa}{M_1 - m_\aa}\right),\quad
- m_\aa \left( 1 - \frac{ \epsilon^2 m_\aa}{M_1 + m_\aa}\right),\quad
M_1 \left( 1 - \frac{ \epsilon^2 M_1^2}{m_1^2 -m^2_\aa}\right)\!\!.\quad\end{aligned}$$ Notice that the two mass eigenvalues at $|m|\simeq m_\aa$ are no longer identically the same due to kinetic mixing with $\lambda_Y$, and this introduces $U(1)_R$ breaking into the hidden sector.
Charged Matter {#Sec: Charged Matter}
--------------
This section adds light, charged matter to the dark sector. The charged matter consists of four chiral superfields $E$, $E^c$, $P$, and $P^c$ that have axial charges under $U(1)_\aa$. The charge assignments of the dark electron and proton are chosen to be chiral to prevent them from acquiring supersymmetric masses in the absence of gauge symmetry breaking. Once the $U(1)_\aa$ gauge symmetry is broken, these states acquire masses at a scale set by $m_\aa$ and $\tan \beta_\aa$. In addition to new matter fields, the gauge sector is extended by a second gauge group $U(1)_\vv$ under which $E$, $E^c$, $P$, and $P^c$ have vector-like charges and which will lead to the formation of hydrogen-like bound states in the hidden sector. In summary the additional matter content has the following charge assignments: $$\begin{aligned}
\begin{array}{|c|cccc|}
\hline
& E& E^c& P & P^c\\
\hline
U(1)_\vv& -1& +1 & + 1& -1\\
U(1)_\aa& +1 & +1 & -1 & -1\\
\hline
\end{array} \end{aligned}$$ The superpotential in Eq. \[eqn:W0\] is augmented by Yukawa terms $$\begin{aligned}
\label{eqn:Wfull}
W = W_0 + W_{\text{Yukawa}} \qquad\qquad W_{\text{Yukawa}} = y_e \Phi^c E E^c + y_p \Phi P P^c + \!\!{\text{ h.c. }}\end{aligned}$$ so that after $U(1)_\aa$ breaking both $E$ and $P$ acquire Dirac masses $$\begin{aligned}
\!\!\!\!W_{\text{Yukawa}} = m_e\!\left( \! 1 + \frac{-\pi + \cot\beta_\aa S^c}{v_\aa}\right) \!E E^c +m_p\!\left(\!1 + \frac{\pi + \tan\beta_\aa S^c}{v_\aa}\right)\!P P^c + \!\!{\text{ h.c. }}\end{aligned}$$ where $$\begin{aligned}
m_e = y_e \sin \beta_\aa v_\aa \qquad \text{and} \qquad m_p = y_p \cos \beta_\aa v_\aa\end{aligned}$$ None of the fields charged under $U(1)_\vv$ acquires a vev, and therefore $U(1)_\vv$ is a massless gauge multiplet.
The interactions of the $U(1)_\aa$ vector superfield $\aa$ with the matter superfields are given by $$\begin{aligned}
K = 2g_{\aa}\left( \aa + \frac{\pi + \pi^\dagger}{\sqrt{2}m_\aa}\right) (E^\dagger E + E^c{}^\dagger E^c - P^\dagger P - P^c{}^\dagger P^c).\end{aligned}$$ Here the interactions of the $\pi$ have been moved from the superpotential to the Kähler potential with the equations of motion. The $\pi$ fields can have subdominant mixing with the Higgs fields of the SSM and mediate subdominant interactions.
The Higgs trilinear coupling, $\lambda$, in Eq. \[eqn:W0\] and the axial gauge coupling, $g_\aa$, are taken to be large enough that the masses in the axial/Higgs sector are of order $\alpha_\vv m_e$ or larger. With this choice of parameters the axial and Higgs sectors decouple, and the low energy limit of the theory is supersymmetric QED with two massive flavors and weakly broken supersymmetry. Because the axial/Higgs sectors respect the $O(2)_R$ symmetry, the arguments of Sec. \[Sec: Wavefunctions Fields\] go through and, in particular, the leading order superspin wavefunctions are as given in Sec. \[eqn:wavefunctions\]. The dominant residual effect of the axial/Higgs sector is to perturb the mass splitting between the hypermultiplet and vector multiplet. These contributions are suppressed through a combination of coupling constants and/or Yukawa suppression. Although the axial $U(1)_\aa$ gauge sector plays a subdominant role in the internal dynamics of the hidden sector, it mediates the dominant coupling to the Standard Model. In particular it mediates supersymmetry breaking, which is the subject of the next section.
Supersymmetry Breaking {#Sec: SSB}
----------------------
Although the hidden sector is supersymmetric at tree level, at the loop level small supersymmetry breaking effects are induced through the kinetic mixing portal to the SSM. This section discusses the strength with which the constituent particles’ masses feel supersymmetry breaking. These soft masses determine the leading supersymmetry breaking effects in the ground state spectrum, as discussed in Sec. \[Sec: SSB Effects\].
The soft parameters to be calculated (see Eq. \[eqn:lpreserve\] and Eq. \[eqn:lbreak\]) are the $U(1)_R$-preserving $\Delta^2_{\tilde{e}}$ and $\Delta^2_{\tilde{p}}$ and the $U(1)_R$-breaking $B_e$ and $B_p$. The largest soft parameters are the $U(1)_R$-preserving ones. If supersymmetry breaking is mediated to the SSM through gauge mediation, then these are given by $$\begin{aligned}
\Delta^2_{\tilde{e} } \simeq \Delta^2_{\tilde{p}} \simeq \frac{\alpha_\aa\epsilon^2}{\alpha'} M^2_{\tilde{E}^c}\end{aligned}$$ where $M_{\tilde{E}^c}$ is the SSM right handed selectron mass. The next largest soft parameters are the $U(1)_R$-breaking $B_{\mu}$-type terms, $B_e$ and $B_p$. To isolate how $U(1)_R$-breaking effects are mediated from the SSM, it is useful to integrate out the bino, which generates the operator $$\begin{aligned}
O_{\slashed{R}} = \lambda_\aa \frac{\epsilon^2 M_1 \square}{ \square + M_1^2} \lambda_\aa +\!\! {\text{ h.c. }}\end{aligned}$$ $B_e$ and $B_p$ are then generated upon insertion of $O_{\slashed{R}}$ in a loop, with a logarithmically enhanced contribution that is the same for both, $$\begin{aligned}
B_e \simeq B_p \simeq B \equiv \frac{\alpha_{\aa}\epsilon^2 M_1}{\pi} \text{log} \frac{\Lambda_{UV}^2}{M_1^2} \end{aligned}$$ where $\Lambda_{UV}$ is the messenger scale. The $B$-terms feed into the $U(1)_\vv$ gaugino mass, which is highly suppressed due to the indirect communication of $U(1)_R$-breaking $$\begin{aligned}
\label{Eq: Vector gaugino mass}
m_{\tilde{\vv}} \sim \frac{ \alpha_\vv B}{4\pi} \sim \frac{\alpha_\aa \alpha_\vv \epsilon^2}{(4 \pi)^2}M_1.\end{aligned}$$ In this model, $\lambda_\vv$ is always light and its mass is smaller than the level splittings induced by supersymmetry breaking, which are of order $m_{\text{soft}}$ $$\begin{aligned}
\frac{m_{\tilde{\vv}}}{m_{\text{soft}}} \sim \frac{\alpha_\vv \alpha' }{(4\pi)^2}\frac{ M_1 m_e }{M_{\tilde{E}^c}^2} \ll 1. \end{aligned}$$ This justifies ignoring the contributions from $m_{\tilde{\vv}}$ to the ground state energy levels.
Sec.\[Sec: SSB Effects\] described how the dominant communication of supersymmetry breaking to the spectrum is through supersymmetry violating perturbations to the rest energies of the constituents. Supersymmetry breaking also introduces several dynamical contributions to bound state spectroscopy from the exchange of particles frfom the axial $U(1)_\aa$ and Higgs sectors. However, these contributions are suppressed for the same reason that the supersymmetric contributions from the axial and Higgs sectors are suppressed.
Benchmark Models {#Sec: Benchmarks}
----------------
This section constructs three benchmark models to illustrate the scales that emerge in the hidden sector. Although doing detailed direct detection phenomenology is outside the scope of this paper, in both cases we aim to construct spectra compatible with iDM phenomenology. In particular we require that the bound states have a mass $m_\text{DM} \sim 100 {\,\mathrm{GeV}}$ and a splitting $\delta \sim 100 {\,\mathrm{keV}}$ between the ground state and the next highest state accessible through axial scattering. Furthermore the (predominantly inelastic) scattering cross section between the ground state and standard model nucleons should be $\OO(10^{-40}) \;\text{cm}^2$.
It is possible to meet these criteria; however, some tension exists between meeting all three criteria simultaneously. In models consistent with these requirements $\omega_-$ or $\varsigma_-$ is the lightest state. The three states to which $\omega_-$/$\varsigma_-$ can upscatter by exchanging an axial photon with a nucleon are $\omega_{1+}$, $\omega_{2+}$, and $v_\mu$. The cross section for $\omega_-$/$\sigma_-$ to upscatter to $v_\mu$ is velocity suppressed and can be ignored in the following. The cross section for $\omega_-$ to upscatter to $\omega_{1+}$ or $\omega_{2+}$ via the operator in Eq.\[eqn:AxialInteraction1\] is given[@Cheung:2009; @Alves:2009nf] by $$\begin{aligned}
\sigma \simeq 64 \pi \frac{ \alpha_\text{EM} \alpha_\aa \epsilon^2 m_\text{N}^2}{m_\aa^4}
=\frac{\alpha_{\text{EM}}m_\text{N}^2}{D_Y^2}\frac{1}{1+\frac{16g_\aa^2\mu_0^4}{\epsilon^2D_Y^2}}
\simeq \frac{4 \times10^{-41}\text{ cm}^2}{1+\frac{16g_\aa^2\mu_0^4}{\epsilon^2D_Y^2}}\frac{(50 {\,\mathrm{GeV}})^4}{D_Y^2}\end{aligned}$$ where $m_\text{N}$ is the mass of the nucleon. In order to fix $\sigma \sim10^{-40} \text{ cm}^2$, one must therefore choose $\mu_0^2\lesssim\epsilon D_Y/g_\aa$. For natural values of the Yukawa couplings, $m_\text{DM}\lesssim v_\aa$, and so from Eq.\[eqn:va\] and the requirement that $m_\text{DM} \sim 100 {\,\mathrm{GeV}}$ follows the constraint that $$\begin{aligned}
\frac{\epsilon}{g_\aa}\gtrsim\frac{(100{\,\mathrm{GeV}})^2}{D_Y}.\end{aligned}$$ Bounds on kinetic mixing [@Hook:2010tw] impose further constraints, requiring $\epsilon\lesssim0.005$ for $m_\aa \sim 1 {\,\mathrm{GeV}}$. Finally, the scale of supersymmetry breaking is proportional to $\epsilon^2 g_\aa^2$ and to get splittings of order $100{\,\mathrm{keV}}$ requires that $\epsilon\sim g_\aa\sim0.005$.
Trying to match the CoGeNT/DAMA anomaly [@Essig:2010ye] with light inelastic dark matter is challenging in this specific model because it is difficult to generate an $\OO(10^{-38}\text{ cm}^2)$ cross section. The primary tension arises from the mediation through the massive axial current. One approach could be to kinetically mix the vector current rather than the axial current. This does not suppress elastic scattering; however, in models of inelastic dark matter where the signal arises from downscattering, rather than upscattering, the elastic rate does not need to be suppressed. This possibility is not pursued further here but illustrates the rich phenomenology possible in composite dark matter models [@Lisanti:2009am].
Following the above logic we choose the following MSSM parameters for all three benchmark models $$\begin{aligned}
\begin{array}{|c|c|c|c|}
\hline
\sqrt{D_Y} & M_1&M_{\tilde{E}^c}&\Lambda_\text{UV}\\
\hline
50{\,\mathrm{GeV}}& 100 {\,\mathrm{GeV}}&900{\,\mathrm{GeV}}&100{\,\mathrm{TeV}}\\
\hline
\end{array} .\end{aligned}$$ The parameters of the dark sector for the three benchmark points are chosen to be $$\begin{aligned}
\begin{array}{|c||c|c|c|c|c|c|}
\hline
\text{Model}& \epsilon & g_\vv & g_\aa & \mu_0 & y_e & y_p\\
\hline
\text{Unmixed}&0.005& 1.5 & 0.004 & 30 {\,\mathrm{GeV}}& 0.15 & 1.2\\
\hline
\text{Mixed}&0.005& 1.1 & 0.004 & 30 {\,\mathrm{GeV}}& 0.25 & 1.4 \\
\hline
\text{Heavy Scalars}&0.005&0.5&0.004&30{\,\mathrm{GeV}}&0.35&1.0\\
\hline
\end{array} .\end{aligned}$$ The first two choices for $g_\vv$ cause $U(1)_\vv$ to hit a Landau pole before the GUT scale. The Landau pole can be avoided by embedding $U(1)_\vv$ into a non-Abelian group, [*e.g.* ]{} $U(1)_\vv \subset SU(2)_\vv$ or $U(1)_\vv\times U(1)_\aa \subset SO(4)$. The rather large values of $\epsilon$ chosen here make for some tension with constraints from BaBar. These constraints may not apply to this model because $\aa_\mu$ may cascade decay through the Higgs sector before decaying into Standard Model particles [@ExperSignatures]. These parameters lead to the following supersymmetric bound state mass scales $$\begin{aligned}
\begin{array}{|c||c|c|c|c|c|c|c|}
\hline
\text{Model}& v_\aa &m_\aa& m_e & m_p & \tan\theta&m_{\text{Prin}}& m_{\text{FS}} \\
\hline
\text{Unmixed}&49 {\,\mathrm{GeV}}& 550{\,\mathrm{MeV}}& 3.0 {\,\mathrm{GeV}}& 53{\,\mathrm{GeV}}&0.24 &46{\,\mathrm{MeV}}& 1500{\,\mathrm{keV}}\\
\hline
\text{Mixed}&49 {\,\mathrm{GeV}}& 550{\,\mathrm{MeV}}& 5.1{\,\mathrm{GeV}}& 62{\,\mathrm{GeV}}&0.29 &22{\,\mathrm{MeV}}& 200{\,\mathrm{keV}}\\
\hline
\text{Heavy Scalars}&49{\,\mathrm{GeV}}&550{\,\mathrm{MeV}}&7.1{\,\mathrm{GeV}}&44{\,\mathrm{GeV}}&0.40&1.2{\,\mathrm{MeV}}&0.48{\,\mathrm{keV}}\\
\hline
\end{array}\end{aligned}$$ Supersymmetry breaking effects are encapsulated in the soft parameters $$\begin{aligned}
\begin{array}{|c||c|c|c|}
\hline
\text{Model} &m_\text{soft}& B & m_{\tilde{\vv}} \\
\hline
\text{Unmixed}& 270{\,\mathrm{keV}}& 6.1{\,\mathrm{eV}}& 0.09 {\,\mathrm{eV}}\\
\hline
\text{Mixed}& 170 {\,\mathrm{keV}}& 6.1 {\,\mathrm{eV}}& 0.05{\,\mathrm{eV}}\\
\hline
\text{Heavy Scalars}&130{\,\mathrm{keV}}&6.1{\,\mathrm{eV}}&0.01{\,\mathrm{eV}}\\
\hline
\end{array} .\end{aligned}$$ The gauge-mediated contribution to $m_{\tilde{\vv}}$ listed here is subdominant to gravity-mediated contributions, which give $$\begin{aligned}
m_{\tilde{\vv}} \sim \frac{F_{\text{susy}}}{M_\text{pl}} \sim 1 {\,\mathrm{eV}}\end{aligned}$$ where $\sqrt{F_{\text{susy}}} \simeq 100 {\,\mathrm{TeV}}$ for these benchmark models.
The resulting spectra are shown in Figure 2. The first parameter point, “Unmixed," realizes the scenario where $B \ll m_\text{soft}\ll m_{FS}$. The various bound states have a mass $ 56 {\,\mathrm{GeV}}$ and the lowest state $\omega_-$ is primarily scalar-scalar. The lowest states accessible by axial photon exchange are the pair of nearly degenerate scalars, $\omega_{1+}$ and $\omega_{2+}$, which are heavier by an amount $\delta = 64 {\,\mathrm{keV}}$. For the second parameter point, “Mixed," the hierarchy of scales is instead $B \ll m_\text{soft}\sim m_\text{FS}$ and there is large mixing between the vector multiplet and hypermultiplet. The bound states have a mass $67 {\,\mathrm{GeV}}$ and an iDM-compatible spectrum is again realized with several states available for upscattering. $\omega_-$ is again the lightest state but the vector $v_\mu$ is now kinematically accessible. Because scattering to $v_\mu$ is velocity suppressed, however, the relevant splitting for iDM is $\delta =80 {\,\mathrm{keV}}$ between $\omega_-$ and $\omega_{1+}$/$\omega_{2+}$. For the third parameter point, “Heavy Scalars," the hierarchy of scales is instead $B \ll m_\text{FS}\ll m_\text{soft}$ and $m_\text{DM} = 51 {\,\mathrm{GeV}}$. Because the hypermultiplet has large selectron components, the hypermultiplet states move upwards and the spectrum is inverted, with $\varsigma_-$ (which is primarily fermion-fermion) and $v_\mu$ as the lightest states. The relevant splitting for iDM is $\delta = 126 {\,\mathrm{keV}}$.
From the results in Sec.\[Sec: U1v\] one can estimate the lifetime of the unstable states in the “Unmixed" benchmark spectrum. One finds $\tau_\VV \sim 10^{-18} \text{ sec}$ for the vector multiplet and $\tau_\HH \sim 10^{-13} \text{ sec}$ for the states in the hypermultiplet. Although the decay formulae from Sec.\[Sec: U1v\] are not directly applicable to the “Mixed" benchmark, the decay rates will be similarly fast, since $\alpha_\vv$ is $\OO(1)$.
Discussion {#Sec: Discussion}
===========
The benchmark models discussed in Sec.\[Sec: Benchmarks\] demonstrate that the non-relativistic supersymmetric bound states discussed in this article can realize a wide range of bound state spectra with a rich hierarchy of scales. In any application of these models to dark matter phenomenology, a number of important issues must be addressed. In particular one needs to examine constraints from direct and indirect detection as well as the implications for early universe cosmology. Although addressing these topics in any detail lies outside of the scope of this paper, in this section we briefly discuss some of the relevant physics. In the following our discussion is limited to the concrete model presented in Sec.\[Sec: Nearly Susy\].
BBN Constraints
----------------
In a realistic model where the dark matter sector is nearly supersymmetric, the hidden $U(1)_\vv$ photon and photino that create the bound states will usually be relativistic at the temperatures relevant for BBN, $T \approx 1$ MeV. This means that they contribute to the energy density of the universe and will modify the successful predictions of standard BBN. Fortunately, the resulting perturbation is sufficiently small to be within observational constraints. This arises because the hidden sector is weakly coupled to the visible sector, typically $\epsilon \lesssim 10^{-2}$, and therefore the dark sector kinetically decouples [@Bringmann:2006mu] from the visible sector at a temperature above the ${\,\mathrm{GeV}}$ scale. The early kinetic decoupling reduces the number of effective degrees of freedom in the dark sector because many Standard Model degrees of freedom become non-relativistic between temperatures of 10 GeV and the QCD phase transition. This generally makes weakly coupled dark sectors with long range Coulombic interactions [@BuckleyFox] safe from BBN constraints [@Ackerman:2008gi].
Recombination
-------------
If dark atoms are relevant for cosmology, then a significant fraction of the supersymmetric dark electrons and protons must recombine into supersymmetric atoms. The recombination of non-supersymmetric hydrogen-like dark matter atoms is studied in [@Kaplan:2009de]. Supersymmetry adds new levels of complexity to the problem. First, supersymmetry introduces new processes where gauginos are emitted in de-excitation processes. Sec. \[Sec: Interactions\] discusses some of these de-excitation processes and, as a rule of thumb, gaugino emission processes are faster if the corresponding gauge boson emission process is a magnetic, spin-flip transition. For non-supersymmetric dark atoms, the most important processes for recombination are the Ly-$\alpha$ decay, the $2s$ double photon decay and the scattering process $e^-+p^+\leftrightarrow H+\gamma$ [@Kaplan:2009de]. None of these are due to spin-flips and so gaugino emission processes are expected to be subdominant.
Even if gaugino emission is subleading as expected, there are new electric transitions between the different, non-degenerate superspin levels with $n=2$ and $n=1$. This additional complexity makes the out of equilibrium problem harder to solve systematically. The numerous new Ly-$\alpha$ lines corresponding to different superspin transitions are sufficiently degenerate that the Doppler broadening smears the energies of the states. For this reason photons from different transitions are indistinguishable implying that the optical depth of Ly-$\alpha$ photons is about the same as in the non-supersymmetric case. Although a detailed analysis is necessary, it appears that the physics of recombination in the supersymmetric case is parametrically the same as in the non-supersymmetric case. In particular for sufficiently large $\alpha_\vv$ recombination is expected to be an efficient process.
Molecules
---------
If supersymmetric atoms can form, it is possible that these atoms may further aggregate into supersymmetric molecules. This section briefly explores this possibility by examining the role that Bose/Fermi statistics plays in atoms and molecules.
The ground state of regular diatomic hydrogen is a $s=0$ state, i.e.the spin wavefunction for the electrons is antisymmetric under particle exchange and the spatial wavefunction is symmetric as it would be for a two-selectron atom. The statistics of the electrons thus does not affect the ground state spatial wavefunction and energy to lowest order. Supersymmetric hydrogen should therefore form diatomic molecules, and the binding energy is approximately the same as for non-supersymmetric hydrogenic systems[@Clavelli:2008zs].
In the Standard Model, further aggregation into molecules larger than $H_2$ is prevented by the Pauli exclusion principle, which forbids more than two electrons from being in the same orbital. In supersymmetric atoms, this aggregation is not forbidden by Pauli because electrons can convert into their scalar superpartners, selectrons. Supersymmetric bound states will share orbitals more effectively and hence are bound more strongly. For non-relativistic molecules composed of $N$ bosonic constituents, the binding energy scales as $$\begin{aligned}
|E_{\text{binding}}|\propto N^{\frac{7}{5}},
\label{eqn:BindingEnergyBound}\end{aligned}$$ in contrast to molecules with fermionic constituents, whose binding energy grows linearly with the number of constituents [@Dyson; @Conlon:1987xt; @Lieb:1988tt]. This means that macroscopic bound states formed from scalar constituents have an enormous binding energy. This suggests a scenario where a fraction of the dark atoms condense into huge “molecules," which may impose additional constraints on supersymmetric atomic dark matter if the formation of macro-molecules is too efficient. One possibility is that the supersymmetric macro-molecules could drive formation of microscopic black holes.
Dark Matter Genesis
-------------------
In order for dark atoms to be a sizeable fraction of the Universe’s dark matter, there needs to be a chemical potential for $U(1)_{e+p}$ generated in the early Universe [@Sakharov]. The most compelling mechanism for generating an asymmetry in the dark matter sectors links the dark matter number density to the baryonic or leptonic number density, for recent work see references in [@Darkogenesis]. The primary novelty with generating a chemical potential for the composite sector is that the minimal gauge invariant operator sourcing $U(1)_{e+p}$ is a dimension 2 operator, [*e.g.*]{} $E P$. This typically means that constructing a renormalizable Lagrangian requires more structure than for elementary dark matter. Normal inelastic dark matter requires $n_b/n_{e+p} {\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$}}1$ and this means that if the $e+p$ asymmetry is to be directly linked to the baryon asymmetry, then the interactions that equilibrate the chemical potentials in the dark sector and Standard Model must freeze out when the dark matter is relativistic. Alternatively, the $e+p$ asymmetry should be generated through a mechanism like the out-of-equilibrium decay of a heavy particle.
Conclusions {#Sec: Conc}
============
Composite dark matter offers a rich phenomenology that has only barely been explored in comparison to models in which dark matter is an elementary particle. Non-relativistic bound states offer one general class of composite dark matter models, and these typically involve a new mass scale that is incorporated either by hand or through a Higgs mechanism. In the later case the Higgs mass is radiatively unstable and supersymmetry is a natural way to stabilize its mass. Supersymmetry breaking can be weakly communicated to this sector, particularly if the interactions of the dark sector with the Standard Model are the dominant link to the supersymmetry breaking sector. If this is the case, then the composite dark matter will form nearly supersymmetric multiplets and the phenomenology of these states can be radically different from the non-supersymmetric case.
Atomic Inelastic Dark Matter [@Kaplan:2009de] was proposed as a model of inelastic dark matter where the hyperfine splitting of the ground state of a hydrogen-like sector is the origin of the inelastic mass splitting. This article created a supersymmetric version of Atomic Inelastic Dark Matter that has many different features from the original, non-supersymmetric model. Most notably, the hyperfine splitting disappears in the supersymmetric limit and the ground state typically contains a scalar-scalar component. This introduces new interactions between dark matter and the Standard Model. The model accommodates spectra and scattering cross sections compatible with iDM phenomenology.
This article has also constructed tools to help in the study of quasi-perturbative supersymmetric bound states and in incorporating supersymmetry breaking into the bound states. These tools illustrate that the form of the weakly broken spectrum and the composition of the various states is in many cases dictated entirely by supersymmetry at leading order. Supersymmetry also imposes strict restrictions on the allowed interactions, which simplifies the matching of effective interaction operators. More generally supersymmetric bound states offer a rich laboratory for studying supersymmetric dynamics and interactions.
Acknowledgements {#acknowledgements .unnumbered}
================
JGW thanks E. Katz for useful conversations during the course of this work. MJ and TR would like to acknowledge helpful conversations with Michael Peskin as well as collaboration with Daniele Alves in early stages of this work. TR is a William R. and Sara Hart Kimball Stanford Graduate Fellow. MJ, SB and JGW are supported by the US DOE under contract number DE-AC02-76SF00515. MJ, SB, TR and JGW receive partial support from the Stanford Institute for Theoretical Physics. JGW is partially supported by the US DOE’s Outstanding Junior Investigator Award and the Sloan Foundation. JGW thanks the Galileo Galilei Institute for their hospitality during the early stages of this work.
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[^1]: The superscript $D$ indicates that the spinor is in the Dirac basis, where $\gamma_0$ is diagonal.
|
---
author:
- |
[**Luke B. Hewitt**]{} [**Tuan Anh Le**]{} [**Joshua B. Tenenbaum**]{}\
`{lbh,tuananh,jbt}@mit.edu`\
Department of Brain and Cognitive Sciences, MIT
bibliography:
- 'bib.bib'
title: 'Learning to learn generative programs with Memoised Wake-Sleep'
---
|
---
abstract: '[**Abstract:**]{} We show that the kinetic approach to statistical mechanics permits an elegant and efficient treatment of fractional exclusion statistics. By using the exclusion-inclusion principle recently proposed \[Phys. Rev. E [**49**]{}, 5103 (1994)\] as a generalization of the Pauli exclusion principle, which is based on a proper definition of the transition probability between two states, we derive a variety of different statistical distributions interpolating between bosons and fermions. The Haldane exclusion principle and the Haldane-Wu fractional exclusion statistics are obtained in a natural way as particular cases. The thermodynamic properties of the statistical systems obeying the generalized exclusion-inclusion principle are discussed.'
address: |
Dipartimento di Fisica - Politecnico di Torino - Corso Duca degli Abruzzi 24, 10129 Torino, Italy\
Istituto Nazionale di Fisica Nucleare, Sezioni di Cagliari e di Torino\
Istituto Nazionale di Fisica della Materia, Unitá del Politecnico di Torino
author:
- 'G. Kaniadakis [^1], A. Lavagno and P. Quarati'
title: Kinetic Approach to Fractional Exclusion Statistics
---
Following Haldane’s formulation [@Hal] of a generalized Pauli exclusion principle, many papers have been recently devoted to the study of fractional exclusion statistics by interpolation of bosonic and fermionic distributions [@Wu]. There is an intrinsic connection between these fractional statistics and the interpretation of the fractional quantum Halleffect [@Lau] and anyonic physics [@Fra; @Com]. Murthy and Shankar [@Murt1] generalized the Haldane statistics to infinite dimensions and showed [@Murt2] that the one dimensional bosons interacting through the two-body inverse square potential $V_{ij}=g(g-1)(x_i-x_j)^{-2}$ of the Calogero-Sutherland model obeys fractional exclusion statistics in the sense of the Haldane interpretation. Using the Thomas-Fermi method, Sen and Bhaduri [@Sen] considered the particle exclusion statistics for the Calogero-Sutherland model in the presence of an external, arbitrary one-body potential. Isakov [@Isak] extended the Calogero-Sutherland model in the case of particles interacting through a generic two-body potential $V_{ij}$ and derived the related exclusion statistics. Rajagopal [@Raja] obtained the von Neumann entropy per state of the Haldane exclusion statistics. Nayak and Wilczek [@Nayak] studied the low-temperature properties, the fluctuations and the duality of exclusions statistics. Murthy and Shankar [@Murt1] also computed, for quasi-particles, in the Luttinger model, the exclusion statistics parameter $g$ related to the exchange statistical parameter $\alpha$ of the quantum phase $e^{i\pi\alpha}$, and showed [@Murt2] that the parameter $g$ can be completely determined by the second virial coefficient in the high temperature limit.
Haldane defined the statistics $g$ of a particle by $$g=-\frac{{\rm d}_{_{N+\Delta N}}-{\rm d}_{_N} }{\Delta N} \ \ ,$$ where $d_{_N}$ is the single-particle Hilbert-space dimension, when the coordinates of $N-1$ particles are kept fixed. Thus the dimension of the Hilbert space for the single particle states is a finite and extensive quantity that depends on the number of particles in the system. One can choose this dimension as $${\rm d}_{_N}=K-g (N-1) \ \ ,$$ and the statistical weight or degeneracy factor is $$W=\frac{({\rm d}_{_N}+N-1) ! }{ N ! \, ({\rm d}_{_N}-1) ! } \ \ .$$ This equation is a simple interpolation between the number of ways of placing $N$ identical bosons or fermions in $K$ single-particle independent states confined to a finite region of matter. The expression of the mean occupation number $n=N/K$ has been obtained in an implicit form by Wu [@Wu]. $$n=\frac{1}{w(E)+g} \ \ ,$$ with $w$ satisfying the relation $$w(E)^g \, [1+w(E)]^{1-g}=e^{\beta (E-\mu)} \ \ .$$ In the special cases $g=0$ and $g=1$, Eq.(4) yields, respectively, the Bose-Einstein (BE) and Fermi-Dirac (FD) distributions.
Ilinski and Gunn [@Ilin] criticize the identity of relations (1) and (3) of Haldane and, using (2), derive a different $W$ constructing a statistical mechanics which is not in complete agreement with Wu statistical mechanics [@Wu].
Acharya and Swamy [@Achar] and Polychronakos [@Poly] have studied, in addition to the Haldane exclusion statistics, new fractional statistics with appealing features as positive probabilities and analytical expressions for all termodynamic quantities. We wish to recall the first work on intermediate statistics published by Gentile at the beginning of 1940 [@Gentile].
In a previous work [@Ka] we have considered the kinetics of particles in a phase space of arbitrary dimensions $D$, obeying an exclusion-inclusion principle. We obtained the statistical distributions of the particle system as the stationary state of a non-linear kinetic equation. A crucial point of this formalism is the definition of the transition probability which can be written in various forms, containing the effects of the inclusion-exclusion principle through the distribution function $n=n(t,{\bf v})$, which is the mean occupation number of the state ${\bf v}$.
Setting $\pi(t,{\bf v} \rightarrow {\bf u})$ the transition probability from the state ${\bf v}$ to the state ${\bf u}$, the evolution equation of the distribution function $n(t,{\bf v})$ can be written as $$\frac{\partial n(t,{\bf v})}{\partial t}=\int [\pi(t,{\bf u}
\rightarrow {\bf v})-
\pi(t,{\bf v} \rightarrow {\bf u})] \, d^{_{^D}} u \ \ .$$ In ref.[@Ka] we postulated the following expression of the transition probability : $$\pi(t,{\bf v} \rightarrow {\bf u})=r(t,{\bf v},{\bf v}-{\bf u}) \,
\varphi[n(t,{\bf v})] \, \psi[n(t,{\bf u})] \ \ ,$$ where $r(t,{\bf v},{\bf v}-{\bf u})$ is the transition rate, $\varphi[n(t,{\bf v})]$ is a function depending on the occupational distribution at the initial state ${\bf v}$ and $\psi[n(t,{\bf u})]$ depends on the arrival state. The function $\varphi(n)$ must obey the condition $\varphi(0)=0$ because the transition probability is equal to zero if the initial state is empty. Furthermore, the function $\psi(n)$ must obey the condition $\psi(0)=1$ because, if the arrival state is empty, the transition probability is not modified. If we choose $\varphi(n)=n$ and $\psi(n)=1$ we obtain the standard linear kinetics. The function $\psi(n)$ defines implicitely the inclusion-exclusion principle enhancing or inhibiting the transition probability.
In this letter we use the kinetic approach outlined above to generate new fractional exclusion statistics. The approach is appropriate both for non-interacting and interacting particles. In the following, we derive general expressions linking the main statistical and thermodynamic quantities to the transition probabilities of the kinetic theory. We obtain implicitely a fractional statistics which interpolates between BE and FD distributions in a single-particle Hilbert space whose dimension is an arbitrary function of $n$. We show that the Haldane-Wu statistics can be derived from this one by considering Brownian particles and by demanding that the Hilbert space dimension be a linear function of $n$. Another particular case within the family of statistics here introduced yields a statistics interpolating among BE, FD and Maxwell-Boltzmann (MB) distributions.
Let us consider Eqs. (6) and (7) and limit the discussion to the one-dimensional velocity space (the extension to a $D$-dimensional phase space is straightforward). We limit ourselves to first neighbors interactions, this being equivalent to truncate the Moyal expansion given by Eq.(7) of Ref. [@Ka] at the second order and we obtain the following, generalized, non-linear Fokker-Planck equation $$\begin{aligned}
\frac{\partial n(t,v)}{\partial t}=\frac{\partial}{\partial v}
\Bigg [\left (J(t,v)+\frac{\partial D(t,v)}{\partial v}\right )
\varphi(n) \psi(n) \ \ \
\nonumber \\+ D(t,v) \left (\psi(n) \frac{\partial
\varphi(n)}{\partial n}-\varphi(n)
\frac{\partial \psi(n)}{\partial n}\right ) \frac{\partial
n(t,v)}{\partial v} \Bigg ] \ \ .\end{aligned}$$ $J(t,v)$ and $D(t,v)$ are the drift and diffusion coefficients, respectively, and are given by the first and the second order moments of the transition rate [@Ka]. Eq.(8) is a continuity equation for the distribution function $n=n(t,v)$ $$\frac{\partial n(t,v)}{\partial t}+\frac{\partial j(t,v,n)}
{\partial v}=0 \ \ ,$$ where the particle current $j=j(t,v,n)$ is given by $$\begin{aligned}
j=-D \, \frac{\varphi(n) \psi (n)}{\overline{(\Delta n)^2}}
\Bigg [\frac{\partial \epsilon}{\partial v} \,
\overline{(\Delta n)^2}
+ \frac{\partial n}{\partial v} \Bigg ] \ \ .\end{aligned}$$ We have defined $$\overline{(\Delta n)^2}=\left \{ \frac{\partial}{\partial n} \log
\left [ \frac{\varphi(n)}{\psi(n)} \right] \right \}^{-1} \ \ ,$$ and $$\frac{\partial \epsilon}{\partial v}=\frac{1}{D(t,v)}\left (J(t,v)
+\frac{\partial D(t,v)}{\partial v}\right ) \ \ .$$ The function $\epsilon =\epsilon (v)$ is an adimensional single particle energy defined up to an additive, arbitrary constant. This energy is appropriate both for non-interacting and interacting particles. Nayak and Wilczek [@Nayak] have examined the particular case $\epsilon (v) \propto v^l$ with $l$ integer. The case $l=2$ corresponds to Brownian particles, the drift and diffusion coefficients being given by $$J=\gamma v, \, \, \, \, \, \,\, \, \, \, \, \,
D=\frac{\gamma}{\beta m} \ \ .$$ The quantity $\gamma$ is a dimensional constant and $\epsilon$ is given by: $\epsilon=\beta (E-\mu)$ with $E=\frac{1}{2} m v^2$ being the kinetic energy. $\beta=1/T$ ($k_{_B}=1$) is the inverse of the temperature and $\mu$ is the chemical potential.
In stationary conditions, the particle current vanishes: $j(\infty,v,n)=0$. Eq.(10) becomes a omogeneous first order differential equation, which can be integrated easily: $$\frac{\psi(n)}{\varphi(n)}=\exp (\epsilon) \ \ .$$ When the functions $\varphi(n)$ and $\psi(n)$ are fixed, Eq.(14) gives the statistical distribution $n=n(\epsilon)$ of the system.
We stress that, if we use as variable the single particle energy $\epsilon$, it is easy to verify that the above-defined function $\overline{(\Delta n)^2}=<n^2>-<n>^2$, is equal to the second order density fluctuation [@Landau] $$\overline{(\Delta n)^2}=- \frac{\partial n}{\partial\epsilon} \ \ .$$ This relation is very important because it reveals that the mean fluctuation of the occupation number is the crucial quantity that determines the equilibrium distribution.
The statistical distribution $n(\epsilon)$ can be obtained by the maximum entropy principle, fixing the total number of particles and energy of the system and using the standard Lagrange multiplier method. Setting ${\cal S}(n)=S(N)/K$ the thermodynamic limit of the entropy per state, we have $$\frac{\partial}{\partial n}\left [{\cal S}(n)-\epsilon \,
n \right ]=0\ \ ,$$
From Eqs. (14) and (16) it is possible to obtain the entropy in terms of the functions $\varphi(n)$ and $\psi(n)$ as $$\frac{\psi(n)}{\varphi(n)}=\exp\left [\frac{\partial {\cal S}(n)}
{\partial n} \right ] \ \ ,$$ or in terms of $\overline{(\Delta n)^2}$ $$\overline{(\Delta n)^2}=-\left [ \frac{\partial ^2 {\cal S}}
{\partial n^2} \right ]^{-1} \ \ .$$ By using the Boltzmann principle with the identification of the entropy ${\cal S}=\log {\cal W}$, it is possible to find the relation between ${\cal W}$ and the functions $\varphi(n)$ and $\psi(n)$ in the following expression $$\frac{\psi(n)}{\varphi(n)}=\exp\left [ \frac{1}{{\cal W}(n)}
\frac{\partial {\cal W}(n)}{\partial n} \right ] \ \ .$$
It is well known that the partition function $\cal{Z}$ per state $${\cal Z}=e^{- \beta\Omega} \ \ ,$$ is related to the mean density $n$ by means of $$n=-\frac{\partial}{\partial \epsilon} \log {\cal Z} \ \ .$$ Taking into account Eqs. (11), (15), (20) and (21), we may write the thermodynamic potential $\Omega$ as a function of $\varphi(n)$ and $\psi(n)$ $$\beta\Omega=\int n \, \frac{\partial}{\partial n}
\log\frac{\psi(n)}{\varphi(n)} \, dn \ \ ,$$ and the density fluctuation as $$\overline{(\Delta n)^2}=-\, n \, \left [ \frac{\partial}{\partial n}
\beta \, \Omega \right ]^{-1} \ \ .$$
Now let us consider the case $\varphi(n)=n$; for the function $\psi(n)$ we choose the particular form $$\psi=\displaystyle{d^{-\frac{\partial d}{\partial n}} \,
(n+d)^{1+\frac{\partial d}{\partial n}}} \ \ .$$ where the function $d=d(n)$ must satisfy the condition $d(0)=1$. In this case, Eq.(14) which defines the statistics, becomes: $$d^{-\frac{\partial d}{\partial n}} \, (n+d)^{1+
\frac{\partial d}{\partial n}} = n \, e^{\epsilon} \ \ .$$ The function ${\cal W}$ becomes $${\cal W}= \frac{(n+ d)^{n+d}}{n^n \, d^d} \ \ ,$$ and the entropy $${\cal S}=- n [ w \log w - (1+w) \log (1+w) ] \ \ ,$$ where $w=d/n$, so we find the general form of the von Neumann entropy per state [@Raja]. The thermodynamic potential $\Omega$ becomes $$\beta\Omega=\left (n \, \frac{\partial d}{\partial n}-d \right ) \,
\log \frac{n+d}{d} \ \ ,$$ and the partition function ${\cal Z}$ can be rewritten as $${\cal Z}=\left ( \frac{n+d}{d} \right )^{d-n\, \frac{\partial d}
{\partial n}} \ \ ,$$ while the fluctuation becomes $$\overline{(\Delta n)^2}=\frac{n \, d \, (n+d)}{\left (d-n \,
\displaystyle{\frac{\partial d}{\partial n}}\right )^2 -n \, d\,
(n+d) \, \displaystyle{ \frac{\partial ^2 d}{\partial n^2}} \,
\log\frac{n+d}{d} } \ \ .$$
The quantity $d$ is an arbitrary function and we can use it to define a family of exclusion statistics. If we select a particular form of $d$, we observe that this one can depend not only on $n$ but also on a parameter $g$: $d=d(g,n)$. In this way Eq.(25) defines, varying $g \in [0,1]$, a fractional statistics. If we require that the statistics interpolates between BE and FD distributions we must set the two conditions: $d(0,n)=1$ and $d(1,n)=1-n$. These two conditions imply that the single-particle Hilbert-space dimension be that of the Bose space when $g=0$ and that of the Fermi one when $g=1$. Let us call $S(N)=K{\cal S}(n)$ and $W(N)$ the entropy and statistical weight of the system of $N$ particles lying in $K$ states. From the Boltzmann principle $S=\log W$ we obtain ${\cal W}=W^{1/K}$. It is easy to see that ${\cal W}$, given by Eq.(26), is the thermodynamic limit of the statistical weight given by Eq. (3), where, this time, ${\rm d}_N={\rm d}(g,N,K)$ is the single-particle Hilbert- space dimension, is an arbitrary function and admits as thermodynamic limit $d(g,n)={\rm d}(g,N,K)/K$. If we consider a statistics interpolating between bosons and fermions, the conditions to which $d=d(g,n)$ must satisfy when $g=0,1$ can be derived in the thermodynamic limit from the two conditions ${\rm d}(0,N,K)=K$ and ${\rm d}(1,N,K)=K-N+1$.
The Haldane-Wu choice of ${\rm d}$ given by Eq.(2) implies $$d=1-g \, n \ \ ,$$ and requires that $\psi (n)$ be given by $$\psi(n)=[1-g\, n]^g \, [1+(1-g) \, n]^{1-g} \ \ .$$ Equation (25) becomes $$[1-g\, n]^g \, [1+(1-g) \, n]^{1-g}=n \, e^{\epsilon} \ \$$ and we obtain, in the case of Brownian particles, the statistics introduced by Haldane [@Hal] and by Wu [@Wu]. The partition function ${\cal Z}$ and the density fluctuation are given by $${\cal Z}=\frac{1+(1-g) n}{1-gn} \ \ .$$ $$\overline{(\Delta n)^2}= n \, (1-g n) \, [1+(1-g) n] \ \ .$$
As a second example of fractional statistics we consider the statistics defined by: $$n=\frac{1}{\exp(\epsilon)-\kappa} \ \ ,$$ studied extensively in ref. [@Achar; @Poly; @Ka]. For $\kappa=-1$, $0$ and $1$ one obtains the FD, MB and BE statistical distributions, respectively. In ref. [@Achar] it is shown that the parameter $\kappa$ is a function of the exchange statistical parameter $\alpha$ appearing in the quantum phase $e^{i \pi\alpha}$. This statistics is generated from $$\psi(n)=1+\kappa n \ \ .$$ In this case the partition function can be written as $${\cal Z}=(1+\kappa n)^{1/\kappa} \ \ ,$$ and it is easy to verify that the density fluctuation $\overline{(\Delta n)^2}$ and the entropy ${\cal S}$ are given by $$\overline{(\Delta n)^2}= n (1+\kappa n) \ \ ,$$ $${\cal S}=\frac{1}{\kappa} (1+\kappa n ) \log (1+\kappa n)- n
\log n \ \ ,$$ while the statistical weight ${\cal W}$ becomes $${\cal W}=\frac{ (1+\kappa n)^{ (1+\kappa n)/\kappa} } { n^n} \ \ ,$$ and can be obtained as the thermodynamic limit of the statistical weight: $$\begin{aligned}
W &=& |\kappa|^N \, \frac{\left (
\displaystyle{\frac{K}{|\kappa|}}\right ) !}
{N ! \, \left ( \displaystyle{\frac{K}{|\kappa|}}-N \right ) ! } \ \ ,
\ \ \kappa <0 \ \ , \\
W &=& \frac{K^N}{N ! } \ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \kappa =0 \ \ , \\
W &=& \kappa^N \, \frac{\left (\displaystyle{\frac{K}{\kappa}}+
N-1\right ) !}
{N ! \, \left ( \displaystyle{ \frac{K}{\kappa}}-1\right ) ! } \ \ ,
\ \ \kappa >0 \ \ .\end{aligned}$$
In conclusion, we have shown that the effects of the exclusion-inclusion principle can be taken into account in the particle kinetics by means of a proper definition of the transition probability. We have derived a non-linear evolution equation which admits as steady solutions the exclusion statistical distributions. We have linked the main thermodynamic properties of the system with the function $\psi (n)$ which generates the exclusion statistics. As first application of the theory we have considered a family of exclusion statistics in which the single-particle Hilbert-space dimension is an arbitrary function of the mean occupational number and contains, as particular case, the Haldane-Wu statistics. As second application we have considered an exclusion statistics interpolating among FD, MB and BE distributions.
F.D.M.Haldane, Phys. Rev. Lett. [**66**]{}, 1529 (1991). Y.S. Wu, Phys. Rev. Lett. [**73**]{}, 922 (1994). R. B. Laughlin, Phys. Rev. Lett. [**50**]{}, 1395 (1983) and in [*The Quantum Hall Effect*]{}, edited by R. Prange and S. Girvin (Springer-Verlag, Heidelberg, 1989). , edited by F. Wilczek (World Scientific, Singapore, 1990). , edited by L. Alvarez-Gaumé, A. Devoto, S. Fubini, C. Trugenberger (North-Holland, Amsterdam, 1993). M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. [**72**]{}, 3629 (1994). M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. [**73**]{}, 3331 (1994). D. Sen and R.K. Bhaduri, Phys. Rev. Lett. [**74**]{}, 3912 (1995). S.B. Isakov, Phys. Rev. Lett. [**73**]{}, 2150 (1994). A.K. Rajagopal, Phys. Rev. Lett. [**74**]{}, 1048 (1995). C. Nayak and F. Wilczek, Phys. Rev. Lett. [**73**]{}, 2740 (1994). K.N. Ilinski and Gunn, preprint HEP-TH-9503233. R. Acharya and P. Narayana Swamy, J. Phys. A: Math. Gen. [**27**]{}, 7247 (1994). A.P. Polychronakos, preprint UUITP-03/95, HEP-TH-9503077. G. Gentile jr, Nuovo Cimento [**17**]{}, 493 (1940). G. Kaniadakis and P.Quarati, Phys. Rev. E [**49**]{}, 5103 (1994). L. D. Landau, E. M. Lifshitz, [*Statistical Physics*]{}, ( Pergamon Press, London, 1959 ).
[^1]: e-mail: Kaniadakis@pol88a.polito.it
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abstract: 'Following the line of ref. [@AS] we propose an improved algorithm which allows to calculate a $D$-dimensional fermion determinant integrating the exponent of $D+1$-dimensional Hermitean bosonic effective action. For a finite extra dimension the corrections decrease exponentially.'
author:
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A.A.Slavnov [^1]\
Steklov Mathematical Institute, Russian Academy of Sciences,\
Vavilov st.42, GSP-1,117966, Moscow, Russia
title: |
[SMI-20/96]{}
Improved Algorithm for Bosonized Fermion Determinant.
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-20mm
Introduction
============
Recently we proposed an algorithm which allows to calculate $D$-dimensional fermion determinants by integrating a $D+1$-dimensional bosonic effective action [@AS]. The following representation for the square of the covariant Dirac operator was obtained: $$\det( \hat{D}+m)^2 =
\label{1}$$ $$= \lim_{L \rightarrow \infty, b \rightarrow
0} \int \exp \{a^4b \sum_{n=-N+1}^N \sum_x [b^{-2}( \phi^*_{n+1}(x)
\phi_n(x)+ \phi^*_n(x) \phi_{n+1}(x)-2 \phi^*_n \phi_n) +$$ $$+i[
\phi^{*}_{n+1}(x) \gamma_5( \hat{D}+m) \phi_n(x)- \phi^{*}_n(x) \gamma_5
(\hat{D}+m) \phi_{n+1}(x)][b]^{-1}+$$ $$+\frac{1}{2}
\phi^{*}_n(D^2-m^2) \phi_n +\frac{i}{ \sqrt{L}}( \phi^*_n(x) \chi(x)+
\chi^*(x) \phi_n(x))] \}d \phi^*_nd \phi_nd \chi^*d \chi .$$ Here $ \phi_n$ are bosonic fields defined on a 4+1 dimensional lattice which have the same spinorial and internal structure as the Dirac fields $ \psi$. The fifth component $t$ to be defined on the one dimensional lattice of the length $L$ with the lattice spacing $b$: $$L=2Nb, \quad -N<n \leq N \label{2}$$ The free boundary conditions in $t$ are assumed: $$\phi_n=0, \quad n \leq -N, \quad n>N \label{3}$$ Lattice covariant derivative is denoted by $D_{ \mu}$ $$D_{\mu} \psi(x)=
\frac{1}{a}[U_{\mu}(x) \psi(x+a_{\mu})- \psi(x)] \label{4}$$ $$\hat{D}=1/2 \gamma_{ \mu}(D_{ \mu}^*-D_{ \mu}) \label{5}$$ (For simplicity we consider naive fermions, but all the construction is extended in a straightforward way to Wilson fermions).
For finite $L$ and $b$ the eq.(\[1\]) should be corrected by the terms $$O(b^2D^2)- \frac{4}{L^2D^2} \sin^2( \frac{DL}{2}) \label{6}$$ where $D$ are eigenvalues of the Dirac operator.
One sees that although in the limit $b \rightarrow 0, L \rightarrow
\infty$ the eq.(\[1\]) is exact, the convergence is not very fast. For numerical simulations it is important to make the corrections as small as possible, so one could use a one-dimensional lattice with relatively small number of sites $N$. Recently there were several publications, mainly based on the Lüsher’s proposal [@ML] for bosonization of fermion determinants, where different aspects of numerical simulations in the bosonized models were discussed [@ML1], [@A] [@JJL]. Further development in ref. [@M], [@BF], [@BFG] lead to algorithms with better convergence properties.
In the present paper I propose a modified algorithm, using essentially the same idea as in ref. [@AS], but providing much better convergence. Instead of polynomial supression of finite size effects as in eq.(\[6\]), new algorithm provides exponential damping $$\exp \{-mL \} \label{7}$$ where $m$ is a bare quark mass.
Improved algorithm for lattice QCD
==================================
Having in mind applications to QCD in this section we consider two fermion flavours interacting vectorially with the Yang-Mills fields $U_{\mu}$. The reason to consider two degenerate flavours is the positivity of the square of the gauge covariant Dirac operator. It is convinient to present the square of fermion determinant in the followi ng form $$\int \exp \{a^4 \sum_{n=1}^2 \sum_x \bar{\psi}_n(x)(
\hat{D}+m) \psi_n(x) \}d \bar{ \psi}d \psi = \det[\gamma_5( \hat{D}+m)]^2=
\label{8}$$ $$= \int \exp \{a^4 \sum_x \bar{\psi}(x)(
\hat{D}^2-m^2) \psi(x) \} d \bar{ \psi}d \psi$$
We again introduce five dimensional bosonic fields $ \phi(x,t)$ with the same spinorial and internal structure as $ \psi(x)$. The notations are as above.
We are going to prove the folowing identity $$\int \exp \{a^4 \sum_x \bar{ \psi}(x)( \hat{D}^2-m^2)
\psi(x) \}d \bar{\psi}d \psi = \label{9}$$ $$= \lim_{L \rightarrow \infty, b \rightarrow 0} \int \exp \{a^4b
\sum_{n=-N+1}^N \sum_x [(b^{-2})( \phi^*_{n+1}(x) \phi_n(x)+ h.c.-2 \phi^*_n
\phi_n) +$$ $$-i \phi^*_{n+1}(x) \gamma_5 \hat{D}L^{-1}(2n+1) \phi_n +
h.c.$$ $$+1/2 \phi^*_n(x) \hat{D}^2L^{-2}(2n+1)^2b^2 \phi_n$$ $$+i( \frac{2m}{ \pi L^5})^{1/4}( \phi^*_n(x) \chi(x)+
\chi^*(x) \phi_n(x))2nb \exp \{-mL^{-1}b^2n^2 \}] \}
d \phi^*_nd \phi_nd \chi^*d \chi .$$ Integration goes over the fields $\phi_n(x)$ satisfying the free boundary conditions (\[3\]). The last term in the exponent in the r.h.s. of eq.(\[9\]) introduces the constraint $$\sum_n \phi_nn \exp \{-mL^{-1}b^2n^2 \}=0 \label{10}$$ Substituting the solution of this constraint to the eq.(\[9\]) one gets the representation for the determinant of the gauge covariant Dirac operator as the path integral of the exponent of the bosonic Hermitean action. Below we present a construction which justifies the eq.(\[9\]) and estimate the corrections due to the finite lattice spacing $b$ and finite lattice size $L$.
Our starting point is the integral $$I= \int \exp \{a^4b
\sum_{n=-N+1}^N \sum_x [(b^{-2})( \phi^*_{n+1}(x) \exp \{-i
\gamma_5 \hat{D}b^2(n+1)^2L^{-1} \} \times \label{11}$$ $$\exp \{i \gamma_5\hat{D}b^2n^2L^{-1} \} \phi_n(x)+ h.c.-2 \phi^*_n
\phi_n) +$$ $$+i( \frac{2m}{ \pi L^5})^{1/4}( \phi^*_n(x) \chi(x)+
\chi^*(x) \phi_n(x))
2nb \exp \{-mL^{-1}b^2n^2 \}] \}
d \phi^*_nd \phi_nd \chi^*d \chi .$$ The operator $ \gamma_5 \hat{D}$ is Hermitean and it’s eigenvalues are real. R.h.s. of eq. (\[11\]) can be written in terms of eigenvectors of the operator $ \gamma_5 \hat{D}$, which are denoted as $ \phi_{\alpha}$: $$I= \int \exp \{b
\sum_{n=-N+1}^N \sum_{\alpha} [(b^{-2})( \phi^{\alpha*}_{n+1} \exp \{-i
D^{ \alpha}b^2(n+1)^2L^{-1} \} \times \label{12}$$ $$\exp \{iD^{ \alpha}b^2n^2L^{-1} \} \phi_n^{\alpha}+ h.c. -2 \phi^{\alpha*}_n
\phi^{\alpha}_n)+$$ $$+i( \frac{2m}{ \pi L^5})^{1/4}( \phi^{\alpha*}_n \chi^{\alpha}+
\chi^{\alpha*} \phi_n^{\alpha})2nb \exp \{-mL^{-1}b^2n^2 \}] \}
d \phi^{\alpha*}_nd \phi^{\alpha}_nd \chi^{\alpha*}d \chi^{\alpha}$$ To calculate the integral (\[12\]) we make the following change of variables: $$\phi_n^{\alpha} \rightarrow \exp \{-iD^{\alpha}n^2b^2L^{-1}
\}\phi_n^{\alpha}, \quad \phi_n^{\alpha*} \rightarrow \exp
\{iD^{\alpha}n^2b^2L^{-1} \} \phi_n^{\alpha*} \label{13}$$ Then the integral (\[12\]) acquires the form $$I= \int \exp
\{b \sum_{n=-N}^{N} \sum_{\alpha}b^{-2}[ \phi^{* \alpha}_{n+1}
\phi^{\alpha}_{n}+ \phi^{* \alpha}_n \phi^{ \alpha}_{n+1} -2 \phi^{*
\alpha}_n \phi^{ \alpha}_n] + \label{14}$$ $$+i(
\frac{2m}{ \pi L^5})^{1/4}( \phi^{\alpha*}_n \exp
\{iD^{\alpha}b^2n^2L^{-1} \}\chi^{\alpha}+
\chi^{\alpha*} \exp \{-iD^{\alpha}b^2n^2L^{-1}\phi_n^{\alpha}) \times$$ $$2nb \exp \{-mL^{-1}b^2n^2 \} \}
d \phi^{\alpha*}_nd \phi^{\alpha}_nd \chi^{\alpha*}d \chi^{\alpha}$$ Now the quadratic form in the exponent does not depend on $D^{\alpha}$ and therefore the corresponding determinant is a trivial constant. So we can calculate the integral by finding the stationary point of the exponent, which is defined by the following equations: $$b^{-2}( \phi^{* \alpha}_{n+1}+ \phi^{*
\alpha}_{n-1}-2 \phi^{* \alpha}_n)+$$ $$i( \frac{2m}{ \pi L^5})^{1/4} \chi^{*
\alpha}2nb \exp \{-(iD^{\alpha}+m)b^2n^2L^{-1} \}
=0, \quad
n \neq -N$$ $$b^{-2}( \phi^{
\alpha}_{n+1}+ \phi^{ \alpha}_{n-1}-2 \phi^{ \alpha}_n)+ \label{15}$$ $$+i( \frac{2m}{ \pi L^5})^{1/4} \chi^{
\alpha}2nb \exp \{(iD^{\alpha}-m)b^2n^2L^{-1} \}
=0, \quad n \neq -N$$ $$b^{-2}( \phi^{* \alpha}_{-N+1}-2 \phi^{*
\alpha}_{-N})+i( \frac{2m}{ \pi L^5})^{1/4} \chi^{*
\alpha}2Nb \exp \{-(iD^{\alpha}+m)b^2N^2L^{-1} \}=0$$ $$b^{-2}( \phi^{ \alpha}_{-N+1}-2 \phi^{
\alpha}_{-N})+i( \frac{2m}{ \pi L^5})^{1/4} \chi^{
\alpha}2Nb \exp \{(iD^{\alpha}-m)b^2N^2L^{-1} \}=0$$ $$\phi_{-N}^{\alpha}= \phi^{ \alpha*}_{-N}=0, \quad \phi^{ \alpha}_{N+1}=
\phi^{ \alpha*}_{N+1}=0$$
These equations are most easily solved for small $b$, when they can be approximated by the differential equations: $$\ddot \phi^{* \alpha} +i( \frac{2m}{ \pi L^5})^{1/4} \chi^{*
\alpha}2t \exp \{-(iD^{\alpha}+m)t^2L^{-1} \}=0\label{16}$$ $$\ddot \phi^{ \alpha} +i( \frac{2m}{ \pi
L^5})^{1/4} \chi^{ \alpha}2t \exp \{(iD^{\alpha}-m)t^2L^{-1} \}=0$$ $$\phi^{\alpha}( \frac{L}{2})= \phi^{\alpha}( -\frac{L}{2})=0, \quad
\phi^{\alpha*}( \frac{L}{2})= \phi^{\alpha*}(- \frac{L}{2})=0$$ The solution of these eq.s is $$\phi^{* \alpha}_{st}=
i \chi^{* \alpha} \frac{(2m \pi L)^{1/4}}{2(m+iD^{\alpha})^{3/2}}
\Phi(tL^{-1/2}(m+iD^{\alpha})^{1/2})+ \label{17}$$ $$+ i \chi^{\alpha*} \frac{(2m)^{1/4}}{( \pi L)^{1/4}(m+iD^{\alpha})^2} \exp
\{ \frac{-L(m+iD^{\alpha})}{4} \}2tL^{-1}-i \chi^{\alpha*} \frac{(2
\pi mL)^{1/4}}{2(m+iD^{\alpha})^{3/2}}$$ $$\phi^{
\alpha}_{st}= i \chi^{ \alpha} \frac{(2m
\pi L)^{1/4}}{2(m-iD^{\alpha})^{3/2}}
\Phi(tL^{-1/2}(m-iD^{\alpha})^{1/2})+ \label{17a}$$ $$+ i \chi^{\alpha} \frac{(2m)^{1/4}}{( \pi L)^{1/4}(m-iD^{\alpha})^2} \exp
\{ \frac{-L(m-iD^{\alpha})}{4} \}2tL^{-1}-i \chi^{\alpha} \frac{(2
\pi mL)^{1/4}}{2(m-iD^{\alpha})^{3/2}}$$ where $ \Phi(x)$ is a Fresnel integral. Substituting this solution into the integral (\[15\]) we get $$I= \int \exp\{- \sum_{\alpha}
\int_{-L/2}^{+L/2}dt \phi^{\alpha*}_{st}(t)\ddot
\phi^{\alpha}_{st}(t) \}d \chi^{\alpha*}d \chi^{\alpha} \label{18}$$ Integrating by parts and using the fact that $
\phi^{\alpha}_{st}(L/2)= \phi^{\alpha}_{st}(-L/2)=0$, we can transform this integral to the form $$I= \int \exp \{
\sum_{ \alpha} \int_{-L/2}^{L/2}dt \dot \phi^{\alpha*}_{st} \dot
\phi^{\alpha}_{st} \} d \chi^{\alpha*} d \chi^{\alpha}= \label{19}$$ $$\int \exp \{- \sum_{\alpha} (\frac{2m}{ \pi L})^{1/2} \frac{ \chi^{\alpha*}
\chi^{\alpha}}{m^2+(D^{\alpha})^2}[ \int_{-L/2}^{L/2}dt \exp
\{-2mL^{-1}t^2 \}+$$ $$+ \frac{4}{L(m^2+(D^{\alpha})^2)}e^{\{- \frac{Lm}{2} \}}+
\frac{2 \exp \{- \frac{L(m+iD^{\alpha})}{4}\}}{(m+iD^{\alpha})L}
\int_{-L/2}^{L/2}dt \exp \{-(m-iD^{\alpha})L^{-1}t^2 \}+$$ $$\frac{2 \exp \{- \frac{L(m-iD^{\alpha})}{4}\}}{(m-iD^{\alpha})L}
\int_{-L/2}^{L/2}dt \exp \{-(m+iD^{\alpha})L^{-1}t^2 \}] \}$$ Integrating over $ \chi$ one gets $$I= \prod_{\alpha}(m^2+(D^{\alpha})^2)(1+O((mL)^{-3/2} \exp \{-
\frac{mL}{2} \})) \label{20}$$ So if $m \sim a^{-1}$, and $L=2Nb$, the corrections are of the order $$O((Nba^{-1})^{-3/2} \exp \{-Nba^{-1} \}) \label{21}$$
At the same time it is easy to show that replacing the sum in the eq.( \[15\]) by the integral over $t$ produces corrections of order $O(b^2a^{-2})$. Therefore taking $N \sim3ab^{-1}$ we can make the corrections due to the finite size of a lattice less than 1 %.
Taking into account that $$( \gamma_5 \hat{D})^2 =-( \hat{D})^2 \label{23}$$ we see that $$I= \det( \hat{D}+m)^2 +O(b^2a^{-2})+O((mL)^{-3/2} \exp \{-
\frac{mL}{2} \}) \label{23}$$ To conclude we note that our equation (\[9\]) is a linearized version of the integral (\[11\]). The exponent in the quadratic form of the action (\[11\]) may be written in the form $$\exp \{-i
\gamma_5 \hat{D}b^2L^{-1}(2n+1) \}, \quad 2nb \leq L \label{2 4}$$ Having in mind that $$|| \gamma_5 \hat{D}|| \leq8a^{-1} \label{25}$$ one sees that $$\frac{|| \gamma_5 \hat{D}b^2(2n+1)||}{L} \leq 8ba^{-1} \label{26}$$ and if $8b<a$ one can replace the exponential in the action (\[11\]) by the first terms of it’s Taylor series, leading to the eq.(\[9\]).
Discussion
==========
We proved that the purely bosonic integral (\[9\]) is equal to the square of the determinant of covariant Dirac operator up to the terms $$O(b^2a^{-2})+O((mL)^{- \frac{3}{2}} \exp \{- \frac{mL}{2} \})\label{27}$$ Convergence of this approximation is much faster than in the ref. [@AS], where the finite size effects were supressed polynomially. For finite lattice spacing $b$ the corrections decrease exponentially when the number of extra fields $N$ increases.To get the same accuracy one needs much smaller number of bosonic fields, which hopefully will simplify Monte-Carlo simulations. It is worthwhile to note that to get exponential damping it is crucial to introduce the exponential factor into the constraint (\[10\]). As for the choice of the quadratic form in the effective bosonic action, there exists a certain freedom. We choose the particular form (\[11\])to simplify analitic calculations. Presumably the choice of the quadratic form done in our previous paper [@AS] is also possible. We also note that contrary to our previous algorithm [@AS] the present construction cannot be reduced to Lüsher’s algorithm with a particular choice of a polynomial due to explicit dependence of the constraint on the mass.
[**Acknowledgements.**]{}\
The main part of this work was done while the author was visiting Max-Planck Institute for Physics in Munich and Universite Paris-Sud Centre d’Orsay. I am grateful to the Theoretical Departments of these Institutes, and in particular to D.Maison and Ph.Boucaud for hospitality and interesting discussions. I thank A.Galli for helpful comments.This researsh was supported in part by Russian Basic Research Fund under grant 96-01-00551.$$~$$
[99]{} [ A.A.Slavnov SMI-28-95 hep-th/ 9512101, M.Lüsher Nucl.Phys. B418 (1994) 637. B.Bunk, K.Jansen, B.Jegerlehner, M.Lüsher, H.Simma, R.Sommer, Nucl.Phys.B (Proc.Suppl.)42 (1995) 49, C.Alexandrou, A.Borelli, Ph. de Forcrand, A.Galli and F.Jegerlehner, Nucl.Phys B456 (1995) 296, K.Jansen, B.Jegerlehner, C.Liu hep-lat/9604016, I.Montvay, DESY 95-192, hep-lat/9510042, A.Borici, Ph.de Forcrand IPS-95-23, hep-lat/9509080, A.Borelli, Ph.de Forcrand, A.Gall i, hep-lat/9602016 ]{}
[^1]: E-mail:$~~$ slavnov@class.mian.su
|
---
address: |
III. Physikalisches Institut, RWTH\
D - 52056 Aachen, Germany\
e-mail: bethke@rwth-aachen.de
author:
- 'S. BETHKE'
title: |
JET PHYSICS AT LEP\
AND\
WORLD SUMMARY OF ${\alpha_{\rm s}}$ [^1]
---
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\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
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${\rm e^+\rm e^-}$ Annihilation Data {#sec:data}
====================================
Over the past few years, a large amount of ${\rm e^+\rm e^-}$ annihilation data in the c. m. energy range from Q $\equiv\sqrt{s} =$ 10 to 189 GeV was accumulated at the CESR, PETRA, PEP, Tristan, LEP and SLC accelerators. The large data samples at LEP-1, which amount to about 4 million hadronic events around the $\z0$ resonance for each of the four LEP experiments, and the most recent data at the highest energies of LEP-2 (a few thousand events per experiment), together with reanalysed PETRA data at lower c.m. energies (about 50.000 hadronic events), provide powerful tools for precise tests of perturbative QCD.
At energies below or at the $\z0$ resonance, respectively at PETRA and at LEP-1, the study of ${\rm e^+\rm e^-}$ annihilation events is rather easy" and straight forward: apart from two-photon processes, the energy and quantum numbers of the hard scattering are well defined and different processes can be identified and selected with only very little backgrounds or biases. At LEP-2, i.e. at energies above the $\z0$ pole, the situation is more complicated:
- The annihilation cross section is orders of magnitude lower than at the $\z0$ pole; see Figure \[fig:sprime\]b.
- Initial state photon radiation reduces the available energy of the hadronic c.m. system, $\sqrt{s'}$, see Figure \[fig:sprime\]a, and causes, together with the resonant cross section around the $\z0$ mass, a large return-to-the-Z" effect. Radiative events can be suppressed requiring a minimum reconstructed ratio of $\sqrt{s'} / \sqrt{s}$ and other kinematic constraints.
- Other processes like ${\rm e^+\rm e^-}\rightarrow W^+W^-$ and ${\rm e^+\rm e^-}\rightarrow \z0 \z0$ emerge above the respective energy thresholds, see the shaded area in Figure \[fig:sprime\]c, causing a certain irreducable background for QCD studies.
In the following sections, recent QCD tests from LEP-1 ($\sqrt{s} \sim
91$ GeV; Section 2), from LEP-2 ($\sqrt{s} \sim 130$ GeV to 189 GeV; Section 3) and from a combination of PETRA and LEP data ($\sqrt{s} \sim 14$ GeV to 172 GeV; Section 4) are presented. The world summary of ${\alpha_{\rm s}}$ is updated in Section 5.
QCD Tests at LEP-1 {#lep1}
==================
${\alpha_{\rm s}}$ from Event Shapes using Optimised ${{\cal O}({\alpha_{\rm s}}^2)}$ QCD
-----------------------------------------------------------------------------------------
The DELPHI collaboration contributed a new measurement of ${\alpha_{\rm s}}$ from oriented event shape distributions at LEP-1 [@d-shape]. 17 different event shape observables are measured as a function of the polar angle of the thrust axis, and ${\alpha_{\rm s}}$ is determined from fits to ${{\cal O}({\alpha_{\rm s}}^2)}$ QCD calculations. As already reported earlier [@sbscale; @d-as; @o-as], good agreement between theory and data can be obtained if both ${\alpha_{\rm s}}$ and the renormalisation scale $\mu$ are determined simultaneously; see Figure \[fig:d-as\].
With optimised renormalisation scales and allowing for scale uncertainties between $0.5\times\mu_{exp}$ and $2\times \mu_{exp}$, consistent results of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ emerge, leading to a combined average of ${{\alpha_{\rm s}}(M_{\rm Z^0})}= 0.117 \pm 0.003$. Both the average and the error, which includes theoretical uncertainties from scale changes as given above, are smaller than those obtained from resummed ${{\cal O}({\alpha_{\rm s}}^2)}$ QCD fits. This is basically due to the choice of the averaging procedure and error definition.[^2] The broad consistency between data and optimised ${{\cal O}({\alpha_{\rm s}}^2)}$ QCD justifies this procedure and suggests to reconsider optimised fixed order perturbation theory as an alternative to resummation which was preferred in the past.
Differences between Quark- and Gluon-Jets
-----------------------------------------
Differences between quark- and gluon-jets were studied quite intensively during the past few years. The aim is to test the basic QCD prediction that hadrons coming from gluon jets should exhibit a softer energy spectrum and a wider transverse momentum distribution than those originating from quark jets, due to the larger colour charge of the gluon. In particular, the ratio $R_{qg}$ of the average multiplicities of hadrons in gluon jets and in quark jets should, for infinite jet energies and in leading order QCD, be $\approx C_A / C_F = 3 / (4/3) = 9/4 =
2.25$.
Experimental procedures to separate quark- from gluon-jets are usually based on vertex tagging of primary b-quark decays. In short, 3-jet like events are selected in which one of the two lower energetic jets is tagged as a b-quark, while the other low energy jet is then taken to be the gluon jet. From first analyses of this type it was found [@o-qgdiff1] that, after correction for misidentified jets, $R_{qg} = 1.27 \pm 0.07$. No QCD calculation for this particular type of analysis exists, such that a direct comparison of this result with theory is not possible.
Theoretical predictions only exist for colour singlet $q\overline{q}$ and $gg$ final states, where however the latter state is not experimentally accessible. In order to perform an analysis closer to theory, OPAL followed a new strategy in which events with a high-energetic gluon-jet recoiling against a (vertex-tagged) $q\overline{q}$ system [@o-qgdiff2] were selected. Such events are relatively rare, leading to about 550 selected gluon jets from OPAL’s LEP-1 data sample.
A comparison of the charged hadron multiplicity distribution of such gluon hemispheres with those of ordinary light quark event hemispheres is shown in Figure \[fig:o-qgdiff\], where a significant difference between quark- and gluon-jets is seen. For a central rapidity range, the hadron multiplicity ratio $R_{qg}$ is found to be $1.87 \pm 0.13$; the remaining difference to the QCD expectation of 2.25 is likely to be explained by finite jet energy effects.
DELPHI has studied the scale dependence of particle multiplicities in quark- and gluon-jets [@d-qgdiff]. Here, gluon-jets and light quark-jets are (anti-)tagged in 3-jet events, and a jet energy scale of $\kappa = E_{jet} {\rm sin} (\theta /2)$, where $\theta$ is the angle between the two lowest energetic jets, is defined. The charged hadron multiplicities for quark- and gluon-jets, as a function of $\kappa$, are diplayed in Figure \[fig:d-qgdiff\]. Also shown is the ratio of multiplicities and the ratio of slopes; the latter being close to the QCD prediction of 2.25.
From these measurements DELPHI determines the ratio $C_A / C_F$ to be $2.27 \pm 0.012$ which is in good agreement with QCD, and in particular with the expected colour charge of the gluon.
QCD Tests at LEP-2 {#sec:lep2}
==================
Hadronic Event Shapes and Running of ${\alpha_{\rm s}}$
-------------------------------------------------------
At each new energy point of LEP-2, all four LEP experiments have extensively studied hadronic event shape distributions and compared the new measurements with the predictions of QCD Monte Carlo models, as well as with analytic QCD calculations. In all cases, up to and including the most recent data at $\sqrt{s} = 189$ GeV, good agreement of data and theory was found, and no significant deviation from the standard expectation was seen. As an example, Figure \[fig:shapes\] shows the thrust-distributions measured by ALEPH [@a-shapes] at LEP-1 and at four energy points of LEP-2, together with a fit to resummed ${{\cal O}({\alpha_{\rm s}}^2)}$ QCD calculations which is in good agreement with the data at all energies.
The L3 collaboration has summarised their measurements [@l3-running] of ${\alpha_{\rm s}}$ from various event shape distributions, at LEP-1, at all LEP-2 energy points, and at hadronic c.m. energies below the $\z0$ pole, from an analysis of radiative events recorded at LEP-1. The data, which are displayed in Figure \[fig:L3-as\], are in very good agreement with the QCD prediction of a running coupling ${\alpha_{\rm s}}(\sqrt{s})$.
Energy Dependence of Charged Particle Production
------------------------------------------------
The energy dependence of particle production was, similarly as hadronic event shapes, continuously monitored by all LEP experiments. The variation of such observables with energy is found to be in good agreement with QCD predictions, and also with the standard" QCD plus hadronisation models. The energy dependence of the average charged hadron multiplicity and of the peak position $\xi^*$ of the $\xi = {\rm ln}(1/x)$ distribution ($x = p /
E_{beam}$) [@l3-running] are shown in Figure \[fig:multiplicities\].
Power Corrections and Energy Dependence of Event Shapes {#sec:powcor}
=======================================================
The energy dependence of mean values of event shape observables can be parametrised by the ${{\cal O}({\alpha_{\rm s}}^2)}$ perturbative QCD prediction [@ert] plus a term including the improved two-loop calculations (the Milan factor“) of power-suppressed 1/Q non-perturbative contributions [@powcor1], the so-called power corrections”. The latter contains the moment $\alpha_0$ of an effective coupling below an infrared scale $\mu_I$, which is expected to be universal for all applicable event shape observables.
A compilation of available data on the mean value of (1-thrust) and of the C-parameter is shown in Figure \[fig:jade-tc\], which is taken from a recent re-analysis of JADE data at PETRA energies [@jade2]. Perturbative QCD plus power corrections is found to give a very good description of the data, with ${{\alpha_{\rm s}}(M_{\rm Z^0})}= 0.118 \pm 0.002 \pm 0.004$ as determined from these data. However, universality of $\alpha_0$ is only found to be satisfied at a level of 30%.
For [*differential*]{} event shape distributions, the power corrections are simply a shift of the perturbative (${{\cal O}({\alpha_{\rm s}}^2)}$) spectra, and these were also studied in a wide c.m. energy range, including the most recent PETRA and LEP data for a total of four event shape observables [@fernandez]. A fit to these data, with ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ and $\alpha_0$ as free parameters for each observable, leads to the results displayed in Figure \[fig:jade-a0as\]. Agreement in both ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ and $\alpha_0$ is obtained, to a good level of accuracy, for two of the observables. However, the jet broadening parameters $B_w$ and $B_t$ deviate significantly [@fernandez] in both ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ and $\alpha_0$.
Most recently, the reason for these deviations was identified and traced to the theoretical predictions [@yuri]; a cure of this problem should soon be available.
World Summary of ${\alpha_{\rm s}}$ {#sec:as}
===================================
Significant determinations of the strong coupling strength, ${\alpha_{\rm s}}$, remain to be a demanding and interesting topic in experimental as well as theoretical study projects in high energy physics. In the following subsections, previous summaries of ${\alpha_{\rm s}}$ measurements [@qcd96; @qcd97] will be updated and a new world average $\wamz$ will be determined. Instead of a complete reference to all available measurements, only the newest results are briefly introduced, and more emphasis is spent on a detailed discussion of the overall [*uncertainty*]{} of $\wamz$, $\dwas$.
Updates and New Results
-----------------------
The results of all significant determinations of ${\alpha_{\rm s}}$, i.e. of all those which are based on QCD calculations which are complete - at least - to next-to-leading order perturbation theory, are summarised in Table \[astab\]. The following entries were added or updated since summer 1997 [@qcd97] (underlined in Table \[astab\]):
- The most recent determination of ${\alpha_{\rm s}}$ from the GLS sum rules, based on new data from $\nu -N$ scattering [@ccfr-gls] is included, replacing the previous result from Chyla and Kataev [@kataev].
- New measurements of ${\alpha_{\rm s}}$ from high statistics studies of vector and axial-vector spectral functions of hadronic $\tau$-decays are available from ALEPH [@a-tau] and OPAL [@o-tau]. As the most complete and precise studies of $\tau$ decays to date, these results are combined and taken to replace earlier results [@old-tau].
- H1 has contributed new determinations of ${\alpha_{\rm s}}$ from (2+1)-jet event rates at HERA [@h1-jet-new], replacing a previous measurement [@h1-jet]. A combination of these new results with a former one from ZEUS [@zeus-jet] is updated in Table \[astab\].
- A new determination of ${\alpha_{\rm s}}$ from $\Upsilon$-decays [@jamin-pich] replaces earlier results [@kobel].
- A recent determination of ${\alpha_{\rm s}}$ from the total ${\rm e^+\rm e^-}$ hadronic cross section measured by CLEO at $\ecm = 10.52$ GeV [@cleo-rhad] is added.
- Determinations of ${\alpha_{\rm s}}$ from JADE data, at $\ecm=35$ and 44 GeV, were updated [@jade2] by the inclusion of another observable, the C-parameter.
- ${\alpha_{\rm s}}$ from the most recent LEP result on $R_l = \frac{\Gamma (Z^0 \rightarrow hadrons)}{\Gamma (Z^0 \rightarrow
leptons)}$ was updated [@lep-ew] (these results are still preliminary).
- LEP results on ${\alpha_{\rm s}}$ from event shapes measured at $\ecm = 183$ and 189 GeV [@lep-shapes] are combined and added to the list (some of these results are still preliminary).
Further interesting and recent developments are a QCD analysis of neutrino deep inelastic scattering data for $xF_3$, in next-next-to-leading order of perturbation theory (NNLO) [@kataev-xf3], resulting in ${{\alpha_{\rm s}}(M_{\rm Z^0})}= 0.118 \pm 0.006$, and a reanalysis of muon deep inelastic scattering data [@newmudis], resulting in ${{\alpha_{\rm s}}(M_{\rm Z^0})}= 0.118 \pm 0.002
\ (stat+syst)$. Both these results are subject to further completion and verification; they are therefore considered to be preliminary and are not included in this summary.
The results for ${\alpha_{\rm s}}(Q)$, given in the $3^{rd}$ row of Table \[astab\], are presented in Fig. \[fig:as-q\]. These results are evolved from the energy scale $Q$, i.e. the typical energy scale of the hard scattering process under study, to the reference energy scale $\mz$, by using the QCD 4-loop beta-function with 3-loop matching at quark pole masses $M_b = 4.7$ GeV and $M_c = 1.5$ GeV [@4-loop], resulting in the values of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ given in the $4^{th}$ row of Table 1. These values of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ are displayed in Fig. \[fig:as-mz\]. The distribution of all ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ results is shown in a scatter plot of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ versus its quoted error, $\Delta{\alpha_{\rm s}}$, and in a frequency distribution of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ (Figure \[fig:as-hist\]).
The world average value of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ and its overall uncertainty as well as the corresponding QCD curves shown in Figs. \[fig:as-q\] and \[fig:as-mz\] will be discussed in the following section.
World Average and Overall Uncertainty of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$
--------------------------------------------------------------------------
In order to average the results of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$, a weigthed mean of the quoted central values is calculated, for all results as well as for subsamples as listed in Table \[aserr\]. The weight of a measurement is taken to be the inverse of the square of its total error.
The central value $\wamz$ does not depend on details of the weighting method; no significant differences are found if, for instance, the simple and unweighted mean is taken. Also, and more important, there are no significant differences between values of $\wamz$ calculated from different subsamples of the data: the averages from the high- and the low-energy data as well as those from ${\rm e^+\rm e^-}$ annihilation, from DIS and from $p\overline{p}$ colliders agree well with each other and with the overall average of $\wamz$ = 0.119, within the respective errors (irrespective of how those errors are defined; see the discussion below).
While the central value of $\wamz$ is remarkably stable and well defined, the determination of its overall uncertainty, $\dwas$, depends very much on the detailed definition of what this uncertainty should be, and how it should be calculated. There are several reasons for this situation:
The errors of most ${\alpha_{\rm s}}$ results are dominated by theoretical uncertainties, which are estimated using a variety of different methods and definitions. The significance of these non-gaussian errors is largely unknown. Furthermore, there are large correlations between different results, due to common theoretical uncertainties, as e.g. in the case of various event shape measurements in ${\rm e^+\rm e^-}$ annihilations. Nothing is known, however, about possible correlations between ${\alpha_{\rm s}}$ determinations from different processes, such as DIS and ${\rm e^+\rm e^-}$ annihilations, or between different procedures and observables used within the same class of processes.
Therefore, in the past, the value of $\dwas$ was often guess-timated", and/or a variety of mathematical methods was applied to obtain a reasonable estimate. Some of these methods will be applied and discussed in the following; the results are summarised in Table \[aserr\]:
----------------------------- --------- ----------- ------------ --------- ------------ ---------
uncorrel. simple rms rms box opt. corr. overall
sample (entries) $\wamz$ $\dwas$ $\dwas$ $\dwas$ $\dwas$ correl.
all (27) 0.1193 0.0012 0.0044 0.0059 0.0049 0.71
$\dwas\le 0.008$ (18) 0.1193 0.0013 0.0038 0.0052 0.0042 0.64
$\dwas\le 0.006$ (7) 0.1190 0.0016 0.0033 0.0041 0.0030 0.49
$\dwas\le 0.004$ (2) 0.1190 0.0021 0.0028 0.0028 0.0022 0.11
only ${\rm e^+\rm e^-}$(15) 0.1210 0.0016 0.0045 0.0059 0.0052 0.77
only DIS(8) 0.1175 0.0025 0.0029 0.0053 0.0061 0.80
only $p\overline{p}$ (3) 0.1156 0.0057 0.0053 0.0072 0.0088 0.69
$Q \le 10$ GeV (9) 0.1184 0.0016 0.0029 0.0045 0.0038 0.69
$Q \ge 30$ GeV (14) 0.1199 0.0020 0.0047 0.0062 0.0060 0.69
----------------------------- --------- ----------- ------------ --------- ------------ ---------
: Average values of $\wamz$ plus averaged uncertainties, for several methods to estimate the latter, and for several subsamples of the available data. \[aserr\]
- For illustrational purposes only, an overall error is calculated assuming that all measurements are entirely uncorrelated and all quoted errors are gaussian. The results are displayed in the $3^{rd}$ column of Table \[aserr\].
- The simple, unweigthed $r.m.s.$ of the mean value of all measurements is calculated and shown in the $4^{th}$ column, labelled simple rms".
- Assuming that each result of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ has a rectangular-shaped rather than a gaussian probability distribution, all resulting weights (the inverse of the square of the total error) are summed up in a histogram, and the resulting $r.m.s.$ of that distribution is quoted as rms box" [@qcd97].
- A correlated error is calculated from the error covariance matrix, assuming an overall correlation factor between all measurements. The correlation factor is adjusted such that the total $\chi^2$ is one per degree of freedom [@schmelling]. The resulting errors and correlation factors are given in the last two columns of Table \[aserr\] (labelled optimised correlation").
All of the methods defined above have certain advantages but also obey inherent problems. The simple rms“ indicates the scatter of all results around their common mean, but does not depend on the individual errors quoted for each measurement. The box rms”, which takes account of the individual errors and of their non-gaussian nature, was criticised to be too conservative an estimate of the overall uncertainty of ${\alpha_{\rm s}}$. The optimised correlation" method is closest to a mathematically appropriate treatment of correlated errors, however - in the absence of a detailed knowledge of these correlations - over-simplifies by the (unphysical ?) assumption of one overall correlation factor, identical to all pairs of measurements. Moreover, the $\chi^2$ calculated from the covariance matrix does not have, if correlations are present, the same mathematical and propabilistic meaning as in the case of uncorrelated data. In the extreme, $\chi^2$ may even be negative.
With these reservations in mind, all three methods do provide some estimate of $\dwas$. Apart from systematic differences in the size of $\dwas$, they all depend on the significance of the data included in the averaging process: in all cases, $\dwas$ is largest if all data are included, and tend to smaller values if the averaging is restricted to results with errors $\Delta{\alpha_{\rm s}}\le \Delta{\alpha_{\rm s}}^{(max)}$, i.e. if only the most significant results are used to calculate $\wamz$ and $\dwas$.
This can be seen from Table \[aserr\], where $\wamz$ and $\dwas$ are also shown for three subsets of data with $\Delta{\alpha_{\rm s}}\le$ 0.008, 0.006 and 0.004. The decrease of $\dwas$ as a function of $\Delta{\alpha_{\rm s}}^{(max)}$ is graphically shown in Figure \[fig:as-err\]. All three estimates of $\dwas$, the simple rms“, the box rms” and the optimised correlation“, decrease from intial values of 0.004 ... 0.006, if all ${\alpha_{\rm s}}$ results are taken into account, to about 0.003 if only the most precise” results are included. Only at the very extreme, taking the two results with the smallest quoted errors, the optimised correlation" method yields an overall error of less than 0.003. Note that, despite the dependence of $\dwas$ on the choice of $\Delta{\alpha_{\rm s}}^{(max)}$, the overall average $\wamz$ does $not$ depend on this selection!
On first sight it seems logical to restrict the determination of $\wamz$, and especially of $\dwas$, to the most significant data, if inclusion of insignificant measurements enlarges $\dwas$. Taken to the extreme, one may even be tempted to quote the one result which carries the smallest quoted error as the final world average value of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ and $\dwas$. However, the errors on ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ estimated in individual studies are, in general, [*lower limits*]{} because unknown and additional systematic effects can only increase the total error. Small systematic errors of single measurements may well be due to ignorance, over-optimism and/or neglection of certain error sources, which may be difficult to judge. Indeed, the errors of the two results with the smallest errors quoted, ${\alpha_{\rm s}}$ from $\tau$-decays and from heavy quark bound states using lattice gauge theory, were often criticised as being overly optimistic.
In this sense, averaging over a number of well understood and commonly accepted measurements of reasonable precision is a safe basis to estimate $\wamz$ and $\dwas$. While there is still a large degree of flexibility to choose the final data set and the procedure to estimate $\dwas$, the choice to select results with $\Delta{{\alpha_{\rm s}}(M_{\rm Z^0})}\le 0.008$ and to determine $\dwas$ using the optimised correlation“ method [@schmelling] seems reasonable and to be neither overly optimistic nor pessimistic. The final world average of ${{\alpha_{\rm s}}(M_{\rm Z^0})}$ is therefore quoted to be, c.f. Table \[aserr\] and Figure \[fig:as-err\], $$\wamz = 0.119 \pm 0.004\ .$$ The central values of 19 out of the 27 measurements listed in Table \[astab\], or equivalently 70%, are inside this error range of $\pm 0.004$, thus being compatible with the expectation of a one standard deviation” interval.
If the result of ${\alpha_{\rm s}}$ which is based on lattice gauge theory [@lgt] is omitted in the averaging process, $\wamz$ and $\dwas$ increase to $0.120 \pm 0.005$. The same is true if the result of the reanalysis [@newmudis] of muon deep inelastic scattering data is used[^3] instead of the one [@virchaux] listed in Table \[astab\].
Note that the value of $\dwas$ = 0.004 is a factor of two larger than the one quoted in the latest edition of the Review of Particle Physics [@pdg98]. The smaller value of 0.002 quoted there corresponds to a (slighly enlarged) r.m.s. assuming all measurements to be [*totally uncorrelated*]{}; an assumption which seems, in view of the results discussed above, unrealistic.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to W. Bernreuther, to O. Biebel, to S. Catani, to A. Kataev and to Yu. Dokshitzer for helpful discussions.
References {#references .unnumbered}
==========
[99]{} DELPHI Collaboration, DELPHI 98-84 CONF 152. S. Bethke, Z. Phys. C43 (1989) 331. DELPHI Collaboration, P. Abreu et al., Z. Phys. C54 (1992) 55 . OPAL Collaboration, P.D. Acton et al., Z. Phys. C55 (1992) 1. OPAL Collaboration, P. Acton et al., Z. Phys. C58 (1993) 387. OPAL Collaboration, K. Ackerstaff et al., Eur. Phys. J. C1 (1998) 479. DELPHI Collaboration, DELPHI 98-78 CONF 146. ALEPH Collaboration, ALEPH 98-025. L3 Collaboration, L3 Note 2304. R.K. Ellis, D.A. Ross and A.E. Terrano, Nucl. Phys. B178 (1981) 421;\
S. Catani and M. Seymour, Phys. Lett. B378 (1996) 287. Yu. Dokshitzer et al., JHEP 05 (1998) 3. P. Fernandez, S. Bethke, O. Biebel, PITHA 98/21, hep-ex/9807007. P. Fernandez, talk at QCD’98, PITHA 98/24, hep-ex/9808005. Yu. Dokshitzer, private communication; talk presented at ICHEP’98. S. Bethke, [*Proc. QCD Euroconference 96*]{}, Montpellier, France, July (1996), Nucl. Phys. (Proc.Suppl.) 54A (1997) 314; hep-ex/9609014. S. Bethke, [*Proc. QCD Euroconference 97*]{}, Montpellier, France, July (1997), Nucl. Phys. B (Proc.Suppl.) 64 (1998) 54; hep-ex/9710030. CCFR Collaboration, J.H. Kim et al., Phys. Rev. Lett. 81 (1998) 3595. J. Chyla, A. Kataev, Phys. Lett. B297 (1992) 385. ALEPH Collaboration, R. Barate et al., Eur. Phys. J. C4 (1998) 409. OPAL Collaboration, K. Ackerstaff et al., CERN-EP/98-102. S. Narison, Phys. Lett. B361 (1995) 121.\
M. Girone, M. Neubert, Phys. Rev. Lett. 76 (1996) 3061. H1 Collaboration, C. Adloff et al., Eur. Phys. J. C5 (1998) 625; C. Adloff et al., DESY-98-087, hep-ex/9807019. H1 Collaboration, T. Ahmed et al., Phys. Lett. B346 (1995) 415. ZEUS Collaboration, M. Derrick et al., Phys. Lett. B363 (1995) 201. M. Jamin and A. Pich, Nucl. Phys. B507 (1997) 334. M. Kobel, Proc. of $XXXVII^{th}$ Rencontre de Moriond, Les Arcs 1992. CLEO Collaboration, R. Ammar et al., Phys. Rev. D57 (1998) 1350. M. Grünewald, D. Karlen, talks at the ICHEP’98, Vancouver. L3 Collaboration, CERN-EP/98-148;\
LEP papers contributed to the ICHEP’98, Vancouver 1998. A. Kataev et al., Phys. Lett. B417 (1998) 374. S. Alekhin, hep-ph/9809544. K.G. Chetyrkin et al., hep-ph/9706430. M. Schmelling, Phys.Scripta 51 (1995) 676. C. Davies et al., Phys. Rev. D56 (1997); hep-lat/9706002. M. Virchaux and A. Milsztajn, Phys. Lett. B274 (1992) 221. Review of Particle Physics, Eur. Phys. J. C3 (1998).
---------------------------------------------- ---------- ------------------------------------- ------------------------------------- -------------------------------- -------------------------- --------
Q
Process \[GeV\] $\alpha_s(Q)$ $ {{\alpha_{\rm s}}(M_{\rm Z^0})}$ exp. theor. Theory
DIS \[pol. strct. fctn.\] 0.7 - 8 $0.120\ ^{+\ 0.010} $^{+0.004}_{-0.005}$ $^{+0.009}_{-0.006}$ NLO
_{-\ 0.008}$
DIS \[Bj-SR\] 1.58 $0.375\ ^{+\ 0.062}_{-\ 0.081}$ $0.121\ ^{+\ 0.005}_{-\ 0.009}$ – – NNLO
1.73 $0.295\ ^{+\ 0.092}_{-\ 0.073}$ $0.114\ ^{+\ 0.010}_{-\ 0.012}$ $^{+0.005}_{-0.006}$ $^{+0.009}_{-0.010}$ NNLO
1.78 $0.339 \pm 0.021$ $0.121 \pm 0.003$ 0.001 0.003 NNLO
DIS \[$\nu$; ${\rm F_2\ and\ F_3}$\] 5.0 $0.215 \pm 0.016$ $0.119\pm 0.005$ $ 0.002 $ $ 0.004$ NLO
DIS \[$\mu$; ${\rm F_2}$\] 7.1 $0.180 \pm 0.014$ $0.113 \pm 0.005$ $ 0.003$ $ 0.004$ NLO
DIS \[HERA; ${\rm F_2}$\] 2 - 10 $0.120 \pm 0.010$ $ 0.005$ $ 0.009$ NLO
10 - 100 $0.118 \pm 0.009$ $ 0.003$ $ 0.008$ NLO
DIS \[HERA; ev.shps.\] 7 - 100 $0.118\ ^{+\ 0.007}_{-\ 0.006}$ $ 0.001$ $^{+0.007}_{-0.006}$ NLO
${\rm Q\overline{Q}}$ states 4.1 $0.223 \pm 0.009$ $0.117 \pm 0.003 $ 0.000 0.003 LGT
4.13 $0.220 \pm 0.027$ $0.119 \pm 0.008 0.001 $0.008$ NLO
$
10.52 $0.20\ \pm 0.06 $ $0.130\ ^{+\ 0.021\ }_{-\ 0.029\ }$ $\ ^{+\ 0.021\ }_{-\ 0.029\ }$ – NNLO
${\rm e^+\rm e^-}$ \[ev. shapes\] 22.0 $0.161\ ^{+\ 0.016}_{-\ 0.011}$ $0.124\ ^{+\ 0.009}_{-\ 0.006}$ 0.005 $^{+0.008}_{-0.003}$ resum
${\rm e^+\rm e^-}$ \[$\sigma_{\rm had}$\] 34.0 $0.146\ ^{+\ 0.031}_{-\ 0.026}$ $0.123\ ^{+\ 0.021}_{-\ 0.019}$ $^{+\ 0.021}_{-\ 0.019} – NLO
$
35.0 $ 0.145\ ^{+\ 0.012}_{-\ 0.007}$ $0.123\ ^{+\ 0.008}_{-\ 0.006}$ 0.002 $^{+0.008}_{-0.005}$ resum
44.0 $ 0.139\ ^{+\ 0.010}_{-\ 0.007}$ $0.123\ ^{+\ 0.008}_{-\ 0.006}$ 0.003 $^{+0.007}_{-0.005}$ resum
${\rm e^+\rm e^-}$ \[ev. shapes\] 58.0 $0.132\pm 0.008$ $0.123 \pm 0.007$ 0.003 0.007 resum
$\p\bar{\p} \rightarrow {\rm b\bar{b}X}$ 20.0 $0.145\ ^{+\ 0.018\ }_{-\ 0.019\ }$ $0.113 \pm 0.011$ $^{+\ 0.007}_{-\ 0.006}$ $^{+\ 0.008}_{-\ 0.009}$ NLO
${\rm p\bar{p},\ pp \rightarrow \gamma X}$ 24.2 $0.137 $0.111\ ^{+\ 0.012\ }_{-\ 0.008\ }$ 0.006 $^{+\ 0.010}_{-\ 0.005}$ NLO
\ ^{+\ 0.017}_{-\ 0.014}$
${\sigma (\rm p\bar{p} \rightarrow\ jets)}$ 30 - 500 $0.121\pm 0.009$ 0.001 0.009 NLO
91.2 $0.122\pm 0.005$ $0.122\pm 0.005$ $ 0.004$ $0.003$ NNLO
${\rm e^+\rm e^-}$ \[ev. shapes\] 91.2 $0.122 \pm 0.006$ $0.122 \pm 0.006$ $ 0.001$ $ resum
0.006$
${\rm e^+\rm e^-}$ \[ev. shapes\] 133.0 $0.111\pm 0.008$ $0.117 \pm 0.008$ 0.004 0.007 resum
${\rm e^+\rm e^-}$ \[ev. shapes\] 161.0 $0.105\pm 0.007$ $0.114 \pm 0.008$ 0.004 0.007 resum
${\rm e^+\rm e^-}$ \[ev. shapes\] 172.0 $0.102\pm 0.007$ $0.111 \pm 0.008$ 0.004 0.007 resum
183.0 $0.109\pm 0.005$ $0.121 \pm 0.006$ 0.002 0.006 resum
189.0 $0.109\pm 0.006$ $0.122 \pm 0.007$ 0.003 0.006 resum
---------------------------------------------- ---------- ------------------------------------- ------------------------------------- -------------------------------- -------------------------- --------
: World summary of measurements of ${\alpha_{\rm s}}$. Underlined entries are new or updated since summer 1997 (DIS = deep inelastic scattering; GLS-SR = Gross-Llewellyn-Smith sum rules; Bj-SR = Bjorken sum rules; (N)NLO = (next-)next-to-leading order perturbation theory; LGT = lattice gauge theory; resum. = resummed next-to-leading order). \[astab\]
[^1]: Presented at the $IV^{th}$ Int. Symp. on Radiative Corrections, Barcelona, Sept. 8-12, 1998.
[^2]: Note that, for instance, a previous study [@o-as] based on 13 observables and using a different procedure to average results and determine the overall error obtained ${{\alpha_{\rm s}}(M_{\rm Z^0})}= 0.122^{+0.006}_{-0.005}$ which, if the same procedure as used in the DELPHI analysis is applied, converts to ${{\alpha_{\rm s}}(M_{\rm Z^0})}= 0.116 \pm
0.003$.
[^3]: The increase of $\dwas$ is due to an artifact of the optimised correlation" method which may increase the overall correlation factor (and thus, the overall error) if individual measurements are closer to their common mean.
|
---
abstract: 'In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions. By modifying Cheng-Yau’s operator $\square$ given in [[@ChengYau77]]{}, we introduce a modified operator $L$ and give new estimates of $L(nH)$ and $\square(nH)$ of such spacelike hypersurfaces. Finally, we give partial generalizations of some Conjectures in locally symmetric Lorentz spaces $L_{1}^{n+1}$.'
author:
- Zhongyang Sun
date: 'Received: date / Accepted: date'
title: Partial generalizations of some Conjectures in locally symmetric Lorentz spaces
---
Introduction
============
Let $L^{n+p}_p$ be an $(n+p)$-dimensional connected semi-Riemannian manifold of index $p$ $(\geqslant0)$. It is called a semi-definite space of index $p$. In particular, $L^{n+1}_1$ is called a Lorentz space. A hypersurface $M^{n}$ of a Lorentz space is said to be spacelike if the metric on $M^{n}$ induced from that of the Lorentz space is positive definite. When the Lorentz space is of constant curvature $c$, we call it Lorentz space form, denote by $\bar{M}_{1}^{n+1}(c)$. When $c>0$, $\bar{M}_{1}^{n+1}(c)=\mathbb{S}_{1}^{n+1}(c)$ is called an $(n+1)$-dimensional de Sitter space; when $c=0$, $\bar{M}_{1}^{n+1}(c)=\mathbb{L}_{1}^{n+1}(c)$ is called an $(n+1)$-dimensional Lorentz-Minkowski space; when $c<0$, $\bar{M}_{1}^{n+1}(c)=\mathbb{H}_{1}^{n+1}(c)$ is called an $(n+1)$-dimensional anti-de Sitter space.
In 1981, it was pointed out by S. Stumbles [[@Stumbles81]]{} that spacelike hypersurfaces with constant mean curvature in arbitrary spacetime come from its relevance in general relativity. In fact, constant mean curvature hypersurfaces are relevant for studying propagation of gravitational radiation. Hence, many geometers studies the complete spacelike hypersurfaces with constant mean curvature $H$ in Lorentz space forms $\bar{M}_{1}^{n+1}(c)$. For instance, A.J. Goddard [[@Goddard77]]{} proposed the following Conjecture:
.2cm [ [**Conjecture 1.**]{}]{} If $M^{n}$ is a complete spacelike hypersurface of de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ with constant mean curvature $H$, then is $M^{n}$ totally umbilical $?$
J. Ramanathan [[@Ramanathan87]]{} proved Goddard’s conjecture for $\mathbb{S}_{1}^{3}(1)$ and $0\leqslant H\leqslant 1$. Moreover, when $H>1$, he also showed that the conjecture is false. When $H^{2}\leqslant c$ if $n=2$ or when $n^{2}H^{2}<4(n-1)c$ if $n\geqslant 3$, K. Akutagawa [[@Akutagawa87]]{} proved that Goddard’s conjecture is true. S. Montiel [[@Montiel88]]{} solved Goddard’s problem without restriction over the range of $H$ provided that $M^{n}$ is compact. There are also many results such as [[@KiKN91; @Oliker92]]{}.
On the other hand, concerning the study of spacelike hypersurfaces with constant scalar curvature in a de Sitter space, H. Li [[@Li97]]{} proposed an interseting problem:
.2cm [ [**Conjecture 2.**]{}]{} If $M^{n} (n\geqslant3)$ is a complete spacelike hypersurface in de Sitter space $\mathbb{S}_{1}^{n+1}(1)$ with constant normalized scalar curvature $R$ satisfying $\frac{n-2}{n}\leqslant R\leqslant1$, then is $M^{n}$ totally umbilical $?$
Recently, F.E.C. Camargo et al. [[@CamargoRL08]]{} proved that Li’s question is true if the mean curvature $H$ is bounded. There are also many results such as [[@BrasilCP01; @ChengS88]]{} and [[@HuSZ07]]{}.
It is natural to study complete or compact spacelike hypersurfaces with constant mean curvature or constant scalar curvature in the more general Lorentz spaces. In 2004, J. Ok Baek, Q.M. Cheng and Y. Jin Suh [[@BaekCY04]]{} studied the complete spacelike hypersurfaces with constant mean curvature $H$ and gave some rigidity theorems in locally symmetric Lorentz spaces $L^{n+1}_1$. Recently, J.C. Liu and Z.Y. Sun [[@Liusun10]]{} studied the complete spacelike hypersurfaces with constant normalized scalar curvature $R$ and obtained some rigidity theorems in locally symmetric Lorentz spaces $L^{n+1}_1$.
In this paper, firstly, we recall that Choi et al. [[@ChoiLS99; @SuhCY02]]{} introduced the class of $(n+1)$-dimensional Lorentz spaces $L_{1}^{n+1}$ of index 1 which satisfy the following conditions for some constants $c_{1}$ and $c_{2}$:
\(i) for any spacelike vector $u$ and any timelike vector $v$ $$K(u,v)=-\frac{c_{1}}{n},
\eqno(1.1)$$
\(ii) for any spacelike vectors $u$ and $v$ $$K(u,v)\geqslant c_{2},
\eqno(1.2)$$ where $K$ denotes the sectional curvature on $L^{n+1}_{1}$.
When $L^{n+1}_{1}$ satisfies conditions (1.1) and (1.2), we will say that $L^{n+1}_{1}$ satisfies condition $(\ast)$.
.2cm [[**Remark 1.1**]{}]{} The Lorentz space form $\bar{M}_{1}^{n+1}(c)$ satisfies condition $(\ast)$, where $-\frac{c_{1}}{n}=c_{2}=c$.
In order to present our theorems, we will introduce some basic facts and notations. Let $\bar{R}_{CD}$ be the components of the Ricci tensor of $L_{1}^{n+1}$ satisfying $(\ast)$, then the scalar curvautre $\bar{R}$ of $L_{1}^{n+1}$ is given by $$\bar{R}=\sum_{A=1}^{n+1}\epsilon_{A}\bar{R}_{AA}
=-2\sum_{i=1}^{n}\bar{R}_{(n+1)ii(n+1)}+\sum_{i,j=1}^{n}\bar{R}_{ijji}=2c_{1}+\sum_{i,j=1}^{n}\bar{R}_{ijji}.$$ It is well known that $\bar{R}$ is constant when the Lorentz space $L_{1}^{n+1}$ is locally symmetric, so $\sum_{i,j=1}^{n}\bar{R}_{ijji}$ is constant. From (2.3) in Section 2, we can define a $P$ such that $$n(n-1)P=n^{2}H^{2}-S=\sum_{i,j=1}^{n}\bar{R}_{ijji}-n(n-1)R.
\eqno(1.3)$$ Hence, when $M^{n}$ is a spacelike hypersurface in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying $(\ast)$, we conclude from (1.3) that the normalized scalar curvature $R$ of $M^{n}$ is constant if and only if $P$ is constant.
Next we will introduce a notion for linear Weingarten spacelike hypersurfaces in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$ as follows:
.2cm [ [**Definition 1.2**]{}]{} Let $M^{n}$ be a spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. We call $M^{n}$ a *linear Weingarten spacelike hypersurface* If $P$ defined by (1.3) and the mean curvature $H$ of $M^{n}$ satisfy the following conditions: $eP+aH=b$, $e^{2}+a^{2}\neq0$, where $e,a,b\in \mathbb{R}$.
.2cm [ [**Remark 1.3**]{}]{} Let $e=0$ and $a\neq0$ in Definition 1.2, a linear Weingarten spacelike hypersurface $M^{n}$ reduces to a spacelike hypersurface with constant mean curvature $H$. Let $a=0$ and $e\neq0$ in Definition 1.2, a linear Weingarten spacelike hypersurface $M^{n}$ reduces to a spacelike hypersurface with constant normalized scalar curvature $R$. Hence, the linear Weingarten spacelike hypersurfaces can be regarded as a natural generalization of spacelike hypersurfaces with constant mean curvature $H$ or with constant normalized scalar curvature $R$ in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$.
In Section 3, by modifying Cheng-Yau’s operator $\square$ given in [[@ChengYau77]]{}, we study complete linear Weingarten spacelike hypersurfaces in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$ and give generalizations of [[@Liusun10 Theorem 1.2(i)]]{} and [[@CamargoRL08 Theorme 1.2]]{}. Thus, we get Theorems 3.6 and 3.9.
In Section 4, by using Cheng-Yau’s operator $\square$ given in [[@ChengYau77]]{}, we study compact linear Weingarten spacelike hypersurfaces in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$ and give generalizations of [[@Li97 Theorme 4.3]]{} and [[@Liusun10 Theorem 1.1]]{}. Then, we obtain Theorems 4.4 and 4.8.
.2cm [ [**Remark 1.4**]{}]{} In this paper, the spacelike hypersurfaces $M^{n}$ in Theorems 3.6-3.9 and Theorems 4.4-4.8 satisfying $P+aH=b$ are linear Weingarten spacelike hypersurfaces in Definition 1.2.
Preliminaries
=============
In this section, we will introduce some basic facts and give estimate the Laplacian $\triangle S$ of the squared length $S$ of the second fundamental form for spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying $(\ast)$. We shall make use of the following convention on the ranges of indices: $1\leqslant A, B, C,\ldots\leqslant n+1;~1\leqslant i, j, k,\ldots\leqslant n$.
We assume that $M^{n}$ is a spacelike hypersurface in Lorentz spaces $L_{1}^{n+1}$. Choose a local field of pseudo-Riemannian orthonormal frames $\{e_1,\ldots,e_{n+1}\}$ in $L_{1}^{n+1}$ such that, restricted to $M^{n}$, $\{e_1,\ldots,e_{n}\}$ are tangent to $M^{n}$ and $e_{n+1}$ is normal to $M^{n}$. That is, $\{e_1,\ldots,e_{n}\}$ are spacelike vectors and $e_{n+1}$ is a timelike vector. Let $\{\omega_{A}\}$ and $\{\omega_{AB}\}$ be the fields of dual frames and the connection forms of $L_{1}^{n+1}$, respectively. Let $\epsilon_{i}=1,\epsilon_{n+1}=-1$, then the structure equations of $L_{1}^{n+1}$ are given by $$\begin{array}{ll}
d\omega_{A}=-\sum\limits_{B}\epsilon_{B}\omega_{AB}\wedge\omega_{B},~~~\omega_{AB}+\omega_{BA}=0,
\\
d\omega_{AB}=-\sum\limits_{C}\epsilon_{C}\omega_{AC}\wedge\omega_{CB}
-\frac{1}{2}\sum\limits_{C,D}\epsilon_{C}\epsilon_{D}\bar{R}_{ABCD}\omega_{C}\wedge\omega_{D}.
\end{array}$$ Here the components $\bar{R}_{CD}$ of the Ricci tensor and the scalar curvature $\bar{R}$ of Lorentz spaces $L_{1}^{n+1}$ are given, respectively, by $$\bar{R}_{CD}=\sum_{B}\epsilon_{B}\bar{R}_{BCDB}, ~~~~~ \bar{R}=\sum_{A}\epsilon_{A}\bar{R}_{AA}.$$ The components $\bar{R}_{ABCD;E}$ of the covariant derivative of the Riemannian curvature tensor $\bar{R}$ are defined by $$\begin{aligned}
\sum_{E}\epsilon_{E}\bar{R}_{ABCD;E}\omega_{E}
=d&\bar{R}_{ABCD}-\sum_{E}\epsilon_{E}(\bar{R}_{EBCD}\omega_{EA}\\
&+\bar{R}_{AECD}\omega_{EB}+\bar{R}_{ABED}\omega_{EC}+\bar{R}_{ABCE}\omega_{ED}).
\end{aligned}$$ We restrict these forms to $M^n$ in $L_{1}^{n+1}$, then $\omega_{n+1}=0$. Hence, we have $\sum_{i}\omega_{(n+1)i}\wedge\omega_{i}=0$. Using Cartan’s lemma, we know that there are $h_{ij}$ such that $ \omega_{(n+1)i}=\sum_jh_{ij}\omega_j$ and $h_{ij}=h_{ji}$, where the $h_{ij}$ are the coefficients of the second fundamental form of $M^{n}$. This gives the second fundamental form of $M^n$, $h=\sum_{i,j}h_{ij}\omega_{i}\otimes\omega_{j}$.
The Gauss equation, components $R_{ij}$ of the Ricci tensor and the normalized scalar curvature $R$ of $M^n$ are given, respectively, by $$R_{ijkl}=\bar{R}_{ijkl}-(h_{il}h_{jk}-h_{ik}h_{jl}),
\eqno(2.1)$$ $$R_{ij}=\sum_{k}\bar{R}_{kijk}-nHh_{ij}+\sum_{k}h_{ik}h_{kj},
\eqno(2.2)$$ $$n(n-1)R=\sum_{i,j}\bar{R}_{ijji}-n^{2}H^{2}+S,
\eqno(2.3)$$ where $H=\frac{1}{n}\sum_jh_{jj}$ and $S=\sum_{i,j}h^{2}_{ij}$ are the mean curvature and the squared length of the second fundamental form of $M^{n}$, respectively.
Let $h_{ijk}$ and $h_{ijkl}$ be the first and the second covariant derivatives of $h_{ij}$, respectively, so that $$\sum_{k}h_{ijk}\omega_{k}=dh_{ij}-\sum_{k}h_{ik}\omega_{kj}-\sum_{k}h_{kj}\omega_{ki},$$
$$\sum_{l}h_{ijkl}\omega_{l}=dh_{ijk}-\sum_{l}h_{ljk}\omega_{li}-\sum_{l}h_{ilk}\omega_{lj}-\sum_{l}h_{ijl}\omega_{lk}.$$ Thus, we have the Codazzi equation and the Ricci identity $$h_{ijk}-h_{ikj}=\bar{R}_{(n+1)ijk},
\eqno(2.4)$$
$$h_{ijkl}-h_{ijlk}=-\sum_{m}h_{im}R_{mjkl}-\sum_{m}h_{jm}R_{mikl}.
\eqno(2.5)$$ Let $\bar{R}_{ABCD;E}$ be the covariant derivative of $\bar{R}_{ABCD}$. Thus, restricted on $M^{n}$, $\bar{R}_{(n+1)ijk;l}$ is given by $$\bar{R}_{(n+1)ijk;l}=\bar{R}_{(n+1)ijkl}+\bar{R}_{(n+1)i(n+1)k}h_{jl}+\bar{R}_{(n+1)ij(n+1)}h_{kl}+
\sum_{m}\bar{R}_{mijk}h_{ml},
\eqno(2.6)$$ where $\bar{R}_{(n+1)ijk;l}$ denotes the covariant derivative of $\bar{R}_{(n+1)ijk}$ as a tensor on $M^{n}$ so that $$\begin{aligned}
\sum_{l}\bar{R}_{(n+1)ijk;l}\omega_{l}
=d&\bar{R}_{(n+1)ijk}-\sum_{l}\bar{R}_{(n+1)ljk}\omega_{li}\\
&-\sum_{l}\bar{R}_{(n+1)ilk}\omega_{lj}
-\sum_{l}\bar{R}_{(n+1)ijl}\omega_{lk}.
\end{aligned}$$ Next we compute the Laplacian $\triangle h_{ij}=\sum_{k}h_{ijkk}$. From (2.4) and (2.5), we have $$\begin{aligned}
\triangle h_{ij}
=&\sum_{k}h_{ikjk}+\bar R_{(n+1)ijk;k}\\
=&\sum_{k}\left(h_{kikj}-\sum_{l}(h_{kl}R_{lijk}+h_{il}R_{lkjk})+\bar
R_{(n+1)i j k ;k}\right).
\end{aligned}$$ From $h_{kikj}=h_{kkij}+\bar R_{(n+1)k i k; j}$, we get $$\triangle h_{ij} =(nH)_{ij}+\sum_k\left(\bar R_{(n+1)ijk;k}+\bar R_{(n+1)kik;j}\right)
-\sum_{k,l}(h_{kl}R_{lijk}+h_{il}R_{lkjk}).
\eqno(2.7)$$ From (2.1) and (2.6) and (2.7), we obtain $$\begin{aligned}
\triangle h_{ij} =&(nH)_{ij}+\sum_k\left(\bar R_{(n+1)ijk;k}+\bar R_{(n+1)kik;j}\right)
-\sum_k(h_{kk}\bar R_{(n+1)ij(n+1)}\\
&+h_{ij}\bar R_{(n+1)k(n+1)k})-\sum_{k,l}(2h_{kl}\bar
R_{lijk}+h_{jl}\bar R_{lkik}+h_{il}\bar R_{lkjk})\\
&-nH\sum_lh_{il}h_{lj}+Sh_{ij}.
\end{aligned}$$ According to the above equation, the Laplacian $\triangle S$ of the squared length $S$ of the second fundamental form $h_{ij}$ of $M^{n}$ is obtained $$\begin{aligned}
\frac{1}{2}\triangle S
=&\sum_{i,j,k}h^2_{ijk}+\sum_{i,j}h_{ij}\triangle h_{ij}\\
=&\sum_{i,j,k}h^2_{ijk}+\sum_{i,j}(nH)_{ij}h_{ij}+
\sum_{i,j,k}\left(\bar R_{(n+1)ijk;k}+\bar R_{(n+1)kik;j}\right)h_{ij}\\
&-\left(\sum_{i,j}nHh_{ij}\bar R_{(n+1)ij(n+1)}+S\sum_k\bar R_{(n+1)k(n+1)
k}\right)\\
&-2\sum_{i,j,k,l}(h_{kl}h_{ij}\bar R_{lijk}+h_{il}h_{ij}\bar R_{lkjk})
-nH\sum\limits_{i,j,l}h_{il}h_{lj}h_{ij}+S^2.
\end{aligned}
\eqno(2.8)$$ Choose a local orthonormal frame field $\{e_{1},\ldots,e_{n}\}$ such that $h_{ij}=\lambda_{i}\delta_{ij}$, where $\lambda_{i}$, $1\leqslant i\leqslant n$, are principal curvatures of $M^{n}$. Estimating the right-hand side of (2.8) by using the curvature conditions $(\ast)$, the following lemma was obtained by J.C. Liu and Z.Y. Sun.
.2cm [ [**Lemma 2.1**]{} ([[@Liusun10 Lemma 2.1]]{}).]{} *Let $M^{n}$ be a spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$, then $$\frac{1}{2}\triangle S \geqslant
\sum_{i,j,k}h^2_{ijk}+\sum_{i}\lambda_{i}(nH)_{ii}
+nc(S-nH^2)+\left(S^2-nH\sum_i\lambda_i^3\right),
\eqno(2.9)$$ where $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$.*
In the following, we will continue to calculate $\triangle S$ for spacelike hypersurfaces in locally symmetry Lorentz spaces satisfying $(\ast)$. Thus, we need the following algebraic Lemma.
.2cm [ [**Lemma 2.2**]{}]{} ([[@AlencarC94; @Okumura74]]{}). *Let $\mu_{1},\ldots,\mu_{n}$ be real numbers such that $\sum_{i}\mu_{i}=0$ and $\sum_{i}\mu^{2}_{i}=B^{2}$, where $B\geqslant0$ is constant. Then $$\left|\sum_{i}\mu^{3}_{i}\right|\leqslant \frac{n-2}{\sqrt{n(n-1)}}B^{3}$$ and equality holds if and only if at least $n-1$ of the $\mu_{i}^{~,}s$ are equal.*
Let $\phi=\sum_{i,j}\phi_{ij}\omega_{i}\otimes\omega_{j}$ be a symmetric tensor defined on $M^{n}$, where $\phi_{ij}=h_{ij}-H\delta_{ij}$. It is easy to check that $\phi$ is traceless. Choose a local orthonormal frame field $\{e_{1},\ldots,e_{n}\}$ such that $h_{ij}=\lambda_{i}\delta_{ij}$ and $\phi_{ij}=\mu_{i}\delta_{ij}$. Let $|\phi|^{2}=\sum_{i}\mu_{i}^{2}.$ A direct computation gets $$|\phi|^{2}=S-nH^{2}=\frac{1}{2n}\sum_{i,j}(\lambda_{i}-\lambda_{j})^{2}.
\eqno(2.10)$$ Hence, $|\phi|^{2}=0$ if and only if $M^{n}$ is totally umbilical. We also get $$\sum_{i}\lambda_{i}^{3}=nH^{3}+3H\sum_{i}\mu_{i}^{2}+\sum_{i}\mu_{i}^{3}.$$ By applying Lemma 2.2 to the real numbers $\mu_{1},\ldots,\mu_{n}$, we obtain $$\begin{aligned}
-nH\sum_{i}\lambda_{i}^{3}
=&-n^{2}H^{4}-3nH^{2}\sum_{i}\mu_{i}^{2}-nH\sum_{i}\mu_{i}^{3}\\
\geqslant&2n^{2}H^{4}-3nSH^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H|(S-nH^{2})^{\frac{3}{2}}.
\end{aligned}
\eqno(2.11)$$ Substituting (2.10) and (2.11) into (2.9), we obtain the following lemma.
.2cm [ [**Lemma 2.3**]{}]{} *Let $M^{n}$ be a spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$, then $$\frac{1}{2}\triangle S \geqslant
\sum_{i,j,k}h^2_{ijk}+\sum_{i}\lambda_{i}(nH)_{ii}
+|\phi|^{2}L_{|H|}(|\phi|),
\eqno(2.12)$$ where $|\phi|^{2}=S-nH^{2}$, $L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2$, $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$.*
Complete linear Weingarten spacelike hypersurfaces in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$
=========================================================================================================================
In this section, according to Cheng and Yau $\square$ given in [[@ChengYau77]]{}, first we introduce a modified operator $L$ acting on any $C^{2}$-function $f$ by $$L (f)=\sum_{i,j}(nH\delta_{ij}-h_{ij})f_{ij}+\frac{(n-1)a}{2}\triangle f,
\eqno(3.1)$$ where $a\in \mathbb{R}$.
Cheng-Yau [[@ChengYau77]]{} gave a lower estimate of $\sum_{i,j,k}h^2_{ijk}$ which is very important in the proof of their theorem. They proved that, for a hypersurface in a space form of constant sectional curvature $c$, if the normalized scalar curvature $R$ is constant and $R\geqslant c$, then $\sum_{i,j,k}h^2_{ijk}\geqslant n^{2}|\nabla H|^{2}$, where $h_{ijk}^{~~~,}s$ are components of the covariant differentiation of the second fundamental form.
For the spacelike hypersurfaces $M^{n}$ in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$, without assumption that the normalized scalar curvature $R$ of $M^{n}$ is constant, we also obtain the estimate $\sum_{i,j,k}h^2_{ijk}\geqslant n^{2}|\nabla H|^{2}$ in the proof of Proposition 3.1.
Next we will prove Propositions 3.1 and 3.3 which will play a crucial role in the proofs of Theorems 3.6 and 3.9.
.2cm [ [**Proposition 3.1**]{}]{} *Let $M^{n}(n\geqslant3)$ be a spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$ and $(n-1)a^{2}+4nb\geqslant0$, where $a,b\in \mathbb{R}$, then $$L(nH)\geqslant|\phi|^{2}L_{|H|}(|\phi|),
\eqno(3.2)$$ where $|\phi|^{2}=S-nH^{2}$, $L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2$, $c=2c_{2}+\frac{c_{1}}{n}>0$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$.*
*Proof* Choose a local orthonormal frame field $\{e_{1},\ldots,e_{n}\}$ such that $h_{ij}=\lambda_{i}\delta_{ij}$. Since $P+aH=b$, it follows from (1.3) that $$n^{2}H^{2}-S=n(n-1)P=-n(n-1)(aH-b).
\eqno(3.3)$$ Noticing that $nH\triangle(nH)=\frac{1}{2}\triangle(nH)^{2}-n^{2}|\nabla H|^{2}$, it follows from (3.1) and (3.3) that $$\begin{aligned}
L (nH)=&\sum_{i,j}(nH\delta_{ij}-h_{ij})(nH)_{ij}+\frac{(n-1)a}{2}\triangle(nH)\\
=&nH\triangle(nH)-\sum_{i}\lambda_{i}(nH)_{ii}+\frac{1}{2}\triangle\left[S-n^{2}H^{2}+n(n-1)b\right]\\
=&\frac{1}{2}\triangle S-n^{2}|\nabla H|^{2}-\sum_{i}\lambda_{i}(nH)_{ii}.
\end{aligned}
\eqno(3.4)$$ Thus, it follows from (2.12) and (3.4) that $$L(nH)\geqslant\sum_{i,j,k}h^2_{ijk}-n^{2}|\nabla H|^{2}+|\phi|^{2}L_{|H|}(|\phi|),
\eqno(3.5)$$ where $|\phi|^{2}=S-nH^{2}$ and $L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2$.
Differentiating formula (3.3) exteriorly yields $2\sum_{i,j}h_{ij}h_{ijk}=2n^{2}HH_{k}+n(n-1)aH_{k}$, then by using Cauchy-Schwarz inequality we have $$4S\sum_{i,j,k}h^{2}_{ijk}\geqslant4\sum_{k}\left(\sum_{i,j}h_{ij}h_{ijk}\right)^{2}
=\left[2n^{2}H+n(n-1)a\right]^{2}|\nabla H|^{2}.
\eqno(3.6)$$ Combining $(n-1)a^{2}+4nb\geqslant 0$ and (3.3), we have $$\begin{aligned}
\left[2n^{2}H+n(n-1)a\right]^{2}-4n^{2}S=&4n^{4}H^{2}+4n^{3}(n-1)aH+n^{2}(n-1)^{2}a^{2}\\
&-4n^{2}\left[n^{2}H^{2}+n(n-1)(aH-b)\right]\\
=&n^{2}\left[(n-1)^{2}a^{2}+4n(n-1)b\right]\\
\geqslant& 0.
\end{aligned}
\eqno(3.7)$$ Thus, we conclude from (3.6) and (3.7) that $$\sum_{i,j,k}h^2_{ijk}\geqslant n^{2}|\nabla H|^{2}.
\eqno(3.8)$$ Consequently, (3.2) follows from (3.5) and (3.8). Finally, the Proposition 3.1 is proved. $\Box$
We also need the following lemma in the proof of Proposition 3.3.
.2cm [ [**Lemma 3.2**]{}]{} ([[@Omori67]]{}). *Let $M^{n}$ be an $n$-dimensional complete Riemannion manifold whose sectional curvature is bounded from below and $F: M^{n}\rightarrow \mathbb{R}$ be a smooth function which is bounded from above on $M^{n}$. Then there exists a sequence of points $\{x_{k}\}\in M^{n}$ such that $$\begin{aligned}
&\lim_{k\rightarrow\infty} F(x_{k})=\sup F,\\
&\lim_{k\rightarrow\infty} |\nabla F(x_{k})|=0,\\
&\lim_{k\rightarrow\infty} \sup\max\{\left(\nabla^{2} F(x_{k})\right)(X, X):~|X|=1\}\leqslant 0.
\end{aligned}$$*
.2cm [ [**Proposition 3.3**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. Suppose that $M^{n}$ has bounded mean curvature $H$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$, $(n-1)a^{2}+4nb\geqslant0$ and $a\geqslant0$, where $a,b\in \mathbb{R}$, then there is a sequence of points $\{x_{k}\}\in M^{n}$ such that $$\begin{aligned}
&\lim_{k\rightarrow\infty} nH(x_{k})=\sup(nH),\\
&\lim_{k\rightarrow\infty} |\nabla(nH)(x_{k})|=0,\\
&\lim_{k\rightarrow\infty} \sup\left(L(nH)(x_{k})\right)\leqslant 0.
\end{aligned}
\eqno(3.9)$$*
*Proof* Choose a local orthonormal frame field $\{e_{1},\ldots,e_{n}\}$ such that $h_{ij}=\lambda_{i}\delta_{ij}$. If $H\equiv0$, the proposition is obvious. Let us suppose that $H$ is not identically zero. By changing the orientation of $M^{n}$ if necessary, we may assume $\sup H>0$. In view of (3.1), $L(nH)$ is given by $$L(nH)=\sum_{i}(nH-\lambda_{i})(nH)_{ii}+\frac{(n-1)a}{2}\sum_{i}(nH)_{ii}.
\eqno(3.10)$$ Since $(n-1)a^{2}+4nb\geqslant0$, it follows from (3.3) that $$\begin{aligned}
(\lambda_{i})^{2}\leqslant S=&n^{2}H^{2}+n(n-1)(aH-b)\\
=&\left[nH+\frac{(n-1)a}{2}\right]^{2}-\frac{(n-1)^{2}a^{2}}{4}-n(n-1)b\\
\leqslant&\left[nH+\frac{(n-1)a}{2}\right]^{2}.
\end{aligned}
\eqno(3.11)$$ Thus, it follows from (3.11) that $$|\lambda_{i}|\leqslant\left|nH+\frac{(n-1)a}{2}\right|.
\eqno(3.12)$$ From (1.2) and (2.2), we have $$\begin{aligned}
R_{ii}=&\sum_{k}\bar{R}_{kiik}-nHh_{ii}+\sum_{k}(h_{ik})^{2}\\
\geqslant&\sum_{k}\bar{R}_{kiik}-\frac{nH}{2}2h_{ii}+(h_{ii})^{2}\\
=&\sum_{k}\bar{R}_{kiik}+(h_{ii}-\frac{nH}{2})^{2}-\frac{n^{2}H^{2}}{4}\\
\geqslant&nc_{2}-\frac{n^{2}H^{2}}{4}.
\end{aligned}
\eqno(3.13)$$ Since $H$ is bounded, it follows from (3.13) that the sectional curvatures of $M^{n}$ are bounded from below. Therefore, we may apply Lemma 3.2 to the function $nH$, obtaining a sequence of points $\{x_{k}\}\in M^{n}$ such that $$\lim_{k\rightarrow\infty} nH(x_{k})=\sup(nH),~~~ \lim_{k\rightarrow\infty} |\nabla(nH)(x_{k})|=0,~~~
\lim_{k\rightarrow\infty} \sup\left(nH_{ii}(x_{k})\right)\leqslant 0.
\eqno(3.14)$$ Since $H$ is bounded, taking subsequences if necessary, we can obtain a sequence of points $\{x_{k}\}\in M^{n}$ which satisfies (3.14) and such that $H(x_{k})\geqslant0$ (by changing the orientation of $M^{n}$ if necessary). Since $a\geqslant0$, it follows from (3.12) that $$\begin{aligned}
0\leqslant nH(x_{k})+\frac{(n-1)a}{2}-|\lambda_{i}(x_{k})|
\leqslant& nH(x_{k})+\frac{(n-1)a}{2}-\lambda_{i}(x_{k})\\
\leqslant& nH(x_{k})+\frac{(n-1)a}{2}+|\lambda_{i}(x_{k})|\\
\leqslant& 2\left[nH(x_{k})+\frac{(n-1)a}{2}\right].
\end{aligned}
\eqno(3.15)$$ Using once more the fact that $H$ is bounded, we can conclude from (3.15) that $\{nH(x_{k})+\frac{(n-1)a}{2}-\lambda_{i}(x_{k})\}$ is non-negative and bounded. By applying $L(nH)$ at $x_{k}$, taking the limit and using (3.14) and (3.15), we obtain $$\begin{aligned}
\lim_{k\rightarrow\infty}\sup\left(L(nH)(x_{k})\right)
&\leqslant \sum_{i}\lim_{k\rightarrow\infty}\sup\left(nH+\frac{(n-1)a}{2}-\lambda_{i}\right)(x_{k})nH_{ii}(x_{k})\\
&\leqslant 0.
\end{aligned}$$ Finally, the Proposition 3.3 is proved. $\Box$
In 2010, J.C. Liu and Z.Y. Sun [[@Liusun10]]{} studied the complete spacelike hypersurfaces with constant normalized scalar curvature $R$ in locally symmetric Lorentz spaces $L^{n+1}_1$ satisfying $(\ast)$ and obtained the following result.
.2cm [ [**Theorem 3.4**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface with constant normalized scalar curvature $R$ in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. Suppose that $M^{n}$ has bounded mean curvature $H$. If the constant $P$ defined by $(1.3)$ satisfies $0\leqslant P\leqslant\frac{2c}{n}$ and $c>0$, where $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$, then $M^{n}$ is totally umbilical.*
In 2008, F.E.C. Camargo, R.M.B. Chaves and L.A.M. Sousa Jr. [[@CamargoRL08]]{} studied the complete spacelike hypersurfaces with constant normalized scalar curvature $R$ in de Sitter spaces $\mathbb{S}_{1}^{n+1}(c)$ and proved the following result.
.2cm [ [**Theorem 3.5**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface with constant normalized scalar curvature $R$ in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$. If the squared length $S$ of the second fundamental form of $M^{n}$ satisfies $$\sup S<2\sqrt{n-1}c$$ and $R\leqslant c$, then $M^{n}$ is totally umbilical.*
In this Section, we study complete linear Weingarten spacelike hypersurfaces in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we give generalizations of Theorems 3.4-3.5 and obtain the following results.
.2cm [ [**Theorem 3.6**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. Suppose that $M^{n}$ has bounded mean curvature $H$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$, $(n-1)a^{2}+4nb\geqslant0$, $a\geqslant0$, $b\leqslant\frac{2c}{n}$ and $c>0$, where $a,b\in \mathbb{R}$, $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$, then $M^{n}$ is totally umbilical.*
.2cm [[**Remark 3.7**]{}]{} When we take $a=0$ in Theorem 3.6, we obtain that $P=b$ is constant and $0\leqslant P\leqslant \frac{2c}{n}$. Hence, Theorem 3.6 is a generalization of Theorem 3.4. If $a=0$ and $L^{n+1}_{1}$ is the de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ in Theorem 3.6, then $-\frac{c_{1}}{n}=c_{2}=c$ and $0\leqslant P=b=c-R\leqslant\frac{2c}{n}$ following from (1.3). At the same time, $0\leqslant P=c-R\leqslant\frac{2c}{n}$ becomes $\frac{n-2}{n}c\leqslant R\leqslant c$ and $R$ is constant. Hence, Theorem 3.6 is also a generalization of the result due to F.E.C. Camargo et al. in [[@CamargoRL08]]{}, saying that a complete spacelike hypersurface $M^{n}$ $(n\geqslant3)$ in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ with constant normalized scalar curvature $R$ satisfying $\frac{n-2}{n}c\leqslant R\leqslant c$ must be totally umbilical provided that $M^{n}$ has bounded mean curvature $H$.
For example, we consider the spacelike hypersurface immersed into $\mathbb{S}_{1}^{n+1}(1)$ defined by $T_{k,r}=\{x\in\mathbb{S}_{1}^{n+1}(1)|-x_{0}^{2}+x_{1}^{2}+\ldots+x_{k}^{2}=-\sinh^{2}r\}$ , where $r$ is a positive real number and $1\leqslant k\leqslant n-1$. $T_{k,r}$ is complete and isometric to the Riemannian product $\mathbb{H}^{k}(1-\coth^{2}r)\times\mathbb{S}^{n-k}(1-\tanh^{2}r)$ of a $k$-dimensional hyperbolic space and an $(n-k)$-dimensional sphere of constant sectional curvatures $1-\coth^{2}r$ and $1-\tanh^{2}r$, respectively. It follows from [[@HuSZ07]]{} that if $k=1$, then $R$ satisfies $0<R=\frac{n-2}{n}(1-\tanh^{2}r)<\frac{n-2}{n}$; similarly, if $k=n-1\geqslant2$, we see that $R=\frac{n-2}{n}(1-\coth^{2}r)<0$. Thus, for any $R$ satisfying $0<R<\frac{n-2}{n}$ and for any $R<0$, we can choose $r$ such that the hypersurfaces $T_{1,r}$ and $T_{n-1,r}$, respectively, are complete, non-totally umbilical and have constant normalized scalar curvature $R$. Hence, when $M^{n}(n\geqslant3)$ is a complete spacelike hyperusrface, the hypothesis that $0\leqslant P\leqslant\frac{2c}{n}$ is essential to umbilicity of $M^{n}$ in Theorems 3.4. Without assumption that $P$ defined by (1.3) is constant in Theorem 3.6. we generalizes the assumption condition $0\leqslant P\leqslant\frac{2c}{n}$ in Theorem 3.4 to more general situations.
*Proof of Theorem* 3.6 If $M^{n}$ is maximal, i.e., $H\equiv0$, according to Nishikawa’s result [[@Nishikawa84]]{}, we know that $M^{n}$ is totally geodesic. We can assume that $H$ is not identically zero. Hence, by Proposition 3.3 we can obtain a sequence of points $\{x_{k}\}\in M^{n}$ such that $$\lim_{k\rightarrow\infty}\sup\left(L(nH)(x_{k})\right)\leqslant0,~~~~~~~
\lim_{k\rightarrow\infty}(nH)(x_{k})=\sup(nH)>0.
\eqno(3.16)$$ From (2.10) and (3.3), we have $$|\phi|^{2}=n(n-1)(H^{2}+aH-b).
\eqno(3.17)$$ In view of $\lim_{k\rightarrow\infty}(nH)(x_{k})=\sup(nH)>0$ and $a\geqslant0$, it follows from (3.17) that $$\lim_{k\rightarrow\infty}|\phi|^{2}(x_{k})=\sup|\phi|^{2}.
\eqno(3.18)$$ Next, we will consider the following polynomial given by $$L_{\sup |H|}(x)=x^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}\sup|H|x+nc-n\sup H^2.$$ We claim that $$L_{\sup |H|}(\sup |\phi|)>0.
\eqno(3.19)$$ Indeed, if $\sup H^{2}<\frac{4(n-1)}{n^{2}}c$, then the discriminant of $L_{\sup |H|}(x)$ is negative. Therefore, we have $L_{\sup |H|}(\sup |\phi|)>0$. Suppose that $\sup H^{2}\geqslant\frac{4(n-1)}{n^{2}}c$. Let $\xi$ be the biggest root of the equation $L_{\sup |H|}(x)=0$, which is positive. We know that $\xi$ is the only one root of $L_{\sup |H|}(x)$ if $\sup H^{2}=\frac{4(n-1)}{n^{2}}c$.
If we can prove that $(\sup |\phi|)^{2}=\sup |\phi|^{2}>\xi^{2}$, then we have $\sup |\phi|>\xi$. Hence, $L_{\sup |H|}(\sup |\phi|)>0$. Since $a\geqslant0$, $b\leqslant\frac{2c}{n}$ and $c>0$, it follows from (3.17) that $$\sup |\phi|^{2}=n(n-1)(\sup H^{2}+a\sup H-b)\geqslant (n-1)(n\sup H^{2}-2c).
\eqno(3.20)$$ By virtue of (3.20), it is straightforward to verify that $$\begin{aligned}
\sup &|\phi|^{2}-\xi^{2}\\
&\geqslant\frac{n-2}{2(n-1)}\left[n^{2}\sup H^{2}-n\sup H\sqrt{n^{2}\sup H^{2}-4(n-1)c}-2(n-1)c\right].
\end{aligned}$$ Thus, $\sup |\phi|^{2}-\xi^{2}>0$ if and only if $$n^{2}\sup H^{2}-n\sup H\sqrt{n^{2}\sup H^{2}-4(n-1)c}-2(n-1)c>0.
\eqno(3.21)$$ Taking into account that the inequality (3.21) is equivalent to $4(n-1)^{2}c^{2}>0$, which is true because of $c>0$. Hence, $\sup |\phi|^{2}-\xi^{2}>0$, which proves our claim.
Evaluating (3.2) at the points $x_{k}$ of the sequence, taking the limit and using (3.16) and (3.18), we obtain that $$\begin{aligned}
0\geqslant& \lim_{k\rightarrow\infty}\sup\left(L(nH)(x_{k})\right)\\
\geqslant&\sup|\phi|^{2}\left(\sup|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}\sup|H|\sup|\phi|+nc-n\sup H^2\right)\\
=&\sup|\phi|^{2}L_{\sup |H|}(\sup |\phi|),
\end{aligned}
\eqno(3.22)$$ where $$L_{\sup |H|}(\sup |\phi|)=\sup|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}\sup|H|\sup|\phi|+nc-n\sup H^2.$$
Therefore, we can conclude from (3.19) and (3.22) that $\sup|\phi|^{2}=0$. That is, $|\phi|^{2}=0$ which shows $M^{n}$ is totally umbilical. This completes the proof of Theorem 3.6. $\Box$
If $L^{n+1}_{1}$ is a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ in Theorem 3.5, then $-\frac{c_{1}}{n}=c_{2}=c$ and $P=c-R$ following from (1.3). Hence, we obtain the following corollary.
.2cm [ [**Corollary 3.8**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$. Suppose that $M^{n}$ has bounded mean curvature $H$. If the normalized scalar curvature $R$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $R-aH=c-b$, $(n-1)a^{2}+4nb\geqslant0$, $a\geqslant0$ and $b\leqslant\frac{2c}{n}$, where $a,b\in \mathbb{R}$, then $M^{n}$ is totally umbilical.*
.2cm [ [**Theorem 3.9**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. Suppose that the squared length $S$ of the second fundamental form of $M^{n}$ satisfies $\sup S<2\sqrt{n-1}c$, where $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$, $(n-1)a^{2}+4nb\geqslant0$ and $a\geqslant0$, where $a,b\in \mathbb{R}$, then $M^{n}$ is totally umbilical.*
*Proof of Theorem* 3.9 First we consider the quadratic form $$D(u,v)=u^{2}-\frac{n-2}{\sqrt{n-1}}uv-v^{2}
\eqno(3.23)$$ and the orthogonal transformation $$\begin{aligned}
\bar{u}&=\frac{1}{\sqrt{2n}}\left[(1+\sqrt{n-1})u+(1-\sqrt{n-1})v\right],\\
\bar{v}&=\frac{1}{\sqrt{2n}}\left[(\sqrt{n-1}-1)u+(\sqrt{n-1}+1)v\right].
\end{aligned}
\eqno(3.24)$$ Using (3.24), we can rewrite (3.23) as follows $$\begin{aligned}
D(u,v)=D(\bar{u},\bar{v})
&=\frac{n}{2\sqrt{n-1}}(\bar{u}^{2}-\bar{v}^{2})\\
&=-\frac{n}{2\sqrt{n-1}}(\bar{u}^{2}+\bar{v}^{2})+\frac{n}{\sqrt{n-1}}\bar{u}^{2}.
\end{aligned}
\eqno(3.25)$$ From (3.24), we have $$u^{2}+v^{2}=\bar{u}^{2}+\bar{v}^{2}.
\eqno(3.26)$$ Take $u=|\phi|$ and $v=\sqrt{n}|H|$. Substituting $u$ and $v$ into (3.23) and using (3.25), we have $$\begin{aligned}
|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2
&=nc+D(|\phi|, \sqrt{n}|H|)\\
&=nc+\frac{n}{2\sqrt{n-1}}(\bar{u}^{2}-\bar{v}^{2})\\
&=nc-\frac{n}{2\sqrt{n-1}}(\bar{u}^{2}+\bar{v}^{2})+\frac{n}{\sqrt{n-1}}\bar{u}^{2}.
\end{aligned}
\eqno(3.27)$$ From (3.26), we have $u^{2}+v^{2}=\bar{u}^{2}+\bar{v}^{2}=|\phi|^{2}+nH^{2}=S$. Hence, it follows from (3.27) that $$|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2\geqslant nc-\frac{n}{2\sqrt{n-1}}S.
\eqno(3.28)$$ If $M^{n}$ is maximal, i.e., $H\equiv0$, according to Nishikawa’s result [[@Nishikawa84]]{}, we know that $M^{n}$ is totally geodesic. We can assume that $H$ is not identically zero. By changing the orientation of $M^{n}$ if necessary, we may assume $\sup H>0$. Since $P+aH=b$, it follows from (1.3) that $S=n^{2}H^{2}+n(n-1)(aH-b)$. Together with the assumption $\sup S<2\sqrt{n-1}c$ and $a\geqslant0$, so we know that $H$ is bounded. Hence, by Proposition 3.3 we can obtain a sequence of points $\{x_{k}\}\in M^{n}$ such that $$\lim_{k\rightarrow\infty}\sup\left(L(nH)(x_{k})\right)\leqslant0,~~~~~~~
\lim_{k\rightarrow\infty}(nH)(x_{k})=\sup(nH)>0.
\eqno(3.29)$$ From (2.10), (3.18) and (3.29), we have $$\lim_{k\rightarrow\infty}S(x_{k})=\sup S.
\eqno(3.30)$$ Combining (3.22), (3.28) and (3.30), we obtain $$\begin{aligned}
0\geqslant& \lim_{k\rightarrow\infty}\sup\left(L(nH)(x_{k})\right)\\
\geqslant&\sup|\phi|^{2}\left(\sup|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}\sup|H|\sup|\phi|+nc-n\sup H^2\right)\\
\geqslant&\sup|\phi|^{2}\left(nc-\frac{n}{2\sqrt{n-1}}\sup S\right).
\end{aligned}
\eqno(3.31)$$ Since $\sup S<2\sqrt{n-1}c$, we conclude from (3.31) that $\sup|\phi|^{2}=0$. That is, $|\phi|^{2}=0$ which shows $M^{n}$ is totally umbilical. This completes the proof of Theorem 3.9. $\Box$
If $L^{n+1}_{1}$ is a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ in Theorem 3.9, then $-\frac{c_{1}}{n}=c_{2}=c$ and $P=c-R$ following from (1.3). Thus, we obtain the following corollary.
.2cm [ [**Corollary 3.10**]{}]{} *Let $M^{n}(n\geqslant3)$ be a complete spacelike hypersurface in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$. Suppose that the squared length $S$ of the second fundamental form of $M^{n}$ satisfies $\sup S<2\sqrt{n-1}c$. If the normalized scalar curvature $R$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $R-aH=c-b$, $(n-1)a^{2}+4nb\geqslant0$ and $a\geqslant0$, where $a,b\in \mathbb{R}$, then $M^{n}$ is totally umbilical.*
.2cm [[**Remark 3.11**]{}]{} Let $a=0$ in Corollary 3.10, we know that $R=c-b$ is constant and $R\leqslant c$. Hence, Corollary 3.10 is a generalization of Theorem 3.5.
Compact linear Weingarten spacelike hypersurfaces in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$
========================================================================================================================
According to Cheng and Yau $\square$ given in [[@ChengYau77]]{}, we introduce a self-adjoint operator $\square$ acting on any $C^{2}$-function $f$ by $$\square (f)=\sum_{i,j}(nH\delta_{ij}-h_{ij})f_{ij}.
\eqno(4.1)$$
In order to prove Theorems 4.4 and 4.8, we need the following proposition.
.2cm [ [**Proposition 4.1**]{}]{} *Let $M^{n}(n\geqslant3)$ be a spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$ and $(n-1)a^{2}+4nb\geqslant0$, where $a,b\in \mathbb{R}$, then $$\square(nH)\geqslant-\frac{1}{2}\triangle \left(n(n-1)R\right)+|\phi|^{2}L_{|H|}(|\phi|),
\eqno(4.2)$$ where $|\phi|^{2}=S-nH^{2}$, $L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2$, $c=2c_{2}+\frac{c_{1}}{n}>0$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$.*
*Proof* Choose a local orthonormal frame field $\{e_{1},\ldots,e_{n}\}$ such that $h_{ij}=\lambda_{i}\delta_{ij}$. Noticing that $nH\triangle(nH)=\frac{1}{2}\triangle(nH)^{2}-n^{2}|\nabla H|^{2}$, it follows from (2.3) and (4.1) that $$\begin{aligned}
\square (nH)=&\sum_{i,j}(nH\delta_{ij}-h_{ij})(nH)_{ij}\\
=&\frac{1}{2}\triangle(nH)^{2}-n^{2}|\nabla H|^{2}-\sum_{i}\lambda_{i}(nH)_{ii}\\
=&-\frac{1}{2}\triangle \left(n(n-1)R\right)+\frac{1}{2}\triangle S-n^{2}|\nabla H|^{2}-\sum_{i}\lambda_{i}(nH)_{ii}.
\end{aligned}
\eqno(4.3)$$ Thus, we conclude from (2.12) and (4.3) that $$\square(nH)\geqslant-\frac{1}{2}\triangle \left(n(n-1)R\right)
+\sum_{i,j,k}h^2_{ijk}-n^{2}|\nabla H|^{2}+|\phi|^{2}L_{|H|}(|\phi|),
\eqno(4.4)$$ where $|\phi|^{2}=S-nH^{2}$ and $L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2$.
Hence, (4.2) follows from (3.8) and (4.4). Finally, the Proposition 4.1 is proved. $\Box$
In 1997, H. Li [[@Li97]]{} studied the compact spacelike hypersurfaces with constant normalized scalar curvature in de Sitter spaces $\mathbb{S}_{1}^{n+1}(c)$ and obtained the following result.
.2cm [ [**Theorem 4.2**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface with constant normalized scalar curvature $R$ in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$. If $\frac{n-2}{n}c\leqslant R\leqslant c$, then $M^{n}$ is totally umbilical.*
In 2010, J.C. Liu and Z.Y. Sun [[@Liusun10]]{} gave a generalization of Theorem 4.2 in a locally symmetric Lorentz space $L^{n+1}_1$ satisfying $(\ast)$ and obtained the Theorem 4.3.
.2cm [ [**Theorem 4.3**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface with constant normalized scalar curvature $R$ in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. If the constant $P$ defined by $(1.3)$ satisfies $0\leqslant P\leqslant\frac{2c}{n}$ and $c>0$, where $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$, then $M^{n}$ is totally umbilical.*
In this Section, we give generalizations of Theorems 4.2-4.3 and get the following results.
.2cm [ [**Theorem 4.4**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$, $(n-1)a^{2}+4nb\geqslant0$, $b\leqslant\frac{2c}{n}$ and $c>0$, where $a,b\in \mathbb{R}$, $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$, then $M^{n}$ is totally umbilical.*
.2cm [[**Remark 4.5**]{}]{} When we take $a=0$ in Theorem 4.4, we obtain $P$ is constant and $0\leqslant P\leqslant\frac{2c}{n}$. Thus, Theorem 4.4 is a generalization of Theorem 4.3.
*Proof of Theorem* 4.4 By using the similar processing as in the proof of Theorem 3.6 on the inequality $L_{\sup |H|}(\sup|\phi|) > 0$, we obtain $$L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2>0.
\eqno(4.5)$$ Since $M^{n}$ is compact and $\square$ is self-adjoint operator, we get $$\int_{M^{n}}\square(nH)dv_{M^{n}}=0.
\eqno(4.6)$$ From (4.2) and (4.6), we get $$0\geqslant\int_{M^{n}}|\phi|^{2}L_{|H|}(|\phi|)dv_{M^{n}},
\eqno(4.7)$$ where $|\phi|^{2}=S-nH^{2}$ and $L_{|H|}(|\phi|)=|\phi|^{2}-\frac{n(n-2)}{\sqrt{n(n-1)}}|H||\phi|+nc-nH^2$.
Hence, we can conclude from (4.5) and (4.7) that $|\phi|^{2}=0$ which shows $M^{n}$ is totally umbilical. This completes the proof of Theorem 4.4. $\Box$
When $L^{n+1}_{1}$ is a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ in Theorem 4.4, we know that $-\frac{c_{1}}{n}=c_{2}=c$ and $P=c-R$ following from (1.3). Thus, we obtain the following corollary.
.2cm [ [**Corollary 4.6**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$. If the normalized scalar curvature $R$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $R-aH=c-b$, $(n-1)a^{2}+4nb\geqslant0$ and $b\leqslant\frac{2c}{n}$, where $a,b\in \mathbb{R}$, then $M^{n}$ is totally umbilical.*
.2cm [[**Remark 4.7**]{}]{} When we take $a=0$ in Corollary 4.6, we obtain that $R=c-b$ is constant and $\frac{n-2}{n}c\leqslant R\leqslant c$. Thus, Corollary 4.6 is a generalization of Theorem 4.2.
.2cm [ [**Theorem 4.8**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface in a locally symmetric Lorentz space $L_{1}^{n+1}$ satisfying $(\ast)$. Suppose that the squared length $S$ of the second fundamental form of $M^{n}$ satisfies $S<2\sqrt{n-1}c$, where $c=2c_{2}+\frac{c_{1}}{n}$ and $c_{1}$, $c_{2}$ are given as in $(\ast)$. If $P$ defined by $(1.3)$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $P+aH=b$ and $(n-1)a^{2}+4nb\geqslant0$, where $a,b\in \mathbb{R}$, then $M^{n}$ is totally umbilical.*
*Proof of Theorem* 4.8 From (3.28) and (4.7), we obtain $$0\geqslant\int_{M^{n}}|\phi|^{2}\left(nc-\frac{n}{2\sqrt{n-1}}S\right)dv_{M^{n}}.
\eqno(4.8)$$ Since $S<2\sqrt{n-1}c$, we can conclude from (4.8) that $|\phi|^{2}=0$ which shows $M^{n}$ is totally umbilical. This completes the proof of Theorem 4.8. $\Box$
When $L^{n+1}_{1}$ is a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ in Theorem 4.8, we know that $-\frac{c_{1}}{n}=c_{2}=c$ and $P=c-R$ following from (1.3). Thus, we obtain the following corollary.
.2cm [ [**Corollary 4.9**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$. Suppose that the squared length $S$ of the second fundamental form of $M^{n}$ satisfies $S<2\sqrt{n-1}c$. If the normalized scalar curvature $R$ and the mean curvature $H$ of $M^{n}$ satisfy the following conditions$:$ $R-aH=c-b$ and $(n-1)a^{2}+4nb\geqslant0$, where $a,b\in \mathbb{R}$, then $M^{n}$ is totally umbilical.*
When we take $a=0$ in Corollary 4.9, we obtain that $R=c-b$ is constant and $R\leqslant c$. Thus, we obtain the following corollary.
.2cm [ [**Corollary 4.10**]{}]{} *Let $M^{n}(n\geqslant3)$ be a compact spacelike hypersurface in a de Sitter space $\mathbb{S}_{1}^{n+1}(c)$ with constant normalized scalar curvature $R$, $R\leqslant c$. If the squared length $S$ of the second fundamental form of $M^{n}$ satisfies $S<2\sqrt{n-1}c$, then $M^{n}$ is totally umbilical.*
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|
---
abstract: |
Via a simulation study we compare the finite sample performance of the deconvolution kernel density estimator in the supersmooth deconvolution problem to its asymptotic behaviour predicted by two asymptotic normality theorems. Our results indicate that for lower noise levels and moderate sample sizes the match between the asymptotic theory and the finite sample performance of the estimator is not satisfactory. On the other hand we show that the two approaches produce reasonably close results for higher noise levels. These observations in turn provide additional motivation for the study of deconvolution problems under the assumption that the error term variance $\sigma^2\rightarrow 0$ as the sample size $n\rightarrow\infty.$\
[*Keywords:*]{} finite sample behavior, asymptotic normality, deconvolution kernel density estimator, Fast Fourier Transform.\
[*AMS subject classification:*]{} 62G07\
author:
- |
Bert van Es\
[Korteweg-de Vries Instituut voor Wiskunde]{}\
[Universiteit van Amsterdam]{}\
[Plantage Muidergracht 24]{}\
[1018 TV Amsterdam]{}\
[The Netherlands]{}\
[vanes@science.uva.nl]{}\
\
[Shota Gugushvili[^1]]{}\
[Eurandom]{}\
[Technische Universiteit Eindhoven]{}\
[P.O. Box 513]{}\
[5600 MB Eindhoven]{}\
[The Netherlands]{}\
[gugushvili@eurandom.tue.nl]{}
title: Some thoughts on the asymptotics of the deconvolution kernel density estimator
---
Introduction
============
Let $X_1,\ldots,X_n$ be i.i.d. observations, where $X_i=Y_i+Z_i$ and the $Y$’s and $Z$’s are independent. Assume that the $Y$’s are unobservable and that they have the density $f$ and also that the $Z$’s have a known density $k.$ The deconvolution problem consists in estimation of the density $f$ based on the sample $X_1,\ldots,X_n.$
A popular estimator of $f$ is the deconvolution kernel density estimator, which is constructed via Fourier inversion and kernel smoothing. Let $w$ be a kernel function and $h>0$ a bandwidth. The kernel deconvolution density estimator $f_{nh}$ is defined as $$\label{fnh} f_{nh}(x)=\frac{1}{2\pi}{\int_{-\infty}^{\infty}}e^{-itx}\frac{\phi_w(ht)\phi_{emp}(t)}{\phi_k(t)}dt=\frac{1}{nh}\sum_{j=1}^n w_h\left(\frac{x-X_j}{h}\right),$$ where $\phi_{emp}$ denotes the empirical characteristic function of the sample, i.e. $$\phi_{emp}(t)=\frac{1}{n}\sum_{j=1}^n e^{itX_j},$$ $\phi_w$ and $\phi_k$ are Fourier transforms of the functions $w$ and $k,$ respectively, and $$w_h(x)=\frac{1}{2\pi}{\int_{-\infty}^{\infty}}e^{-itx}\frac{\phi_w(t)}{\phi_k(t/h)}dt.$$ The estimator was proposed in @carroll1 and @carroll2 and there is a vast amount of literature dedicated to it (for additional bibliographic information see e.g. @vanes2 and @vanes1).
Depending on the rate of decay of the characteristic function $\phi_k$ at plus and minus infinity, deconvolution problems are usually divided into two groups, ordinary smooth deconvolution problems and supersmooth deconvolution problems. In the first case it is assumed that $\phi_k$ decays algebraically and in the second case the decay is essentially exponential. This rate of decay, and consequently the smoothness of the density $k,$ has a decisive influence on the performance of . The general picture that one sees is that smoother $k$ is, the harder the estimation of $f$ becomes, see e.g. @fan1.
Asymptotic normality of in the ordinary smooth case was established in @fan2, see also @fan4. The limit behaviour in this case is essentially the same as that of a kernel estimator of a higher order derivative of a density. This is obvious in certain relatively simple cases where the estimator is actually equal to the sum of derivatives of a kernel density estimator, cf. @vanes0.
Our main interest, however, lies in asymptotic normality of in the supersmooth case. In this case under certain conditions on the kernel $w$ and the unknown density $f,$ the following theorem was proved in @fan3.
\[thmanfan\] Let $f_{nh}$ be defined by . Then $$\label{anfan}
\frac{\sqrt{n}}{s_n}(f_{nh}(x)-\ex[f_{nh}(x)])\convd {\mathcal N}(0,1)$$ as $n\rightarrow\infty.$ Here either $s_n^2=(1/n)\sum_{j=1}^n Z_{nj}^2,$ or $s_n^2$ is the sample variance of $Z_{n1},\ldots,Z_{nn}$ with $Z_{nj}=(1/h)w_h((x-X_j)/h).$
The asymptotic variance of $f_{nh}$ itself does not follow from this result. On the other hand @vanes2, see also @vanes1, derived a central limit theorem for where the normalisation is deterministic and the asymptotic variance is given.
For the purposes of the present work it is sufficient to use the result of @vanes1. However, before recalling the corresponding theorem, we first formulate conditions on the kernel $w$ and the density $k.$
\[condw\] Let $\phi_w$ be real-valued, symmetric and have support $[-1,1].$ Let $\phi_w(0)=1,$ and assume $\phi_w(1-t)=At^{\alpha}+o(t^{\alpha})$ as $t\downarrow 0$ for some constants $A$ and $\alpha\geq 0.$
The simplest example of such a kernel is the sinc kernel $$\label{sinckernel}
w(x)=\frac{\sin x}{\pi x}.$$ Its characteristic function equals $\phi_{w}(t)=1_{[-1,1]}(t).$ In this case $A=1$ and $\alpha=0.$
Another kernel satisfying Condition \[condw\] is $$\label{fankernel}
w(x)=\frac{48\cos x}{\pi x^4}\left(1-\frac{15}{x^2}\right)-\frac{144\sin x}{\pi x^5}\left(2-\frac{5}{x^2}\right).$$ Its corresponding Fourier transform is given by $\phi_{w}(t)=(1-t^2)^3 1_{[-1,1]}(t).$ Here $A=8$ and $\alpha=3.$ The kernel was used for simulations in @fan3 and its good performance in deconvolution context was established in @delaigle1.
Yet another example is $$\label{wandkernel}
w(x)=\frac{3}{8\pi}\left(\frac{\sin (x/4)}{x/4}\right)^4.$$ The corresponding Fourier transform equals $$\phi_{w}(t)=2(1-|t|)^3 1_{[1/2,1]}(|t|)+(6|t|^3-6t^2+1)1_{[-1/2,1/2]}(t).$$ Here $A=2$ and $\alpha=3.$ This kernel was considered in @wand and @delaigle1.
Now we formulate the condition on the density $k.$
\[condk\] Assume that $\phi_k(t)\sim C
|t|^{\lambda_0}\exp\left[-|t|^{\lambda}/\mu\right]$ as $|t|\rightarrow \infty,$ for some $\lambda>1,\mu>0,\lambda_0$ and some constant $C.$ Furthermore, let $\phi_k(t)\neq 0$ for all $t\in{\mathbb{R}}.$
The following theorem holds true, see @vanes1.
\[thman\] Assume Conditions \[condw\] and \[condk\] and let $\ex [X^2]<\infty.$ Then, as $n\rightarrow\infty$ and $h\rightarrow 0,$ $$\frac{\sqrt{n}}{h^{\lambda(1+\alpha)+\lambda_0-1} e^{{1}/{(\mu
h^\lambda})}}\,(f_{nh}(x)-\ex [f_{nh}(x)])\convd {\mathcal
N}\left(0,\frac{A^2}{2\pi^2}\left(\frac{\mu}{\lambda}\right)^{2+2\alpha}(\Gamma(\alpha+1))^2\right).$$ Here $\Gamma$ denotes the gamma function.
The goal of the present note is to compare the theoretical behaviour of the estimator predicted by Theorem \[thman\] to its behaviour in practice, which will be done via a limited simulation study. The obtained results can be used to compare Theorem \[thmanfan\] to Theorem \[thman\], e.g.whether it is preferable to use the sample standard deviation $s_n$ in the construction of pointwise confidence intervals (computation of $s_n$ is more involved) or to use the normalisation of Theorem \[thman\] (this involves evaluation of a simpler expression). The rest of the paper is organised as follows: in Section \[simulations\] we present some simulation results, while in Section \[conclusions\] we discuss the obtained results and draw conclusions.
Simulation results {#simulations}
==================
All the simulations in this section were done in Mathematica. We considered three target densities. These densities are:
1. density \# 1: $Y\sim {\mathcal N}(0,1);$
2. density \# 2: $Y\sim \chi^2(3);$
3. density \# 3: $Y\sim 0.6{\mathcal N}(-2,1)+0.4{\mathcal N}(2,0.8^2).$
The density \# 2 was chosen because it is skewed, while the density \# 3 was selected because it has two unequal modes. We also assumed that the noise term $Z$ was ${\mathcal N}(0,0.4^2)$ distributed. Notice that the noise-to-signal ratio $\operatorname{NSR}=\var[Z]/\var[Y] 100\%$ for the density \# 1 equals $16\%,$ for the density \# 2 it is equal to $2.66\%,$ and for the density \# 3 it is given by $3\%.$ We have chosen the sample size $n=50$ and generated $500$ samples from the density $g=f\ast k.$ Notice that such $n$ was also used in simulations in e.g. @delaigle3. Even though at the first sight $n=50$ might look too small for normal deconvolution, for the low noise level that we have the deconvolution kernel density estimator will still perform well, cf. @wand. As a kernel we took the kernel . For each model that we considered, the theoretically optimal bandwidth, i.e. the bandwidth minimising $$\label{mise}
\operatorname{MISE}[f_{nh}]=\ex \left[{\int_{-\infty}^{\infty}}(f_{nh}(x)-f(x))^2dx \right],$$ the mean-squared error of the estimator $f_{nh},$ was selected by evaluating for a grid of values of $h_k=0.01k,k=1,\ldots,100,$ and selecting the $h$ that minimised $\operatorname{MISE}[f_{nh}]$ on that grid. Notice that it is easier to evaluate by rewriting it in terms of the characteristic functions, which can be done via Parseval’s identity, cf. @carroll2. For real data of course the above method does not work, because depends on the unknown $f.$ We refer to @delaigle3 for data-dependent bandwidth selection methods in kernel deconvolution.
Following the recommendation of @delaigle2, in order to avoid possible numerical issues, the Fast Fourier Transform was used to evaluate the estimate . Several outcomes for two sample sizes, $n=50$ and $n=100,$ are given in Figure \[fig1\]. We see that the fit in general is quite reasonable. This is in line with results in @wand, where it was shown by finite sample calculations that the deconvolution kernel density estimator performs well even in the supersmooth noise distribution case, if the noise level is not too high.
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In Figure \[fig2\] we provide histograms of estimates $f_{nh}(x)$ that we obtained from our simulations for $x=0$ and $x=0.92$ (the densities \# 1 and \# 2) and for $x=0$ and $x=2.04$ (the density \# 3). For the density \# 1 points $x=0$ and $x=0.92$ were selected because the first corresponds to its mode, while the second comes from the region where the value of the density is moderately high. Notice that $x=0$ is a boundary point for the support of density \# 2 and that the derivative of density \# 2 is infinite there. For the density \# 3 the point $x=0$ corresponds to the region between its two modes, while $x=2.04$ is close to where it has one of its modes. The histograms look satisfactory and indicate that the asymptotic normality is not an issue.
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Our main interest, however, is in comparison of the sample standard deviation of at a fixed point $x$ to the theoretical standard deviation computed using Theorem \[thman\]. This is of practical importance e.g. for construction of confidence intervals. The theoretical standard deviation can be evaluated as $$\operatorname{TSD}=\frac{A \Gamma(\alpha+1)
h^{\lambda+\alpha+\lambda_0-1}e^{1/(\mu
h^{\lambda})}}{\sqrt{2n\pi^2}}\left(\frac{\mu}{\lambda}\right)^{1+\alpha},$$ upon noticing that in our case, i.e. when using kernel and the error distribution ${\mathcal
N}(0,0.4^2),$ we have $A=8,\alpha=3,\lambda_0=0,\lambda=2,\mu=2/0.4^2.$ After comparing this theoretical value to the sample standard deviation of the estimator $f_{nh}$ at points $x=0$ and $x=0.92$ (the densities \# 1 and \# 2) and at points $x=0$ and $x=2.04$ (the density \# 3), see Table \[table1\], we notice a considerable discrepancy (by a factor $10$ for the density \# 1 and even larger discrepancy for densities \# 2 and \# 3). At the same time the sample means evaluated at these two points are close to the true values of the target density and broadly correspond to the expected theoretical value $f\ast w_h(x).$ Note here that the bias of $f_{nh}(x)$ is equal to the bias of an ordinary kernel density estimator based on a sample from $f,$ see e.g. @fan1.
$f$ $h$ $\hat{\mu}_1$ $\hat{\mu}_2$ $\hat{\sigma}_1$ $\hat{\sigma}_2$ $\sigma$ $\tilde{\sigma}$
------ ------ --------------- --------------- ------------------ ------------------ ---------- ------------------
\# 1 0.24 0.343 0.252 0.0423 0.039 0.429 0.072
\# 2 0.18 0.066 0.389 0.035 0.067 0.169 0.114
\# 3 0.25 0.074 0.159 0.025 0.037 0.512 0.068
: \[table1\] Sample means $\hat{\mu}_1$ and $\hat{\mu}_2$ and sample standard deviations $\hat{\sigma}_1$ and $\hat{\sigma}_2$ evaluated at $x=0$ and $x=0.92$ (densities \# 1 and \# 2) and $x=0$ and $x=2.04$ (the density \# 3) together with the theoretical standard deviation $\sigma$ and the corrected theoretical standard deviation $\tilde{\sigma}$. The bandwidth is given by $h.$
To gain insight into this striking discrepancy, recall how the asymptotic normality of $f_{nh}(x)$ was derived in @vanes1. Adapting the proof from the latter paper to our example, the first step is to rewrite $f_{nh}(x)$ as $$\label{asnrm1}
\frac{1}{\pi h}\int_{0}^1 \phi_w(s) \exp[s^{\lambda}/(\mu h^{\lambda})]ds\frac{1}{n}\sum_{j=1}^n \cos\left(\frac{x-X_j}{h}\right)+\frac{1}{n}\sum_{j}^n \tilde{R}_{n,j},$$ where the remainder terms $\tilde{R}_{n,j}$ are defined in @vanes1. Then by estimating the variance of the second summand in , one can show that it can be neglected when considering the asymptotic normality of as $n\rightarrow\infty$ and $h\rightarrow 0.$ Turning to the first term in , one uses the asymptotic equivalence, cf. Lemma 5 in @vanes1, $$\label{asnrm2}
\int_{0}^1 \phi_w(s) \exp[s^{\lambda}/(\mu h^{\lambda})]ds \sim A \Gamma (\alpha+1) \left(\frac{\mu}{\lambda}h^{\lambda}\right)^{1+\alpha} e^{1/(\mu h^{\lambda})},$$ which explains the shape of the normalising constant in Theorem \[thman\]. However, this is precisely the point which causes a large discrepancy between the theoretical standard deviation and the sample standard deviation. The approximation is good asymptotically as $h\rightarrow0,$ but it is quite inaccurate for larger values of $h.$ Indeed, consider the ratio of the left-hand side of with the right-hand side. We have plotted this ratio as a function of $h$ for $h$ ranging between $0$ and $1,$ see Figure \[ratioplot\]. One sees that the ratio is close to $1$ for extremely small values of $h$ and is quite far from $1$ for larger values of $h.$ It is equally easy to see that the poor approximation in holds true for kernels and as well, see e.g. Figure \[ratioplot\], which plots the ratio of both sides of for the kernel .
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This poor approximation, of course, is not characteristic of only the particular $\mu$ and $\lambda$ that we used in our simulations, but also holds true for other values of $\mu$ and $\lambda.$
Obviously, one can correct for the poor approximation of the sample standard deviation by the theoretical standard deviation by using the left-hand side of instead of its approximation. The theoretical standard deviation corrected in such a way is given in the last column of Table \[table1\]. As it can be seen from the table, this procedure led to an improvement of the agreement between the theoretical standard deviation and its sample counterpart for all three target densities. Nevertheless, the match is not entirely satisfactory, since the corrected theoretical standard deviation and the sample standard deviation differ by factor $2$ or even more. A perfect match is impossible to obtain, because we neglect the remainder term in and $h$ is still fairly large. We further notice that the concurrence between the results is better for $x=0$ than for $x=0.92$ for densities \# 1 and \# 2, and for $x=2.04$ than for $x=0$ for the density \# 3. We also performed simulations for the sample sizes $n=100$ and $n=200$ to check the effect of having larger samples. For brevity we will report only the results for density \# 2, see Figure \[chinlarge\] and Table \[chitable\], since this density is nontrivial to deconvolve, though not as difficult as the density \# 3. Notice that the results did not improve greatly for $n=100,$ while for the case $n=200$ the corrected theoretical standard deviation became a worse estimate of the sample standard deviation than the theoretical standard deviation. Explanation of this curious phenomenon is given in Section \[conclusions\].
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$n$ $h$ $\hat{\mu}_1$ $\hat{\mu}_2$ $\hat{\sigma}_1$ $\hat{\sigma}_2$ $\sigma$ $\tilde{\sigma}$
-------- ------ --------------- --------------- ------------------ ------------------ ---------- ------------------
\# 100 0.17 0.063 0.393 0.025 0.051 0.108 0.090
\# 200 0.15 0.052 0.402 0.023 0.049 0.070 0.084
: \[chitable\] Sample means $\hat{\mu}_1$ and $\hat{\mu}_2$ and sample standard deviations $\hat{\sigma}_1$ and $\hat{\sigma}_2$ evaluated at $x=0$ and $x=0.92$ for the density \# 2, together with the theoretical standard deviation $\sigma$ and the corrected theoretical standard deviation $\tilde{\sigma}$.
Furthermore, note that $$\var\left[\frac{1}{\sqrt{n}}\sum_{j=1}^n \cos\left(\frac{x-X_j}{h}\right)\right]\rightarrow \frac{1}{2}$$ as $n\rightarrow \infty$ and $h\rightarrow0,$ see @vanes1. This explains the appearance of the factor $1/2$ in the asymptotic variance in Theorem \[thman\]. One might also question the goodness of this approximation and propose to use instead some estimator of $\var[\cos((x-X)h^{-1})],$ e.g. its empirical counterpart based on the sample $X_1,\ldots,X_n.$ However, in the simulations that we performed for all three target densities (with $n$ and $h$ as above), the resulting estimates took values close to the true value $1/2.$ E.g. for the density \# 3 the sample mean turned out to be $0.502298,$ while the sample standard deviation was equal to $0.0535049,$ thus showing that there was insignificant variability around $1/2$ in this particular example. On the other hand, for other distributions and for different sample sizes, it could be the case that the direct use of $1/2$ will lead to inaccurate results.
Next we report some simulation results relevant to Theorem \[thmanfan\]. This theorem tells us that for a fixed $n$ we have that $$\label{fanexpr}
\frac{\sqrt{n}}{s_n}(f_{nh}(x)-\ex[f_{nh}(x)])$$ is approximately normally distributed with zero mean and variance equal to one. Upon using the fact that $\ex[f_{nh}(x)]=f\ast
w_h(x),$ we used the data that we obtained from our previous simulation examples to plot the histograms of and to evaluate the sample means and standard deviations, see Figure \[fanhisto\] and Table \[fantable\]. One notices that the concurrence of the theoretical and sample values is quite good for the density \# 1. For the density \# 2 it is rather unsatisfactory for $x=0,$ which is explainable by the fact that in general there are very few observations originating from the neighbourhood of this point. Finally, we notice that the match is reasonably good for the density \# 3, given the fact that it is difficult to estimate, at the point $x=2.04,$ but is still unsatisfactory at the point $x=0.$ The latter is explainable by the fact that there are less observations originating from the neighbourhood of this point. An increase in the sample size ($n=100$ and $n=200$) leads to an improvement of the match between the theoretical and the sample mean and standard deviation at the point $x=0$ for the density \# 2, see Figure \[chifan\] and Table \[chifantable\], however the results are still largely inaccurate for this point. In essence similar conclusions were obtained for the density \# 3. These are not reported here.
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$f$ $h$ $\hat{\mu}_1$ $\hat{\mu}_2$ $\hat{\sigma}_1$ $\hat{\sigma}_2$
------ ------ --------------- --------------- ------------------ ------------------
\# 1 0.24 -0.046 -0.093 0.953 1.127
\# 2 0.18 -3.984 -0.084 17.2 1.28
\# 3 0.25 -0.768 -0.141 4.03 1.63
: \[fantable\] Sample means $\hat{\mu}_1$ and $\hat{\mu}_2$ and sample standard deviations $\hat{\sigma}_1$ and $\hat{\sigma}_2$ evaluated at $x=0$ and $x=0.92$ (densities \# 1 and \# 2) and $x=0$ and $x=2.04$ (the density \# 3).
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$n$ $h$ $\hat{\mu}_1$ $\hat{\mu}_2$ $\hat{\sigma}_1$ $\hat{\sigma}_2$
----- ------ --------------- --------------- ------------------ ------------------
100 0.17 -1.33 -0.015 9.89 1.31
200 0.15 -1.02 -0.015 6.36 1.58
: \[chifantable\] Sample means $\hat{\mu}_1$ and $\hat{\mu}_2$ and sample standard deviations $\hat{\sigma}_1$ and $\hat{\sigma}_2$ evaluated at $x=0$ and $x=0.92$ for the density \# 2 for two sample sizes: $n=100$ and $n=200.$
Note that in all three models that we studied the noise level is not high. We also studied the case when the noise level is very high. For brevity we present the results only for the density \# 1 and for sample size $n=50.$ We considered three cases of the error distribution: in the first case $Z\sim {\mathcal N}(0,1),$ in the second case $Z\sim {\mathcal N}(0,2^2)$ and in the third case $Z\sim {\mathcal N}(0,4^2).$ Notice that the $\operatorname{NSR}$ is equal to $100\%,$ $400\%$ and $1600\%,$ respectively. The simulation results are summarised in Figures \[highnoise\] and \[fanhighnoise\] and Tables \[tablehighnoise\] and \[fantablehighnoise\]. We see that the sample standard deviation and the corrected theoretical standard deviation are in better agreement among each other compared to the low noise level case. Also the histograms of the values of look better. On the other hand the resulting curves $f_{nh}$ were not too satisfactory when compared to the true density $f$ in the two cases $Z\sim {\mathcal N}(0,1),$ and $Z\sim {\mathcal N}(0,2^2)$ (especially in the second case) and were totally unacceptable in the case $Z\sim {\mathcal N}(0,4^2).$ This of course does not imply that the estimator is bad, rather the deconvolution problem is very difficult in these cases.
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$\operatorname{NSR}$ $h$ $\hat{\mu}_1$ $\hat{\mu}_2$ $\hat{\sigma}_1$ $\hat{\sigma}_2$ $\sigma$ $\tilde{\sigma}$
---------------------- ------ --------------- --------------- ------------------ ------------------ ---------- ------------------
100% 0.36 0.294 0.236 0.046 0.045 0.057 0.075
400% 0.59 0.214 0.189 0.053 0.053 0.046 0.076
1600% 0.89 0.150 0.156 0.279 0.289 0.251 0.342
: \[tablehighnoise\] Sample means $\hat{\mu}_1$ and $\hat{\mu}_2$ and sample standard deviations $\hat{\sigma}_1$ and $\hat{\sigma}_2$ together with theoretical standard deviation $\sigma$ and corrected theoretical standard deviation $\tilde{\sigma}$ evaluated at $x=0$ and $x=0.92$ for the density \# 1 for three noise levels: $\operatorname{NSR}=100\%,$ $\operatorname{NSR}=400\%$ and $\operatorname{NSR}=1600\%$.
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$\operatorname{NSR}$ $h$ $\hat{\mu}_1$ $\hat{\mu}_2$ $\hat{\sigma}_1$ $\hat{\sigma}_2$
---------------------- ------ --------------- --------------- ------------------ ------------------
100% 0.36 -0.038 -0.098 1.091 1.228
400% 0.59 -0.079 -0.134 1.155 1.193
1600% 0.89 -0.015 0.035 1.027 1.086
: \[fantablehighnoise\] Sample means $\hat{\mu}_1$ and $\hat{\mu}_2$ and sample standard deviations $\hat{\sigma}_1$ and $\hat{\sigma}_2$ of evaluated at $x=0$ and $x=0.92$ for the density \# 1 for two noise levels: $\operatorname{NSR}=400\%$ and $\operatorname{NSR}=1600\%$.
Finally, we mention that results qualitatively similar to the ones presented in this section were obtained for the kernel as well. These are not reported here because of space restrictions.
Discussion {#conclusions}
==========
In the simulation examples considered in Section \[simulations\] for Theorem \[thman\], we notice that the corrected theoretical asymptotic standard deviation is always considerably larger than the sample standard deviation given the fact that the noise level is not high. We conjecture, that this might be true for the densities other than \# 1, \# 2 and \# 3 as well in case when the noise level is low. This possibly is one more explanation of the fact of a reasonably good performance of deconvolution kernel density estimators in the supersmooth error case for relatively small sample sizes which was noted in @wand. On the other hand the match between the sample standard deviation and the corrected theoretical standard deviation is much better for higher levels of noise. These observations suggest studying the asymptotic distribution of the deconvolution kernel density estimator under the assumption $\sigma\rightarrow 0$ as $n\rightarrow\infty,$ cf.@delaigle0, where $\sigma$ denotes the standard deviation of the noise term.
Our simulation examples suggest that the asymptotic standard deviation evaluated via Theorem \[thman\] in general will not lead to an accurate approximation of the sample standard deviation, unless the bandwidth is small enough, which implies that the corresponding sample size must be rather large. The latter is hardly ever the case in practice. On the other hand, we have seen that in certain cases this poor approximation can be improved by using the left-hand side of instead of the right-hand side. A perfect match is impossible to obtain given that we still neglect the remainder term in . However, even after the correction step, the corrected theoretical standard deviation still differs from the sample standard deviation considerably for small sample sizes and lower levels of noise. Moreover, in some cases the corrected theoretical standard deviation is even farther from the sample standard deviation than the original uncorrected version. The latter fact can be explained as follows:
1. It seems that both the theoretical and corrected theoretical standard deviation overestimate the sample standard deviation.
2. The value of the bandwidth $h,$ for which the match between the corrected theoretical standard deviation and the sample standard deviation become worse, belongs to the range where the corrected theoretical standard deviation is larger than the theoretical standard deviation. In view of item 1 above, it is not surprising that in this case the theoretical value turns out to be closer to the sample standard deviation than the corrected theoretical value.
The consequence of the above observations is that a naive attempt to directly use Theorem \[thman\], e.g. in the construction of pointwise confidence intervals, will lead to largely inaccurate results. An indication of how large the contribution of the remainder term in can be can be obtained only after a thorough simulation study for various distributions and sample sizes, a goal which is not pursued in the present note. From the three simulation examples that we considered, it appears that the contribution of the remainder term in is quite noticeable for small sample sizes. For now we would advise to use Theorem \[thman\] for small sample sizes and lower noise levels with caution. It seems that the similar cautious approach is needed in case of Theorem \[thmanfan\] as well, at least for some values of $x.$
Unlike for the ordinary smooth case, see @bissantz, there is no study dealing with the construction of uniform confidence intervals in the supersmooth case. In the latter paper a better performance of the bootstrap confidence intervals was demonstrated in the ordinary smooth case compared to the asymptotic confidence bands obtained from the expression for the asymptotic variance in the central limit theorem. The main difficulty in the supersmooth case is that the asymptotic distribution of the supremum distance between the estimator $f_{nh}$ and the true density $f$ is unknown. Our simulation results seem to indicate that the bootstrap approach is more promising for the construction of pointwise confidence intervals than e.g. the direct use of Theorems \[thmanfan\] or \[thman\]. Moreover, the simulations suggest that at least Theorem \[thman\] is not appropriate when the noise level is low.
[34]{}
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[^1]: The research of this author was financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). Part of the work was done while this author was at the Korteweg-de Vries Instituut voor Wiskunde in Amsterdam.
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[**Recent progress in simulating galaxy formation from the largest to the smallest scales**]{}
Claude-André Faucher-Giguère
[**Abstract**]{}
Galaxy formation simulations are an essential part of the modern toolkit of astrophysicists and cosmologists alike. Astrophysicists use the simulations to study the emergence of galaxy populations from the Big Bang, as well as problems including the formation of stars and supermassive black holes. For cosmologists, galaxy formation simulations are needed to understand how baryonic processes affect measurements of dark matter and dark energy. Owing to the extreme dynamic range of galaxy formation, advances are driven by novel approaches using simulations with different tradeoffs between volume and resolution. Large-volume but low-resolution simulations provide the best statistics, while higher resolution simulations of smaller cosmic volumes can be evolved with more self-consistent physics and reveal important emergent phenomena. I summarize recent progress in galaxy formation simulations, including major developments in the past five years, and highlight some key areas likely to drive further advances over the next decade.
Cosmology now has a standard model, in which of most of the mass is dark matter, the acceleration of the universe is due to dark energy, and in which tiny density perturbations in the early universe were seeded by inflation. In this “$\Lambda$ cold dark matter” ($\Lambda$CDM) model, described by just six parameters, baryons make up just five percent of the present-day energy density .
Although the physical nature of baryons is much better understood than dark matter and dark energy, how primordial fluctuations eventually evolved into the galaxies that we use to map the universe in visible light remains a challenging problem at the frontiers of modern astrophysics. There are a few reasons that make galaxy formation one of the most active areas of astrophysical research today. These can be loosely grouped into two categories: astrophysics and cosmology.
Astrophysicists want to know how galaxies formed and how they evolved because the diverse astronomical phenomena involved are interesting in their own right. For example, astrophysicists seek to understand the origins of galaxy properties, such as their masses, sizes, and colors, and why correlations between different properties (so-called “scaling relations”) are observed. Astrophysicists are also interested in how galaxies came to be because it provides the context for understanding other problems, such as how stars and black holes formed in galaxies.
For cosmologists, the details of how galaxies assembled may not be of prime interest. However, cosmologists must know enough about galaxy formation physics to understand how their measurements are affected by how baryons interact with the dark sector. Cosmologists have so far been able to get away with a relatively crude understanding of how galaxies formed, usually relying on simulations containing only dark matter [@2010ApJ...715..104H], but this is changing. Indeed, upcoming experiments aiming to measure the equation of state of dark energy to better than one percent, such as the Large Synoptic Survey Telescope (LSST), the Euclid mission, and the Wide Field Infrared Survey Telescope (WFIRST), will require modeling baryonic processes with much greater accuracy. In particular, exploiting the statistical power of the weak lensing signal will require modeling the non-linear matter power spectrum at the level of one percent or better for scales corresponding to comoving wavenumbers $0.1 \lesssim k \lesssim 10$ $h$ Mpc$^{-1}$, where previous simulations have shown that baryonic effects can range from $\approx 1$ to $>10\%$ [@2011MNRAS.415.3649V]. Since baryonic processes can substantially change the profiles of individual dark matter halos, they are also proving critical to constraining the properties of dark matter via dynamical measurements of galaxies, a point that I return to below.
[**Challenges and recent successes of large-volume simulations**]{}
Given the standard cosmology, the recipe for simulating galaxy formation is in principle straightforward: start with the right mix of dark matter, dark energy, and baryons, then integrate all the relevant evolution equations. The problem is of course that this brute force approach is well out of reach of computational capabilities, and this will remain the case for decades to come. Alternatives include semi-analytic techniques, in which baryonic processes are approximated with analytic prescriptions “painted on” dark matter-only simulations [@1991ApJ...379...52W; @2001MNRAS.320..504S], and semi-empirical methods in which observed galaxy populations are mapped to simulated dark matter distributions [@2013ApJ...770...57B; @2013MNRAS.428.3121M; @2013MNRAS.435.1313H].
In what follows, I focus on recent progress using cosmological hydrodynamic simulations. Such simulations follow the coupled dynamics of dark matter and baryons starting from $\Lambda$CDM initial conditions. Simulations with volume sufficient to capture representative portions of the universe cannot resolve the interstellar medium (ISM) of galaxies in significant detail and are far from resolving the formation of stars. Such simulations typically have mass resolution $\sim 10^{6}$ M$_{\odot}$ and force resolution $\sim$1 kpc. These simulations therefore rely critically on “subgrid” models to capture processes internal to galaxies. To a large extent, advances in galaxy formation modeling are currently driven by the design and application of better subgrid models for the variety of crucial processes that cannot be explicitly resolved in cosmological simulations, such as star formation and stellar feedback, and supermassive black hole growth and active galactic nucleus (AGN) feedback.
The subgrid processes and their effects on resolved scales could in principle be so complex that they could not be captured by a manageable set of subgrid models. The first generations of cosmological hydrodynamic simulations in fact failed in many respects to produce realistic galaxy populations. They produced galaxies that were too massive and too compact [@1994MNRAS.267..401N]. From earlier analytic and semi-analytic work [@1974MNRAS.169..229L; @1986ApJ...303...39D; @1991ApJ...379...52W], it was known that stellar feedback was important to produce realistic galaxy populations. Supernovae (SNe), in particular, can drive galaxy-scale outflows that eject gas from galaxies before it has time to turn into stars.
The first attempts to include stellar feedback in cosmological simulations revealed that it is highly non-trivial. For example, when SNe are modeled by adding thermal energy to surrounding gas (“thermal feedback”), the feedback is inefficient because the energy is rapidly radiated away. This is one form of the “overcooling problem,” and is due to the fact that at the relatively coarse resolution of cosmological simulations, the energy from individual SNe is generally not sufficient to heat the gas enough to avoid rapid cooling. This is because the low resolution makes it difficult to resolve a multiphase ISM, where cooling times would be longer in the unresolved hot and tenuous gas phase. Several different methods have been developed to circumvent the overcooling problem. One method (“delayed cooling”) is to temporarily turn off radiative cooling to increase the efficiency of energy conversion into kinetic motion [@2001ApJ...555L..17T; @2006MNRAS.373.1074S]. A variant (“stochastic heating”) keeps cooling on at all times, but temporarily stores feedback energy until a certain minimum heating temperature is reached [@2012MNRAS.426..140D]. Another method (“hydrodynamically decoupled winds”) is to directly prescribe the desired velocity and mass loading of galactic winds [@2003MNRAS.339..312S; @2006MNRAS.373.1265O].
Using such methods, several groups showed that stellar feedback models could be adjusted in ways that produce galaxy properties and demographics in much better agreement with observations, at least for galaxies of mass comparable to the Milky Way ($\sim L^{\star}$) or less [@2010MNRAS.406.2325O; @2011ApJ...742...76G], corresponding to dark matter halos of mass $M_{\rm h} \lesssim 10^{12}$ M$_{\odot}$. This confirmed that star formation-driven galactic winds could plausibly reconcile $\Lambda$CDM with observed galaxy populations. Simulations with galactic winds also enabled important advances in our understanding of how heavy elements synthesized in stars and stellar explosions were dispersed in the intergalactic medium.
These tentative successes stimulated much subsequent modeling of feedback in galaxy simulations, but it was recognized that the results were sensitive to model assumptions and thus that clear gaps remained in our understanding of how galaxies evolved. One influential simulation project, called OWLS (“OverWhelmingly Large Simulations”), demonstrated the dependence of simulation results on subgrid prescriptions particularly clearly by exploring more than fifty variations [@2010MNRAS.402.1536S].
More recently, the trend has been to tune parameters of the subgrid prescriptions so that the simulations match certain basic observational constraints. The most basic constraint that all the simulations aim to reproduce is the galaxy stellar mass function, but additional properties such as galaxy sizes can break degeneracies between different models [@2015MNRAS.450.1937C]. Two recent large projects, Illustris [@2014MNRAS.444.1518V] and EAGLE [@2015MNRAS.446..521S], have followed this approach and produced simulated galaxy populations in boxes $\sim100$ Mpc on a side. In many respects, these recent simulations approximate observations well. In both simulations, stellar feedback is key to regulating star formation in galaxies below $L^{\star}$, but feedback from supermassive black holes must also be included to explain the properties of the most massive galaxies.
The fact that different semi-analytic models [@2003ApJ...599...38B; @2006MNRAS.365...11C] and cosmological simulations can explain galaxy stellar masses with the same basic ingredients is encouraging and suggests that feedback from stars and black holes are common elements of successful models, a point highlighted in a recent review of galaxy formation models . On the other hand, the fact that different variants of how the feedback is modeled produce similar galaxy mass distributions tells us that we have not yet converged on a unique theory of galaxy formation.
Fortunately, there are ways of distinguishing between different models. Once tuned to match basic observed properties, the simulations can be tested by comparing them with observations which were not used in the tuning. I highlight two sets of observations of particular significance for galaxy evolution.
The first is the color distribution of galaxies. Galaxies are observed to have a bimodal color distribution, the “blue cloud” and the “red sequence.” The Illustris and EAGLE simulations were not tuned to match the color distribution of galaxies, so comparing with the observed color distribution is an important test. The original Illustris simulation predicted increasingly red colors with increasing galaxy mass, in qualitative agreement with observations, but rather large quantitative differences relative to observations from the Sloan Digital Sky Survey [@2014MNRAS.444.1518V]. A more recent version of the Illustris simulation, IllustrisTNG, was designed in part to produce a better match to the observed galaxy color distribution [@2018MNRAS.475..624N]. The EAGLE galaxies match the observed color distribution as a function of stellar mass about as well as IllustrisTNG [@2017MNRAS.470..771T], but both simulations appear to produce slightly too much residual star formation in some of the most massive galaxies and thus underpredict the observed tail of red galaxies.
The second is predictions for the gaseous halos of galaxies, known as the circum-galactic medium (CGM). Observations of the CGM (typically using quasar absorption lines, but also increasingly in emission) are powerful discriminants of galaxy formation theories because they directly probe the inflows and outflows that regulate galaxy growth. Comparing simulations with CGM observables has been an active area of research in the last few years, stimulated by the availability of rich data sets at both low and high redshifts. So far, the results of comparisons with observations have been mixed: they reveal both agreements and disagreements from which we are learning the limitations of the current models [@2015MNRAS.449..987F; @2016MNRAS.462.2440T; @2017MNRAS.465.2966S]. Because the CGM provides a large number of different observational constraints (including absorption strengths and kinematics in different ions), it will continue to be a very fruitful approach to test galaxy formation models.
Even though they generally do not agree perfectly with observations and have their limitations, cosmological simulations provide extremely rich data sets that can be mined to provide insights into a wide array of science questions, ranging from the origins of galaxy morphologies to the chemical evolution of galaxies to the effects of galaxy evolution on the cosmic distribution of dark matter [@2015MNRAS.454.1886S; @2016MNRAS.456.1235S; @2018MNRAS.475..676S].
[**Bridging cosmological and sub-galactic scales**]{}
In parallel to the developments summarized above, another line of research in galaxy formation modeling has gained momentum in the past few years. Until recently, detailed studies of star formation were largely decoupled from cosmological models because of the large separation of physical scales. Using a “zoom-in” approach, in which the large-scale cosmological environment is included at low resolution but in which the resolution is highly refined around galaxies of special interest, it is becoming possible to resolve scales comparable to individual star-forming regions. As a result, it is possible develop finer-scale subgrid models for galaxy formation simulations that are more directly tied to our understanding how stellar feedback acts on small scales. By tying subgrid models to constraints on the small-scale physics, we can break degeneracies between theories that agree on larger scales.
A compromise of highly refined simulations is that they cannot match the galaxy statistics provided by larger volume but lower resolution simulations. Nevertheless, several factors have motivated researchers to pursue cosmological zoom-in simulations and other types of highly resolved models. First, some problems simply require higher resolution. These include resolving low-mass dwarf galaxies and the detailed internal structure of more massive galaxies. As dark matter-dominated systems, dwarf galaxies are important laboratories for constraining the properties of dark matter using astronomical observations. New observational facilities such as the Atacama Large Millimeter Array (ALMA), the James Webb Space Telescope (JWST) to be launched next year, and increasingly sophisticated integral field spectrographs on ground-based telescopes are mapping interstellar gas and stellar populations at high resolution in both large and small galaxies. Making full use of these observational capabilities requires simulations that resolve as much of the dynamical, thermodynamic, and chemical processes operating in galaxies as possible.
Second, developing more explicit subgrid models for zoom-in simulations has stimulated fruitful cross fertilization between the fields of galaxy formation and star formation. Galaxy formation modelers are starting to draw more directly on the vast body of work on the physics of star-forming regions in constructing subgrid models [@2012ApJ...759L..27P; @2012ApJ...745...69K; @2016ApJ...829..130R]. At the same time, researchers working on star formation physics can use galaxy formation simulations to include more realistic boundary conditions in their models [@2016MNRAS.460.2297R]. The strengthening of ties between these two subfields of astrophysics has already enabled rapid progress, including some important advances that will be summarized below.
Third, researchers hope that the results of zoom-in simulations can be coarse grained to develop better subgrid models for large volume containing thousands of galaxies [@2016MNRAS.462.3265D]. Such large-volume simulations will remain necessary to compute several important quantities of interest to both astrophysicists and cosmologists, including galaxy clustering, gravitational lensing by cosmic structures, microwave background anisotropies, and the Ly$\alpha$ forest. Large-volume simulations are also the best tool to capture the full range of galaxy evolution pathways. By studying the emergent outcomes of better resolved galaxy models anchored to higher resolution feedback models, simulators aim to reduce the number of parameters that must be tuned to reproduce observed galaxy populations.
A priori, it is not clear that zoom-in simulations have sufficient resolution to meaningfully increase the predictive power of galaxy formation models. For example, state-of-the-art zoom-in simulations of Milky Way-mass galaxies have baryonic resolution elements of mass $\sim 10^{4}-10^{5}$ M$_{\odot}$ and spatial resolution $\sim10-100$ pc [@2013MNRAS.428..129S; @2015ApJ...804...18A; @2016ApJ...827L..23W], with on-going efforts aiming to improve these resolution parameters by one order of magnitude. By contrast, resolving the formation of individual stars would require a mass resolution better than 1 M$_{\odot}$. Moreover, the turbulent ISM has structure on scales orders of magnitude smaller than will be resolvable for the foreseeable future.
The significance of the latest generation of cosmological zoom-in simulations is that they are starting to resolve a few key characteristic scales critical for capturing how stellar feedback operates in galaxies. In particular, zoom-in simulations of dwarf galaxies are now routinely evolved with resolution elements of mass $\lesssim 500$ M$_{\odot}$ [@2017MNRAS.471.3547F] and are thus are often able to resolve the cooling radius of individual SN remnants (SNRs) in the ISM, corresponding to a swept up mass $M_{\rm cool} \approx 1,000$ M$_{\odot}$ (weakly dependent on ambient medium density and metallicity) [@2015MNRAS.450..504M]. This is also sufficient to resolve the ISM into different star-forming regions. Together, these factors allow the simulations to much more accurately predict how SNe deposit energy and momentum in the ISM. Simulations of SN feedback have demonstrated that the clustering of SNe, inherited from the clustering of star-forming regions, is important to correctly model how different SNRs overlap and merge into large bubbles of hot gas [@2016MNRAS.456.3432G; @2017MNRAS.470L..39F]. These hot bubbles can vent out of galaxies and appear important to generate galaxy-scale outflows carrying enough mass and energy to explain observed galactic winds.
Even in today’s state-of-the-art zoom-in simulations, individual SNRs are typically not well resolved in higher-mass galaxies, but simulators have begun to adopt new solutions to the overcooling problem anchored to well-resolved SNR models. One solution, independently proposed by several groups [@2014MNRAS.445..581H; @2015MNRAS.450..504M; @2015ApJ...802...99K; @2015MNRAS.451.2900K], is to inject at the resolution scale of cosmological zoom-ins both the thermal energy *and* radial momentum that each SNR would have had on that scale if its evolution had been resolved in the simulation. Injecting momentum in addition to thermal energy is important because the momentum of an SNR is boosted by an order of magnitude during the Sedov-Taylor phase. In practice, this is done by calibrating the momentum and residual thermal energy to the results of higher resolution SNR simulations. Another approach is to bypass modeling individual SNRs and instead use a subgrid model calibrated to match the properties of superbubbles produced by clustered SNe [@2014MNRAS.442.3013K].
Many recent galaxy formation simulations have also begun to incorporate approximations for stellar feedback processes other than SNe, including radiation, stellar winds, and cosmic rays [@2013ApJ...770...25A; @2013MNRAS.434.3142A; @2014MNRAS.445..581H; @2014ApJ...797L..18S; @2017ApJ...836..204N]. Although usually energetically subdominant relative to SNe, these other processes can be important because they couple differently to the ambient medium, they have different time dependencies, and they can interact non-linearly with one another. Simulations of massive galaxies are furthermore beginning to incorporate models of supermassive black hole growth and feedback that are increasingly anchored to high-resolution models of the small-scale physics [@2017MNRAS.472L.109A; @2017MNRAS.470.4530W].
![ The star formation histories are arranged in decreasing order of halo mass at redshift $z=0$, which is labeled at the top left of each panel. These simulations predict that all galaxies have bursty star formation histories at high redshift. The more massive galaxies settle into a time-steady mode of star formation at lower redshift but the dwarf galaxies continue to be bursty all the way to $z=0$. The approximate transition redshift $z\sim 1$ between bursty and time-steady star formation in massive galaxies is indicated by the grey bands. This transition in star formation variability corresponds to a gas morphology evolution from chaotic to a well-ordered disc configuration (see Fig. \[fig:gas\_morphology\]). The bursty star formation predicted by high-resolution simulations like these has important implications ranging from dark matter halos in dwarf galaxies to the growth of supermassive black holes. Adapted from [@2018MNRAS.473.3717F]. []{data-label="fig:FIRE_SFHs_norm_NA"}](FIRE_SFHs_norm_NA){width="60.00000%"}
[**Recent successes and predictions of zoom-in simulations**]{}
Despite the approximations used, the latest generation of cosmological zoom-in simulations has produced promising results. One large zoom-in simulation campaign, in which I have played a role, is the FIRE project (for “Feedback In Realistic Environments”). In the FIRE simulations, individual SNe are resolved in time and modeled by injecting both energy and momentum, as described above. The FIRE simulations also include approximations for photoionization, radiation pressure, and stellar winds, following the energetics and time dependencies from a standard stellar population synthesis model [@1999ApJS..123....3L]. In these simulations, star formation is self-regulated by stellar feedback [@2014MNRAS.445..581H] and galaxy-scale outflows emerge from the collective action of feedback processes acting on small scales [@2015MNRAS.454.2691M].
Encouragingly, the FIRE simulations (and the more recent FIRE-2 variants using a new hydrodynamics solver) do a reasonable job of reproducing the observationally-inferred relationship between stellar mass and dark matter halo mass over more than seven orders of magnitude in stellar mass, up to $\sim L^\star$. In these simulations, a Kennicutt-Schmidt relation between the star formation rate (SFR) surface density and gas surface density ($\Sigma_{\rm SFR}$ vs. $\Sigma_{\rm g}$) roughly consistent with observations [@1998ApJ...498..541K] also emerges from regulation by stellar feedback. These results from the FIRE simulations are significant because the subgrid models for stellar feedback were anchored to the physics of SNR evolution and the energetics for the feedback mechanisms were not adjusted to match observed galaxy masses. Moreover, the simulations did not switch off hydrodynamic interactions or gas cooling to increase the efficiency of feedback processes.
Because there is large variance in how galaxies evolve, even at fixed final mass, the modest samples of galaxies simulated using the zoom-in technique (typically ranging from a single main galaxy to at most a few dozen halos) do not allow the kind of rigorous statistical comparisons with observed galaxy populations possible with large-volume simulations. Moreover, zoom-in simulations are generally evolved with resolution that increases with decreasing galaxy mass, since finer resolution elements can be afforded for lower-mass systems. As a result, numerical convergence has not yet been demonstrated uniformly across the full range of galaxy masses simulated with zoom-ins. There could therefore remain significant discrepancies with observations that would become clearer with larger and/or more uniform simulation samples. Rather than matching observations “within the error,” arguably the most important contribution of high-resolution simulations like those of the FIRE project is in making predictions for emergent behaviors unanticipated from large volume studies. In this respect, the FIRE simulations have produced some important predictions that were indeed unexpected, including by this author. Such predictions can be used to test the high-resolution simulations, and have stimulated new lines of research.
![. Magenta shows cold molecular or atomic gas ($T \lesssim 1,000$ K), green shows warm ionized gas ($10^{4} \lesssim T \lesssim 10^{5}$ K), and red shows hot gas ($T\gtrsim10^{6}$ K). The gas is very clumpy and dynamic at high redshift (a; redshift $z=3.4$) and only later settles into a well-ordered rotating disc similar to spiral galaxies observed in the nearby universe (b; $z=0$). The transition in gas morphology occurs in tandem with the transition from bursty to time-steady star formation (Fig. \[fig:FIRE\_SFHs\_norm\_NA\]). []{data-label="fig:gas_morphology"}](gas_morphology_compressed){width="99.00000%"}
One key prediction of the FIRE simulations concerns the character of the star formation histories (SFHs) of galaxies. In most large-volume simulations to date, SFHs are relatively smooth in time and roughly determined by a competition between cosmological inflows and galactic winds [@2012MNRAS.421...98D]. In contrast, the FIRE simulations predict SFHs that are much more time variable at high redshift, as well as in dwarf galaxies all the way to the present time (see Fig. \[fig:FIRE\_SFHs\_norm\_NA\]). Such bursty star formation is not unique to the FIRE simulations but appears generic in simulations that restrict star formation to high-density regions of the ISM [@2012MNRAS.422.1231G; @2013MNRAS.429.3068T; @2015MNRAS.451..839D]. When the ISM is resolved into high-density clumps, star formation bursts can occur in rapid gravitational collapse events. The variability can be further enhanced by the explosive response of stellar feedback to local bursts of star formation. Figure \[fig:gas\_morphology\] shows how the emergence of well-ordered galactic discs correlates with the transition from bursty to time-steady star formation in massive galaxies. The predictions for SFR variability can be tested by measuring the SFRs of galaxies using light in bands that probe different timescales (e.g., recombination lines powered by young, massive stars vs. continuum emission including light from older stellar populations).
Several cosmological zoom-in simulations using different subgrid approximations have shown that bursty stellar feedback can transfer enough energy to the dark matter in the inner kiloparsec of dwarf galaxies to turn the cusps predicted by pure cold dark matter models into cored profiles [@2010Natur.463..203G; @2013MNRAS.429.3068T; @2014MNRAS.437..415D; @2015MNRAS.454.2981C]. This effect is maximized in halos of mass $M_{\rm h} \sim 10^{10}-10^{11}$ M$_{\odot}$ as a result of a competition between the energy available from SN feedback and the depth of the gravitational potential. In contrast, galaxy formation simulations with smoother SFHs and standard cold dark matter do not produce such cored dark matter distributions [@2014MNRAS.444.3684V; @2016MNRAS.457.1931S]. If observationally-inferred cores [@2004MNRAS.351..903G; @2008AJ....136.2648D] are confirmed, e.g. by ruling out possible systematic effects in the modeling [@2017arXiv170607478O], knowing whether star formation is sufficiently bursty to explain the cores using baryonic effects will be critical to determine whether modifications to the standard cold dark matter paradigm are necessary.
In more massive galaxies, bursty star formation has important implications for the growth of supermassive black holes and the emergence of galaxy-black hole scaling relations, such as the relation between black hole and stellar bulge masses ($M_{\rm BH}-M_{\rm bulge}$) [@1998AJ....115.2285M]. For example, the FIRE simulations show that repeated gas ejection events by driven by bursty stellar feedback at early times can continuously deplete galactic nuclei of gas and delay the growth of central black holes relative to scaling relations observed in the local universe, and similar results have been found in other simulations as well [@2017MNRAS.468.3935H; @2017MNRAS.472L.109A].
[**Conclusion and outlook**]{}
Galaxy formation is far from solved, but the last five years have seen major advances in modeling using cosmological hydrodynamic simulations. These advances are enabling new insights into the variety of baryonic processes involved and their emergent outcomes. Large-volume hydrodynamic simulations are for the first time matching observed galaxy demographics at a level comparable to finely-tuned semi-analytic models, while higher resolution simulations are starting to resolve the ISM of individual galaxies and are making new testable predictions. Future progress will continue to be driven by both large volume and high resolution simulations. In fact, the synergy between the two approaches is likely to grow stronger as the high-resolution simulations are used to refine the subgrid models used in large volumes, and as these large volumes are exploited to investigate the implications for cosmology and large galaxy samples. Before closing, I highlight some key areas where progress will be particularly fruitful going forward.
First, approaches to coarse grain the physics captured in high-resolution models into subgrid prescriptions remain somewhat [*ad hoc*]{} and it would be highly beneficial to develop more systematic methods. Second, the physical processes included in current simulations are incomplete and often rely on crude approximations. In this area, rapid progress is already underway using simulations that include combinations of magnetic fields, cosmic ray transport, radiation-hydrodynamics, and detailed chemistry networks, but the complexity of the problem guarantees that this line of investigation will remain open for the foreseeable future. Third, most simulation codes do not take full advantage of the supercomputing facilities available today. This is especially the case for highly zoomed in simulations, which often only scale well to a few hundred or thousand compute cores (out of hundreds of thousands cores on national supercomputers accessible to scientists). Moreover, the largest supercomputers increasingly rely on acceleration by graphics processing units (GPUs) or other many-core technology, but most current simulation codes are not yet designed to benefit substantially from these hardware accelerators. To some degree, progress in galaxy formation simulations has therefore been limited by the capabilities of the simulation codes and it will be important to improve both their scaling and hardware acceleration to make use of upcoming exascale facilities, which will become available in the next few years. This will be needed not only to simulate the astrophysics at significantly higher resolution and with a richer set of physical processes included self-consistently, but also to evolve hydrodynamic simulations with volume of multiple cubic gigaparsecs and trillions of resolution elements. Such simulations will be necessary to exploit the full information content of wide-field sky surveys of the next decade.
Correspondence and requests for materials should be addressed to the author. The author declares no competing financial interests.
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abstract: 'Splines are a popular and attractive way of smoothing noisy data. Computing splines involves minimizing a functional which is a linear combination of a fitting term and a regularization term. The former is classically computed using a (weighted) L2 norm while the latter ensures smoothness. Thus, when dealing with grid data, the optimization can be solved very efficiently using the DCT. In this work we propose to replace the L2 norm in the fitting term with an L1 norm, leading to automatic robustness to outliers. To solve the resulting minimization problem we propose an extremely simple and efficient numerical scheme based on split-Bregman iteration combined with DCT. Experimental validation shows the high-quality results obtained in short processing times.'
address: ' Department of Electrical and Computer Engineering, Duke University.[^1]\'
author:
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bibliography:
- 'mtepper.bib'
title: 'L1 Splines for Robust, Simple, and Fast Smoothing of Grid Data'
---
Introduction
============
Smoothing a dataset consists in finding an approximating function that captures important patterns in the data, while disregarding noise or other fine-scale structures. Let $y \in {{\mathbb R}}^{n_1 \times \dots \times n_m} \rightarrow {{\mathbb C}}$ be an $m$-dimensional discrete signal, where $n_j$ ($1 \leq j \leq d$) is the domain of $y$ along the $j$-th dimension. We can model $y$ by $$y = \hat{y} + r ,
\label{eq:model}$$ where $r$ represents some noise and $\hat{y}$ is a smooth function. A very common regularization choice is to enforce $C^2$ continuity, in which case $\hat{y}$ is called a (cubic) spline. Smoothing $y$ relies upon finding the best estimate of $\hat{y}$ under the proper smoothness and noise assumptions. We can approximate $\hat{y}$ by minimizing an objective functional $${\mathrm{F}}(z) = {\mathrm{R}}_y (z) + s {\mathrm{P}}(z) ,$$ where ${\mathrm{R}}_y (z)$ is a data fitting term, defined by the distribution of $r$, ${\mathrm{P}}(z)$ is a regularization term, and $s$ is a scalar that determines the balance between both terms. Such scalar parameter can be automatically derived using Bayesian or MDL techniques, as will be later shown.
For clarity, we will describe in depth the case $m=1$, where we have $n=n_1$ samples. We extend the results later to the general $m$-dimensional case.
Let us begin by explaining the smoothing term. The $C^2$ continuity requirement leads to define ${\mathrm{P}}(z) = {\left\| Dz \right\|_{2}}^ 2$, $D$ being a discrete *second-order* differential operator, defined $\forall i, 2 \leq i \leq n-1$, by $$\begin{aligned}
D_{i, i-1} &= \tfrac{2}{h_{i-1} (h_{i-1} + h_{i})} ,\\
D_{i, i} &= \tfrac{-2}{h_{i-1} \ h_{i}} ,\\
D_{i-1, i} &= \tfrac{2}{h_{i} (h_{i-1} + h_{i})} ,\end{aligned}$$ where $h_i$ represents the step, or sampling rate, between $y_i$ and $y_{i+1}$. Assuming repeating border elements, that is, $y_0 = y_1$ and $y_{n+1} =y_n$, gives $D_{1, 1} = -D_{1, 2} = -1/h_1^2$, and $D_{n, n-1} = -D_{n, n} = -1/h_{n-1}^2$.
Regarding the fitting term, the classical assumption is that the noise $r$ in Equation () has Gaussian distribution with zero mean and unknown variance, which leads to setting ${\mathrm{R}}_y (z) = {\left\| z - y \right\|_{2}}^2$. Smoothing then can be formulated as the least-squares regression $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| z - y \right\|_{2}}^2 + s {\left\| Dz \right\|_{2}}^ 2 .$$ For clarity, we call the estimate $\hat{y}$ obtained with this method an L2 spline.
It is a well known fact that least squares estimates for regression models are highly non-robust to outliers. Although there is no agreement on a universal and formal definition of an outlier, it is usually regarded as an observation that does not follow the patterns in the data. Notice that smoothing should produce an estimate $\hat{y}$ taking into account only important patterns, that is, the inliers, in the data. In this sense, the L2 formulation cannot correctly handle outliers by itself.
In order to solve this problem, we propose to take a different assumption on the distribution of the noise $r$ in Equation (). By choosing a distribution with fatter tails than the Gaussian distribution, the derived estimator will correctly handle outliers. We thus assume that $r$ follows a Laplace distribution with zero mean and unknown scale parameter, a common practice in other problems as we will further discuss below. This leads to the fitting term ${\mathrm{R}}_ y(z) = {\left\| z - y \right\|_{1}}$ and the regression then becomes $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } \ {\left\| z - y \right\|_{1}} + s {\left\| Dz \right\|_{2}}^2 .$$ We call the estimate $\hat{y}$ obtained with this formulation an L1 spline.
Let us point out that the use of L1 fitting terms for solving inverse problems is not new. For example, in 2001, Nikolova proved the theoretic pertinence of using L1 fitting terms for image denoising [@nikolova02]. Other interesting works have addressed this approach for total variation image denoising [@alliney97; @chan2004; @aujol06; @nikolova12] or total variation optical flow [@zach07; @wedel09; @raket11] (this robustness-to-outliers type of ideas was previously introduced in the context of optical flow by Black and Anandan [@black91]). Recall that the total-variation regularization term involves first-order derivatives, while the proposed L1 splines, on the other hand, involve second-order derivatives.
Regularly sampled signals are extremely common in practice, and their analysis becomes easier and faster. In particular, we follow the most common choice when dealing with discrete $m$-dimensional data, which is assuming a “rectangular” Cartesian sampling pattern. When the sampling is isotropic, i.e., “square,” we refer to this type of data as grid data.
We developed an iterative algorithm for computing L1 splines, based on split-Bregman iteration [@goldstein09], that is specially suited for the case of grid data. This algorithm is extremely fast, both in running time and in the number of iterations until convergence. It is also outstandingly simple, making the implementation completely straightforward.
The remainder of the paper is structured as follows. In Section \[sec:L2splines\], we overview a fast algorithm to compute L2 splines and robust L2 splines, a modification of the least squares regression that allows to handle outliers. In Section \[sec:L1splines\], we present the proposed algorithm for computing L1 splines. Then, in Section \[sec:results\], we show results obtained with L1 splines, systematically outperforming its L2 and robust L2 counterparts in the presence of outliers. We also show that the proposed computational algorithm is very efficient. Finally, in Section \[sec:conclusions\] we provide some concluding remarks.
Smoothing splines {#sec:L2splines}
=================
As aforementioned, the classical assumption is that the noise $r$ in Equation () follows a Gaussian distribution with zero mean and unknown variance. This leads to solve the least-squares regression $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| z - y \right\|_{2}}^2 + s {\left\| Dz \right\|_{2}}^ 2 .
\label{eq:LSformulation}$$ Since both terms are differentiable, we obtain $$\hat{y} = (I + s {D^\mathrm{T}} D)^{-1} y .
\label{eq:LSsolution}$$
Garcia proposed a very efficient method for dealing with regularly sampled data [@garcia10]. Assuming that the data are equally spaced, that is, without loss of generality $\forall i, h_i = 1$, we obtain $$D =
\begin{pmatrix}
-1 & 1 \\
1 & -2 & 1 \\
& \ddots & \ddots & \ddots \\
&& 1 & -2 & 1 \\
&&& 1 & -1
\end{pmatrix} .$$ An eigendecomposition of $D$ yields $D = U \varLambda {U^\mathrm{T}}$, where $\varLambda$ is a diagonal matrix containing the eigenvalues of $D$, given by [@yueh05] $$\varLambda_{i,j} =
\begin{cases}
-2 + 2 \cos((i-1) \pi / n) , & \text{if $i=j$;} \\
0 , & \text{otherwise.}
\end{cases}$$ Since $U$ is a unitary matrix, we can write Equation () as $$\hat{y} = U (I + s \varLambda^2)^{-1} {U^\mathrm{T}} y .
\label{eq:LSsolutionDecomposed}$$ Let us define the matrix $\varGamma = (I + s \varLambda^2)^{-1}$. Trivially, $$\varGamma_{i, j} =
\begin{cases}
\left[ 1 + s (-2 + 2 \cos((i-1) \pi / n))^2 \right]^{-1} , & \text{if $i=j$;} \\
0 , & \text{otherwise.}
\end{cases}
\label{eq:varGamma}$$ Following Strang [@strang99] and Garcia [@garcia10], let us observe that ${U^\mathrm{T}}$ is a DCT-II matrix and $U$ is an inverse DCT-II matrix. Then, Equation () can be expressed as $$\hat{y} = {\mathrm{DCT^{-1}}} (\varGamma \ {\mathrm{DCT}} (y)) ,
\label{eq:LSsolutionDCT}$$ where ${\mathrm{DCT}}(\cdot)$ and ${\mathrm{DCT^{-1}}}(\cdot)$ stand for the DCT-II and inverse DCT-II functions. Equation () provides a fast and simple algorithm for computing L2 splines.
Robust estimation. {#sec:robustL2}
------------------
Often in practice there are in $y$ some values $y_i$ that could not be observed (or recorded) for some reason. We would like to be able to handle such cases in such a way that the missing values are inferred from the ones that can be observed. Let $W$ be an $n \times n$ diagonal matrix such that $W_{i, i}$ represents a weight assigned to observation $i$. $W$ is defined by $$W_{i, i} =
\begin{cases}
0 &\text{if datapoint $i$ is missing;} \\
\rho &\text{otherwise.}
\end{cases}$$ where $\rho$ is some arbitrary constant in $(0, 1]$; in practice, and without loss of generality, we set $\rho=1$. We can then solve $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| W^{1/2} (z - y) \right\|_{2}}^2 + s {\left\| Dz \right\|_{2}}^ 2 ,
\label{eq:WLSformulation}$$ which will simply omit the missing points from the computation of the residual while the regularizer will still have a smoothing effect over both present and missing points. Equation () acts as an impainting algorithm, filling the missing values in such a way that continuity between filled values and smoothed ones is preserved. The minimization of Equation () gives $$(I + s {D^\mathrm{T}}D) \hat{y} = W (y - \hat{y}) + \hat{y} .$$ This leads to the iterative procedure $$\hat{y}^{k+1} = (I + s {D^\mathrm{T}}D)^{-1} \left( W \left( y - \hat{y}^k \right) + \hat{y}^k \right) ,$$ which, similarly to Equation (), becomes $$\hat{y}^{k+1} = {\mathrm{DCT}}^{-1} \left( \varGamma \ {\mathrm{DCT}} \left( W \left(y - \hat{y}^k \right) + \hat{y}^k \right) \right) .
\label{eq:WLSsolutionDCT}$$
On a different note, real data often present observations that lie abnormally far from their “true” value, i.e., that do not appear to follow the pattern of the other data points. The main drawback of the penalized least squares formulation Equation () is its sensitivity to these outliers. To address this issue, weights can be assigned to every point, as in Equation (), such that outliers exert less influence during the estimation process. In this case, the weights are iteratively refined during the estimation process using robust estimators for the mean and variance of the data. Defining these estimators is a complex problem by itself. For details about how $W$ can be set and updated for added robustness to outliers, refer to Garcia’s work [@garcia10].
L1 splines {#sec:L1splines}
==========
In this section we introduce a different splines formulation in order to handle outliers in the data. We assume that the noise $r$ in Equation () follows a Laplace distribution with zero mean and unknown scale parameter, which leads to ${\mathrm{R}}_ y(z) = {\left\| z - y \right\|_{1}}$. The regression then becomes $$\min_{z} \ {\left\| z - y \right\|_{1}} + s {\left\| Dz \right\|_{2}}^2 .
\label{eq:L1formulation}$$
Goldstein and Osher [@goldstein09] proposed a very elegant and efficient algorithm for solving the L1 constrained problem (related to a number of very efficient optimization algorithms, e.g., see [@combettes11]) $$\min_u {\left\| \Phi(u) \right\|_{1}} + {\mathrm{H}}(u) .$$ For this, they consider the equivalent problem $$\min_u {\left\| d \right\|_{1}} + {\mathrm{H}}(u) \quad \text{s.t.} \quad d = \Phi(u) ,$$ which they first convert it into the unconstrained problem $$\min_u {\left\| d \right\|_{1}} + {\mathrm{H}}(u) + \tfrac{\lambda}{2} {\left\| d - \Phi(u) \right\|_{2}}^2 .$$ In this form, the penalty function does not accurately enforce the constraint for small $\lambda$. The constraint is enforced by letting $\lambda \rightarrow \infty$. However, another solution for this new formulation is found by using the following two-phase algorithm $$\begin{aligned}
\left( u^{k+1}, d^{k+1} \right) &= {\underset{u}{\operatorname{argmin}}\ } {\left\| d \right\|_{1}} + {\mathrm{H}}(u) +
\tfrac{\lambda}{2} {\left\| d - \Phi(u) - b^k \right\|_{2}}^2 ,\\
b^{k+1} &= b^k + \left( \Phi(u^{k+1}) - d^k \right) .\end{aligned}$$ This algorithm is often denoted in the literature as split-Bregman iteration. This class of algorithms has several nice theoretical properties and has successfully been applied to several problems in practice such as image restoration [@osher05], image denoising [@xu07], compressed sensing [@yin08], and image segmentation [@goldstein10]; see also [@combettes11] and references therein.
We use this technique for solving Equation (). We begin by setting $\Phi(z) = z-y$ and $H(z) = s {\left\| Dz \right\|_{2}}^2$, which leads to the problem $$\min_{z, d} \ {\left\| d \right\|_{1}} + s {\left\| Dz \right\|_{2}}^ 2 \quad \text{s.t.} \quad d = z - y .$$ We then transform it into the unconstrained form $$\min_{z, d} \ {\left\| d \right\|_{1}} + s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d - z + y \right\|_{2}}^2 ,
\label{eq:L1formulationExtended}$$ and the Bregman iteration simply takes the form $$\begin{aligned}
\left( z^{k+1}, d^{k+1} \right) & = {\underset{z, d}{\operatorname{argmin}}\ } \ {\left\| d \right\|_{1}} + s {\left\| Dz \right\|_{2}}^ 2 +
\tfrac{\lambda}{2} {\left\| d - z + y - b^k \right\|_{2}}^2 ,
\label{eq:splitBregman1}\\
b^{k+1} & = b^k + (z^{k+1} - y - d^{k+1}) . \label{eq:splitBregman2}\end{aligned}$$
Because of the splitting of the L1 and L2 components in the functional (), we can perform this minimization efficiently by iteratively minimizing with respect to $z$ and $d$ separately, $$\begin{aligned}
z^{k+1} &= {\underset{z}{\operatorname{argmin}}\ } \ s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d^k - z + y - b^k \right\|_{2}}^2 \label{eq:splitBregman1_1} , \\
d^{k+1} &= {\underset{d}{\operatorname{argmin}}\ } \ {\left\| d \right\|_{1}} + \tfrac{\lambda}{2} {\left\| d - z^{k+1} + y - b^k \right\|_{2}}^2 \label{eq:splitBregman1_2} .\end{aligned}$$ For minimizing Equation () we set $\tilde{y} = d^k + y - b^k$ and $\tilde{s} = 2s / \lambda$. We obtain $$\min_{z} \ \tilde{s} {\left\| Dz \right\|_{2}}^ 2 + {\left\| \tilde{y} - z \right\|_{2}}^2 .$$ This is a classical L2 spline and can be minimized using Equation (), as already explained. The optimal value of $d$ in Equation () can be explicitly computed using shrinkage operators, $$\begin{aligned}
d^{k+1} &= {\underset{d}{\operatorname{argmin}}\ } \ {\left\| d \right\|_{1}} + \tfrac{\lambda}{2} {\left\| d - z^{k+1} + y - b^k \right\|_{2}}^2 \nonumber \\
&= {\mathrm{Shrink}} (z^{k+1} - y + b^k , 1/\lambda) ,\end{aligned}$$ where $$\begin{aligned}
{\mathrm{Shrink}}(v, \gamma) =
\begin{pmatrix}
{\mathrm{shrink}}(v_1, \gamma) \\
\vdots \\
{\mathrm{shrink}}(v_j, \gamma) \\
\vdots \\
{\mathrm{shrink}}(v_m, \gamma)
\end{pmatrix}
\intertext{and}
{\mathrm{shrink}}(x, \gamma) = \frac{x}{|x|} \max(|x|-\gamma, 0) .
\label{eq:shrinkage}\end{aligned}$$ We thus obtain a very efficient algorithm for computing L1 splines, combining DCT and shrinkage operators.
On a different note let us mention that Equation () can also be interpreted [@mateos12] as a relaxation of $$\min_{z, d} {\left\| d \right\|_{0}} + s {\left\| Dz \right\|_{2}}^2 + \tfrac{\lambda}{2} {\left\| d-z+y \right\|_{2}}^2 ,
\label{eq:L0formulation}$$ where the L0 norm is replaced by its (convex) L1 counterpart. In this case the underlying model for $y$ is $y = \hat{y} + r + d$, where $r$ is zero-mean Gaussian noise and $d$ represents the “oulier” noise. Under this assumptions, $d$ practically becomes an “indicator function” of the presence of (sparse) outliers (see also [@black91]). Besides the different angle in the derivation of the model, our approach differs from [@mateos12] in two very important points. First, we use split-Bregman iteration by introducing the variable $b^k$ in the optimization procedure, see equations () and (). In [@mateos12] Equation () is first solved using direct alternate minimization over $z$ and $d$, and then Equation () is solved via non-convex minimization using the previous solution as a starting point. Second, considering the grid structure, we use the DCT approach to solve Equation (), instead of the classical Cholesky decomposition. The combination between split-Bregman and DCT results in a sound and fast algorithm for computing L1 splines on grid data.
Handling missing data
---------------------
In the classical L2 formulation, a diagonal binary weighting matrix $W$ is used to cope with missing values (see Section \[sec:robustL2\] for details). Let us denote by $w$ the diagonal of $W$. Let us first define $$\begin{aligned}
{\left\| z \right\|_{1, w}} &= \sum_{\substack{i=1 \\ w_i = 1}}^{m} |z_i|
& \text{and} &&
{\left\| z \right\|_{2, w}} &= \left( \sum_{\substack{i=1 \\ w_i = 1}}^{m} z_i^2 \right)^{1/2} .\end{aligned}$$ We then pose Equation () as $$\begin{gathered}
\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| (z - y) \right\|_{2, w}}^2 + s {\left\| Dz \right\|_{2}}^ 2 ,\end{gathered}$$ and equivalently extend Equation () as $$\begin{gathered}
\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| (z - y) \right\|_{1, w}} + s {\left\| Dz \right\|_{2}}^ 2 .\end{gathered}$$ This leads to $$\min_{z, d} \ {\left\| d \right\|_{1, w}} + s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d - z + y \right\|_{2, w}}^2 .$$ Then the split-Bregman iteration can be written as $$\begin{aligned}
z^{k+1} &= {\underset{z}{\operatorname{argmin}}\ } \, s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d^k - z + y - b^k \right\|_{2,w}}^2 , \label{eq:WsplitBregman1_1} \\
d^{k+1} &= {\underset{d}{\operatorname{argmin}}\ } \, {\left\| d \right\|_{1,w}} + \tfrac{\lambda}{2} {\left\| d - z^{k+1} + y - b^k \right\|_{2,w}}^2 , \label{eq:WsplitBregman1_2} \\
b^{k+1} &= b^k + (z^{k+1} - y - d^{k+1}) . \label{eq:WsplitBregman2}\end{aligned}$$ Equation () can be solved using Equation (). Solving Equation () amounts to performing a shrinkage operation on the dimensions where $w$ equals 1. In Equation (), it suffices to update the dimensions of $b^k$ where $w$ equals 1.
Handling multidimensional data
------------------------------
Let us now return to the general case of $m$-dimensional data. Following Garcia [@garcia10], we extend Equation () as $$\hat{y} = {\mathrm{DCT}}^{-1}_m \left( \varGamma^m \circ {\mathrm{DCT}}_m (y) \right) ,
\label{eq:LSsolutionDCTd}$$ where ${\mathrm{DCT}}_m(\cdot)$ and ${\mathrm{DCT}}^{-1}_m(\cdot)$ stand for the $m$-dimensional DCT-II and inverse DCT-II functions, and $\circ$ denotes the Schur (element-wise) product. Notice that the multidimensional DCT is simply a composition of one-dimensional DCTs along each dimension. Extending Equation (), $\varGamma^m$ is an $m$-th order tensor defined by $$\varGamma^m = 1^m \div \left( 1^m + s \varLambda^m \circ \varLambda^m \right) ,
\label{eq:varGammad}$$ where $1^m$ is an $m$-th order tensor of ones, and $\div$ denotes the element-wise division. Finally, $\varLambda^m$ is an $m$-th order tensor, defined by $$\varLambda^m_{i_1, \dots, i_m} = \sum_{j=1}^{d} \left( -2 + 2 \cos \frac{(i_j - 1) \pi}{n_j} \right) .
\label{eq:varLambdad}$$ where $n_j$ denotes the size of $\varLambda^m$ along the $j$-th dimension.
**The algorithm and its complexity.** The pseudocode for the general $m$-dimensional case is presented in Algorithm \[algo:l1spline\]. Let us analyze its complexity. The DCT and inverse DCT require $O(n \log n)$ operations, where $n=\prod_{1 \leq j \leq m} \, n_j$. The remaining operations are linear in $m$. The overall complexity of the algorithm is then $O \left( N_o N_i (m + n \log n) \right)$, where $N_o$ and $N_i$ are, respectively, the number of outer-loop and inner-loop iterations in Algorithm \[algo:l1spline\]. Notice that Goldstein and Osher [@goldstein09] recommend to perform only one inner-loop iteration for achieving optimal efficiency. Thus, we set $N_i = 1$ for all experiments. We will later see that in many cases the algorithm converges quickly ($N_o$ can be very small then). The algorithm’s complexity is thus dominated by the computation of the DCT and inverse DCT. Of course, these standard operations can be easily computed using GPU, speeding-up the execution by several orders of magnitude.
compute $\varGamma^m$ according to equations () and (). $d^1 \gets 0$, $b^1 \gets 0$, $k \gets 1$
$z^{k+1} \gets {\mathrm{DCT}}^{-1}_m (\varGamma^m \circ {\mathrm{DCT}}_m (d^k + y - b^k))$ $\displaystyle d^{k+1} \gets {\mathrm{Shrink}} (z^{k+1} - y + b^k , 1/ \lambda)$ $b^{k+1} \gets b^k + (z^{k+1} - y - d^{k+1})$ $k \gets k+1$ $z^k$
Experimental results {#sec:results}
====================
For all experiments we adhere to the following setup:
1. using generalized cross validation, we find the best estimate $\hat{s}$ for $s$ for the robust L2 formulation (problem ());
2. we then find L2 splines (problem ()), robust L2 splines (problem ()),[^2] and/or L1 splines (problem ()), setting $s = \hat{s}$.
This protocol allows us to show that, even when $s$ is chosen to fit optimally the robust L2 formulation, the proposed method provides better estimates. For the L1 formulation, in Equation (), we simply set $\lambda=\min(s,1)$ for all examples. We recall that $N_i$ is set to 1. We also set $\varepsilon = 10^{-3}$ (see Algorithm \[algo:l1spline\]) and additionally limit the maximum number of outer iterations to a hundred. The algorithm stops when any of the two conditions is met.
Fig. \[fig:spline\_1D\_16\] presents two one-dimensional examples. We depict the original signal $\hat{y} \in [1, \dots, n] \rightarrow {{\mathbb R}}$, where $n=2^{16}$. We observe the signal $y = \hat{y} + r_1$, where $r_1$ is Gaussian noise. Some points $y_j$ ($1 \leq j \leq n$) are further contaminated with uniform noise $r_2$, where $r_2 \in [ a, \dots, b ]$, such that $y_j = \min (\max (\hat{y}_j + r_1 + r_2, a), b)$. The points affected by $r_2$ are depicted in red and the remaining ones in green. In the top row, $a = -5, b = 5$; and in the bottom row, $a = 0, b = 5$. In both examples, only the L1 spline is correct. The classical and robust L2 splines are both unable to correctly recover the original data in the corrupted part.
Fig. \[fig:errorPlot\] shows the evolution of the relative error as the number of iterations increases for the example in Fig. \[fig:spline\_1D\_16\] (top row). As we can observe, the proposed algorithm is able to converge quickly, reaching a precision of $10^{-3}$ in less than 20 iterations.
![The relative error (in logarithmic scale) as the iterations progress when computing the first example in Figure \[fig:spline\_1D\_16\]. The algorithm is able to quickly decrease the error during the first 20 iterations.[]{data-label="fig:errorPlot"}](errorPlot.pdf){width=".5\columnwidth"}
Fig. \[fig:timeDistribution\] depicts the relative time-cost of each operation during the execution of the proposed method (Fig. \[fig:spline\_1D\_16\], top row). Computing the DCT and the inverse DCT covers more than 84% of the total running time. Implementing these standard operations in GPU would boost the performance of the algorithm by orders of magnitude.
![Percentage of the execution time spent in each operation when computing the first example in Figure \[fig:spline\_1D\_16\]. Clearly, the vast majority of time is spent in DCT or inverse DCT operations.[]{data-label="fig:timeDistribution"}](timeChart.pdf){width=".5\columnwidth"}
In Fig. \[fig:spline\_2D\_n30\] we present a two-dimensional example. We depict the original signal $\hat{y} \in [1, \dots, 256]^2\rightarrow [-6.5497, 8.1054]$ in Fig. \[fig:spline\_2D\_n30\_original\], and we add two types of noise: first, Gaussian noise $r_1$ with zero mean and variance $\sigma^2=2$ (Fig. \[fig:spline\_2D\_n30\_noisy\]), and then uniform noise $r_2$ in the interval $[ -5 \cdot \max (\hat{y} + r_1), \dots, 5 \cdot \max (\hat{y} + r_1) ]$ (Fig. \[fig:spline\_2D\_n30\_corrupted\]). Again, only the L1 spline correctly recovers the original signal.
-- -- --
-- -- --
We next test the proposed algorithm with a climate time-series provided by the Met Office Hadley Centre [@brohan06].[^3] The dataset contains the evolution of global average land temperature anomaly (in ) with respect to the 1961-1990 average temperature. The results, which confirm an upward trend in the second half of the 20th century, are shown in Fig. \[fig:temp\].
We also test on this dataset the effect of varying the parameters $s$ and $\lambda$, see Fig. \[fig:params\]. As in the classical L2 formulation, $s$ has a direct impact on the obtained result, controlling the degree of smoothness of the solution, see Fig. \[fig:params\_s\]. On the contrary, Fig. \[fig:params\_lambda\] shows that the newly introduced parameter $\lambda$ is very stable and provides very similar results in a wide range $(\lambda \in [0.1, 100])$. This stability allows us to fix its value to $\lambda=1$ for all the experiments in this work.
Another interesting example is presented in [@mateos12]. The dataset consists of power consumption measurements (in kW) for a government building, collected every fifteen minutes from July 2005 to October 2010. As in [@mateos12], we downsample the data by a factor of four, yielding one measurement per hour, and use only a subset of the whole data. The results are displayed in Fig. \[fig:load\].
![Power consumption measurements (in kW) for a government building [@mateos12] with zoom-in details on the bottom.[]{data-label="fig:load"}](spline_1D_load-a.pdf "fig:"){width=".2\columnwidth"} ![Power consumption measurements (in kW) for a government building [@mateos12] with zoom-in details on the bottom.[]{data-label="fig:load"}](spline_1D_load-b.pdf "fig:"){width=".2\columnwidth"} ![Power consumption measurements (in kW) for a government building [@mateos12] with zoom-in details on the bottom.[]{data-label="fig:load"}](spline_1D_load-c.pdf "fig:"){width=".2\columnwidth"}
We also test in a synthetic example the ability to recover signals with sharp transitions, see Fig. \[fig:square\]. In this case we use a simple piece-wise constant function. We can observe clear overshoot (plus ringing) effects on the L2 and robust L2 splines. The robust L2 spline also results in transitions with less vertical slopes, creating a bluring effect. With the L1 spline we obtain a much better reconstruction, with almost non-existent overshooting.
This very same effect can be observed in real examples, see Fig. \[fig:119082\]. When approximating images with splines, some structure is lost by blur and some structure is artificially created by overshooting and ringing. This can be observed in Figs. \[fig:119082\_original\] and \[fig:119082\_noise\], were the difference between the original image and the image estimated by robust L2 splines exhibits structure. Observe, however, that almost no structure in the difference is visible when the reconstruction is performed using L1 splines.
[@c@c@]{}
Application to range data
-------------------------
In this section we perform smoothing of depth data obtained with a Kinect camera. This kind of data is particularly challenging because:
- it presents relatively smooth areas separated by sharp transitions,
- edges are highly noisy, that is, edge pixels oscillate over time between foreground and background, and
- it contains missing data, which appear for two different reasons: (1) the disparity between the IR projector and the IR camera produces “shadows,” and (2) the depth cannot be recovered in areas where the IR pattern is not clearly observable (e.g., because they receive direct sunlight or interference from another Kinect).
We use splines to interpolate and denoise these data, showing the advantage of L1 splines over its robust L2 counterpart. The displayed images are part of the LIRIS human activities dataset [@harl2012].
In the first example, shown in Fig. \[fig:kinect2D\], we use a single depth frame (with standard Kinect resolution of $640 \times 480$). The missing data are represented in black, while depth data goes from red to yellow as depth increases. Both, the L1 spline and the robust L2 spline are able to interpolate the missing data with reasonable values. Notice, however, that the latter exhibits, as aforementioned, overshooting and ringing (clearly perceived in the 1D profile). These effects are much milder in the L1 reconstruction.
[ccc]{}
In Fig. \[fig:kinect3D\_original\] we can clearly observe that the position of the missing data is not consistent across frames. We can integrate data from several frames to achieve more accurate interpolations, by performing 3D reconstructions. Thus, in this example, we treat depth data as a 3D signal (2D + time), by considering three consecutive frames. The data dimensionality is then $640 \times 480 \times 3$. A full depth video can be smoothed by using 3D splines as a sliding-window type of filter. The robust L2 spline again presents a noisier behavior and with significative overshooting. On the other hand, the L1 spline is much smoother in smooth areas while correctly preserving abrupt transitions.
**Running times.** We present in Table \[tab:runningTime\] the running-time and number of iterations until convergence for every example in this work. The time of the robust L2 and the L1 splines is comparable. All code is written in pure Matlab, with no C++ or mex optimizations. All experiments were run on a MacBook Pro with a 2.7GHz Intel Core i7 processor. Finally, note that in most cases the algorithm converges in less than twenty outer-iterations (recall that $\varepsilon=10^{-3}$). In the example in Fig. \[fig:spline\_2D\_n30\], the maximum number of iterations (100) is reached with a final error of $10^{-2.8}$.
-- ------------------------------- --------------------------- ------------------ ------- --------
Robust L2 spline
Time Time Iters.
Fig. \[fig:spline\_1D\_16\] $2^{20}$ 4.931 3.590 7
Fig. \[fig:spline\_2D\_n30\] $256 \times 256$ 0.541 0.982 100
Fig. \[fig:temp\] 163 0.007 0.045 72
Fig. \[fig:load\] 501 0.010 0.010 11
Fig. \[fig:square\] $2^{10}$ 0.035 0.007 6
Fig. \[fig:119082\_original\] $321 \times 481$ 1.167 0.758 17
Fig. \[fig:119082\_noise\] $321 \times 481$ 1.143 0.873 20
Fig. \[fig:kinect2D\] $480 \times 640$ 0.911 0.266 2
Fig. \[fig:kinect3D\] $480 \times 640 \times 3$ 7.511 2.202 3
-- ------------------------------- --------------------------- ------------------ ------- --------
: Execution times (in seconds) and number of iterations until convergence of the proposed algorithm for the different experiments performed in this work.[]{data-label="tab:runningTime"}
Conclusions {#sec:conclusions}
===========
We have presented a new method for robustly smoothing regularly sampled data. We do this with modified splines, where we replace the classical L2-norm in the fitting term by an L1-norm. This automatically handles outliers, thus obtaining a robust approximation.
We also presented a new technique, using split-Bregman iteration, for solving the resulting optimization problem. The algorithm is extremely simple and easy to code. The method converges very quickly and has a small memory footprint. It also makes extensive use of the DCT, thus being straightforward to implement in GPU. These characteristics make this method very suitable for large-scale problems.
Acknowledgment {#acknowledgment .unnumbered}
==============
Work partially supported by NSF, ONR, NGA, ARO, DARPA, and NSSEFF. We thank Dr. Gonzalo Mateos for kindly providing the power consumption dataset.
[^1]: This work was partially done while the authors were with the Department of Electrical and Computer Engineering, University of Minnesota.
[^2]: Code available at <http://www.mathworks.com/matlabcentral/fileexchange/25634-robust-spline-smoothing-for-1-d-to-n-d-data>.
[^3]: Data are available in <http://hadobs.metoffice.com/crutem3/diagnostics/global/nh+sh/annual>.
|
---
abstract: 'The electronic structure of Bernal-stacked graphite subject to a tilted magnetic field is studied theoretically. The minimal nearest-neighbor tight-binding model with the Peierls substitution is employed to describe the structure of Landau levels. We show that while the orbital effect of the in-plane component of the magnetic field is negligible for massive Dirac fermions in the vicinity of a the $K$ point of the graphite Brillouin zone, at the $H$ point it leads to the experimentally observable splitting of Landau levels, which grows approximately linearly with the in-plane field intensity.'
address: |
Institute of Physics, Academy of Science of the Czech Republic,v.v.i.,\
Cukrovarnická 10, 162 53 Prague 6, Czech Republic
author:
- 'Nataliya A. Goncharuk and Ludvík Smrčka'
title: 'Tight-binding description of Landau levels of graphite in tilted magnetic fields'
---
[*Keywords*]{}: graphite electronic structure, tilted magnetic field, splitting of Landau levels.
Introduction {#Introduction}
============
The recent attention paid to graphene monolayers has been motivated by their unusual twodimensional (2D) Dirac energy spectrum of electrons. In Bernal-stacked graphene multilayer composed of weakly coupled graphene sheets, the interlayer interaction converts the 2D electron energy spectrum of graphene into the three-dimensional (3D) spectrum of graphite. The electronic structure of 3D graphite subject to magnetic fields perpendicular to $x-y$ planes of graphene layers were extensively studied a long time ago, see, e.g., Refs. [@McClure_1956; @McClure_1960; @Inoue_1962; @Wallace_1972; @Dresselhaus_1974; @Nakao_1976].
The application of the tilted magnetic field $\vec{B}=(0, B_y, B_z)$ is a standard method used to distinguish between 2D and 3D electron systems, as in 3D systems the orbital effect of the in-plane magnetic field component should be observable. We will study this problem theoretically using a simple tight-binding quantum mechanical model of the graphite electron structure.
Various approaches were employed previously to study the influence of the tilted magnetic fields.
The Fermi surfaces of metal single crystals were investigated by measurements of the de Haas-van Alphen effect in tilted magnetic fields. The interpretation of experiments relies on the quasiclassical Onsager-Lifshitz quantization rule [@Onsager_1952; @Lifshitz_1956], the Fermi surface is reconstructed from the periods of magneto-oscillations which are proportional to angular-dependent extremal-cross-sections perpendicular to the direction of the tilted magnetic field.
In semiconductor superlattices the quasiclassical interpretation of data measured in tilted magnetic fields fails, as reported in Refs. [@Chang_1982; @Stormer_1986; @Jaschinski_1998; @Nachtwei_1998; @Kawamura_2001; @Goncharuk_2007]. In these papers the observed quantum effects are attributed to the shift of centers of $k$-space orbits in neighboring quantum wells by $|e|B_y/d$ in the $k_x$-direction, where $d$ is the distance between quantum wells. In real space this means that the in-plane magnetic-field length $\ell_y=\sqrt{\hbar/|e|B_y}$ should become comparable with $d$ to reach the visible effect [@Dingle_1978].
Besides semiconductor superlattices, other layered materials with much shorter interlayer distances were also investigated in tilted magnetic fields. Different versions of angular magnetoresistance oscillations (AMRO) were studied both experimentally and theoretically in low-dimensional quasi-2D and quasi-one-dimensional organic conductors (see, e.g.,Refs. [@Kartsovnik_1988; @Kajita_1989; @Yamaji_1989; @Yagi_1990; @Lebed_1989; @Osada_1991; @Danner_1994; @Chashechkina_1998; @Lee_1998] and references therein), and also in intercalated graphite [@Iye_1994; @Enomoto_2006]. On the theory side, the high Landau level (LL) filing factors and weak interlayer interaction were considered in iterpretation the data.
In pristine graphite this problem has been touched on by two recent theoretical articles.
The graphene multilayer energy spectrum in magnetic fields parallel to the layers was described quantum-mechanically in Ref. [@Pershoguba_2010], as the standard theory of AMRO in tilted magnetic fields was not applicable due to the relatively strong interlayer interaction (in comparison with the intercalated graphite) between graphene sheets.
The LLs in the bilayer graphene in magnetic fields of arbitrary orientations were calculated analytically in Ref. [@Hyun_2010].
Both papers conclude that a very strong in-plane field component is necessary to induce an observable effect on the electronic structure. Indeed, to reach $\ell_y$ comparable with the distance between graphene layers in graphite, the magnetic field $B_y = 5865$ T would be necessary.
In this paper we make use of the specific features of the LL structure in two nonequivalent neighboring graphene sheets in graphite, and show that at the $H$ point of the graphite hexagonal Brillouin zone the application of the tilted magnetic field leads to experimentally observable splitting of LLs of the order of several meV.
Model {#Model}
=====
Bulk graphite is composed of periodically repeated graphene bilayers formed by two nonequivalent Bernal-stacked graphene sheets, as shown in Fig. \[Fig1\]. There are two sublattices, $A$ and $B$, on each sheet and, therefore, four atoms in a unit cell. The distance between the nearest atoms $A$ and $B$ in a single layer is $1.42\,$ [Å]{}, the interlayer distance between nearest atoms $A$ is $d=3.35\,$[Å]{}.
![(Color online) The lattice structure of graphite. The unit cell is a green parallelepiped.[]{data-label="Fig1"}](Fig1.eps){width="0.85\linewidth"}
To describe the graphite band structure, we employ the minimal nearest-neighbor tight-binding model, introduced by Koshino et al. in Ref. [@Koshino_2008]. This model is reduced Slonczewski, Weiss and McClure (SWM) model. The tight-binding Hamiltonian $\mathcal{H}$ includes only two instead of seven tight-binding parameters, the intralayer interaction $\gamma_0=3.16$ eV between the nearest atoms $A$ and $B$ in the plane, and the interlayer interaction $t=0.39$ eV between the nearest atoms $A$ out of plane.
While the reduced SWM model is not appropriate, e.g., for a Fermi surface description, this model has been successfully applied in the theoretical papers [@Pershoguba_2010; @Hyun_2010], and used to describe recent magneto-optical measurements on graphite in Refs. [@Henriksen_2008; @Nicholas_2009; @Orlita_2009; @Orlita_2010; @Ubrig_2011]. It has been shown in Refs. [@Orlita_2009; @Orlita_2010; @Ubrig_2011] that the transitions between Landau levels originating from the $K$ and $H$ points of the graphite Brillouine zone can be understood within a simple picture of an effective bilayer with a coupling strength enhanced twice in comparison to a true graphene bilayer, $2t$, (which is definitely the effect of a superlattice) and an effective graphene monolayer. These solid arguments are in agreement with the theoretical model based on the reduced SWM model we develop in our manuscript.
In this model the wave functions are expressed via four orthogonal components $\psi^A_j$, $\psi^B_j$, $\psi^A_{j+1}$, $\psi^B_{j+1}$, which are, in zero magnetic field, Bloch sums of atomic wave functions over the lattice sites of sublattices $A$ and $B$ in individual layers $j$.
The continuum approximation is used in the vicinity of the $H - K - H$ axis of the graphite hexagonal Brillouin zone, for small $\vec{k}=\left(k_x,k_y\right)$ measured from the axis. Then the electron wave length is larger than the distance between atoms, and the non-zero matrix elements of $\mathcal{H}$ can be written as $$\begin{aligned}
\mathcal{H}^{AB}&=\hbar v_F(k_x+ik_y),\\
\mathcal{H}^{BA}&=\hbar v_F(k_x-ik_y),\\
\mathcal{H}^{AA}&=t.\end{aligned}$$ The Fermi velocity, $v_F$, is defined by $\hbar v_F
= \sqrt{3}a\gamma_0/2$, and will be used as an intralayer parameter instead of $\gamma_0$ in the subsequent consideration.
The effect of the arbitrary oriented magnetic field, $\vec{B}=(0, B_y,
B_z)$, can be conveniently introduced into the zero-field Hamiltonian by the Peierls substitution. If we choose the vector potential in the Landau form $\vec{A}=(B_yz-B_zy, 0, 0)$, the substitution will read $$\hbar k_x \rightarrow \hbar k_x-|e|B_zy+|e|B_yjd,
\label{eqkx}$$ where an integer number $j$ indicates the graphite layer number. Consequently, the matrix elements, $\mathcal{H}^{AB}$ and $\mathcal{H}^{BA}$, become layer dependent in a tilted magnetic field, $$\begin{aligned}
\mathcal{H}_j^{AB}=v_F(k_x-|e|B_zy+|e|B_yjd+ik_y)=v_F\Pi_j,
\label{eqhab1}\\
\mathcal{H}_j^{BA}=v_F(k_x-|e|B_zy+|e|B_yjd-ik_y)=v_F\Pi_j^{\ast}.
\label{eqhab2}\end{aligned}$$
Making use of the above approximations, the Schrödinger equation involving all layers $j$, leads to the following system of equations $$\begin{aligned}
v_F\Pi_{j}^{\ast}\psi_{j}^B-E\psi_{j}^A+
t\,\psi_{j-1}^A+t\,\psi_{j+1}^A=0,
\label{e1}\\
-E\psi_j^B+v_F\Pi_{j}^{\ast}\psi_{j}^A=0,
\label{e2}\\
v_F\Pi_{j+1}^{\ast}\psi_{j+1}^B-E\psi_{j+1}^A+
t\,\psi_{j}^A+t\,\psi_{j+2}^A=0,
\label{e3}\\
-E\psi_{j+1}^B+v_F\Pi_{j+1}\psi_{j+1}^A=0.
\label{e4}\end{aligned}$$ Note that the structure of $\mathcal{H}$ allows to express the function $\psi_j^B$ via the function $\psi_j^A$ from the same layer, and we are thus left with two interlayer equations for $\psi_j^A$ and $\psi_{j+1}^A$ $$\begin{aligned}
(v_F^2\Pi_j \Pi_j^{\ast}-E^2)\psi_j^A+
t\, E(\psi_{j-1}^A+\psi_{j+1}^A )=0,
\label{EQU1}
\\
(v_F^2\Pi_{j+1}^{\ast} \Pi_{j+1}-E^2)\psi_{j+1}^A+
t\, E(\psi_{j}^A+\psi_{j+2}^A )=0.
\label{EQU2}\end{aligned}$$
Zero-field case
===============
It follows from the condition of periodicity in the $z$-direction that $\psi_{j+1}^A$ and $\psi_j^A$ can be written as $$\psi_{j+1}^A=e^{ik_zd(j+1)}\phi_1^A,\quad \psi_j^A=e^{ik_zdj}\phi_2^A,
\label{eqperio}$$ where $\phi_1^A$ and $\phi_2^A$ denote the 2D wave functions in two non-equivalent layers of the graphite unite cell, and $k_z$ is restricted to the first Brillouin zone, $-\pi/2<k_zd<\pi/2$.
For $\vec{B}=0 $ Eqs. (\[EQU1\], \[EQU2\]) are transformed to $$\begin{aligned}
(\hbar^2 v_F^2k^2 - E^2)\phi_2^A + 2 t\,E\cos(k_zd)\phi_1^A=0,\\
(\hbar^2 v_F^2k^2 - E^2)\phi_1^A + 2 t\,E\cos(k_zd)\phi_2^A=0,\end{aligned}$$ and from here the four eigenvalues are obtained $$E^{\pm,\pm} = \pm \mathcal{T} \pm \sqrt{\mathcal{T}^2+\hbar^2 v_F^2k^2},$$ where $$\mathcal{T}=t\cos(k_zd)
\label{T}$$ denotes the $k_z$-dependent coupling of two graphene sheets.
Perpendicular magnetic field
============================
In the perpendicular magnetic field, $B_y = 0$, the system remains periodic in the $z$-direction, and Eq. (\[eqperio\]) is still valid. To find a proper form of $\mathcal{H}^{AB}$ and $\mathcal{H}^{BA}$ we introduce the perpendicular magnetic field length, $\ell_z^2 = \hbar/(|e|B_z)$, the centre of a cyclotron orbit, $y_0=\ell_z^2k_x$, the dimensionless variable, $\eta =
(y-y_0)/\ell_z$, and the perpendicular-magnetic-field-dependent parameter, $\mathcal{B} = 2\hbar |e|B_zv_F^2$. Then, in our notation, $$\begin{aligned}
\mathcal{H}^{AB}&=v_F\Pi=-\sqrt{\mathcal{B}}\,a^{\dag}=
-\sqrt{\frac{\mathcal{B}}{2}}\left(-\frac{\partial}{\partial\eta}+
\eta\right),\\
\mathcal{H}^{BA}&=v_F\Pi^{\ast}=-\sqrt{\mathcal{B}}\,a=
-\sqrt{\frac{\mathcal{B}}{2}}
\left(\frac{\partial}{\partial\eta}+\eta\right),\end{aligned}$$ $a^{\dag}$ and $a$ being raising and lowering operators, respectively.
With help of these expressions, Eqs. (\[EQU1\]) and (\[EQU2\]) can be written as $$\begin{aligned}
\left[\frac{\mathcal{B}}{2}\left(-\frac{\partial^2}{\partial\eta^2}
+\eta^2+1\right)-E^2\right]\phi_2^A + 2\mathcal{T}E\phi_1^A=0,
\label{eq1B}\\
\left[\frac{\mathcal{B}}{2}\left(-\frac{\partial^2}{\partial\eta^2}
+\eta^2-1\right)-E^2\right]\phi_1^A + 2\mathcal{T}E\phi_2^A=0.
\label{eq2B}\end{aligned}$$ It is obvious from these equations that $\phi_1^A$ and $\phi_2^A$ are closely related to the eigenfunctions of the harmonic oscillator, $\varphi_n(\eta)$. Assuming $$\phi_1^A=\frac{1}{L_x}e^{ik_xx}\sum_{n'=0}^{\infty}A_{1,n'}
\varphi_{n'}(\kappa),\quad
\phi_2^A=\frac{1}{L_x}e^{ik_xx}\sum_{n'=0}^{\infty}A_{2,n'}
\varphi_{n'}(\kappa),$$ and having in mind that $$\left(-\frac{\partial^2}{\partial\eta^2}
+\eta^2\right)\varphi_n(\eta)=(2n+1)\varphi_n(\eta),$$ we get $$\begin{aligned}
\left[\mathcal{B}(n+1)-E^2\right] A_{2,n}+
2 E \sum_{n'=0}^{\infty}\mathcal{T}_{n,n'}A_{1,n'}=0,\label{eq1p}
\\
\left[\mathcal{B}n-E^2\right] A_{1,n}+
2 E \sum_{n'=0}^{\infty}\mathcal{T}_{n,n'}A_{2,n'}=0,\label{eq2p}\end{aligned}$$ where $T_{n,n'}$ is defined by $$\mathcal{T}_{n,n'}=
\mathcal{T}\int_{-\infty}^{+\infty}\varphi_{n}(\eta)\varphi_{n'}(\eta)d\eta
=\mathcal{T}\delta_{n,n'}.
\label{t}$$ It follows from (\[t\]) that only the LLs with the same quantum numbers $n$ (but with the different energies) are coupled and we arrive to $$\begin{aligned}
\left[\mathcal{B} (n+1)-E^2\right]A_{2,n}+2\mathcal{T}_{n,n}EA_{1,n}=0,
\label{e1-Bz}\\
\left(\mathcal{B} n-E^2\right)A_{1,n}+2\mathcal{T}_{n,n}EA_{2,n}=0.
\label{e2-Bz}\end{aligned}$$ Solving Eqs. (\[e1-Bz\], \[e2-Bz\]) yields the eigenenergies $$E_{n}^{\pm,\pm}=\pm
\sqrt{2\mathcal{T}_{n,n}^2+\mathcal{B}\left(n+\frac{1}{2}\right)
\pm \sqrt{4\mathcal{T}_{n,n}^4
+4\mathcal{T}_{n,n}^2\mathcal{B}(n+\frac{1}{2})+
\frac{\mathcal{B}^2}{4}}},$$ which are presented in Fig. \[Fig2\].
![(Color online) Landau subbands of graphite, $E_{n}^{\pm,-}$ and $E_{n}^{\pm,+}$ (denoted as $E_{1,n}^{\pm}$ and $E_{2,n}^{\pm}$ at the $H$ point) subject to perpendicular magnetic field $B_z=5$ T, as a function of $k_z$ along the K-H-K pass of the Brillouin zone. The $k_z$ dependence of LLs in the vicinity of the $H$ point restricted by the rectangular box will be compared with those subject to the tilted magnetic field, as shown in Fig. \[Fig6\].[]{data-label="Fig2"}](Fig2.eps){width="0.95\linewidth"}
As the densities of states of the above Landau subbands have singularities at the points $K$ and $H$, $k_zd=0$ and $k_zd=\pi/2$, respectively, we concentrate on the field dependence of levels corresponding to these points.
At the $K$ point the eigenenergies are given by $$E_{n}^{\pm,\pm}=\pm\sqrt{2 t^2+\mathcal{B}
\left(n+\frac{1}{2}\right)\pm\sqrt{4 t^4+4t^2\mathcal{B}(n+\frac{1}{2})+
\frac{\mathcal{B}^2}{4}}},$$ i.e., the subbands are equivalent to those of a graphene bilayer with the coupling constant doubled: $2t$ instead of $t$.
![(Color online) LLs of graphite at the $H$ point of the Brillouin zone, $E_{1,n}^{\pm}$ and $E_{2,n}^{\pm}$, in the perpendicular magnetic field, $B_z$.[]{data-label="Fig3"}](Fig3.eps){width="0.95\linewidth"}
At the $H$ point the eigenenergies read $$E_{n}^{\pm,\pm}=\pm\sqrt{\mathcal{B}\left(n+\frac{1}{2}\right)\mp
\frac{\mathcal{B}}{2}}.$$ Due to the effectively vanishing inter-layer coupling the spectrum corresponds to the Dirac fermions. The coefficients $A_{1,n}$ and $A_{2,n}$ in Eqs. (\[e1-Bz\], \[e2-Bz\]) are equal to $1$ and this implies that the corresponding wave functions are localized either in the layer $1$ or in the layer $2$. To emphasize it, we will write the eigeneneries as $$\begin{aligned}
E_{1,n}^{\pm}&=E_{n}^{\pm,-}=\pm\sqrt{\mathcal{B}n},\\
E_{2,n}^{\pm}&=E_{n}^{\pm,+}=\pm\sqrt{\mathcal{B}(n+1)}.\end{aligned}$$ The energy spectrum of LLs at the $H$ point is presented in Fig. \[Fig3\].
We also denote the wave functions $\phi_{1(2)}^A$ and $\phi_{1(2)}^B$ as $|n\rangle_{1(2)}^A$ and $|n\rangle_{1(2)}^B$ to stress that they are the envelope wave functions of atomic wave functions $A$ and $B$ in the layers 1 and 2, as shown in TABLE \[Table1\].
Let us mention that $E_{2,n}^{\pm}=E_{1,n+1}^{\pm}$, i.e., we are left with pairs of degenerated LLs with different quantum numbers $n$ but the same eigenenergies, with the wave functions localized in two different layers.
---------------------------------- ------------------------------------------
Energy {$\phi_1^A,\,\,\,\phi_1^B,\,\,\,\,
\phi_2^A,\,\,\,\,\phi_2^B$}
$E_1^+=\sqrt{\mathcal{B}n}$ $\{|n\rangle_1^A,-|n-1\rangle_1^B,
0\,\,,0\,\}$
$E_1^-=-\sqrt{\mathcal{B}n}$ $\{|n\rangle_1^A, \,\,\,|n-1\rangle_1^B,
0\,\,,0\,\}$
$E_2^+=\sqrt{\mathcal{B}(n+1)}$ $\{0,\,\,0\,,
|n\rangle_2^A,-|n+1\rangle_2^B\}$
$E_2^-=-\sqrt{\mathcal{B}(n+1)}$ $ \{0,\,\,0\,,
|n\rangle_2^A, \,\,\,|n+1\rangle_2^B\}$
---------------------------------- ------------------------------------------
: Energies and wave functions at the graphite $H$ point in perpendicular magnetic fields
\[Table1\]
Tilted magnetic field
=====================
While the previous two paragraphs summarized the already published theories devoted to $\vec{B}=0$ and $\vec{B}= (0,0,B_z)$, here we present new results for $\vec{B}= (0,B_y,B_z)$.
In tilted magnetic fields the off-diagonal matrix elements, $\mathcal{H}_j^{AB}$ and $\mathcal{H}_j^{BA}$, given by Eqs. (\[eqhab1\], \[eqhab2\]) remain layer dependent and take the form $$\begin{aligned}
\mathcal{H}_j^{AB}&=v_F\Pi_j=-\sqrt{\frac{\mathcal{B}}{2}}
\left(-\frac{\partial}{\partial\eta}+\eta-j\eta_d\right),\\
\mathcal{H}_j^{BA}&=v_F\Pi_j^{\ast}=-\sqrt{\frac{\mathcal{B}}{2}}
\left(\frac{\partial}{\partial\eta}+\eta-j\eta_d\right),\end{aligned}$$ where the small dimensionless parameter $$\eta_d = \frac{B_y}{B_z}\frac{d}{\ell_z}
%= \frac{\ell_z d}{\ell_y^2}$$ means the shift of the cyclotron orbit center in the $j$-layer due to the in-plane component of the magnetic field, $B_y$. Fig. \[Fig4\] illustrates the $B_z$ dependence of $\eta_d$ for various tilt angles of the magnetic field.
Note that graphite subject to tilted magnetic fields is no longer periodic in the $z$-direction, but becomes periodic in the direction of the tilted magnetic field. To take into account the shift of the cyclotron orbits, we apply the approach developed in Ref. [@Goncharuk_2005] for semiconductor superlattices.
![(Color online) The dimensionless parameter $\eta_d$ as a function of the perpendicular component of the magnetic field, $B_z$, calculated for various tilt angles of the magnetic field $\vec{B}$. []{data-label="Fig4"}](Fig4.eps){width="0.8\linewidth"}
Accordingly, the Eqs. (\[EQU1\]) and (\[EQU2\]) are modified to $$\begin{aligned}
\left[\frac{\mathcal{B}}{2}\left(-\frac{\partial^2}{\partial\eta^2}
+\left(\eta-j\eta_d\right)^2+1\right)-E^2\right]\psi_j^A+
t\, E\left(\psi_{j-1}^A+\psi_{j+1}^A\right)=0,
\label{eq1t}\\
\left[\frac{\mathcal{B}}{2}\left(-\frac{\partial^2}{\partial\eta^2}
+\left(\eta-(j-1)\eta_d\right)^2-1\right)-E^2\right]\psi_{j+1}^A+\nonumber\\
+t\, E\left(\psi_{j}^A+\psi_{j+2}^A\right)=0.
\label{eq2t}\end{aligned}$$
The new periodicity implies that $\psi_{j}^A$ and $\psi_{j+1}^A$ can be written as $$\psi_{j}^A=\, e^{i k_z d j}\phi_1^A\left(\eta+j\eta_d\right),\quad
\psi_{j+1}^A=\, e^{i k_z d(j+1)}\phi_2^A\left[\eta+(j+1)\eta_d\right],
\label{eqperiot}$$ where again $-\pi/2\leq k_z d\leq \pi/2$. Here the wave functions $\phi_1^A\left(\eta+j\eta_d\right)$ and $\phi_2^A\left(\eta+j\eta_d\right)$ are associated with cyclotron orbits in two layers. Introducing the shift operator by $$\phi\left(\eta+\eta_d\right)=e^{i\kappa\eta_d}\phi(\eta),\,\,\,\
\kappa=-i\partial/\partial\eta,$$ and employing the $\kappa$-representation, two interlayer Eqs. (\[eq1t\]) and (\[eq2t\]) can be given the form similar to Eqs. (\[eq1B\], \[eq2B\]) for the perpendicular magnetic field $$\begin{aligned}
\left[\frac{\mathcal{B}}{2}\left(-\frac{\partial^2}{\partial\kappa^2}
+\kappa^2+1\right)-E^2\right]\phi_2^A+2\widetilde{\mathcal{T}}E\phi_1^A = 0 ,
\label{main-eq1}\\
\left[\frac{\mathcal{B}}{2}\left(-\frac{\partial^2}{\partial\kappa^2}
+\kappa^2-1\right)-E^2\right]\phi_1^A+2\widetilde{\mathcal{T}}E\phi_2^A = 0,
\label{main-eq2} \end{aligned}$$ but with the new coupling $\widetilde{\mathcal{T}}$ which depends, due to new periodicity, not only on $k_z$ but also on both components $B_z$ and $B_y$ of the arbitrary oriented magnetic field via $\eta_d$, $$\widetilde{\mathcal{T}}(\kappa)=t\cos{(\kappa\eta_d+k_zd)}.
\label{T'}$$
The Eqs. (\[main-eq1\]) and (\[main-eq2\]) represent the main result of this paper and in the following we will discuss the possible methods of their solutions.
Expressing again $\phi_1^A$ and $\phi_2^A$ with a help of (\[eq1p\]) and (\[eq2p\]) we arrive to $$\begin{aligned}
\left[\mathcal{B}(n+1)-E^2\right] A_{2,n}+
2 E \sum_{n'=0}^{\infty}\widetilde{\mathcal{T}}_{n,n'}A_{1,n'}=0,\label{eq1pt}
\\
\left[\mathcal{B}n-E^2\right] A_{1,n}+
2 E \sum_{n'=0}^{\infty}\widetilde{\mathcal{T}}_{n,n'}A_{2,n'}=0,\label{eq2pt}\end{aligned}$$ where $$\widetilde{\mathcal{T}}_{n,n'}=
\int_{-\infty}^{\infty}\varphi_{n}(\kappa)\widetilde{\mathcal{T}}(\kappa)
\varphi_{n'}(\kappa)d\kappa.
\label{T'int}$$ The integrals (\[T’int\]) can be evaluated analytically and expressed via the generalised Laguerre polynomials (see, e.g., Ref. [@Ryzhik]). The Eqs. (\[eq1pt\]) and (\[eq2pt\]) define the matrix which should be diagonalized. The nonzero coupling of LLs with different $n$ allows us to conclude that the degeneracy of LLs at the $H$ point will be removed, and the LLs with different $n$ avoid to cross. Then the standard approach is to solve the secular equation numerically, the minor complication being that the matrix elements depend on the energy.
Here we prefer to obtain analytic results by the lowest order perturbation theory application and treating $\eta_d$ as a small parameter.
A simple trigonometric relation and a series expansion restricted to terms linear in $\eta_d$ imply $$\begin{aligned}
\widetilde{\mathcal{T}}(\kappa)&=t\cos(k_zd)\cos{(\kappa\eta_d)}-
t\sin{(k_zd)}\sin{(\kappa\eta_d)}\approx\nonumber\\
&\approx t\cos(k_zd) -t\sin{(k_zd)}\eta_d\kappa \cdots,
\label{T'-1}\end{aligned}$$ and $$\widetilde{\mathcal{T}}_{n,n'} = t\cos(k_zd)\delta_{n,n'}-
t\eta_d\sin{(k_zd)}\int_{-\infty}^{\infty}\varphi_{n}(\kappa)
\kappa\varphi_{n'}(\kappa)d\kappa.
\label{T'int1}$$ The integrals in Eq. (\[T’int1\]) can be easily evaluated using the relation (see, e.g., Ref. [@Blokhincev]) $$\kappa\varphi_{n}(\kappa)=\sqrt{\frac{n+1}{2}}\varphi_{n+1}(\kappa)+
\sqrt{\frac{n}{2}}\varphi_{n-1}(\kappa).$$
Let us pay attention to the field dependence of LLs at the most interesting $K$ and $H$ points of the graphite Brillouin zone.
At the $K$ point $k_zd=0$, and, consequently, $\cos(k_zd)=1$, $\sin(k_zd)=0$. Then $\widetilde{\mathcal{T}}_{n,n'}$ reduces to $$\widetilde{\mathcal{T}}_{n,n'} =t\delta_{n,n'}.$$ This coupling corresponds to the effective bilayer subject to the perpendicular field discussed above. Thus, we have found that corrections induced by $B_y$ are very small and of the order $\eta_d^2$. This is in agreement with conclusions presented in Ref. [@Hyun_2010].
The field dependence at the $H$ point, $k_zd=\pm \pi/2$, is more interesting. In perpendicular magnetic fields the coupling between layers disappears, and, as presented above, we obtained the LLs corresponding to graphene Dirac fermions, namely $E_{1,n}^{\pm}=\pm\sqrt{\mathcal{B}n}$ for the first layer, and $E_{2,n}^{\pm}=\pm\sqrt{\mathcal{B}(n+1)}$ for the second layer, $n=0,1,2,\cdots$.
In the magnetic field of an arbitrary direction the interlayer interaction is not reduced to zero, but remains finite. The non-zero matrix elements $\widetilde{\mathcal{T}}_{n,n'}$ can be written as $$\widetilde{\mathcal{T}}_{n,n+1}=\widetilde{\mathcal{T}}_{n+1,n} =
t \eta_d \sqrt{\frac{n+1}{2}},\quad
\widetilde{\mathcal{T}}_{n,n-1}=\widetilde{\mathcal{T}}_{n-1,n} =
t \eta_d \sqrt{\frac{n}{2}}.$$
![(Color online) Splitted by tilting the magnetic field with $\varphi = 20^o$ LLs of graphite at the $H$ point, $E_{n}^{\pm,\pm}$, as a function of the perpendicular component of the magnetic field, $B_z$. The dotted lines correspond to results of numerical calculations.[]{data-label="Fig5"}](Fig5.eps){width="0.95\linewidth"}
The small perturbation $\eta_d$ couples the states $|n\rangle_2^A$ with $|n+1\rangle_1^A$ and $|n-1\rangle_1^A$. Among them the states $|n\rangle_2^A$ and $|n+1\rangle_1^A$ are degenerated, i.e., they belong to the same unpertubed eigenvalues $\pm\sqrt{\mathcal{B}(n+1)}$. Consequently, at least the lowest order perturbation approach suitable to remove the degeneracy must be applied, which yields equations $$\begin{aligned}
\left[\mathcal{B}(n+1)-E^2\right] A_{2,n}+
2 E \widetilde{\mathcal{T}}_{n,n+1} A_{1,n+1}=0,\label{eq1pp}
\\
\left[\mathcal{B}(n+1)-E^2\right] A_{1,n+1}+
2 E \widetilde{\mathcal{T}}_{n+1,n}A_{2,n}=0.\label{eq2pp}\end{aligned}$$ The secular equation derived from Eqs. (\[eq1pp\], \[eq2pp\]) reads $$\left[\mathcal{B}(n+1)-E^2\right]^2 - 4 E^2
\widetilde{\mathcal{T}}^2_{n,n+1}=0,
\label{sec-eq}$$ and from here we get the four eigenenergies $$E_{n+1}^{\pm,\pm}=\pm \widetilde{\mathcal{T}}_{n,n+1}\pm
\sqrt{\mathcal{B}(n+1)+\widetilde{\mathcal{T}}^2_{n,n+1}},
\quad n = 0,1,2,\cdots
\label{LLs-Hpoint}$$
The eigenenergies, $E_{0}^{\mp}$ and $E_{0}^{\pm}$, steming from $E_{1,0}^{\pm}$, remain the same as in the perpendicular magnetic field. In that case the degeneracy is not removed.
Lifting of LL degeneracy by the tilted magnetic field in LLs with $n>0$ is shown in Fig. \[Fig5\]. The LL splitting is of the order of several meV, and it grows with the tilt angle, i.e., with the in-plane magnetic field component, $B_y$.
The corresponding eigenfunctions calculated with the same level of accuracy are presented in TABLE \[Table2\]. They are mixed from wave functions of both layers with an equal weight.
To test the accuracy of the above approximations we have calculated the eigenvalues numerically with the larger basis $|n\rangle_1^A, |n+1\rangle_1^A, |n+2\rangle_1^A,
|n\rangle_2^A, |n+1\rangle_2^A, |n+2\rangle_2^A$ instead of the minimal one $|n+1\rangle_1^A, |n\rangle_2^A$. At the $H$ point we have found only negligible quantitative corrections to the results obtained analytically, as presented in Fig. \[Fig5\]. Similarly, including the higher order expansion in $\eta_d$ does not influence the results for the chosen range of angles and magnetic fields.
The above basis allows to calculate also the $k_z$ dependence in the vicinity of the $H$ point defined roughly by the rectangle in Fig. \[Fig2\]. The most interesting feature is the development of additional local extrema near the $H$ point, which are more pronounced for LLs with higher $n$. The same is true for minigaps open at the crossing points of LLs, as mentioned in the previous paragraphs. The results are shown in Fig. \[Fig6\].
------------- ---------------------------------------------------------
Energy {$\phi_1^A,\,\,\,\,\,\,\phi_1^B,\,\,\,\,\,\,\,\phi_2^A,
\,\,\,\,\,\,\,\phi_2^B$}
$E_n^\mp$ $\{|n\rangle_1^A,
-|n-1\rangle_1^B,-|n+1\rangle_2^A,|n+2\rangle_2^B\}$
$E_n^{--}$ $\{|n\rangle_1^A,|n-1\rangle_1^B,
-|n+1\rangle_2^A,-|n+2\rangle_2^B\}$
$E_n^{++}$ $\{|n+1\rangle_1^A,-|n\rangle_1^B,
|n\rangle_2^A,-|n+1\rangle_2^B\}$
$E_n^{\pm}$ $\{|n+1\rangle_1^A,|n\rangle_1^B,
|n\rangle_2^A,|n+1\rangle_2^B\}$
------------- ---------------------------------------------------------
: Energies and wave functions in tilted magnetic fields at the graphite $H$ point
\[Table2\]
![(Color online) Graphite Landau subbands, depicted by the rectangular box in Fig. \[Fig2\], as a function of $k_z$ nearby the $H$ point. Dashed curves are graphite LLs, $E_{1,n}^{+}$ and $E_{2,n}^{+}$, in the perpendicular magnetic field $B_z = 5$ T, $\varphi =0^{\circ}$. Solid curves are LLs of graphite, $E_{n}^{-/+}$ and $E_{n}^{+/+}$, subject to the tilted magnetic field with the perpendicular component $B_z = 5$ T and the tilt angle $\varphi = 20^{\circ}$.[]{data-label="Fig6"}](Fig6.eps){width="0.95\linewidth"}
In general, our approach must fail for magnetic fields close to the in-plane orientation, $B_y \gg B_z$, as $\eta_d \rightarrow \infty$ for $B_z \rightarrow 0$, and the expansion in powers of $\eta_d$ is no longer acceptable.
Also the perturbation theory is less appropriate for states with large $n$, as the energy difference between neighboring LLs is smaller then for states with small $n$ and, moreover, the interlayer coupling matrix elements, $\widetilde{\mathcal{T}}_{n,n\pm 1}$, increase with $\sqrt{n}$. The limits of the numerical approach are not clear at present, but we should have in mind that from the experimental point of view the angles with almost in-plane orientation are not so interesting due to the mosaic structure of most graphite crystals.
Conclusion
==========
Based on the simple nearest-neighbor tight-binding quantum mechanical model, we presented the calculation method of the band structure of Bernal-stacked graphite subject to tilted magnetic fields. We applied the lowest order perturbation theory to obtain analytic solutions of the formulated equations, the accuracy of which was later checked by the simplified numerical calculation. The special attention has been paid to the field dependence of the LLs at the $K$ and $H$ points of the graphite Brillouin zone where the density of states exhibits van Hove singularities in the perpendicular magnetic field. We have found that at the $K$ point, where the electron structure in the perpendicular magnetic field reminds strongly that of the bilayer graphene, the influence of the in-plane component of the magnetic field is negligible. On the other hand, at the $H$ point, where the electron structure mimics the behavior of the Dirac fermions, the application of the tilted magnetic field leads to the splitting of LLs. This splitting is of the order of several meV, which is an experimentally observable value, and it grows with increasing of the in-plane component of the magnetic field.
Acknowledgements
================
The authors benefited from discussions with Milan Orlita. The support of the European Science Foundation EPIGRAT project (GRA/10/E006), AV CR research program AVOZ10100521 and the Ministry of Education of the Czech Republic project LC510 is acknowledged.\
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abstract: 'Stable topological defects of light (pseudo)scalar fields can contribute to the Universe’s dark energy and dark matter. Currently the combination of gravitational and cosmological constraints provides the best limits on such a possibility. We take an example of domain walls generated by an axion-like field with a coupling to the spins of standard-model particles, and show that if the galactic environment contains a network of such walls, terrestrial experiments aimed at detection of wall-crossing events are realistic. In particular, a geographically separated but time-synchronized network of sensitive atomic magnetometers can detect a wall crossing and probe a range of model parameters currently unconstrained by astrophysical observations and gravitational experiments.'
author:
- 'M. Pospelov'
- 'S. Pustelny'
- 'M. P. Ledbetter'
- 'D. F. Jackson Kimball'
- 'W. Gawlik'
- 'D. Budker'
title: 'How do you know if you ran through a wall? '
---
[*Introduction.*]{} Despite a remarkable success of the Standard Model in describing all phenomena in particle physics, the cosmology presents a formidable puzzle, with dark energy and dark matter - two substances of unknown origin - comprising about 75% and 20% of the Universe’s energy budget. Last decades have seen a dramatic expansion of all experimental programs aimed at clarifying the nature of dark matter (DM) and dark energy (DE). While many widely ranging theories of dark matter exist, most of the experimental efforts go into searches of dark matter of some particle physics variety, producing upper limits on the DM-atom interaction strength. Tests of DE models occur on cosmological scale, showing so far its consistency with the cosmological constant.
The purpose of this Letter is to show that a new class of objects, stable topological defects (such as monopoles, cosmological strings or domain walls), that will contribute both to the DM and DE, can be searched for and studied with the global network of synchronized atomic magnetometers. To be more specific, we consider an example of a domain wall network built from the axion-like fields. Our focus on axion-like fields and the pseudoscalar interaction of these fields with matter, is motivated by the theoretical considerations of “technical naturalness", that allow preserving the lightness of the pseudoscalar fields despite a significant strength of interaction with matter. Observable effects of light pseudoscalar particles can vary considerably, depending on their mass $m_a$. We refer the reader to a sample of literature on the subject, covering a wide range of $m_a$ from $10^{-33}$ to $10^5$ eV [@sample].
Scalar-field potentials with some degree of discrete symmetries admit domain-wall-type solutions interpolating between domains of different energy-degenerate vacua [@aDW]. In these models, initial random distribution of the scalar field in the early Universe leads to the formation of domain-wall networks as the Universe expands and cools. For QCD-type axions, if stable, such domain walls could lead to disastrous consequences in cosmology by storing too much energy [@aDW]. For an arbitrary scalar field, where parameters of the potential are chosen by hand, the “disaster" can be turned into an advantage. Indeed, over the years there were several suggestions how a network of domain walls could be a viable candidate for DM or DE [@Press; @mura].
Herein, we revisit a subset of these ideas from a pragmatic point of view. We would like to address the following questions: (1) if a network of domain walls formed from axion-like fields exists in our galaxy, what are the chances for an encounter between the Solar system and a pseudoscalar domain wall? and (2) how could the event of a domain-wall crossing the Earth be experimentally determined? Given gravitational constraints on the average energy density of such walls and constraints on the coupling of axion-like fields to matter [@spin; @Raffelt], it is not obvious that the allowed parameter range would enable detection. Yet we show in this Letter that there is a realistic chance for the detection of the domain walls, even when the gravitational and astrophysical constraints are taken into account. This goal can be achieved with correlated measurements from a network of optical magnetometers with sensitivities exceeding $1~{\rm pT}/\sqrt{\rm Hz}$, placed in geographically distinct locations and synchronized using the global positioning system (GPS).
[*Physics of light pseudoscalar domain walls.*]{} We start by considering the Lagrangian of a complex scalar field $\phi$, invariant under $Z_N$-symmetry, $\phi\to\exp(i2\pi k/N)\phi$, where $k$ is an integer. We choose the potential in such a way that it has $N$ distinct minima $$\begin{aligned}
\label{ZN}
{\cal L}_\phi = |\partial_\mu \phi|^2 - V(\phi) ;~~ V(\phi) = \frac{\lambda}{S_0^{2N-4}}
\left| 2^{N/2} \phi^N - S_0^N\right|^2\!\!,\end{aligned}$$ where $S_0$ has dimension of energy and $\lambda$ is dimensionless.
Choosing $\phi = 2^{-1/2}S\exp(i a/S_0)$ to parameterize the scalar field, we find that the potential $V(\phi)$ is minimized for the following values of $S$ and $a$, $$S=S_0;~~ a=S_0\times \left\{ 0; ~ \frac{2\pi}{N};~\frac{4\pi}{N};...~\frac{2\pi(N-1)}{N}
\right\}.
\label{minima}$$ Freezing the Higgs mode to its minimum, $S=S_0$, produces the effective Lagrangian for the $a$ field, $$\label{La}
{\cal L}_a = \frac12 (\partial_\mu a)^2 - V_0 \sin^2\left(\frac{Na}{2S_0} \right),$$ with $V_0 = 4\lambda S_0^4$. The spatial field configuration $a({\bf r})$ interpolating between two adjacent minima represents a domain-wall solution. A network of intersecting domain walls is possible for $N\geq 3$. The solution for a domain wall along the $xy$-plane that interpolates between $a=0$ and $2\pi S_0/N$ neighboring vacua with the center of the wall at $z=0$ takes the following form, $$\label{wall}
a(z) = \frac{4S_0}{N}\times \arctan\left[\exp(m_az)\right]; ~~ \frac{da}{dz} = \frac{2S_0m_a}{N\cosh(m_az)}.$$ The characteristic thickness of the wall $d$ is determined by the mass $m_a$ of a small excitation of $a$ around any minimum, $d\sim 2/m_a$. The mass $m_a$ can be expressed in terms of the original parameters of the potential, $m_a = N S_0^{-1}(V_0/2)^{1/2}= (2\lambda)^{1/2}NS_0$. Owing to the fact that $V(\phi)$ can have many different realizations other than (\[ZN\]), we shall use solution (\[wall\]) as an example, rather than a generic domain-wall profile for $N\geq3$. The important parameters are the gradient of the field inside the wall, $m_aS_0/N$, and $m_a$, which determines the wall thickness.
[*Gravitational and astrophysical constraints.*]{} From the macroscopic point of view at distance scales much larger than $d$, the wall can be characterized by its mass per area, referred to as tension, $$\sigma = \frac{\rm Mass}{\rm Area} = \int dz \left| \frac{da}{dz}\right|^2= \frac{8S_0^2 m_a}{N^2}.$$ The network of domain walls will have an additional distance-scale parameter $L$, an average distance between walls, or a characteristic size of a domain. This parameter is impossible to calculate without making further assumptions about the mechanisms of wall formation and evolution. We treat it as a free variable and constrain the maximum energy density of the domain walls, $\rho_{\rm DW} \sim \sigma/L$ in the neighborhood of the Solar System by the dark-matter energy density, $\rho_{\rm DM} \simeq 0.4~ {\rm GeV/cm}^{3}$, $$\label{rho}
\rho_{\rm DW} \leq\rho_{\rm DM} \Longrightarrow \frac{S_0}{N} \leq 0.4{\rm~TeV} \times
\left[\frac{ L}{\rm 10^{-2}~ly } \times \frac{\rm neV}{m_a} \right]^{1/2}.$$ This constraint implies some flexible evolution of the domain-wall network and the possibility for them to build up their mass inside galaxies. We consider such the constraint as the most conservative, i.e. giving the most relaxed bound on $\rho_{\rm DW}$. If the network of domain walls is “stiff" and its density inside galaxies is not enhanced relative to an average cosmological value, then a stronger constraint can be derived by requiring that domain walls provide a (sub)dominant contribution to the dark-energy density, $\rho_{\rm DW} \leq\rho_{\rm DE}$, where $\rho_{\rm DE} \simeq 0.4\times 10^{-5}$ GeV/cm$^3$ [@Peebles2003]. In that case the constraint on $S_0/N$ is strengthened by $\sim 300$. Our choice of the normalization for $L$ and $m_a$ in (\[rho\]) is suggested by the requirement of having wall crossings within $\sim$10 yr with relative velocity of $v=10^{-3}c$ typical for galactic objects, and having the signal duration in excess of 1 ms. This choice can be self-consistent within the cosmological scenario for the formation of the domain-wall network from randomly distributed initial $a_{\rm in}$, assuming that the network is “frustrated", and exhibits $\rho_{\rm DW} \sim R^{-1}$ scaling, where $R$ is the cosmological scale factor. As a word of caution, we add that the numerical simulations of domain walls in some scalar field theories have shown much faster redshifting of $\rho_{\rm DW}$, and never achieved the frustrated state [@Sousa:2009is]. In light of this, some unorthodox cosmological/astrophysical scenarios for the formation of domain walls may be required.
We consider two types of pseudoscalar coupling of the field $a$ with the axial-vector current of a standard-model fermion, $J^{\mu} = \bar \psi\gamma_\mu\gamma_5 \psi$, $$\begin{aligned}
\label{lin}
{\cal L}_{\rm lin} = J^{\mu} \times i\phi
\overleftrightarrow{\partial}_\mu \phi^* \times \frac{1 }{S_0f_a } \, \longrightarrow
J^{\mu} \times \frac{\partial_\mu a }{f_a }
\\
{\cal L}_{\rm quad} =J^{\mu}\times\partial_\mu V(\phi) \times
\frac{4S_0^2}{ (f_a'N)^2V_0 } \,\longrightarrow
J^{\mu} \times \frac{\partial_\mu a^2 }{(f'_a)^2}
\label{quad}\end{aligned}$$ where the arrows show the reduction of these Lagrangians at the minima of $V(a)$, and $f_i,f_i'$ are free parameters of the model with dimension of energy. The normalization is chosen in a way to make connection with axion literature. The derivative nature of these interactions softens problems with “radiative destabilization" of $m_a$. It is also important that the effective energy parameters normalizing all higher dimensional interactions in (\[lin\]) and (\[quad\]) are assumed to be above the weak scale. Both ${\cal L}_{\rm lin} $ and ${\cal L}_{\rm quad} $ lead to the interaction of spins ${\bf s}_i$ of atomic constituents and the gradient of the scalar field, $$\begin{aligned}
H_{\rm int} &=& \sum_{i=e,n,p}
%\frac{2}{f_i}
2{\bf s}_i\cdot [f_i^{-1}\nabla a +(f'_i)^{-2} \nabla a^2],
\label{Hint}\end{aligned}$$ For light scalars of interest, the astrophysical bounds limit $|f_{n,p,e}| > 10^{9} $ GeV [@Raffelt], while bounds on quadratic $\partial_\mu a^2$ interactions are [*significantly weaker*]{}, $f'_i > 10~{\rm TeV}$ [@OP]. In what follows we will derive the signal from $f_i$ in (\[Hint\]), and then generalize it to the $f'_i$ case.
[*Spin signal during the wall crossing.*]{} The principles of sensitive atomic magnetometry are, for example, described in Ref. [@Budker2007Optical]. A typical device would use paramagnetic atomic species such as K, Cs, or Rb by themselves or in combination with diamagnetic atoms whose magnetic moments are generated by nuclear spin (e.g., the spin-exchange-relaxation-free \[SERF\] $^3$He-K magnetometer described in Ref. [@Romalis]). Specializing (\[Hint\]) for the case of two atomic species, $^{133}$Cs in the $F=4$ state and $^3$He in the $F=1/2$ state, we calculate the energy difference $\Delta E$ between the $F_z=F$ and $F_z=-F$ states in the middle of the wall, $$\begin{aligned}
\nonumber
H_{\rm int} = \frac{{\bf F\cdot \nabla} a}{F f_{\rm eff}};~f_{\rm eff}^{-1}({\rm Cs}) = \frac{1}{f_e}-\frac{7}{9f_p};
~ f_{\rm eff}^{-1}({\rm He})=\frac{1}{f_n};\\
\Delta E=\frac{ 4S_0m_a}{Nf_{\rm eff}}\simeq 10^{-15}\, {\rm eV} \!\times\! \frac{m_a}{\rm neV}\!\times\!
\frac{10^9\,\rm GeV}
{f_{\rm eff}}\!\times\! \frac{S_0/N}{0.4\, \rm TeV},\nonumber\\
\label{DeltaE}\end{aligned}$$ In these formulae we assumed that the nuclear spin is mostly due to unpaired neutron ($^3$He) or $g_{7/2}$ valence proton ($^{133}$Cs), and one can readily observe complementary sensitivity to $f_i$ in two cases. We can express these results in terms of the equivalent “magnetic field" inside the wall using $\mu{\bf B}_{\rm eff} {\bf F}/F =\nabla a {\bf F}/(Ff_{\rm eff})$ identification, where $\mu$ is the nuclear magnetic moment. The magnitude of $B_{\rm eff}$ (direction is impossible to predict) is given by $$\begin{aligned}
%B_{\rm eff} &\simeq& \frac{ |\nabla a|}{f_n \times 2.0 \mu_N}\nonumber \\
B_{\rm eff}^{\rm max}\simeq \frac{m_a}{\rm neV}\times \frac{10^9\,\rm GeV}{f_{\rm eff}}\times \frac{S_0/N}{0.4\, \rm TeV}
\times\left\{\begin{array}{c}
10^{-11}\, {\rm T} ~({\rm Cs})\\
-10^{-8}\, {\rm T} ~({\rm He})
\end{array}\right.\!\!\!,\nonumber\\
\label{eq:Beff}\end{aligned}$$ and the larger equivalent field strength for $^3$He originates from its smaller magnetic moment. The couplings and wall parameters in Eq. (\[eq:Beff\]) are normalized to the maximum allowed values from Eq. (\[rho\]). The duration of the signal is given by the ratio of wall thickness to the transverse component of the relative Earth-wall velocity, $$\Delta t \simeq \frac{d}{v_\perp}= \frac{2}{m_a v_\perp} = 1.3 \,{\rm ms}\times \frac{\rm neV}{m_a} \times \frac{10^{-3}}{v_\perp/c}.$$ Such crossing time can easily be in excess of the Cs magnetometer response time $t_r$, and we can combine the $B_{\rm eff}^{\rm max}$ and $\Delta t$ into a signal factor ${\cal S} = B_{\rm eff}^{\rm max} (\Delta t)^{1/2}$ to be directly compared to experimental sensitivity, $$\begin{aligned}
{\cal S}
\simeq \frac{0.4\, \rm pT}{\sqrt{\rm Hz}}\times \frac{10^9\,\rm GeV}{f_{\rm eff}}\times \frac{S_0/N}{0.4\, \rm TeV}\times
\left[\frac{m_a}{\rm neV} \frac{10^{-3}}{v_\perp/c}\right]^{1/2} \nonumber\\
\leq \frac{0.4\, \rm pT}{\sqrt{\rm Hz}}
\times \frac{10^9\,\rm GeV}{f_{\rm eff}}\times \left[ \frac{ L}{\rm 10^{-2}~ly }\frac{10^{-3}}{v_\perp/c}\right]^{1/2}\!\!\!\!\!,\;\;\;\;\;\;
\label{S}\end{aligned}$$ where in the inequality we used the gravitational constraint from Eq. (\[rho\]). The maximally allowed value for the signal ($\sim {\rm pT/\sqrt{Hz}}$), after taking into account the gravitational and astrophysical constraints, exceeds capabilities of modern magnetometers that can deliver fT/$\sqrt{\rm Hz}$ sensitivity [@Budker2007Optical]. For the $^3$He-K SERF magnetometer, the more appropriate figure of merit would be the tipping angle of the helium spin after the wall crossing, assuming that the typical crossing time is below the dynamical response time. Taking the spins to be oriented parallel to the wall, we calculate this angle to be $$\Delta \theta = \frac{4 \pi S_0}{v_\perp Nf_{\rm eff}} \simeq 5\times 10^{-3}\,{\rm rad}\times \frac{10^9\,\rm GeV}{f_{\rm eff}}\times
\frac{10^{-3}}{v_\perp/c}\times \frac{S_0/N}{0.4\, \rm TeV}.
\label{angle}$$ This could be far in excess of 10-nrad tipping angles that can be experimentally detected [@Kornack2005Nuclear]. Thus, both types of magnetometers offer ample opportunities for a realistic detection of the wall-crossing events. So far we have used the galactic constraints (\[rho\]), $\rho_{\rm DW} \leq\rho_{\rm DM} $. It is noteworthy that even if the energy density of walls in the galaxy does not exceed cosmological dark-energy density, i.e. $\rho_{\rm DW} \leq\rho_{\rm DE} $, the expected signal can reach $\Delta \theta \sim 10^{-5}$ rad and ${\cal S} \sim {\rm fT/\sqrt{Hz}}$, which is still a realistic signal for detection with the best magnetometers. It is remarkable that a possible domain-wall component of DE can, in principle, be detected in the laboratory.
Going over to $f'$ couplings, we notice that the structure of the signal is different: $B_{\rm eff}^{\rm max}$ now changes sign, vanishing in the middle of the wall. Taking $B_{\rm eff}^{\rm max}$ at $a= S_0\pi/(2N)$ inside the wall, and skipping intermediate states in a similar derivation, our sensitivity formulae (\[eq:Beff\]) and (\[S\]) are modified according to the following substitution, $$\begin{aligned}
\frac{10^9~{\rm GeV}}{ f_{\rm eff} }\longrightarrow 0.6\times 10^4 \times \left(\frac{10~{\rm TeV}}{f'_{\rm eff}}\right)^2 \times \frac{S_0/N}{0.4~{\rm TeV}},\end{aligned}$$ where again $f'$ is normalized on its minimum allowed value. One can observe a dramatic increase in the possible signal due to a much weaker astrophysical constraints on ${\cal L}_{\rm quad}$. In Fig. 1, we plot the experimental accessible parameter space in terms of characteristic time between wall crossing events, $T=L/(10^{-3} c)$, and strength of the coupling constants, $f$ and $f'$, fixing $m_a = 10^{-9}~eV,v_\perp/c=10^{-3}$ for concreteness, and saturating either DM or DE density constraints. We assume that the magnetometer sensitivity is ${\cal S} = {\rm fT/\sqrt{Hz}}$. The light(dark) shaded areas indicate the coupling range that can be realistically probed with the magnetometer network when DM(DE) constraints are saturated, by imposing all constraints and additionally requiring $T<10$ yr. One can see that the large part of the parameter space is accessible, and for the case of ${\cal L}_{\rm quad}$ even the DE constraint can allow for a detectable signal with $T<1$ yr.
[*Network of synchronized magnetometers.*]{} While a single magnetometer is sensitive enough to detect a domain-wall crossing, due to the rarity of such events it would be exceedingly difficult to confidently distinguish a signal from false positives induced by occasional abrupt changes of magnetometer-operation conditions, e.g., magnetic-field spikes, laser-light-mode jumps, etc. A global network of synchronized optical magnetometers is an attractive tool to search for galactic/cosmological domain walls, as it would allow for efficient vetoes of false domain-wall crossing events.
Ideally, one would require $n\geq 5$ magnetometer stations in such a network. The difference in timing $t_i$ of a putative signal is related to the transverse velocity and the unit normal vector to the wall, ${\bf n}$, $t_i-t_j = {\bf L}_{ij}\cdot {\bf n} v_\perp^{-1}$, where ${\bf L}_{ij}$ are the three-vectors of the relative positions of magnetometers $i$ and $j$. Four stations are required to specify magnetometer-defined 3D system of coordinates, and three time intervals between four $t_i$ will enable to unambiguously determine the three-vector ${\bf n}v_\perp^{-1}$. This makes the predictions for the timing of the event at the fifth station, $t_5$, which can be used as a tool for rejecting accidental backgrounds. Consider a network of similar magnetometers with fast response time separated by distances of $O(300 ~\rm km)$ operating during a long period ${\cal T}\sim {\rm yr}$. Suppose that $\tau$ is an average time between accidental spikes in the background above certain value $B_{\rm eff}^0$ that cannot be distinguished from the signal. Then the probability of having four events in four different stations within time intervals corresponding to the typical wall travel time from station to station, $t_{\rm trav} \sim l/v \sim {\rm s}$, is $P_{1234}\sim{\cal T }t_{\rm trav} ^3 \tau^{-4}$, where we take ${\cal T}\gg \tau \gg t_{\rm trav}$. To have this probability below one, one should achieve $\tau > 100\,{\rm s}$. If indeed four accidental background spikes lead to false signals in four stations within $t_{\rm trav}$, the domain wall interpretation will predict the event in the fifth station within a narrow window of the wall crossing $\Delta t\sim {\rm ms}$, and the probability of this to happen due to accidental background is $P_{12345} \sim (\Delta t/\tau)P_{1234}$, or less than $10^{-5}$ for $\tau \sim 100\,{\rm s}$. Increasing the number of stations will enable to search for weaker signal $B_{\rm eff}^0$, and tolerate shorter $\tau$ [@Abbott2004].
Recently we set up a prototype for the magnetometer network consisting of two magnetometers operated in magnetically shielded environments located in Kraków, Poland and Berkeley, USA (a separation distance of about 9000 km). One of the magnetometers (Kraków) is based on nonlinear magneto-optical rotation [@Pustelny2008Magnetometry], while the other magnetometer (Berkeley) is a SERF device [@Ledbetter2008Spin].
The magnetometers achieved comparable sensitivities of 10 fT/Hz$^{1/2}$, which can be further improved upon optimization. The expected parameters of the signal, $\Delta t \sim$ 1 ms and the minimum time-separation between the events $\Delta t_{\rm trav} \sim 30\,{\rm s}$, can be precisely determined using a GPS time source (for more details see Ref. [@Pustelny2012Global]). We have recently performed proof-of-principle experiments [@Pustelny2012Global] demonstrating the ability to correlate the signals of two magnetometers. In particular, we demonstrated significant reduction of noise and rejection of false-positive events present in magnetometer signals. The measurements proved the feasibility of correlated magnetic-field measurements opening avenues for further investigations involving more magnetometers.
[*Summary*]{}. We have shown that a network of modern magnetometers offers a realistic chance for detecting the event of an axion-type domain-wall crossing and can probe parts of the parameter space where such walls can contribute significantly to dark matter/dark energy.
The authors are grateful to N. Afshordi, A. Arvanitaki, A. Derevianko, J. Brown, S. Carroll, M. Kozlov, V. Flambaum, M. Kamionkowski, and M. Hohensee for discussions. This work was supported in part by the NSF and Polish TEAM program of the Foundation for Polish Science. SP is the scholar of the Polish Ministry of Science and Higher Education within the Mobility Plus program.
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---
abstract: 'We present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the $H^1$- and the $L^2$-norm on hierarchical hybrid grids. For the pre-asymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion ($1.6\cdot10^{11}$) degrees of freedom executed on 14310 compute cores demonstrates the efficiency of the new scaling approach.'
author:
- 'S. Bauer[^1]'
- 'D. Drzisga[^2]'
- 'M. Mohr'
- 'U. Rüde[^3]'
- 'C. Waluga'
- 'B. Wohlmuth'
bibliography:
- 'literature.bib'
title: '[A stencil scaling approach for accelerating matrix-free finite element implementations]{}[^4]'
---
matrix-free, finite-elements, variable coefficients, stencil scaling, variational crime analysis, optimal order a priori estimates
65N15, 65N30, 65Y20
Introduction {#sec:intro}
============
Problem setting and definition of the scaling approach {#sec:frame}
======================================================
Variational crime framework and a priori analysis {#sec:analysis}
=================================================
Guaranteed uniform coercivity
=============================
Reproduction property and primitive concept
===========================================
Cost of a work unit {#sec:cost}
===================
Numerical accuracy study and run-time comparison {#sec:num}
================================================
[^1]: Dept. of Earth and Environmental Sciences, Ludwig-Maximilians-Universit[ä]{}t M[ü]{}nchen
[^2]: Institute for Numerical Mathematics (M2), Technische Universit[ä]{}t M[ü]{}nchen
[^3]: Dept. of Computer Science 10, Friedrich-Alexander-Universit[ä]{}t Erlangen-N[ü]{}rnberg
[^4]: Submitted to the editors .
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[<span style="font-variant:small-caps;">**Material Laws and numerical Methods in applied superconductivity\
**</span>]{}
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[**ZAGUAN: Theses Digital Repository\
University of Zaragoza (Spain)**]{}
> [*Thesis submitted for the Degree of Doctor in Physics.\
> University of Zaragoza, Spain (2012).\
> *]{}
>
> [Author: Harold Steven Ruiz Rondan\
> ]{}
>
> [Advisor: Dr. Antonio Badía Majós\
> ]{}
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[**Material Laws And Numerical Methods In Applied Superconductivity\
**]{}
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[**Harold Steven Ruiz Rondan**]{}
  
[**Department of Condensed Matter Physics, University of Zaragoza.\
Spanish National Research Council (CSIC), Institute of Materials Science of Aragón (ICMA)**]{}
> [***Every step of my career\
> makes me feel proudest\
> of my family.\
> To them.***]{}
0.2cm
**Preface** {#preface .unnumbered}
===========
One century has elapsed since the discovery of superconductivity by Heike Kamerlingh Onnes, opening a new world of significant applications in technologies ranging from electric power devices such as motors and generators, large magnet systems such as those needed in storage rings for particle accelerators, and electricity transmission in power lines. As it is well-known, the technological usage of any superconducting material is based upon its ability to carry and maintain a current with no applied voltage whatsoever, i.e., with an almost negligible loss of energy even in those cases when the superconductor is subjected to strong enough applied magnetic fields. Although electrical currents can flow with a negligible loss of energy maintaining the superconductor in an appropriate temperature environment, superconductivity can be destroyed by the effect of a sufficiently intense magnetic field or the flow of a current density exceeding a critical value. Indeed, most of the technological applications of the superconductors are directly linked to their magnetic properties, and in particular in the way that they expell the magnetic fields. This fact leads to the classification of superconductors in two different kinds. On the one hand, *Type-I* superconductors are mainly characterized by a unique curve for the maximal applied magnetic field which a superconductor is able to expell before the sudden transition to the normal state occurs. On the other hand, *Type-II* superconductors are characterized by a new phase or “mixed-state” where the transition from the superconducting state to the normal state allows the existence of bundles of magnetic flux penetrating the sample (vortices), before reaching the sudden transition to the normal state. This remarkable property allows to preserve the superconducting state with the advantage of sustaining much higher magnetic fields, and therefore carrying much higher current densities. However, this ability is directly related to the pinning efficiency of a given material as the motion of vortices produces a high dissipation of energy which in turn can lead to the *quench* of the superconducting state. It is worth noting that all the superconductors, from metal-alloys to cuprates, fullerenes, $MgB_{2}$, iron-based systems that have been discovered along the last 60 years, are *Type-II* superconductors, and consequently almost all the actual superconducting technology is based on these kind of materials. Thus, since the vortices must be pinned by the underlying crystallographic structure and the presence of different kind of defects, the knowledge of the electromagnetic properties and laws governing the pinning of vortices becomes a crucial but not trivial issue for the understanding and developing of devices in the framework of applied superconductivity.
In spite of significant theoretical and practical interest, from the macroscopical point of view, the material laws and the electromagnetic properties of type-II superconductors still deserve attention, and currently no book exists that covers all the aspects about this topic in full depth. This thesis attempts to contribute with some novel numerical methods in applied superconductivity, including a comprehensive discussion of the different mechanisms involved in the vortex dynamics.
The book has been structured in three parts with sequential chapters increasing the level of complexity, both from the mathematical point of view and as concerns to the underlying phenomena. On the one hand, a general critical state theory for type-II superconductors with magnetic anisotropy, its computation, implications, and consequently some applications for particular problems, is what the first and second part try to convey. On the other hand, some microscopical aspects of the superconductivity have been also considered and the attained results have been compiled along the third part of this thesis.
In detail, the first part of this book is devoted to the study of the electromagnetic properties of type-II superconductors in the critical state regime. After an introductory Chapter 1, which reviews the classical statements of the critical state theory and derived approaches, Chapter 2 focuses on the variational theory for critical state problems and the establishment of a general material law for 3D critical states with an associated magnetic anisotropy and the underlying physics for the mechanism of flux depinning and cutting. Then, a technical but important issue arises and is covered in Chapter 3: how to deal with large-scale nonlinear minimization problems such as those presented in the general critical state theory for applied superconductivity, but in a personal computer. A well defined structure for the minimization functional, constraints, bounds, and preconditioners, based upon a set of FORTRAN packages, solve this problem.
Hopefully, at the end of the first part, the reader will feel either attracted or at least intrigued by the scope of our theory and methods. In this sense, the second part of this book is devoted to sketch some of the main results obtained along this line, i.e., we show some examples where we have implemented our general critical state theory whose impact affects not only the understanding of the physical properties of a superconducting system but also at its potential applications.
In Chapter 4, the advantages of the variational method are emphasized focusing on its numerical performance that allows to explain a wide number of physical scenarios. In particular, we present a thorough analysis of the underlying effects derived of the three dimensional magnetic anisotropy and different material laws (*or models*) which allow us to treat with the flux depinning and cutting mechanisms.
Chapter 5 deals with the study of the longitudinal transport problem (the current is applied parallel to some bias magnetic field) in type II superconductors. In particular, for the slab geometry with three dimensional components of the local electromagnetic quantities, the complex interaction between shielding and transport is solved. On the one hand, based on a simplified analytical method for 2D configurations, and on the other hand, based on a wide set of numerical studies for general scenarios (3D), it is shown that an outstanding inversion of the current flow in a surface layer, and the remarkable enhancement of the current density by their compression towards the center of the sample, are straightforwardly predicted when the physical mechanisms of flux depinning and consumption (via line cutting) are recalled. In addition, a number of striking collateral effects, such as local and global paramagnetic behavior, are predicted.
Chapter 6 addresses to a comprehensive study of the electromagnetic response of superconducting wires subjected to diverse configurations of transverse magnetic field and/or longitudinal transport current. In particular, we have performed a wide set of numerical experiments dealing with the local and global effects underlying to the distribution of field and current for a straight, infinite, type II superconducting wire, it immersed in an oscillating magnetic field applied perpendicular to its surface ($\textbf{B}_{0}$), and the simultaneous action of an AC transport current ($I_{tr}$). Thus, in a first part we have introduced the theoretical framework of this problem focusing on the numerical advantages of our variational method. Likewise, we provide a thorough discussion about some of the main macroscopic quantities which may be experimentally measured, such as the magnetization curve and the hysteretic AC loss, as well as on the local behavior of the electromagnetic quantities **E**, **B**, and **J**. Three different regimes of excitation have been considered: (*i*) Isolated electromagnetic excitations, where only the action of $\textbf{B}_{0}$ or $I_{tr}$ is considered, (*ii*) Synchronous electromagnetic sources, where the concomitant action of $\textbf{B}_{0}$ and $I_{tr}$ shows a unique oscillating phase and frequency, and (*iii*) Asynchronous electromagnetic sources, where $\textbf{B}_{0}$ and $I_{tr}$ do not show the same oscillating frequency and therefore are out-phase. The underlying effects of considering premagnetized wires under the above mentioned regimes are also considered. Thus, several striking effects as the strong localization of the local density of power loss, a distinct low-pass filtering effect intrinsic to the wire’s magnetic response, exotic magnetization loops, increases and decreases of the hysteretic AC loss by power supplies with double frequency effects, and significant differences between the widely used approximate formulae and the actual AC loss numerically calculated, have been detected and explained.
The last part of this dissertation concerns our contribution to another aspect of superconductivity. By means of a specific integral method applied to spectroscopic data, we have been able to draw some conclusions on the influence of the Electron-Phonon (E-Ph) coupling mechanism in cuprate superconductors. More specifically, we have focused on the analysis of high-resolution angle resolved photoemission spectroscopies (ARPES) in several families of cuprate superconductors. Although relying on solid (and sophisticated) techniques in the realm of quantum theory, we describe a phenomenological procedure that allows to obtain relevant physical parameters concerning the E-Ph interaction.
Thus, in chapter \[ch-7\], we introduce a novel theoretical model which allows a quite general explanation of the so-called nodal *kink effect* observed in ARPES, for several doping levels in the cuprate families $La_{2-x}Sr_{x}CuO_{4}$, $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$, and $YBa_{2}Cu_{3}O_{6+x}$.
Finally, in an effort to clarify the influence of the E-Ph coupling mechanism to the boson mechanism which causes the pair formation in the superconducting state, chapter \[ch-8\] addresses the study of the superconducting thermodynamical quantities, $T_{c}$, the ratio gap $2\Delta_{0}/k_{B}T_{c}$, and the zero temperature gap $\Delta_{0}$, for a wide set of natural and empirical equations.
In reading this book, we want to remark that each one of its parts have its own introduction and concluding sections, and also the list of references to the literature have been placed forward. In addition, a small glossary can be found at the end of this book.
Hopefully, this thesis may serve to bring a bigger community interested in the world of superconductivity, either in the application of their macroscopical properties or the understanding of their microscopical ground.
February 2012, Zaragoza - Spain.
{#section .unnumbered}
\[Intro-P1\] **Introduction** {#intro-p1-introduction .unnumbered}
-----------------------------
The high interest concerning the investigation of the macroscopic magnetic properties of type-II superconductors in the mixed state is markedly associated with its relevance to technological and industrial applications achieving elevated transport currents with no discernible energy dissipation. It relates to a wide list of physical phenomena concerning the physics of vortices, which may be basically analyzed in terms of interactions between the flux lines themselves (lattice elasticity and line cutting), and interactions with the underlying crystal structure averaged by the so-called flux pinning mechanism.
In a mesoscopic description of real type-II superconductors, the distribution of vortices may be simplified through a mean-field approach for a volume containing a big enough number of vortices and making use of an appropriate material law incorporating the intrinsic properties of the material. This picture of coarse-grained fields, i.e.: magnetic induction ${\bf B}\equiv
\langle{\bf b}\rangle$, electric current density ${\bf J\equiv
\langle{\bf j}\rangle}$ and electric field ${\bf E\equiv \langle{\bf
e}\rangle}$, allows to state the problem of the driving force due to the currents circulating in the superconducting sample and their balance with the limiting pinning force acting on the vortex lattice so as to prevent destabilization and the consequent propagation of dissipative states. Per unit volume, this reads ${\bf J}\times{\bf B}={\bf F}_p$ (or $J_{\perp}B =
F_p$). The underlying concept behind this balance condition is already a classical subject well known as the critical state model by Charles P. Bean [@P1-Bean_1964]. In this simple, but brilliant model, the response of the superconducting sample is provided by assuming that the electrical current density vector $\textbf{J}$ (oriented perpendicular to the direction of the local magnetic field vector ${\bf B}$) compensates with the pinning force, and then, it is constrained by a threshold value $\textbf{J}_{c}$ which defines a local critical state for the array of magnetic flux lines. Thus, in view of Faraday’s law, external field variations are opposed by the maximum current density $J_{c}$ within the material, and after the changes occur, $J_{c}$ persists in those regions which have been affected by an electric field. Although, such a model allows to capture the main features of the magnetic response of superconductors with pinning at low frequencies and temperatures, through the minimal mathematical complication, the stronger limitation of Bean’s ansatz is that one can just apply it to vortex lattices composed by parallel flux lines perpendicular to the current flow, and unless for highly symmetric situations **J** does not necessarily satisfy the condition $\textbf{J}=\textbf{J}_{\perp}$. In fact, a proper theory for the critical state must allow the coexistence of nonparallel flux lines. Thus, rotations of **B** can lead to entanglement and recombination of neighboring flux lines which brings a component of the current density along the local magnetic field, $\textbf{J}_{\parallel}$. This component generates distortions which also become unstable when a threshold value $J_{c\parallel}$ is exceeded, giving way to the so-called flux cutting phenomenon.
When the conditions $J_{\parallel}=J_{c\parallel}$ and $J_{\perp}=J_{c\perp}$ become active, the so-called double critical state appears [@P1-Clem_DCSM]. In simple words, this upgraded theory ([*double critical state model*]{} or DCSM) generalises the one-dimensional concept introduced by Bean [@P1-Bean_1964] to anisotropic scenarios for the material law in terms of the natural concepts $E_{\parallel}(J_{\parallel})$ and $E_{\perp}(J_{\perp})$ [@P1-Ruiz_PRB_2009; @P1-Ruiz_SUST_2010; @P1-Ruiz_SUST_2011]. From the mathematical point of view, the critical state problem can be understood as finding the equilibrium distribution for the circulating current density $\textbf{J}(\textbf{r})$ defined by the conditions $J_{\parallel}\leq J_{c\parallel}$ and $J_{\perp}\leq J_{c\perp}$, both consistent with the Maxwell equations in quasistationary form, and under continuity boundary conditions that incorporate the influence of the sources. Being a quasi-stationary approach, the critical state is customarily stated without an explicit role for the transient electric field. Thus, Faraday’s law is implicitly used through Lenz’ law by selecting the actual value $\pm
J_{c}$ or $0$ that minimizes flux variations when solving Ampere’s law $\nabla\times \textbf{B}=\mu_{0}\textbf{J}$ along the process. Customarily, one also considers situations where the local components of the magnetic field $\textbf{H}(\textbf{r})$ along the superconductor (SC) are much higher than the lower critical field $H_{c1}$ and well below $H_{c2}$ to allow the use of the linear relation $\textbf{B}=\mu_{0}\textbf{H}$.
Within this picture of the electromagnetic problem, in this first *part* of the book we introduce the important definitions and concepts of those topics behind the critical state theory, extending its scope for three dimensional cases with help of numerical methods in the framework of the variational formalism for optimal control problems. We want to emphasize that although it is not our intent to develop a comprehensive study of these mathematical topics, we will show common mathematical techniques which are found to be particularly useful in applied superconductivity. The reader is referred to the references [@P1-Badia_PRL_2001; @P1-Badia_PRB_2002; @P1-Jackson; @P1-Mayergoyz; @P1-Arfken; @P1-Pontryagin; @P1-Leitmann; @P1-Knowles] for a more thorough discussion of this material.
Chapter 1 is devoted to introduce the theoretical background that justifies the critical state concept as a valid constitutive law for superconducting materials. First, the critical state is described by the classical differential formalism of the Maxwell equations, and then, the prescribed magnetoquasistationary approximation is thoroughly discussed.
In chapter 2, our proposed general critical state theory is developed in two parts. Firstly, the critical state problem is posed in terms of an equivalent optimal control problem with variational statements, i.e., the classical Maxwell equations are translated to the variational formalism where a simpler set of integral equations with boundary conditions is to be solved by a minimization procedure. Is to be noticed, that despite of the fact that the reader can feel more familiar to the differential formalism, the numerical solution of the differential set of equations is much more cumbersome than minimizing an integral functional. Secondly, the underlying vortex physics is posed in terms of a quite general material law for type-II superconductors with magnetic anisotropy, which characterizes the conducting behavior in terms of the threshold values for the current density and the physical mechanisms of flux depinning and cutting.
Finally, chapter 3 covers the basic facts related to the computational method adopted for the solution of general critical state problems such as those tackled in the following part of this book. Here, no attempt is made to scrutinize through the FORTRAN packages for large scale nonlinear optimization. Instead, the presentation of this chapter must be understood as a schematic tool for dealing with a wide variety of problems in applied superconductivity.
\[ch-1\] **General Statements Of The Critical State**
=====================================================
### \[ch-1-1\] *1.1 The Critical State In The Maxwell Equations Formalism* {#ch-1-1-1.1-the-critical-state-in-the-maxwell-equations-formalism .unnumbered}
The fundamental concept on which the critical state theory relies is that, in many cases, the experimental conditions allow to analyze the evolution of the system in a magnetoquasisteady (MQS) regime of the time-dependent Maxwell equations accompanied by material constitutive laws, ${\bf H}({\bf
B})$, ${\bf D}({\bf E})$, and ${\bf J}({\bf E})$. Thus, Faraday’s and Ampere’s laws represent a coupled system of time evolution field equations $$\begin{aligned}
\label{Eq.1.1}
\partial_{t}{\bf B}=-\nabla\times{\bf E}\quad , \quad
\partial_{t}{\bf D}=\nabla\times{\bf H}-{\bf J} \, ,\end{aligned}$$ which together determine the distribution of supercurrents within the sample. Here, the induced transient electric field is determined through an appropriate material relation ${\bf J}({\bf E})$, and is used to update the profile of $\bf
J$.
Notice that, as equilibrium magnetization is usually neglected in the critical-state regime, one is enabled to use the relation $\textbf{B}=\mu_{0}\textbf{H}$, so that there are no average surface currents.Furthermore, as the magnetic fields of interest are some fraction of the critical transition magnetic field $H_{c2}$ that is much greater than the penetration field $H_{c1}$, the distribution of vortices and the corresponding supercurrents will be thermodynamically favoured to go into the superconductor, such that a ramp in the magnetic field is induced by the external excitation within the interval $[t,t+dt]$, see Fig. \[Figure\_1\_1\](a). Thus, as a consequence of a very fast diffusion (elevated flux flow resistivity), the electric field quickly adjusts to a constant value along the excitation interval, and once the magnetic field ramp stops, $E$ goes back to zero again.
The [*readjusting*]{} vertical bands are considered a second order effect and allow for charge separation and recombination, according to the specific ${\bf E}({\bf J})$ model \[see Fig. \[Figure\_1\_1\](b)\]. Therefore, we are allowed to model the flux as entering the superconductor at zero field cooling, where the electric field arises when some [*critical*]{} condition for the volume current density is reached ($J_c$ in the 1D representation). Then, corresponding to the MQS limit, the electric field instantaneously increases to a certain value determined by the rate of variation of the magnetic field and then goes back to zero.
By taking divergence in both sides of the Faraday’s and Ampere’s laws, and recalling integrability (permutation of space and time derivatives) it leads to the additional conditions $$\begin{aligned}
\label{Eq.1.2}
\partial_{t}(\nabla\cdot{\bf B})=0 \quad ,\quad
\partial_{t} (\nabla\cdot{\bf D}) + \nabla\cdot{\bf J} = 0 \, .\end{aligned}$$ Within this picture, the remaining Maxwell equations can be interpreted as “*spatial initial conditions*” for Eq. (\[Eq.1.2\]) which are defining the existence of conserved electric charges, i.e., $$\begin{aligned}
\label{Eq.1.3}
\nabla\cdot{\bf B}(t=0) = 0 \quad ,\quad \nabla\cdot {\bf D} (t=0) = \rho
(t=0)\, .\end{aligned}$$
In this sense, the set of equations (\[Eq.1.1\]), upon substitution of $\bf
H$, $\bf D$ and $\bf J$ through the constitutive laws, and with appropriate initial conditions, uniquely determine the evolution profiles ${\bf
B}({\bf r},t)$ and ${\bf E}({\bf r},t)$.
![\[Figure\_1\_1\] (a) Schematic representation the time dependence of the electromagnetic fields within the MQS regime. (b) Pictorial drawing of the critical state model in terms of a one dimensional ${E}({J})$ law.](Figure_1_1.pdf){width="100.00000%"}
Notice that, for [*slow*]{} and [*uniform*]{} sweep rates of the external excitations (magnetic field sources and/or transport current), the transient variables $\bf E$, $\bf D$ and $\rho$ are small, and proportional to $\dot{\bf B}$, whereas $\ddot {\bf B}$, $\dot {\bf E}$ and $\dot {\rho}$ are negligible. Thus, the main hypothesis within the MQS regime is that the [*displacement*]{} current densities $\partial _t{\bf D}$ are much smaller than $\bf J$ in the bulk and vanish in a first order treatment. This causes a crucial change in the mathematical structure of the Maxwell equations: Ampere’s law is no longer a time evolution equation, but becomes a purely spatial condition. It reads as $$\begin{aligned}
\label{Eq.1.4}
\nabla\times {\bf B} \simeq \mu_{0}{\bf J} \, ,\end{aligned}$$ with approximate integrability condition $\nabla\cdot{\bf J}\simeq 0$.
In the MQS limit, Faraday’s law is the unique time evolution equation. Then, one can find the evolution profile ${\bf B}({\bf r},t)$ from $$\begin{aligned}
\label{Eq.1.5}
\partial_{t}{\bf B}=-\nabla\times{\bf
E}=-\nabla\times\left[\rho(\textbf{J})~\mu_{0}\nabla\times{\bf
B}\right] \, .\end{aligned}$$ Here, $\rho ({\bf J})$ plays the role of a nonlinear and possibly nonscalar resistivity that should properly incorporate the physics of the threshold and dissipation mechanisms associated to the flux depinning and flux cutting mechanisms.
We want to mention that, although the *B-formulation* in Eq. (\[Eq.1.5\]) is definitely the most extended one, the possibilities of *E-formulations* [@P1-Barret_2006], *J-formulations* [@P1-Wolsky_2008], or a vector potential oriented theory (*A-formulation*) [@P1-Campbell_2009], in which the dependent variables are the fields ${\bf E}$, ${\bf J}$, or $\textbf{A}$ respectively, have also been exploited by several authors.
### \[ch-1-2\] *1.2 The Critical State Regime And The MQS Limit* {#ch-1-2-1.2-the-critical-state-regime-and-the-mqs-limit .unnumbered}
In spite of the seeming simplicity of the MQS approach ($\partial_{t}\textbf{D}\approxeq0$), we want to emphasize that the numerical procedure to solve a critical state problem is closely linked to the consequences of having assumed this limit. Below, two of the most relevant consequences of the MQS limit are highlighted.
1. Notice that, making use of the conductivity law through its inverse function ${\bf E}({\bf J})$, the successive field penetration profiles within the superconductor may be obtained by the finite-difference expression of Faraday’s law, $$\begin{aligned}
\label{Eq.1.6}
\frac{B_{l+1}-B_{l}}{\delta t}=-\nabla\times{\bf
E}~\left(\mu_{0}\textbf{J}_{l+1}\approx\nabla\times\textbf{B}_{l+1}\right) \, .\end{aligned}$$ Here we have assumed an evolutionary discretization scheme, where $\textbf{B}_{l}$ stands for the local magnetic field induction at the time layer $l\delta t$, and the current density profiles are related to some magnetic diffusion process that takes place when the local condition for critical state $\textbf{J}(\textbf{r})\leq \textbf{J}_{c}(\textbf{r})$ is violated. On the other hand, the constitutive law ${\bf D}({\bf E})$ which is not used in Eq. (\[Eq.1.6\]), plays no role in the evolution of the magnetic variables $\textbf{B}_{l+1}$ and $\textbf{J}_{l+1}$, which means that the magnetic “*sector*” is decoupled from the charge density profile because the coupling term (charge recombination) has disappeared. In this sense, notice that the local profile $\textbf{B}_{l+1}$ can be solved in terms of the previous field distribution $\textbf{B}_{l}$ and the boundary conditions at time layer $(l+1)\delta t$.
2. As the initial conditions must fulfill the Ampere’s law $\nabla\times\textbf{B}_{l}=\mu_{0}\textbf{J}_{l}$ as well as $\nabla\cdot\textbf{B}_{l}=0$ and $\nabla\cdot\textbf{J}_{l}=0$, only the inductive component of $\textbf{E}$ (given by $\nabla\times {\bf E}_{\rm ind} =
- \dot {\bf B}$, $\nabla\cdot {\bf E}_{\rm ind} = 0$) determines the evolution of $\bf B$ (Faraday’s law). At this point, the conducting law in its inverse formulation ${\bf E}({\bf J})$ seems show certain ambiguity, as far as two different material laws related by ${\bf E}_2 ({\bf J}) = {\bf E}_1 ({\bf
J}) + \nabla\Phi ({\bf J})$ determine the same magnetic and current density profiles. Going into some more detail, whereas for the complete Maxwell equations statement, the potential component of the electric field ($\nabla\times{\bf E}_{\rm pot} = 0$, $\epsilon _0 \nabla\cdot {\bf E}_{\rm pot} = \rho$), is coupled to $\bf B$ and ${\bf E}_{\rm ind}$ through the $\dot {\bf D}$ term (which contains both inductive and potential parts), within the MQS limit it is irrelevant for the magnetic quantities. In fact, one is enabled to include the presence of charge densities without contradiction with the condition $\nabla\cdot {\bf J} \simeq 0$ by means of inhomogeneity or nonlinearity in the $\textbf{E}(\textbf{J})$ relation. Then one has that the condition $\nabla\cdot\textbf{J}=0$ does not imply $\nabla\cdot\textbf{E}=0$. The charge density ${\rho}$ can be understood as a parametrized charge of [*static*]{} character as far as $\dot
{\rho}$ is neglected. As indicated above, once the magnetic variables are computed, one has the freedom to modify the “*electrostatic sector*” if necessary by the rule ${\bf E}({\bf J})+\nabla\Phi$ while still maintaining the values of $\bf B$ and $\bf J$. This invariance can be of practical interest as far as the “electrostatic” behavior in the critical state regime is still under discussion, it because of the inherent difficulties in the direct measurement of transient charge densities [@P1-Ruiz_SUST_2011; @P1-Joos_2006; @P1-Clem_2011_PRB; @Clem_2011_SUST].
\[ch-2\] **Variational Theory for Critical State Problems**
===========================================================
### \[ch-2-1\] *2.1 General Principles Of The Variational Method* {#ch-2-1-2.1-general-principles-of-the-variational-method .unnumbered}
As we have mentioned before, the numerical solution of the critical state problem from the differential formalism of the Maxwell equations may be cumbersome. One possibility for making the resolution of this system affordable is to find an equivalent variational statement of Eq. (\[Eq.1.6\]). Then, one can avoid the integration of these set of differential equations by *inversion* of a set of Euler-Lagrange equations $$\begin{aligned}
\label{Eq.2.1}
\mu_{0}\textbf{J}_{l+1}-\nabla\times\textbf{B}_{l+1}=0\, ,\end{aligned}$$ and $$\begin{aligned}
\label{Eq.2.2}
\mu_{0}\nabla\times\textbf{p}_{l}+\textbf{B}_{l+1}-\textbf{B}_{l}=0\, ,\end{aligned}$$ for arbitrary variations of the Lagrange multiplier (i.e., $\delta\textbf{p}_{l}$), and the time-discretized local magnetic induction field (i.e., $\delta\textbf{B}_{l+1}$).Eventually, the Lagrange multiplier, $\textbf{p}_{l}$, will be basically identified with the electric field of the problem.
Going into more detail, let us consider a small path step $\delta t$, from some initial profile of the magnetic field $\textbf{B}_{l}(\textbf{r})$ to a final profile $\textbf{B}_{l+1}(\textbf{r})$, and the corresponding $\textbf{J}_{l}(\textbf{r})$ and $\textbf{J}_{l+1}(\textbf{r})$. Defining $\Delta\textbf{B}=\textbf{B}_{l+1}-\textbf{B}_{l}$, both configurations can be considered to be connected by a steady process performing a small linear step, such that $\textbf{B}_{l+1}=\textbf{B}_{l}+s\Delta\textbf{B}$ with $s\in[0,1]\delta t$. Recalling that the initial condition fulfills Ampere’s law $\nabla\times\textbf{B}_{l}=\mu_{0}\textbf{J}_{l}$, as well as $\nabla\cdot\textbf{B}_{l}=0$ and $\nabla\cdot\textbf{J}_{l}=0$, the time-averaged Lagrange density (over whole space) is $$\begin{aligned}
\label{Eq.2.3}
{\cal L}
=\frac{1}{2}|\Delta\textbf{B}|^{2}+\textbf{E}\cdot(\nabla\times\textbf{B}_ { l+1
}
-\textbf{J}_{l+1})\delta t \, .\end{aligned}$$ Thus, the physically admissible Lagrangian multipliers in the critical state regime must satisfy the condition $$\begin{aligned}
\label{Eq.2.4}
\textbf{p}=\textbf{E}_{cs}\delta t \, ,\end{aligned}$$ where the critical state electric field $(\textbf{E}_{cs})$ must be properly defined by the imposed material law $\textbf{E}(\textbf{J})$.
However, concerning the “[*unknown parameter*]{}” $\textbf{J}_{l+1}$, as far as it is not allowed to take arbitrary values, we cannot impose arbitrary variations as it is customary for the typical steady condition of the Euler-Lagrange equations. Instead, an Optimal Control-like Maximum principle equivalent to a maximal projection rule $\textbf{\^{E}}\cdot\textbf{J}$ must be used (see Refs. [@P1-Badia_PRL_2001; @P1-Badia_PRB_1998]). For a more comprehensive review of the optimal control theory which can be understood as a generalization of the variational calculus, the interested reader is directed to see, for instance, Refs. [@P1-Pontryagin; @P1-Leitmann; @P1-Knowles].
In simple terms, the optimal control concept introduces a geometrical picture of the material law for the boundary conditions of the vector $\textbf{J}$ that may be of much help when discussing the idea of a general critical state theory. Summarizing, it is necessary to declare that there must be a region $\Delta_{\textbf{r}}$ within the **J** space (possibly oriented according to the local magnetic field $\textbf{\^{B}}$, and/or also depending on $|\textbf{B}|$ and **r**) such that nondissipative current flow occurs when the condition $\textbf{J}\in\Delta_{\textbf{r}}$ is verified. Thus, the minimum of the Lagrangian must be sought within the set of current density vectors fulfilling ${\bf J}\in\Delta_{\textbf{r}}$, i.e.: $\textbf{J}_{l+1}$ is determined by the condition $$\begin{aligned}
\label{Eq.2.5}
{\rm Min}\{ {\cal L} \}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}\equiv{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\, .\end{aligned}$$ Notice that an $\textbf{E}(\textbf{J})$ law is needed in addition to Eq. \[Eq.2.3\]. Thus, together with the concept of a very high dissipation when **J** is driven outside $\Delta_{\textbf{r}}$ by some nonvanishing electric field, Eq. \[Eq.2.5\] suffices to determine the relation between the directions of **J** and **E**. Notice that the maximal shielding condition is equivalent to the maximum projection rule, it means that the orthogonality condition of the electric field direction with the surface of $\Delta_{\textbf{r}}$ is recalled, and the Lagrange multiplier can be straightforwardly identified with the electric field of the problem, i.e., $$\begin{aligned}
\label{Eq.2.6}
{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{r}}}\equiv{\rm
Max}\{\textbf{E}\cdot\textbf{J}\}|_{\textbf{J}\in\Delta_{\textbf{r}}} \, .\end{aligned}$$ Notice also that Ampere’s law is imposed \[Eq. (\[Eq.2.1\])\] through the Lagrange multiplier, while the discretized version of Faraday’s law \[Eq. (\[Eq.2.2\])\] is derived as an Euler-Lagrange equation for the variational problem, so that absolute consistency with the Maxwell equations is obtained. In fact, maximal global (integral) shielding is achieved through a maximal local shielding rule \[Eq. (\[Eq.2.6\])\] that reproduces the elementary evolution of $\partial_{t}{\bf J}$ for a perfect conductor with restricted currents. Thus, in practice, if one explicitly introduces Ampere’s law $(\nabla \textbf{B}_{l+1}=\mu_{0}\textbf{J}_{l+1})$, minimization is made in terms of $$\begin{aligned}
\label{Eq.2.7}
{\cal F}[\textbf{J}_{l+1}]={\rm
Min}\left\{ -\int_{\Re^{3}}\textbf{E}\cdot\textbf{J}_{l+1} \right\} \, ,\end{aligned}$$ and the minimum is sought over $\textbf{J}_{l+1}\in\Delta_{r}$ for a fixed $\textbf{E}$. However, as we have mentioned before, special attention must be payed to the feasible ambiguity of the function $\textbf{E}(\textbf{J})$ as it can lead to fake values of the variables $\textbf{J}$.
Likewise, the straightforward equivalence between the convex functionals for Eqs. (\[Eq.2.5\]) and (\[Eq.2.7\]) allows to establish an equivalent minimization principle in terms of the general definitions $$\begin{aligned}
\label{Eq.2.8}
\textbf{B}=\nabla\times\textbf{A} \, ,\end{aligned}$$ and $$\begin{aligned}
\label{Eq.2.9}
\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi \, ,\end{aligned}$$ by imposing the material law $\textbf{E}(\textbf{J})$ through the Lagrange multiplier $\textbf{p}_{l+1}$. Thus, the minimization problem turns to find out the invariant gauge conditions $\nabla\varPhi_{l+1}$ and $\textbf{J}_{l+1}\in\Delta_{\textbf{r}}$ for a given function $\textbf{A}[\textbf{J}]$, in such manner that $$\begin{aligned}
\label{Eq.2.10}
{\cal F}[\textbf{B}_{l+1},\nabla \varPhi]
={\rm Min}\int_{\Re^{3}}&&
\frac{1}{2}|\Delta\textbf{B}|^{2}-\Delta\textbf{A}\cdot(\nabla\times\textbf{B}_{
l+1}-\textbf{J}_{l+1})\\
&&-\nabla\varPhi(\nabla\times\textbf{B}_{l+1}-\textbf{J}_{l
+1})\delta t \, . \nonumber\end{aligned}$$ We call the readers’ attention to notice that the uncoupling of the electromagnetic potentials can be accomplished by exploiting the arbitrariness involved in the definition of $\textbf{A}$. In fact, since **B** is defined through Eq. (\[Eq.2.8\]) in terms of **A**, the vector potential is arbitrary to the extent that the gradient of some scalar function can be added. Therefore, the “*magnetic sector*” could be decoupled of the “*electric sector*” if the physical admissible states in the time-averaged Lagrange density $L$ are invariant gauge of the Lagrange multipliers **p**. As a consequence, if the problem is such that there are no intrinsic electromagnetic sources, $\varPhi\equiv0$ (for type-II superconductors it means absence of transport current), a proper choice of **A** should satisfy the Coulomb gauge $(\nabla\cdot\textbf{A}\equiv0)$. In this sense, by using the Laplace equation, the second term in Eq. (\[Eq.2.10\]) is reduced to $\Delta\textbf{A}\cdot\delta_{t}^{2}\textbf{A}$ meanwhile the third term have vanished by assuming $\varPhi\equiv0$. Then, as the MQS approximation relies in assume that the electric field quickly adjusts to a constant value along the interval $[t+\delta t]$, for enough small time steps $\delta t$ (see Fig. \[Figure\_1\_1\]) the action of $\Delta\textbf{A}\cdot\delta_{t}^{2}\textbf{A}$ may be neglected, and therefore the solution of the critical state problem can be also achieved from the functional for the magnetic sector: $$\begin{aligned}
\label{Eq.2.11}
{\cal F}[\textbf{B}(\cdot)]={\rm Min}\int_{\Re^{3}}
\frac{1}{2}|\Delta\textbf{B}|^{2} \, .\end{aligned}$$ Recall that, the minimization principle is based on a discretization of the path followed by the external sources, meaning that it is an approximation to the continuous evolution whose accuracy increases as the step diminishes.
Moreover, we must emphasize that the derived functionals \[Eqs. (\[Eq.2.7\]) & (\[Eq.2.11\])\] are in matter of fact fully equivalents, as long as the minimization procedure accomplishes the boundary conditions imposed by the prescribed sources and the material law $\textbf{J}\in\Delta_{\textbf{r}}$. Thus, in those cases when an intrinsic electromagnetic source must be considered, i.e., $\nabla\varPhi\neq0$, the global set of variables must me constrained by the prescribed conditions. For example, if the superconductor is carrying a transport current $I_{tr}$ flowing through the surface $s$, one has to mandatorily consider the external constraint $$\begin{aligned}
\label{Eq.2.12}
\int_{s}\textbf{J}\cdot\hat{\textbf{n}}ds=I_{tr} \, \end{aligned}$$ and further update the distribution of current to satisfy the physical condition $\textbf{E}\cdot\textbf{J}=0$ (at those points where the magnetic flux does not vary), by means the use of a *calibrated* potential $\textbf{\~{A}}$. Thus, one of the advantage of the formulation in Eq. (\[Eq.2.11\]) is that the number of variables can be reduced avoiding to include the intrinsic variables associated to $\varPhi$, accordingly to the statement $\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi\equiv\partial_{t}
\textbf{\~{A}}$
Concluding, for 3D problems, it must be emphasized that the introduced minimization principle can be applied for any shape of the superconducting volume $\Omega$ as well as for any general restriction (material law) for the current density $\textbf{J}_{l+1}\in\Delta_{\textbf{r}}$. Different possibilities for the material law are described in the following chapter. It is also to be noticed that the searching of the minimum for the allowed set of current densities must fulfill the intrinsic condition $\nabla\cdot\textbf{J}=0$ to be consistent with charge conservation in the quasi-steady regime. Further, from the numerical point of view, the advantage of the variational formulation in Eqs. (\[Eq.2.7\]) & (\[Eq.2.11\]) is that one can avoid the integration of the equivalent partial differential equations and straightforwardly minimize the discretized integral by using a numerical algorithm for constrained minimization (see Chapter \[ch-3\]). This fact represents an important advantage in the performance and power of the numerical methods applied to the design of superconducting devices, where symmetry arguments can allow further simplifications and correspondingly faster numerical convergence.
### \[ch-2-2\] *2.2 The Material Law: SCs with magnetic anisotropy* {#ch-2-2-2.2-the-material-law-scs-with-magnetic-anisotropy .unnumbered}
In this section, we will continue our discussion of the critical state theory which still needs the explicit inclusion of a material law $\textbf{J}(\textbf{E})$ that dictates the magnetic response of a superconducting sample for a given external excitation. For simplicity, we start with an overview of the material law for 1D systems, that will be gradually generalized until a 3D formulation is reached.
***2.2.1 Onto The 1D Critical States***
\[ch-2.2.1\] For our purposes, it is sufficient to recall that the basic structure of the critical state problem (Fig. \[Figure\_1\_1\]) relates to an experimental graph within the $\{V , I \}$ plane that basically contains two regions defined by the critical current value $I_{c}$ as follows:
1. $-I_c \leq I \leq I_c$ with perfect conducting behavior, i.e.: $V=0$ and $\partial _t I = 0$.
2. For $I \gtrsim I_c$, the curve is characterized by a high ${\partial_{I} V}$ slope (and antisymmetric for $I \lesssim - I_c$). Further steps, with $I$ increasing above the critical value $I_c$, i.e., the eventual transition to the normal state, may be neglected for slow sweep rates of the external sources, which produce moderate electric fields.
Within the local description of the electromagnetic quantities involved in the superconducting response, different models have been used for the corresponding $E \leftrightarrow J$ graph, the most popular being
1. The [*power law*]{} model \
$E=\alpha\;{\rm sgn}(J)\left({|J|}/{J_c}\right) ^n$, with $\alpha$ a constant and $n$ high.
2. The [*piecewise continuous linear*]{} approximation \
$E=0$ for $|J|\leq J_c$, and $E=\beta\;{\rm sgn}(J)(|J|-J_c)$ for $|J|> J_c$, $\beta$ having a high value.
3. [*Bean’s model*]{} \
$J$ constant for $E=0$, and $J={\rm
sgn}(E)J_c$ for $E\neq 0$.
In some treatments, the first or second models are implemented, in order to transfer a full ${\bf E}({\bf J})$ law to the Maxwell equations. Further, notice that Bean’s model may be obtained from the other representations through the limiting cases $n\to\infty$ and $\beta\to\infty$ respectively.
The well known experimental evidence of a practical sweep rate independence for magnetic moment measurements (unless for high frequency alternating sources or at elevated temperatures) allows the use of the clearest Bean’s model because the critical state problem is no longer a time-dependent problem, but a path-dependent one, meaning that the trajectory of the external sources ${\bf H}_{0}$ uniquely determines the magnetic evolution of the sample. This makes an important difference when one compares to more standard treatments, as far as Faraday’s law is not completely determined from the path [@P1-Ruiz_PRB_2009]. Strictly speaking, one has $$\begin{aligned}
\label{Eq.2.13}
\Delta{\bf B} = - \nabla\times [{\bf E}\Delta t] \,\, ,\end{aligned}$$ with $\Delta t$ (and therefore $|{\bf E}|$) depending on the external sources. i.e., the absence of an intrinsic time constant gives way to the arbitrariness in the time scale of the problem.
Furthermore, in the actual 1D applications of Beans’s model, Faraday’s law is not strictly solved and **E** is absent from the theory. It is just the sign rule (the [*vectorial*]{} part of the material law), that is used to integrate Ampere’s law. Notice that such sign rule corresponds to a maximal shielding response against magnetic vector variations, and thus, determines the selection of $J= \pm J_c$.
Regarding the direction of **E**, in “1D” problems one has ${\bf
J}\parallel {\bf E}$ and both orthogonal to $\bf B$, such that the physical threshold related to a maximum value of the force balancing the magnetostatic term ${\bf J}\times{\bf B}$ gives place to the material law $$\begin{aligned}
\label{Eq.2.14}
J_{\perp}={\rm sgn}(E_{\perp})J_{c\perp} \qquad{\rm for}\qquad E_{\perp}\neq
0\,\, .
$$ Here, $E_{\perp}$ stands for the component of ${\bf E}$ along the direction ${\bf B}\times ({\bf J}\times{\bf B})$, and the material law falls in a “1D” scalar condition which describes the physical mechanism of vortex depinning.
At this point, the constitutive relation for the critical state describes the underlying physics for the coarse-grained fields in homogeneous type-II superconductors. However, it is well known that the coarse-grained behavior approach straightforwardly depends on the manufacturing process of the superconducting sample as far as inclusion of impurities, magnetic defects, or deformation of their cristal structure imposes the local coupling of $J_{c}$ with the intrinsic variation of the magnetic field B. Thus, for practical purposes we emphasize that the theoretical framework developed in this book is fully general, with caution of suggest to the experimentalist the need of an apriori measurement of the dependence $\textbf{J}_{c}[\textbf{r},\textbf{B},T]$ at least in those cases where the condition $\textbf{J}\perp \textbf{B}$ can be asserted. Henceforth, the implementation for a particular superconductor can be carried out.
***2.2.2 Towards The 3D Critical States***
\[ch-2.2.2\]
In the case of superconductors with anisotropy of the critical current, the description of their magnetic behavior requires the development of approaches more sophisticated than 1D-Bean’s model. The main issue is that, in general, the parallelism of ${\bf E}$ and ${\bf J}$ and their perpendicularity to ${\bf B}$ are no longer warranted. Then, the [*sign rule*]{} of Eq. (\[Eq.2.14\]) does not suffice for determining the solution of Eq. (\[Eq.2.6\]), and the optimal control condition with $\textbf{J}\in\Delta_{\textbf{r}}$ ([*a vectorial rule*]{}) must be invoked.
The simplest assumption that translates the critical state problem to 3D situations was already issued by Bean in Ref. [@P1-Bean_1970]. It has been called the ***isotropic critical state model*** (ICSM) and generalizes 1D Bean’s law to $$\label{Eq.2.15}
{\bf J} =J_{c}\,\hat {\bf E}\quad{\rm if} \quad E \neq 0 \, ,$$ i.e., the region $\Delta_{\textbf{r}}$ becomes a sphere. Noticeably, in spite of lacking a solid physical basis, thanks to its mathematical simplicity, this qualitative model has been widely used by several authors for reproducing a number of experiments with rotating and crossed magnetic fields [@P1-Badia_PRL_2001; @P1-Badia_PRB_2002; @P1-ICSM_app]. In any case, one could argue that statistical averaging over a system of entangled flux lines within a random pinning structure might be responsible for the isotropization of $\Delta_{\textbf{r}}$.
On the other hand, the general statement of the critical state in terms of well accepted physical basis was firstly introduced by John R. Clem [@P1-Clem_DCSM], and it is currently known as the ***double critical state model*** (DCSM). In particular, this theory assumes two different critical parameters, $J_{c\parallel}$ and $J_{c\perp}$ acting as the thresholds for the components of ${\bf J}$ parallel and perpendicular to ${\bf B}$ respectively. Notice that, $J_{c\perp}$ relates to the flux depinning threshold induced by the Lorentz force on flux tubes ($\textbf{J}\times\textbf{B}$), while the additional $J_{c\parallel}$ is imposed by a maximum gradient in the angle between adjacent vortices ($\textbf{B}\cdot\nabla\gamma=J_{\parallel}$) before mutual cutting and recombination occurs \[see Fig. \[Figure\_2\_1\] (a)\]. In brief, the DCSM may be expressed by the statement $$\begin{aligned}
\label{Eq.2.16}
\left\{
\begin{array}{ll}
{\bf J}_{\parallel} & =J_{c\parallel}\;\hat {\bf u}
\\
{\bf J}_{\perp} & =J_{c\perp}\,\hat {\bf v}
\end{array}
\right. \, ,\end{aligned}$$ being $\textbf{\^{u}}$ the unit vector for the direction of **B**, and $\textbf{\^{v}}$ a unit vector in the perpendicular plane to **B**.
Within the DCSM, the region $\Delta_{\textbf{r}}$ is a cylinder with its axis parallel to $\bf B$, and a rectangular longitudinal section in the plane defined by the unit vectors $\hat{\bf B},\hat{\bf J}_{\perp}$ \[see Fig. \[Figure\_2\_1\] (b)\]. The edges of the region $\Delta_{\textbf{r}}$ introduce a criterion for classifying the CS configurations into:
1. T zones or flux transport zones ($J_{\perp}=J_{c\perp}$; $J_{\parallel}<J_{c\parallel}$) where the flux depinning threshold has been reached (${\bf J}$ belongs to the horizontal sides of the rectangle),
2. C-zones or flux cutting zones ($J_{\parallel}= J_{c \parallel}~;~J_{\perp}<J_{c \perp}$) where the cutting threshold has been reached (${\bf J}$ belongs to the vertical sides of the rectangle),
3. CT zones ($J_{\parallel} = J_{c \parallel}$ and $J_{\perp} = J_{c \perp}$) where both $J_{\parallel}$ and $J_{\perp}$ have reached their critical values (corners of the rectangle), and
4. O zones depicted the regions without energy dissipation (the current density vector belongs to the interior of the rectangle).
![\[Figure\_2\_1\] [*Pane*]{} (a), Top: Schematic representation of the local relative orientations of [**B**]{} and [**J**]{}. Also sketched is the direction of the magnetic field at some neighboring point, at an angle $\gamma$. The vectors ${\bf B}$, ${\bf B}'$ and ${\bf J}$ do not necessarily lie at the same plane. The current is decomposed into its parallel and perpendicular components, i.e.: ${\bf J}={\bf
J}_{\parallel}+{\bf J}_{\perp}$. Bottom: the [*perfect conducting*]{} region within the plane perpendicular to $\textbf{B}$. An induced electric field is shown. Initially (${\bf
J}_{0}$), the high dissipation region is touched, but almost instantaneously [**J**]{} shifts along the boundary, reaching a point where the condition ${\bf
E}\perp\partial\Delta_{\textbf{r}}$ is fulfilled. Anisotropy within the plane is allowed. [*Pane*]{} (b): Geometric interpretation of the DCSM. [**J**]{} is constrained to the boundary of a rectangular region. T, C and CT states are related to the horizontal and vertical sides, and to the corners. Coupling between the components $J_{c\parallel}$ and $J_{c\perp}$ is envisaged by the EDCSM (dotted red ellipse). [*Pane*]{} (c): Our generalization of the material law for critical state problems or SDCST. Several regions are shown from the degree of the superelliptical functions ($n=1,2,3,4,6,10,20,40,\infty$) and $\chi=J_{c\parallel}/J_{c\perp}=1$.](Figure_2_1.pdf){width="100.00000%"}
Notice that $J_{c\parallel}$ and $J_{c\perp}$ are determined from different physical phenomena, and their values may be very different (in general $J_{c\parallel}>J_{c\perp}$ or even $J_{c\parallel}\gg J_{c\perp}$). Nevertheless, the coupling of parallel and perpendicular effects has been longer recognized by the experiments [@Clem_2011_SUST; @P1-Boyer_80] and, for instance, may be included in the theory by the condition $J_{c\parallel}=K B J_{c\perp}$ with $K$ a material dependent constant.
Recalling that the mesoscopic parameters $\textbf{J}_{c}$ are related to averages over the flux line lattice, interacting activation barriers for the mechanisms of flux depinning and cutting are expected and this may give place to deformations of the boundary $\partial\Delta_{\textbf{r}}$ \[see Fig. \[Figure\_2\_1\](b)\]. Thus, validated in those cases where a good agreement with the experiments is achieved, the theoretical scenario can be enlarged by a number of alternative approaches that focus on different aspects of the vast number of experimental activities in this field, e.g. one can identify the so-called:
1. *Isotropic critical state models (ICSM)* [@P1-Ruiz_PRB_2009; @P1-Ruiz_SUST_2010; @P1-ICSM_app]\
$J^{2}=J_{\parallel}^2+J_{\perp}^2 \leq J_{c}^2$
2. *Elliptical double critical state models (EDCSM)* [@P1-Ruiz_PRB_2009; @P1-EDCSM_app]\
$J_{\parallel}^2/J_{c\parallel}^2+J_{\perp}^2/J_{c\perp}^2\leq1$
3. *T critical state model (TCSM)* [@P1-Ruiz_PRB_2009; @P1-Ruiz_SUST_2010; @P1-Ruiz_SUST_2011; @P1-Brandt_2007; @P1-Ruiz_PRB_2011]\
$J_{\parallel}$ unbounded $\forall$ $J_{\perp}\leq J_{c\perp}$ .
Remarkably, the whole set of models have been recently unified by us in Ref. [@P1-Ruiz_SUST_2010] within a continuous two-parameter theory that poses the critical state problem in terms of geometrical concepts within the $J_{\parallel}-J_{\perp}$ plane (see Fig. \[Figure\_2\_1\] (c)). To be specific, in this framework, we have shown that by the application of our variational statement [@P1-Ruiz_PRB_2009], one is able to specify almost any critical state law by means of an integer index $n$, that accounts for the smoothness of the $J_{\parallel}(J_{\perp})$ relation, and a certain [*bandwidth*]{} characterizing the magnetic anisotropy ratio $\chi\equiv
J_{c\parallel}/J_{c\perp}$. This and the variational formalism introduced above constitutes the so-called ***Smooth Double Critical State Theory*** (SDCST), which allows to elucidate the relation between diverse physical processes and the actual material law.
Mathematically, the material law introduced in our general theory for the critical state problem or SDCST is based upon the idea that either material or extrinsic anisotropy can be easily incorporated by prescribing a region $\Delta_{\textbf{r}}$ where the physically admissible states of $\textbf{J}$ are hosted as limiting cases of a smooth expression defined by the two-parameter family of superelliptic functions, $$\label{Eq.2.17}
\left(\frac{J_{\parallel}}{J_{c\parallel}}\right)^{2n}+\left(\frac{J_{\perp}}{J_
{c\perp}}\right)^{2n}\leq 1.$$ We call the readers’ attention to the fact that an index $n=1$ and a bandwidth defined by $\chi\equiv J_{c \parallel}/J_{c \perp}=1$ correspond to the standard ICSM [@P1-ICSM_app]. On the other hand, when one assumes enlarged bandwidth (i.e.: $\chi>1$), the region $\Delta_{\textbf{r}}$ of the SDCST becomes the standard EDCSM introduced by Romero-Salazar and Pérez-Rodríguez [@P1-EDCSM_app]. When the bandwidth $\chi$ is extremely large, i.e., $J_{c\parallel}\gg J_{c
\perp}$, one recovers the so-called $T-$zones treated by Brandt and Mikitik [@P1-Brandt_2007]. Rectangular regions strictly corresponding to the DCSM [@P1-Clem_DCSM] are obtained for the limit $n\rightarrow\infty$ and arbitrary $\chi$. Finally, allowing $n$ to take values over the positive integers, a wide scenario describing anisotropy effects is envisioned \[Fig. \[Figure\_2\_1\](c)\]. Such regions will be named after superelliptical and their properties can be understood in terms of the rounding (or smoothing) of the corners for the DCSM.
\[ch-3\] **Computational Method**
=================================
In chapter 2.1 we have mentioned that the minimization functionals \[Eq. (\[Eq.2.7\]) or Eq. (\[Eq.2.11\])\] may be transformed so as to get a practical vector potential formulation. In turn, the resulting formulation can be expressed in terms of the so-called magnetic inductance matrices which allows a clearest identification of the set of elements playing some role in the minimization procedure. In this chapter, we shall discuss how to implement the above statements for general critical states in the framework of the computational methods for large scale nonlinear optimization problems.
Being more specific, in Eq. (\[Eq.2.11\]) the integrand $\frac{1}{2}
(\Delta{\bf B}) ^2$ can be rewritten as $\frac {1}{2} (\Delta{\bf B})\cdot
(\nabla \times \Delta{\bf A})$, and manipulated to get $\frac {1}{2} (\Delta{\bf A})\cdot (\nabla \times
\Delta{\bf B})$ plus a divergence term, fixed by the external sources at a distant surface. Now, the integral is restricted to the superconducting sample volume $\Omega$, because $\nabla \times \Delta{\bf B} = \mu _0 \Delta{\bf J}$ is only unknown within the superconductor. In addition, assuming that local sources such as an injected transport current may be introduced as an external constraint, and with the boundary condition that $\textbf{A}$ goes to zero sufficiently fast as they approach infinity, the vector potential can be expressed as: $$\label{Eq.3.1}
\Delta{\bf A} = \Delta{\bf A}_{0}
+ \frac {\mu _0 }{4 \pi} \int _\Omega \frac {\Delta{\bf J}}{|{\bf r} - {\bf
r}'|}
d^3 {\bf r}' \, .$$
This transforms ${\cal F}$ into a double integral over the body of the sample, i.e.: $$\begin{aligned}
\label{Eq.3.2}
{\cal F}[\textbf{A}(\cdot),J\in\Delta_{\textbf{r}}]= &&\frac {8\pi}{\mu_0}\int
_\Omega \Delta{\bf A}_{0}\cdot {\bf J}_{l+1}({\bf r})d^3 {\bf r}
\nonumber\\&&
+\int\!\int _{\Omega \times \Omega}
\frac {{\bf J}_{l+1}({\bf r}')
\cdot [{\bf J}_{l+1}({\bf r}) - 2 {\bf J}_{l}({\bf r})]}{|{\bf r} - {\bf
r}'|}d^3 {\bf r}d^3 {\bf r}'\;\end{aligned}$$
As a consequence, only the unknown current components within the superconductor $(\textbf{J}_{l+1})$ appear in the computation so reducing the number of unknown variables. At this point let me emphasize that Eq. (\[Eq.3.2\]) can be applied for any shape of the superconducting volume $\Omega$ as well as for any physical constraint (material law $\Delta_{\textbf{r}}$) for the local current density $\textbf{J}_{l+1}$, and further for any condition defined by the external sources ($\textbf{A}_{0}$). Above this, minimization must ensure the charge conservation condition by searching the minimum for the allowed set of current densities fulfilling $\nabla\cdot\textbf{J}=0$.
On the other hand, also it may be noticed that the double integral in Eq. (\[Eq.3.2\]) can be (*eventually*) identified as the Neumann formula once it has been transformed into filamentary closed circuits. A noteworthy fact is that regarding the superconducting volume, the coefficients of the intrinsic inductance matrices are straightforwardly independent of time and consequently, they appear in the root problem before going to minimize the functional. Indeed, the proper description of the inductance coefficients directly depends on the geometry of the superconductor and the boundary conditions defined by the dynamics of the external electromagnetic sources, where any symmetry of the problem allows further simplifications and correspondingly faster numerical convergence. To be specific, upon discretization in current elements ($I_{i}=J_{i}s_{i}$), the minimization functional for critical state problems bears the algebraic structure $$\begin{aligned}
\label{Eq.3.3}
{\cal
F}[I_{l+1}]=\frac{1}{2}\sum_{i,j}I_{i,l+1}M_{ij}I_{j,l+1}-\sum_{i,j}I_{i,l}M_{
ij } I_ {j,l+1}+\sum_{i}I_{i,l+1}\Delta A_{0}(M_{0}) \, ,\end{aligned}$$ with $\{I_{i,l+1}\}$ the set of unknown currents at specific circuits for the problem of interest, $M_{ij}$ their intrinsic [*inductance coupling coefficients*]{}, and $M_{0}$ the inductance matrices associated to the external sources $A_{0}$.
Corresponding to the critical state rule ${\bf J}\in\Delta_{\textbf{r}}$, in order to minimize Eq. (\[Eq.3.3\]) each value $J_i$ must be constrained. Thus, as it was described in chapter \[ch-2\].2.2, we have found that a number of constraints related to physically meaningful critical state models may be expressed in the algebraic form $$\begin{aligned}
\label{Eq.3.4}
{\rm
F}_{\alpha}\left(\sum_{i}I_{i}C_{ij}^{\alpha}I_{j}\right)\leq f_{0\alpha} \,
~\forall ~ j\end{aligned}$$ with $f_{0}$ some constant representing the physical threshold, and ${\rm F}_{\alpha}(\cdot)$ an algebraic function based upon a coupling matrix $C_{ij}^{\alpha}$ whose elements depend on the physical model. For example, in the simpler cases (isotropic models), the constraints correspond to assume the matricial elements $C_{ij}=\delta_{ij}$, and the physical threshold $f_{0}={J}_{c}^{2}$.
For simplicity, most technical procedures related to the introduction of intricate models and either depict the minimization functional in terms of the inductance coefficients (including those for external sources) will be left as matter of study of the following chapters (Part \[Part\_2\]). In return, below we present a thorough analysis of the computational tools handled for critical state problems at large scale.
With the purpose of obtaining a minimal understanding about how a critical state problem can be tackled from the numerical point of view, the computational method is sketched in the flow charts of figures \[Figure\_3\_1\] & \[Figure\_3\_2\].
![\[Figure\_3\_1\] Flow chart describing the preparation and management of the input elements for the objective function.](Figure_3_1.pdf){height="13cm" width="13cm"}
The first step is designing a grid which will allows to describe the superconducting volume $\Omega$ as a set of elements $\delta\Omega_{i}$, each of them characterized by a well defined current density flowing along the coordinates $\textbf{r}_{i}$. Then, the matrices for the intrinsic inductance coefficients between the elements $\textbf{J}(\textbf{r}_{i})$ and $\textbf{J}(\textbf{r}_{j})$ for all the set of possible couples $(\textbf{r}_{i},\textbf{r}_{j})\in\Omega$ must be calculated and stored on disk. As the mesh of points $(\textbf{r}_{i},\textbf{r}_{j})$ can be considerably large, we suggest take advantage on the matricial formalism provided by Matlab$^{\textregistered}$ and their own language for storage data. Once the spatial elements playing some role into the functional have been properly defined, the temporal *sector* must be introduced by means enough small path steps of the external electromagnetic sources, i.e., the experimental conditions must be connected by the finite difference expressions such a $\Delta\textbf{B}_{0}=\textbf{B}_{l+1}-\textbf{B}_{l}$, where the associated distribution of currents $\textbf{I}_{l+1}$ plays the role of unknown. To be specific, in those cases where the superconductor is subjected to an external magnetic field $\textbf{B}_{0}$, additional inductance matrices ($M_{0}$) must be introduced according to the definition $$\label{Eq.3.5}
\textbf{A}_{0}
(\textbf{B}_{0},\textbf{r}_{j})=\textbf{B}_{0}\times\textbf{r}_{ j } \, .$$ On the other side, the vector potential $\textbf{A}_{0}$ not only allows to define the contribution at the local potential $\textbf{A}$ produced by an external magnetic field ($\textbf{A}_{{\rm B}}$), rather it also allows consider the coupling with another materials such as ferromagnets.
Before going within the minimization procedure, it has to be noticed that those cases considering a transport current along the superconducting sample must be understood as a problem where the minimization variables are required to satisfy a set of auxiliary constraints \[see Eq. (\[Eq.2.12\])\] under the global critical state condition $I_{tr}\leq I_{c}$. Also, as the values for the elements $\textbf{I}_{l}(\textbf{r})$ are assumed to be known in advance, the linear elements into the argument of the functional (objective function) can be calculated before minimizing.
At this point, it is probably worthwhile to argue on what we mean by the computational method for minimization of an objective function. Firstly, this notion is clearly computer dependent, as the size of large scale problems can require a substantial amount of memory and store. Moreover, what is large in a personal computer can be significantly different from what is large on a super computer. The first machine just to have a smaller memory and storage than the second one, and therefore has more difficulty handling problems involving a large amount of data. Secondly, the size of the objective function strongly depends on the structure and the mathematical formulation of the problem and exploiting it is often crucial if one wants to obtain an answer efficiently. The complexity of this structure is often a central key in assessing the size of a problem. For example, for linear objective functions (not our case) it is possible to solve pretty large size problems (say four million variables). However, the objective function for problems in applied superconductivity is in general highly nonlinear and, for instance, the quadratic terms suggest to reduce the number of variables in a root square factor (say two thousand variables). One advisable possibility for reducing the number of elements in the objective function is subdividing the problem into loosely connected subsystems, i.e., all the internal operations which do not depend of the minimization variables must be preallocated to a well structured data (Figure. \[Figure\_3\_1\]). Lastly, an efficient algorithm for nonlinear optimization problems must be either invoked or built. Fortunately, nowadays there is a significant amount of available software with standard optimization tools which allow a faster foray in this matter [@P1-Matlab; @P1-Lancelot].
We must call reader’s attention on the fact that efficient algorithms for small-scale problems (in the sense that, assuming infinite precision, quasi-Newton methods for unconstrained optimization are invariant under linear transformations) do not necessarily translate into efficient algorithms for large scale problems. Perhaps the main reason is that, in order to be able to handle large problems with a high accuracy, the structure of the objective function and the minimization algorithms have both of them to be enough simple and tractable to avoid a wasting of time in the scaling of variables for the inner iteration subproblem (the minimization itself) and the finding of an optimum value for each one of the variables with respect to the remaining variables sought. In this context, one of the most powerful algorithms for large and nonlinear constrained optimization problems, known as LANCELOT, has been developed by the professors Andrew Conn (IBM corporation, USA), Nick Gould (Rutherford Appleton Laboratory, UK), Philippe Toint (Facultés Universitaries Notre-dame de la Paix, Belgium), and Dominique Orban (Ecole Polytechnique de Montreal, Canada) [@P1-Lancelot]. The wide number of optimization techniques provided by this package and their flexibility in handle and storage of large amounts of variables, make this program a clever choice for tackle highly complicated systems as those described by Eq. (\[Eq.3.2\]).
![\[Figure\_3\_2\] Flow chart describing the main structure of the computational method implemented along this book.](Figure_3_2.pdf){height="13cm" width="13cm"}
A thorough study of the minimization techniques and the computational language allocated in this package is far away of the purpose of this thesis. However, the structure of a general problem can be understood via the flow chart in figure \[Figure\_3\_2\]. In brief, the minimization functional is translated into a suite of FORTRAN procedures for minimizing an objective function, where the minimization variables are required to satisfy a set of auxiliary constraints and possibly internal bounds. Here, the major advantage of LANCELOT is the use of a Standard Input Format (SIF) as a unified method for communicating numerical data and FORTRAN subprograms with any optimization algorithm. Thus, when an optimization problem (minimizing or maximizing a sought of variables) is specified in the SIF decoder, one is required to write one or more files in ordered sections which accomplish the role of introduce the set of preconditioners for the objective function.
Once the set of input data has been structured accordingly to the number of variables and further on the temporal dependence of the experimental conditions (see Fig. \[Figure\_3\_1\]), one is enabled to predefine a set of input cards allowing the knowledgeable user to specify a priori known limits on the possible values of the objective function, as well as on the specific optimization variables, accuracy parameters, and scaling factors (see Fig. \[Figure\_3\_2\]). Then, the minimization functional or so-called objective function is subdivided in a set of groups, whose purpose is twofold: On the one hand, the linear and nonlinear (*quadratic*) elements for the minimization procedure are identified in a fore. Likewise, the specification of analytical first derivatives is optional, but recommended whenever possible. The SIF decoder allows also include the Hessian matrix of the objective function, if the second-order partial derivatives of the whole set of minimization variables are known,otherwise the derivatives of the nonlinear element functions can be approximated by some finite difference method.
Actually, the full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithmscan be straightforwardly called by LANCELOT where at least ten different minimization algorithms have been already implemented and coded according to the standard input format provided by the SIF decoder [@P1-Lancelot]. Notwithstanding, the solution obtained by LANCELOT may be compromised if finite difference approximations are used, it has been our experience that, once understood, the programming language of SIF is in fact quite efficient for problem specifications, in such manner that for objective functions correctly written, and constraint functions well defined, any method can efficiently reach to the solution sought. In this sense, additional groups may be announced to make up the objective function by including an “starting point” for the envisaged solution (if it is more or less known, or by defect it is equals to zero) or, for introducing additional constraints (external functions conditioning the system) as it is the case when the superconductor implies a flow of transport current.
It may happen that a specific problem uses variables or general constraints whose numerical values are widely different in magnitude, causing significant difficulties in the numerical convergence. However, LANCELOT also gives the chance of incorporate a list of scaling factors which are applied to the general constraints and variables separately before the optimization commences, allowing a clearest handling of the group elements in highly nonlinear problems as those herein considered. Thus, assuring a good convergence, whether single or double precision,is implying in turn to modify the experimental time stepping and either, the accuracy parameters such as, the number of iterations allowed, the constraint and gradient accuracy, the penalty parameter and the trust region for the optimizing.
Finally, in applied superconductivity, a set of complementary programs have to be developed in an effort to provide a comprehensive understanding of the temporal evolution of the electromagnetic quantities. For example, if the set of minimization variables corresponds to the local profiles of current density $\textbf{J}_{i}(\textbf{r}_{i})$, additional codes must to be used for calculating $\textbf{A}_{i}(\textbf{r})$, $\textbf{B}_{i}(\textbf{r})$, and $\textbf{E}_{i}(\textbf{r})$ in the whole $\Re^{3}$-space. Thus, although integrated quantities such as the magnetic moment $\textbf{M}(\textbf{J}_{i},\textbf{r}_{i})$ may be revealing a smooth trend despite the use of a poor numerical accuracy, it is of utter importance testing the numerical convergence by calculating the local profiles for the electromagnetic quantities concerning to derived quantities. In this sense, the following part of this book is devoted to the reliable solution of some interesting problems in applied superconductivity, where the critical state statement falls into a large scale optimization problem.
{#section-1 .unnumbered}
{#section-2 .unnumbered}
In summary, in this part we have shown that the critical state theory for the magnetic response of type-II superconductors may be built in a quite general framework and in turn it may be solved by several means. As our interest is to deal with highly nonlinear problems at large-scale, we have emphasized in the performance of variational methods and computational techniques for solving problems on personal computers.
We remark that the basic concepts underlying our generalization of the critical state theory can be identified as follows:
1. The critical state theory bears a Magneto Quasi Steady (MQS) approximation for the Maxwell equations in which $\ddot{\textbf{B}}$, $\dot{\bf E}$ and $\dot{\rho}$ are second order quantities and consequently, the displacement current densities $\dot{\textbf{D}}$ are much smaller than $\textbf{J}$ in the bulk and vanish in a first order treatment. This means that the [*magnetic flux dynamics*]{} can be entirely described by the finite-difference expression of Faraday’s law $$\begin{aligned}
\label{Eq.3.6}
\Delta\textbf{B}=-\nabla\times(\textbf{E}\delta t) \, ,\end{aligned}$$ where the physically admissible states must accomplish the MQS Ampere’s law, i.e., $\nabla\times\textbf{B}=\mu_{0}\textbf{J}$. Here, the inductive part of ${\bf E}$ may be introduced through Faraday’s law, whereas the role of electrostatic quantities is irrelevant for the magnetic sector. In other words, [**E**]{} may be modified by a gradient function (${\bf E}\to{\bf
E}+\nabla\phi$) with no effect on the magnetic response.
2. In type-II superconductors, the law that characterizes the *conducting behavior* of the material may be written in terms of thresholds values for the current density constrained to a geometrical region $(\textbf{J}\in\Delta_{\textbf{r}})$ which suffices to determine the relation between the directions of **E** and **J**. Thus, **E** is no longer an unknown variable but rather plays the role of a parameter to be adjusted in a direct algebraic minimization, i.e., $$\begin{aligned}
\label{Eq.3.7}
{\rm Min}\{L\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}\equiv{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\equiv{\rm
Max}\{\textbf{E}\cdot\textbf{J}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\, .\end{aligned}$$ In physical terms, the material “*reacts*” with a maximal shielding rule when electric fields are induced, and a perfect conducting behavior characterizes the magnetostatic equilibrium when external variations cease. In fact, is to be noticed that the above representation can be understood as the macroscopic counterpart of the underlying vortex physics. Thus, recalling that, in type II superconductors an incomplete isotropy for the limitations of the current density relative to the orientation of the local magnetic field arises from the different physical conditions of current flow either along or across the Abrikosov vortices, one may talk about magnetically induced anisotropy where the physical barriers of flux depinning and cutting are customarily depicted by the condition $\textbf{J}\leq\textbf{J}_{c}\in\Delta_{\textbf{r}}$. The evolution from one magnetostatic configuration to another occurs through the local violation of this condition, i.e.: ${\bf
J}\notin\Delta_{\textbf{r}}$ ($\textbf{J}>\textbf{J}_{c}$). However, owing to the high dissipation, an almost instantaneous response may be assumed, represented by a [*maximum shielding*]{} rule in the form ${\rm Max}\{{\bf
J}\cdot\hat{\bf E}\}\left.\right|_{{\bf J}\in\Delta_{\textbf{r}}}$.
3. With the aim of offering a meaningful reduction of the number of variables, we have shown that the problem can be simplified by solving a minimization functional with a underlying structure based upon inductance matrices \[see Eq. (\[Eq.3.3\])\]. In particular, the mutual inductance representation with ${\bf J}({\bf
r})$ as the unknown, offers two important advantages:
\(i) intricate boundary conditions and infinite domains are avoided, and
\(ii) the transparency of the numerical statement and its performance (stability) are outlined.
Then, the quantities of interest (flux penetration profiles and magnetic moment) are obtained by integration.
4. Most popular models for critical state problems have been generalized in our so-called *smooth double critical state theory* (SDCST) for anisotropic material laws [@P1-Ruiz_SUST_2010]. This theory relies on our variational framework for general critical state problems [@P1-Ruiz_PRB_2009] that allows us to incorporate the above-mentioned physical structure in the form of mathematical restrictions for the circulating current density. Two fundamental material-dependent quantities play key roles in this theory $(J_{c\parallel},J_{c\perp})$ related to the flux cutting and flux depinning thresholds. Notoriously, the boundary condition for the material law $\textbf{J}\in\Delta_{\textbf{r}}$ and the mutual interaction between the critical thresholds have been described in a quite general picture, based upon the relation between the coupling parameters $\chi\equiv J_{c\parallel}/J_{c\perp}$ and the smoothing index $n$ of the *superelliptical* condition $(J_{\parallel}/J_{c\parallel})^{2n}+(J_{\perp}/J_{c\perp})^{2n}\leq1$.
Hence, our SDCST cover a wide range of laws:
\(i) the isotropic model ($\chi^{2}=1$, $n=1\Rightarrow \Delta_{\textbf{r}}$ is a circle),
\(ii) the elliptical model ($\chi^{2}>1$, $n=1\Rightarrow \Delta_{\textbf{r}}$ is an ellipse),
\(iii) the rectangular model ($\chi^{2}\geq 1$, $n\to\infty\Rightarrow
\Delta_{\textbf{r}}$ is a rectangle),
\(iv) the infinite band model ($\chi^{2}\rightarrow\infty$), and
\(v) else others with smooth magnetic anisotropy ($\chi^{2}\geq 1$, $n\in{\rm N}
\geqslant 1 \Rightarrow \Delta_{\textbf{r}}$ is a rectangle with smoothed corners).
Finally, let me emphasize that the scope of our theory is rather beyond the actual examples treated in the following part of this thesis. On the one side, we have shown that the critical state concept allows arbitrariness in the presence of electrostatic charge and potential, and one could simply upgrade the models by the rule ${\bf E}\to{\bf E}+\nabla\phi$ if necessary. For instance, a scalar function $\phi$ may be introduced if the direction of ${\bf E}$ has to be modified respect to the maximum shielding rule in the MQS limit. On the other side, the extension of the theory to arbitrary sample geometries is intrinsically allowed by the mutual inductance representation. Thus, this first part has laid necessary groundwork for attacking general critical state problems in 3D geometry.
[100]{}
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TM}}$ A. R. Conn, N. I. M. Gould and , *LANCELOT, a FORTRAN, package for large-scale nonlinear optimization* (Springer Berlag, New York, 1992). **Latest version**: *GALAHAD* (LANCELOT B) 2011 documentation at http://www.galahad.rl.ac.uk/index.html IEEE standard 754 provides definitions for levels of precision in computational platforms. In our case, the objective function is able to handle numerical values between the range $[2.225073858507201{\rm E}^{-308},1.797693134862316{\rm E}^{308}]$ with a precision about 15 decimal digits. Anything outer overflows to an infinite $(\pm{\rm Inf})$ or does not represent a real number (NaN). ]{}
{#section-3 .unnumbered}
\[Intro-P2\] **Introduction** {#intro-p2-introduction .unnumbered}
-----------------------------
In the first part of this book the magnetic flux dynamics of type-II superconductors within the critical state regime has been posed in a generalized framework, by using a variational theory supported by well established physical principles and quite general numerical methods. The equivalence between the variational statement and more conventional treatments, based on the solution of the differential Maxwell equations together with appropriate conductivity laws have been stated. On other side, in an effort to explore new physical scenarios devoted to convey the advantages of the variational statement, in this part we present a thorough analysis of several problems of recognized importance for the development and physical understanding of intrinsic phenomena linked to the technological application of type-II superconductors.
In particular, Chapter \[ch-4\] is devoted to present the extensions of the so-called double critical state model to three dimensional configurations in which either flux transport (T-states), cutting (C-states) or both mechanisms (CT-states) occur. Firstly, we show the features of the transition from T to CT states. Secondly, we focus on our generalized expression for the flux cutting threshold in 3D systems and show its relevance in the slab geometry. Recall that, our method has allowed us to unify a number of conventional models describing the complex vortex configurations in the critical state regime. Thus, in this chapter several material laws already included in our generalized SDCST are compared to each other so as to weigh out the inherent influence of the magnetic anisotropy and the coupling between the flux depinning and cutting mechanisms. This is done by using different initial configurations (diamagnetic and paramagnetic) of a superconducting slab in 3D magnetic field, which allow to show that the predictions of the SDCST range from the collapse to zero of transverse magnetic moment in the isotropic model to nearly force-free configurations in which paramagnetic values can arbitrarily increase with the applied field for magnetically anisotropic current-voltage laws.
Chapter \[ch-5\] addresses the study of several intriguing phenomena for the transport current in type II superconductors. In particular, we present an exhaustive study of the electromagnetic response for the so-called longitudinal transport problem (current is applied parallel to the external magnetic field) in the slab geometry. On the one hand, we will introduce a simplified analytical model for a 2D configuration of the electromagnetic quantities. Then, based upon numerical studies for general scenarios (3D) we will go beyond the analytical models, and in general, it will shown that a remarkable inversion of the current flow in a surface layer may be predicted under a wide set of experimental conditions, including modulation of the applied magnetic field either perpendicular or parallel (longitudinal) to the transport current density. On the other hand, according to our SDCST where the magnetic anisotropy of the superconducting material obeys a geometrical region enclosed by a superelliptical function for the current density vector, a thorough characterization of the underlying mechanism of flux cutting and depinning has been performed. Thus, the intriguing occurrence of negative current patterns and the enhancement of the transport current flow along the center of the superconducting sample are reproduced as a straightforward consequence of the magnetically induced internal anisotropy. Moreover, we establish that the maximal transport current density allowed by the superconducting sample after compression towards the center of the sample, is related to the maximal projection of the current density vector onto the local magnetic field or material law. Also, it will be shown that a high correlation exists between the evolution of the transport current density and the appearance of striking collateral effects, such as local and global paramagnetic structures in terms of the applied longitudinal magnetic field. Finally, the elusive measurement of the threshold value for the cutting current component ($J_{c\parallel}$) is suggested on the basis of local measurements of the transport current density.
Finally, chapter 6 is devoted to introduce a thorough study of the electromagnetic response, either local or global, of straight infinite superconducting wires in the critical state regime under the action of diverse configurations of transverse magnetic field and/or longitudinal transport current. A comprehensive theoretical framework for the physical concepts underlying the temporal evolving of the electromagnetic quantities and the production of hysteretic losses is in a fore. Thus, along this line, and for the numerical implementation of our numerical statement, we have considered three different excitation regimes which are focused on the electromagnetic response of a superconducting wire with cylindrical cross-section: (*i*) Isolated electromagnetic excitations, in which only the action of an external source of oscillating transverse magnetic field, $\textbf{B}_{0}$, or an impressed AC transport current, $I_{tr}$, is conceived. (*ii*) Synchronous oscillating excitations, which deals with the simultaneous action of $\textbf{B}_{0}$ and $I_{tr}$ for experimental situations wherein both sources are showing the same oscillating features (identical phase and frequency). Eventually, in (iii) asynchronous excitation sources, we have addressed to most intricate configurations where the oscillating sources are out of phase by assuming that one of them sources is connected to a power supply with a double frequency than the other. The temporal dynamics of the assorted electromagnetic quantities, such as the local profiles of current density $\textbf{J}_{i}$, the lines of magnetic field (isolevels of the vector potential $\textbf{A}$), the vector components of the magnetic flux density $\textbf{B}$, the local density of power dissipation $\textbf{E}_{i}\cdot \textbf{J}_{i}$, the magnetic moment curves $\textbf{M}$, and the hysteretic AC losses $L$, are shown for each one of the above mentioned cases including a wide set of amplitudes for the oscillating excitations. Striking differences between the actual hysteretic losses (predicted by numerical methods) and the regular approximation formulas with the concomitant action of both sources are highlighted. Also quite interesting magnetization loops with exotic shapes non connected to Bean-like structures are outlined. An outstanding low pass filtering effect intrinsic to the magnetic response of the system, and a strongest localization of the heat release is envisioned for systems subjected to synchronous excitations. Furthermore, contrary to the generalized assumption that asynchronous sources may attain reductions in the hysteretic losses, we show that as a consequence of considering double frequency effects, noticeably increase of the hysteretic losses may be found.
\[ch-4\] **Type-II SCs With Intrinsic Magnetic Anisotropy**
===========================================================
As stated above, a rather complete description of irreversible phenomena in type-II superconductors at a macroscopic level is done through the SDCST framework by the application of our variational statement [@P2-Ruiz_PRB_2009] and further use of an appropriate material law $\textbf{J}(\textbf{E})$ [@P2-Ruiz_SUST_2010]. Essentially, our concept is to define the material law in terms of a geometrical region $\Delta_{\textbf{r}}(\textbf{J})$ within the $J_{\parallel}-J_{\perp}$ plane, such that nondissipative current flow occurs when the condition $\textbf{J}=\textbf{J}_{\parallel}+\textbf{J}_{\perp}\in\Delta_{\textbf {r}}$ is verified. In contrast, a very high dissipation is to be assumed when $\textbf{J}$ is driven outside $\Delta_{\textbf{r}}$. Is of utter importance to recall that the material law encodes the mechanism related to the breakdown of magnetostatic equilibrium as well as the dissipation modes operating in the transient from one state to the other. Thus, our scheme allows to translate the DCSM physics [@P2-Clem_DCSM] onto a region of currents defined in the $\Re^{3}$–*space* (3D) by a cylinder with its axis parallel to the local magnetic field $\textbf{B}$, and a rectangular longitudinal section in the plane defined by the vectors $\textbf{\textbf{J}}_{\parallel}=J_{c
\parallel}\textbf{\^{u}}$ and $\textbf{J}_{\perp}=J_{c
\perp}\textbf{\^{v}}$, being $\textbf{\^{u}}$ the unit vector for the direction of **B**, and $\textbf{\^{v}}$ a unit vector in the perpendicular plane to **B** (see Figure \[Figure\_4\_1\]).
Is to be noticed that in 2D problems with in-plane currents and magnetic field, the current density region straightforwardly coincides with the above mentioned longitudinal section ($\Delta_{\textbf{r}}=\Delta_{{\rm p}}$). We recall that, in this scheme the parts of the sample where the local profiles of the current density **J** have reached the boundary $J_{c\perp}$ (the flux depinning threshold) are customarily called flux transport zones ($J_{\perp}
= J_{c \perp}~;~J_{\parallel}<J_{c \parallel}$), and the profiles satisfying this condition are called T-states. They are represented by points in a horizontal band. Physically, the flux lines are migrating while basically retaining their orientation. On the other hand, regions where only the cutting threshold is active are denoted as flux cutting zones ($J_{\parallel}
= J_{c \parallel}~;~J_{\perp}<J_{c \perp}$) or simply as C-states. They are represented by points in a vertical band. In those regions where both mechanisms have reached their critical values are defined as CT zones ($J_{\parallel} =
J_{c \parallel}$ and $J_{\perp} = J_{c \perp}$) or CT-states. The current density vector belongs to the corners of a rectangle. Finally, the regions without energy dissipation are called O zones, and the current density vector belongs to the interior of the rectangle.
![\[Figure\_4\_1\] *Left*: The critical current restriction is represented by a cylindrical region $\Delta_{\textbf{r}}$ around the local magnetic field axis (length $2J_{c\parallel}$ and diameter $2J_{c\perp}$), $\Delta_{p}$ is the projection on the plane (x,y), $\alpha$ is the angle between the in-plane field projection and the x-axis, and $\theta$ is the angle between the field and the $z$ axis. *Right*: Geometrical representation of some of the material-law models depicted into the SDCST. The width and height of the region is controlled by the anisotropy parameter $\chi^{2}=J_{c\parallel}^{2}/J_{c\perp}^{2}$, and a smoothing index $n$.](Figure_4_1.pdf){width="100.00000%"}
In this chapter, and corresponding to the material laws depicted in the right side of figure \[Figure\_4\_1\], we will show that the variational statement may be used to predict the magnetic response of type-II superconductors with 3D anisotropy. In a first part, we will give the details related to the mathematical statement of the general critical state in a three dimensional slab geometry, i.e., both in-plane and perpendicular magnetic field components are applied to an infinite slab and varied in a given fashion. Then, the second and third part are devoted to apply the theory and predict the magnetic structure for the limiting cases, i.e., on one hand, the isotropic model ($\chi^2=1,n=1$) and on the other hand, the infinite width-band model ($\chi^2\rightarrow\infty$) or model of *T-states*. Subchapter \[ch-4\].4 is devoted to explore the set of effects associated to the DCSM hypothesis ($\chi^2\geq1,n\rightarrow\infty$). A wide range of applied fields will be considered, and our results compared to those with possible analytical approaches. Finally, as we are highly interested in knowing and understanding the role played by the physical mechanisms of flux depinning and flux cutting, the last part addresses different physical scenarios by means of different smooth double critical state models paying special attention to the influence of the smoothing index $n$ and the widthband $\chi$. In any case, smooth models have to be considered as related to a number of experiments that one could not explain within [*piecewise continuous*]{} models [@P2-Voloshin_1997; @P2-Fisher_2000; @P2-Voloshin_2001; @P2-Fisher_1997; @P2-Voloshin_2010; @P2-Fisher_2000_b] or the previous ones. In addition, appendix 1 explores the concept of critical angle gradient in 3D systems, as an alternative model to deal with anisotropic systems in the slab symmetry.
### \[ch-4-1\] *4.1 3D variational statement in slab geometry* {#ch-4-1-4.1-3d-variational-statement-in-slab-geometry .unnumbered}
In this subchapter, we derive a specific variational formulation in superconducting slabs for eventual 3D local field configurations (figure \[Figure\_4\_1\]), i.e., both in plane and transverse local magnetic field components emerge as derived effects of the flux depinning and flux cutting mechanisms. To be specific, we will consider an infinite slab, cooled under the assumption of an initial state defined by a uniform vortex lattice perpendicular to the external surfaces (i.e., a constant magnetic field $H_{z0}$), and then subjected to a certain process for the applied parallel field (i.e., $[H_{x0}(t),H_{y0}(t)]$) as is indicated in Fig. \[Figure\_4\_2\].
Recalling the symmetry properties of the electromagnetic quantities, one can describe the problem as a stack of current layers parallel to the sample’s surface, in such manner that the slab occupies the space $|z|\leq a$. Thus, it suffices to discretize the upper half, i.e.: $0\leq z_{i} \leq a$ as symmetry (or antisymmetry) conditions may be applied, and the position independence for a given value of $z_{i}$ ensures a divergenceless ${\bf J}$. Notice also that, within this approximation, one has to include two components of ${\bf J}$ within each layer, i.e.: $[J_{x}(z_{i}),J_{y}(z_{i})]$. At this point it would be worth mentioning that in order to simplify the mathematical statements we shall normalize the electrodynamic quantities by defining $\textbf{h}\equiv\textbf{H}/J_{c\perp}a$, $\textbf{j}\equiv\textbf{J}/J_{c\perp}$, and $\texttt{z}\equiv z/a$. Recall that one may assume the numerical value $J_{c\perp}$ as known *a priori* or obtained from experiment. In turn, our problem will be described in terms of $N_{s}$ discretized layers of equal thickness $\delta$ ($z_{i}=\delta\, i\;,\; \delta\equiv a/N_{s}$), each one characterized by a current density function $\textbf{j}(z_{i})=\textbf{j}_{x}(z_{i})+\textbf{j}_{y}(z_{i})$ distributed along $|z_{i}|\leq N_{i} a/N_{s}$. Eventually, for each layer, the unknown variables entering the minimization procedure may be defined accordingly to sheet currents, $I_{i,l+1}^{x}\equiv \delta j_{x}(z_{i},t=l+1)$ and $I_{i,l+1}^{y}\equiv \delta j_{y}(z_{i},t=l+1)$.
![\[Figure\_4\_2\] One the one hand, we show a pictorial illustration of the slab geometry with a perpendicular magnetic component $H_{z0}$ \[*sketch (a)*\]. On the other hand, schematics of the time dependence of the applied magnetic fields in the diamagnetic and paramagnetic configurations are depicted \[*sketches (b) and (c), respectively*\] ](Figure_4_2.pdf){width="100.00000%"}
Then, a straightforward application of Ampère’s law allows to express the penetrating magnetic field along the $x-axis$ as the sums over the layers: $$\begin{aligned}
\label{Eq.4.1}
h_{x}(z_{i})\equiv h_{i}^{x}=-\sum_{j>i}I_{j}^{y}-I_{i}^{y}/2 \, .\end{aligned}$$ Similarly, the local profiles for the longitudinal magnetic field component $h_{y}(z_{i})$ can be evaluated from, $$\begin{aligned}
\label{Eq.4.2}
h_{y}(z_{i})\equiv h_{i}^{y}=\sum_{j>i}I_{j}^{x}+I_{i}^{x}/2 \, .\end{aligned}$$ Following the concept introduced in the previous chapter \[see Eq. (\[Eq.3.3\])\], the following form of the objective function over the current sheets arises $$\begin{aligned}
\label{Eq.4.3}
{\cal
F}[I_{l+1}]=&&~\frac{1}{2}\sum_{i,j}I_{i,l+1}^{x}{\rm
M}_{ij}^{x}I_{j,l+1}^{x}-\sum_{i,j}
{\rm I}_{i,l}^{x}{\rm M}_{ij}^{x}I_{j,l+1}^{x}\nonumber \\
&&+\frac{1}{2}\sum_{i,j}I_{i,l+1}^{y}{\rm M}_{ij}^{y}I_{j,l+1}^{y}
-\sum_{i,j} {\rm I}_{i,l}^{y}{\rm M}_{ij}^{y}I_{j,l+1}^{y}\nonumber \\
&&-\sum_{i}I_{i,l+1}^{y}(i-1/2)({\rm h}_{0,l+1}^{x}-{\rm h}_{0,l}^{x})\nonumber
\\
&&+\sum_{i}I_{i,l+1}^{x}(i-1/2)({\rm h}_{0,l+1}^{y}-{\rm h}_{0,l}^{y}) \, .\end{aligned}$$
In an effort to provide an easier understanding of the above functional, we stress that only the physical quantities playing the role of unknowns are shown in italics. Then, we may straightforwardly identify the quadratic and linear groups for the minimization procedure. In detail, the sheet currents $I_{l+1}^{x}$ and $I_{l+1}^{y}$ represent the unknown variables to be minimized as a given initial state $[{\rm I}_{l}^{x},{\rm I}_{l}^{y}]$ is connected by the steady processes $\Delta {\rm h}_{0}^{x}={\rm
h}_{0,l+1}^{x}-{\rm h}_{0,l}^{x}$ and $\Delta {\rm h}_{0}^{y}={\rm
h}_{0,l+1}^{y}-{\rm h}_{0,l}^{y}$, and their mutual inductance matrices ${\rm
M}_{ij}^{x}$ and ${\rm M}_{ij}^{y}$. It is to be noticed that the index $l$ is introduced to indicate time discretization, i.e., $I_{i}(l+\delta
t)-I_{i}(t)\equiv I_{i,l+1}-I_{i,l}$. When this index is omitted, it will be meant that the element is time independent, i.e., it is valid for any step $l$ and calculated as an external input for the objective function (see Figs. \[Figure\_3\_1\] & \[Figure\_3\_2\]).
On the other hand, recall that the inductance matrices are directly linked to the design of a grid representing the *location* of variables into the superconducting volume. Thus, in the slab symmetry the circuits are just layers made up of straight lines along the $x$ and $y$ axis, and $\{I_{i}^{x},I_{i}^{y},\forall i\in\Omega\}$ is a compact notation for the whole set. Then, for our discretized array of layers the reader can check that a straightforward substitution of the squared components of the magnetic field entering the expression in Eq. (\[Eq.2.11\]) in terms of Eqs. (\[Eq.4.1\]) & (\[Eq.4.2\]), leads to the following formulas for the mutual inductance coupling elements: $$\begin{aligned}
\label{Eq.4.4}
{\rm M}_{ij}^{x}={\rm M}_{ij}^{y} & \equiv & 1+2\left[{\rm
min}\left\{ i,j\right\}\right]\quad \forall\; i\neq j
\nonumber\\
{\rm M}_{ii}^{x}={\rm M}_{ii}^{y}& \equiv & 2\left(\frac{1}{4}+i-1\right)\end{aligned}$$ Notice that inductive coupling only occurs between $x$ and $y$ layers separately, and the corresponding coefficients are identical.
Finally, we stress that minimization has to be performed under a prescribed material law $\textbf{J}\in\Delta_{\textbf{r}}$ (i.e., some of the geometrical regions depicted in Figure \[Figure\_4\_1\]), and ${\cal F}$ turns a new minimization functional for each different time step $(l=1,2,...)$. Specifically, the *three dimensionality* of the local magnetic field vector is controlled by the threshold values for the physical mechanisms responsible of the depinning and cutting of the vortices, i.e., the critical values $J_{c\parallel}$ and $J_{c\perp}$. Thus, in order to understand the three dimensionality of the vector $\textbf{J}$ one has to consider the polar decomposition $$\begin{aligned}
\label{Eq.4.5}
{\bf J}_{i}={\bf J}_{i}^{\parallel}+{\bf J}_{i}^{\perp\alpha}+{\bf
J}_{i}^{\perp{\theta}}\, ,
$$ with the parallel, azimuth and polar components of ${\bf J}_{i}$ defined in terms of the magnetic field direction $\textbf{\^{H}}_{i}$. After some simple algebraic operations in a Cartesian coordinate system, the following expressions are obtained for such components:
1. The current component parallel to $\textbf{\^{h}}_{i}$ or so-called cutting current component $I_{i}^{\parallel}$: $$\begin{aligned}
\label{Eq.4.6}
I_{i}^{\parallel}=\frac{h_{i}^{x}I_{i}^{x}+h_{i}^{y}I_{i}^{y}}{\left[(h_{i}^{x}
)^{2}+(h_{i}^{y})^{2}+(h_{i}^{z})^{2}\right]^{1/2}}
\, .\end{aligned}$$
2. The component of ${\bf I}$ perpendicular to the plane defined by the vectors $\textbf{\^{z}}$ and $\textbf{\^{h}}$ or so-called azimuthal current component $$\begin{aligned}
\label{Eq.4.7}
I_{i}^{\perp
\alpha}=\frac{-h_{i}^{y}I_{i}^{x}+h_{i}^{x}I_{i}^{y}}{\left[(h_{i}^{x})^{2}+(h_{
i}^{y})^{2}\right]^{1/2}}
\, .\end{aligned}$$
3. The component of ${\bf I}$ perpendicular to $\textbf{\^{h}}$ and contained in the plane defined by the vectors $\textbf{\^{z}}$ and $\textbf{\^{h}}$ or so-called polar current component $I_{\perp \theta}$: $$\begin{aligned}
\label{Eq.4.8}
I_{i}^{\perp
\theta}=\frac{h_{i}^{z}(h_{i}^{x}I_{i}^{x}+h_{i}^{y}I_{i}^{y})}{\left\{\left[(h_
{ i }^{x}
)^{2}+(h_{i}^{y})^{2}+(h_{i}^{z})^{2}\right]~
\left[(h_{i}^{x})^{2}+(h_{i}^{y})^{2}\right]\right\}^{1/2} } \, .\end{aligned}$$
Thus, as an example, within the framework of the DCSM hypothesis one has to invoke the conditions $$\begin{aligned}
\label{Eq.4.9}
\left(1-(h_{i}^{x})^{2}\right)
(I_{i}^{x})^{2}+\left(1-(h_{i}^{y})^{2}\right)(I_{i}^{y})^{2}-2h_{i}^{x}h_{i}^{y
} I_{i}^{x}I_{i}^{y} \leq {\rm I}_{c\perp}^{2} \, ,\end{aligned}$$ and $$\begin{aligned}
\label{Eq.4.10}
(h_{i}^{x} I_{i}^{x})^{2} + (h_{i}^{y} I_{i}^{y})^{2}+
2h_{i}^{x}h_{i}^{y}I_{i}^{x} I_{i}^{y} \leq {\rm I}_{c\parallel}^{2} \, . \end{aligned}$$
In summary, the objective function is constrained by the group of functions defined by Eqs. (\[Eq.4.9\]) & (\[Eq.4.10\]), and their minimization provides the magnetic response of the superconductor by means a collection of discretized current elements for the planar sheets of current density $[j_{i}^{x},j_{i}^{y}]$ at the time steps $l+1=1,2,3,...$.
### \[ch-4-2\] *4.2 Isotropic predictions in “3D” configurations* {#ch-4-2-4.2-isotropic-predictions-in-3d-configurations .unnumbered}
![\[Figure\_4\_3\] In the diamagnetic configuration and the isotropic model ($\chi^{2}=1$,$n=1$), we show the profiles for the local magnetic field component $h_{x}[z,h_{y}(a)]$ and their corresponding current-density profiles $j_{y}[z,h_{y}(a)]$, starting from a *first*-time step defined by $h_{x}(a)=1.1$ and $h_{z0}=1.5$. The current component $j_{x}[z,h_{y}(a)]$ and the cutting component $j_{\parallel}[z,h_{y}(a)]$ are also shown. The curves are labeled according to the component of longitudinal magnetic field at the slab surface, and correspond to the values $h_{y}(a)=0.040, 0.2, 0.4,
0.6, 0.8, 1.0, 1.2, 1.6, 2.0, 4.0,$ and $8.0$.](Figure_4_3.pdf){height="5cm" width="13cm"}
Below, we show the theoretical predictions for the region or material law defined by the region $(\chi^{2}=1,n=1)$ or isotropic model (see Fig. \[Figure\_4\_1\]), along the magnetization processes indicated in Figure \[Figure\_4\_2\].
Starting from a fully penetrated state with a magnetic field applied perpendicular to the slab surfaces ($h_{z0}$), i.e., a lattice of parallel vortices is assumed to nucleate parallel to the $z$ axis within the sample, one configures either a diamagnetic or a paramagnetic critical state by sweeping an applied parallel component $h_{x0}$ (thus inducing $j_y$). For example, if in a first temporal branch $(t<t')$ the material is subjected to an increasing magnetic field $h_{x}$ by means of $\Delta h_{x}(a)$, time path steps are characterized by tilted flux lines which penetrate the specimen until an equilibrium distribution is achieved (diamagnetic). Then, if the external magnetic field is subsequently lowered, thereby reducing the retaining magnetic pressure, flux lines migrate out of the sample until the equilibrium is restored (paramagnetic).
Eventually, at $t=t'$ an increasing ramp in the other longitudinal field component $h_{y0}$ is switched on and thus, an electric field $E_{x}$ arises at a surface layer of the superconductor which produces a current density $j_{x}$ that will screen the excitation. Then, owing the restrictions on the current density vector ${\bf j}$ introduced by the material law, the local component $j_{y}$ is affected and the corresponding local magnetic field $h_{x}(z)$ is pushed towards the center of the sample in the diamagnetic case (see Figure. \[Figure\_4\_3\]) or towards the external surface in the paramagnetic one (see Figure. \[Figure\_4\_4\]).
In detail, figures \[Figure\_4\_3\] & \[Figure\_4\_4\] show how the local component $h_{x}(0)$ increases (diamagnetic case) or it reduces (paramagnetic case) until the specimen is fully penetrated to satisfy the condition $h_{x}(z) = h_{x}(a)\;\forall~z$, i.e., $j_{y}(z)\rightarrow0$ as $h_{y}$ increases. As a consequence, no sign reversal in the induced currents is predicted. Two important features of this model are to be remarked.
![\[Figure\_4\_4\] Same as figure \[Figure\_4\_3\], but in the paramagnetic configuration illustrated in figure \[Figure\_4\_2\]. Here, the curves are labeled according to the values $h_{y}(a)=0.040, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 2.0,
3.0, 4.0, 5.0, 6.0, 7.0,$ and $8.0$.](Figure_4_4.pdf){height="5cm" width="13cm"}
On the one hand, it has to be recalled that by the symmetry conditions invoked before, we may assert that, for long loops, the contribution coming from the U turn at the far ends, exactly equals the contribution of the long sides. This may be shown starting from the condition $\nabla\cdot \textbf{J}=0$ (no sources) that allows us to consider the current-density distribution as a collection of loops and ensures the equality of the integrals over $z j_{x}$ and $z j_{y}$. Thus, the magnetic moment components per unit area may be obtained by numerical integration of the current density along the slab thickness, i.e., $$\begin{aligned}
\label{Eq.4.11}
\textbf{M}=\int_{-a}^{a}\textbf{z}\times\textbf{j}dz \, .\end{aligned}$$ Then, the saturation of the current components $j_{x}(z)$ and $j_{y}(z)$, indicates that the isotropic hypothesis ($j_{c}=j_{c\parallel}=j_{c\perp}$ i.e., $\chi^{2}=1, n=1$) is providing a straightforward explanation of the observed magnetization collapse [@P2-ICSM_app]. Furthermore, by comparison between the Figs. \[Figure\_4\_3\] & \[Figure\_4\_4\] with their corresponding magnetization curves (see curves in green in Fig. \[Figure\_4\_5\]), we also noticed that the magnetization collapse is obtained simultaneous to the monotonic reduction of the *cutting* current density or $j_{\parallel}$.
On the other hand, another related phenomenon, the so-called [*paramagnetic peak effect*]{} of the magnetic moment can not be foreseen by the isotropic material law. As a consequence, more sophisticated models have to be either invoked and revalidated by the study of the components of cutting and depinning for the current density. It has to be mentioned, that in previous works far away of the DCSM hypothesis, this observation was explored in terms of the so-called two velocity electrodynamic model as a crude approximation for the real dynamics in a flux line lattice (see details in Ref. [@P2-2hydrodynamic_model])
### \[ch-4-3\] *4.3 T-states in “3D” configurations* {#ch-4-3-4.3-t-states-in-3d-configurations .unnumbered}
In this section we continue the previous discussion, but now assuming a region with an infinite band-width $(\chi\rightarrow\infty)$ or so called model for T-states. Recalling that the magnetic field $\textbf{h}$ is measured in units of the physically relevant penetration field $J_{c\perp}a$ then, numerical experiments with $h_{z0}=0.1$, $h_{z0}=1.5$, and $h_{z0}=10$ will cover the range of interest.
![\[Figure\_4\_5\] The magnetic moment $M_{x}$ (solid lines) and $M_{y}$ (dashed lines) per unit area as a function of the applied magnetic field component $h_{y0}$ in the diamagnetic (left pane) and paramagnetic (right pane) initial configurations of Fig. \[Figure\_4\_2\]. Several cases are shown accordingly to the field intensities: $h_{x0}=1.1$ together with $h_{z0}=0.1$ (black), $h_{z0}=1.5$ (blue), and $h_{z0}=10.0$ (red), for the T-states model. Also, the corresponding curve for $h_{x0}=1.1$ and $h_{z0}=1.5$ for the isotropic model (green) are shown. Units are $j_{c\bot}a$ for $h$ and $j_{c\bot}a^{2}$ for $M$.](Figure_4_5.pdf){height="6cm" width="13cm"}
In figure \[Figure\_4\_5\] we display our results for the magnetic moment components per unit area under the experimental conditions depicted in figure \[Figure\_4\_2\]. The plots indicate the following features:
1. In general, a saturation is reached for $M_{y}(h_{y0})$, as compared to the eventual linear increase of $M_{x}(h_{y0})$ for the highest values of $h_{y0}$.
2. The higher $h_{z0}$, the sooner the saturation is reached.
3. Increasing $h_{z0}$ rapidly diminishes the slope of $M_{x}(h_{y0})$.
4. In the paramagnetic case, a minimum is observed (more evidently for $M_x$, and more visible for $h_{z0}=1.5$), that is smoothed either for the higher or lower values of this field component.
![\[Figure\_4\_6\] Magnetic field components $h_{x}(z)$ (left) and $h_{y}(z)$ (right) corresponding to the current-density profiles for the T-state limit in the diamagnetic configuration with $h_{z0}=1.5$ (top) and $h_{z0}=0.1$ (bottom) respectively.](Figure_4_6.pdf){height="10cm" width="13cm"}
Here, we want to call readers’ attention on the fact that our results for moderate perpendicular fields ($h_{z0}=1.5$ and $h_{z0}=10$) are in perfect agreement with the differential equation approach provided by Brandt and Mikitik in Ref. [@P2-Brandt_2007] (for more details see Ref. [@P2-Ruiz_PRB_2009]). For these cases, the underlying flux penetration profiles fully coincide with our calculations. However, in Ref. [@P2-Brandt_2007] the low field region was uncovered. Thus, here we will show the exotic behavior of the field and current-density profiles for the low field regime (e.g., $h_{z0}=0.1$) in comparison with the local electromagnetic behavior for moderate fields (e.g., $h_{z0}=1.5$).
![\[Figure\_4\_7\] Same as Fig. \[Figure\_4\_6\] but for the paramagnetic configuration](Figure_4_7.pdf){height="10cm" width="13cm"}
Figures \[Figure\_4\_6\] & \[Figure\_4\_7\] respectively display the behavior of the in-plane magnetic field components \[$h_{x}(z)$,$h_{y}(z)$\] in the diamagnetic and paramagnetic cases. On the one hand, in the diamagnetic case we notice that for moderate fields, the local magnetic field $h_{x}(z)$ is monotonically pushed towards the center of the sample with a nearly homogeneous distribution of the cutting current component $j_{\parallel}$ (see figure \[Figure\_4\_8\]). In an analogous manner, for the paramagnetic case the dynamics of the local component $h_{x}(z)$ is also related to the dynamics of the cutting current component. Thus, in a first stage the array of vortices closer to the center of the sample shows a decrease of the local component $h_{x}(z)$ until the full penetration state for the applied magnetic field $h_{y}(z)$ is achieved. Then, the cutting condition $j_{\parallel}\neq0$ is warranted for the whole sample. Interestingly, once the center of the sample reaches the cutting condition a fast change of sign in the slope of $j_{\parallel}$ is envisaged which straightforwardly corresponds to the change of sign in the slope of the magnetic moment $M_{x}$ or *peak effect*. In turn, it leads to a second stage which is mainly characterized by an array of vortices with a monotone increase of the components $h_{y}(z)$ and $h_{x}(z)$ under the boundary condition for the initial state (e.g., in our cases we have assumed $h_{x}(a)=1.1$ for t=t’, *see also Fig.* \[Figure\_4\_2\]).
On the other hand, it is to be noticed that the effects induced by the consideration of an unbounded cutting component are rather less simple for the low field regime (e.g., $h_{z0}=0.1$). Indeed, a steep variation of $h_{y}$ occurs for the inner region of the sample, corresponding to large values of $j_{\parallel}$ essentially dominated by $j_{x}$. On the contrary, $h_{x}$ displays a small slope, which relates to the condition $j_{\perp}=1$ (essentially, $j_{\perp}\approx j_{y}$ in the inner region).
![\[Figure\_4\_8\] Profiles of the parallel current component $j_{\parallel}$ for the T-state hypothesis “$J_{c\parallel}\to\infty$ and $J_{c\perp}=1.0$”. The curves are labeled according to the applied field $h_{y0}$ (at right), assuming $h_{z0}=1.5$. The diamagnetic (left pane) and paramagnetic (right pane) cases are shown. In the paramagnetic case, the profiles of $J_{\parallel}$ for $h_{y0}=1.35,\, 1.7,\,2.0,\, 2.3,\, 2.6,\, 3.0$ are shown as an inset, and correspond to the sign change in the slope of the magnetic moment $M_{x}$ (see figure \[Figure\_4\_5\]).](Figure_4_8.pdf){height="8cm" width="13cm"}
More into detail, Fig. \[Figure\_4\_9\] displays the behavior of the projection of the current density onto the direction of the magnetic field ($j_{\parallel}$) under the ansatz of a T-state structure for $h_{z0}=0.1$. It is apparent that the full penetration of the T-state perturbation requires a high field component ($h_{y0}\approx 18$ and $h_{y0}\approx 30$ for the diamagnetic and paramagnetic cases respectively), and a very high ratio $J_{\parallel}/J_{c\perp}\equiv j_{\parallel}$ ($\approx
180$ for the diamagnetic case and $\approx 340$ for the paramagnetic one). Notice that until these values are reached, one has $J_{\parallel}=0,
J_{\perp}=1$ for the inner part of the sample, and a certain distribution $J_{\parallel}(z)$ for the outer region. We also recall a somehow complex structure with one or two minima in between the surface of the sample and the point reached by the perturbation. Interestingly, when $h_{y0}$ grows, the minimum becomes very flat, corresponding to a nearly constant value of $j_{\parallel}$. From the physical point of view, the minimum basically represents the region where ${\bf h}$ rotates so as to accommodate the penetration profile ${\bf h}(z)$ to the previous state of magnetization $(h_{x},0,h_{z0})$. From the point of view of Faraday’s law, this takes place as quickly as possible so as to minimize flux variations.
Finally, we want to call readers’ attention on two derived facts from this model. On the one hand, is to be noted that within the T-states model the magnetization collapse does not take place at least for perpendicular fields lower than $h_{z0}=10$. Indeed, a tiny slope on the magnetization moment curve is still present (see Fig. \[Figure\_4\_5\]). On the other hand, as consequence of the non constrained cutting component, there is no restriction on the longitudinal component of the current density that increases arbitrarily towards the center of the sample (see Figs. \[Figure\_4\_6\] & \[Figure\_4\_7\]). Thus, strictly speaking, this model can not be consider as physically admissible although some of the experimental evidences may be reproduced.
![\[Figure\_4\_9\] Profiles of the component $j_{\parallel}$ for the limit $J_{c_{\parallel}}\to\infty$ (T-state) with $h_{z0}=0.1$. In all cases the perpendicular current profiles satisfy $J_{\perp}=J_{c\perp}=1.0$. The diamagnetic (left) and paramagnetic (right) cases are shown. Left: inset (a) shows a zoom of $j_{{||}}$ for the first profiles of $h_{y_{0}}$. Inset (b) schematically shows the evolution of the vector $\bf J$ as function of its parallel and perpendicular components. Bottom: inset (a) shows a zoom of $j_{{||}}$ for the first profiles of increasing $h_{y0}$. Inset (b) shows the magnetic moment components ($M_{x}$, $M_{y}$) per unit area as a function of $h_{y0}$.](Figure_4_9.pdf){height="7cm" width="13cm"}
### \[ch-4-4\] *4.4 CT-states in “3D” configurations* {#ch-4-4-4.4-ct-states-in-3d-configurations .unnumbered}
Below, we show the theoretical predictions derived by choosing a rectangular region for the material law $\Delta_{r}$ with a finite bandwidth $\chi$ or so-called DCSM (i.e., $\chi\geq1$ and $n\rightarrow\infty$ within our SDCST). In order to keep the previous sequence of results and the underlying ideas, the same numerical experiments depicted in Fig. \[Figure\_4\_2\] will be analyzed.
Before going into detail, let us recall that the cases analyzed in the previous two sections directly correspond to the lower and higher limits of the double critical state approach, and in consequence, any material law displayed between them will be characterized by intermediate profiles for the electromagnetic quantities (more details in Refs. [@P2-Ruiz_PRB_2009; @P2-Ruiz_SUST_2010]). Thus, we can summarize the rich phenomenology encountered by means a thorough analysis of the magnetic moment curves $M_{x}(h_{y0})$ and $M_{y}(h_{y0})$, and the local profiles for the cutting current component $j_{\parallel}(z)$.
![\[Figure\_4\_10\] Magnetic moment curves per unit area ($M_{x},M_{y}$) as a function of the applied field $h_{y0}$ for the experimental configurations displayed in Fig. \[Figure\_4\_2\]. Shown are the diamagnetic (left) and paramagnetic (right) cases for $h_{x0}=1.1$ and the moderate perpendicular field $h_{z0}=1.5$. The T-state curves ($j_{c\parallel}\gg 1$) are shown for comparison with the DCSM-cases: $j_{c_{||}}^{2}=$3.0, 2.7, 2.3, 2.0, 1,7, 1.3, 1.0. The insets show the particular case $j_{c_{||}}^{2}=2.0$ in the region where the transition T$\rightarrow$CT is visible.](Figure_4_10.pdf){height="7cm" width="13cm"}
Firstly, we show the corrections to $M_x$ and $M_y$ both for the diamagnetic and paramagnetic cases either at a moderate perpendicular field (Figure \[Figure\_4\_10\]) or at a low perpendicular field (Figure \[Figure\_4\_11\]), when the DCSM region corresponds to the aspect ratio values $\chi^{2}=1.0,
1.3, 1.7, 2.0, 2.3, 2.7$ and $3.0$.
On the one hand, for a moderate perpendicular field (e.g., $h_{z0}=1.5$), it is noticeable that the limitation in $j_{c\parallel}$ produces a [*corner*]{} in the magnetic moment dependencies $M_{x,y}(H_{y0})$, which establishes the departure from the [*master curve*]{} defined by the T-state model. The corner in $M_{x}$ and $M_{y}$ appears at some characteristic field $h_{y0}^{*}$ that increases with $\chi$, eventually disappearing within the region of interest. Thus, the higher value of $h_{y0}$ for which the corner can not be observed ($h_{y0}\approx 3$ in the conditions depicted into the plots of Fig. \[Figure\_4\_10\]), defines the acting threshold of the T-state model. The fine structure of the corner is shown in the insets of Fig. \[Figure\_4\_10\]. Notice that, indeed, the deviation from the master curve takes place in two steps, being the second one that really defines the corner.
![\[Figure\_4\_11\] Same as figure \[Figure\_4\_10\], but in the lower perpendicular field regime $h_{z0}=0.1$. Here, the following DCSM-cases: $J_{c_{\parallel}}^{2}=$ 11.0, 9.0, 7.5, 6.0, 4.5, 3.0, 2.0, and 1.0, are shown. ](Figure_4_11.pdf){height="7cm" width="13cm"}
On the other hand, for a low perpendicular field (e.g., $h_{z0}=0.1$), the general trends in the CT-state corrections do not very much differ from those at moderate field values. However, some distinctive features are worth to be mentioned for the $M_{x,y}(H_{y0})$ curves (Figure \[Figure\_4\_11\]). To start with, we recall that the corner structure that defines the separation of the [*CT-curves*]{} from the [*master T-state behavior*]{} is different. Thus, as one can notice in Fig. \[Figure\_4\_11\], it is only for the higher values of the parameter $\chi^{2}$ that the separations take place abruptly. In particular, a smooth variation occurs for $\chi^{2}<6$ in all cases. Also noticeable is the change in the behavior of the initial part of the $M_{x}(h_{y0})$ curves for the paramagnetic case. Recall that the minimum observed for the moderate field patterns ($h_{z0}=1.5$) has now disappeared (this can be already detected for the T-states). Significantly, one can observe that by decreasing $\chi^{2}$, $M_{x}$ develops a nearly flat region at the low values of $h_{y0}$. Physically, this means that the initial $h_{x}(z)$ profile is basically unchanged. For the lowest values of $\chi^2$ this can take place over a noticeable range of applied fields $h_{y0}$ [@P2-Ruiz_PRB_2009].
![\[Figure\_4\_12\] Profiles of the parallel currents $j_{\parallel}$ for the rectangular region hypothesis or DCSM with $J_{c\parallel}^{2}=2$ and $J_{\perp}=J_{c\perp}=1.0$. The curves are labeled according to the applied field $h_{y0}$ (at right), assuming $h_{z0}=1.5$. The diamagnetic (left) and paramagnetic (right) cases are shown.](Figure_4_12.pdf){height="7cm" width="13cm"}
Secondly, with the aim of providing a fair understanding on how the T-states break down for the 3D configurations studied in this chapter, in Figure \[Figure\_4\_12\] we have plotted the local profiles of $j_{\parallel}(z)$ for a moderate perpendicular field $h_{z0}=1.5$ while the external magnetic field $h_{y0}$ is increased. The left pane shows the process of saturation in which $j_{\parallel}$ reaches the value $j_{c\parallel}$ for the diamagnetic initial condition. Analogously, the right pane shows their corresponding behavior but for the paramagnetic initial condition. To allow a physical interpretation of these profiles, we have introduced the following notation: $cT$ denotes that $j_{\parallel}$ has reached the limit $j_{c\parallel}$ only partially within the sample, while $CT$ means that $j_{\parallel}$ equals $j_{c\parallel}$ for the whole range $0\leq z\leq a$. For the [*partial penetration*]{} cT-states, we additionally distinguish between the so-called cT$^{(1)}$ and cT$^{(2)}$ phases. As one can see in the plot, cT$^{(1)}$ means that $j_{\parallel}$ penetrates *linearly* from the surface until the limitation is reached somewhere within the sample. For the diamagnetic case, the profile stops at the actual value $j_{c\parallel}$. However, for the paramagnetic case, the structure is more complex. Thus, $j_{\parallel}$ penetrates linearly until a [*linear increase*]{} (towards the center) curve is reached. This structure is followed until the contact between both lines reaches the surface. Then, the so-called cT$^{(2)}$ region appears. The cutting current component $j_{\parallel}$ has reached the threshold value $j_{c\parallel}$ at the surface, and the whole $j_{\parallel}$ curve “pivots” around this point until the full CT-state is reached. We call the readers’ attention that the initial separations of the magnetic moment from the T-state master curves take place as soon as a cT-state is obtained. Further, the corners can be clearly assigned to the instant at which the full CT state appears.
![\[Figure\_4\_13\] *Left pane:* Profiles of the parallel ($J_{||}$) and perpendicular ($J_{\perp}$) current densities in the diamagnetic configuration at a low perpendicular field $h_{z0}=0.1$, for the rectangular region or DCSM law with $J_{c\parallel}^{2}=2.0$ and $J_{c_{\perp}}=1.0$. *Central pane:* Same as above, but the profiles for the paramagnetic configuration are shown. *Right pane:* Magnetic field components $h_{x}(z)$ (solid-lines) and $h_{y}(z)$ (dashed-lines) corresponding to the above mentioned paramagnetic configuration. For clarity, the $h_{y}(z)$ profile corresponding to $H_{y0}=1.40$ has been labeled accordingly. All the curves follow the same color scale convention corresponding to the values of the applied field $h_{y0}$. In left pane, inset (a) schematically shows the CT structure of the full penetration regime in the diamagnetic case. The CT-C structure behavior of ${\bf J}$ for the paramagnetic case is shown in the inset (b). ](Figure_4_13.pdf){height="6cm" width="13cm"}
Also interesting are the peculiarities of the cutting and depinning components of the current density penetration profiles for low values of the perpendicular field $h_{z0}$. They can be observed in Figs. \[Figure\_4\_13\] & \[Figure\_4\_14\], which both reveal new physical mechanisms that do not appear for the moderate perpendicular field values. Once more, the first observation is that the appearance of the corner in the magnetic moment straightforwardly relates to the current density profiles. Thus, for the lower values of $\chi$ (no corner present), the profile $j_{\parallel}$ displays a rather simple structure, basically jumping from $0$ to $j_{c\parallel}$ at some point within the sample (Fig. \[Figure\_4\_13\]). On the contrary, for the higher values of $\chi$ (those displaying a corner in $M_{xy}$) the evolution of the cutting profiles $j_{\parallel}(z)$ is much more complex (Fig. \[Figure\_4\_14\]). Let us analyze these plots in more detail:
Firstly, Fig. \[Figure\_4\_13\] shows the cutting profiles $j_{\parallel}(z)$ both for the diamagnetic and paramagnetic cases with a DCSM region characterized by the parameter $\chi^{2}=2$. It is to be noticed that, in both cases, the step-like structure with $J_{\parallel}=0$ in the inner part and $J_{\parallel}=J_{c\parallel}$ in the periphery evolves until the [*full penetration*]{} state $J_{\parallel}=J_{c\parallel}\;,\;\forall\; z$ is reached. However, an outstanding fact is that in the paramagnetic case, for the first time along the exposition of this chapter we have met a set of conditions that produce an excursion of $j_{\perp}$, i.e., the customary condition $J_{\perp}=J_{c\perp}$ is violated during the process of increasing $h_{y0}$. To be specific, $J_{\perp}$ starts from the condition $J_{\perp}=J_{c\perp}$, given by the initial process in $h_{x0}$. Then, a basically linear decrease from some inner point towards the surface occurs, with an eventual reduction to a nearly null value at some regions within the sample (*C-states are basically provoked*). Further increase of $h_{y0}$ produces a new CT-state. This behavior is shown in a pictorial form within the insets of Fig. \[Figure\_4\_13\]. Recall that the average current density sharply transits from a T-state ($J_{\perp}=J_{c\perp}\,,\,J_{\parallel}=0$) to the CT-state ($J_{\perp}=J_{c\perp}\,,\,J_{\parallel}=J_{c\parallel}$) for the diamagnetic case, while a T $\to$ C $\to$ CT evolution happens for the initial paramagnetic conditions. This behavior allows a physical interpretation in terms of the evolution of the magnetic field profiles. Thus, as stated before, the cases with small $\chi$ are characterized by a nearly frozen profile in $h_x$, as shown in right pane of Fig. \[Figure\_4\_13\]. Then the structure of $h_{x}(z)$ and $h_{y}(z)$ is basically a cross between two straight lines, where the crossing point coincides with the minimum in $J_{\perp}(z)$. Thus, recalling the interpretation of the perpendicular component of the current density \[Eqs. (\[Eq.4.7\]) & (\[Eq.4.8\])\], the minimum should be expected as $h_{x}^{2}+h_{y}^{2}$ has a very small variation around the crossing point of the two families of nearly parallel lines.
![\[Figure\_4\_14\] *Left pane:* Profiles of $j_{||}$ for the diamagnetic case within the rectangular DCSM with $\chi^{2}=7.5$ and $h_{z0}=0.1$. In all cases, one gets $J_{\perp}=J_{c\perp}=1.0$. *Right pane:* The corresponding magnetic moment components ($M_{x},M_{y}$) as a function of $h_{y0}$ are shown. The evolution from the initial full penetration T state to the final full penetration CT state takes place in three steps that are classified according to the structure along the sample width, by means the defined states: cT$^{(1)}\equiv$ T-CT, cT$^{(2)}\equiv$ T-CT-T, cT$^{(3)}\equiv$ CT-T and eventually CT. ](Figure_4_14.pdf){height="7cm" width="13cm"}
On the other hand, the details about the behavior of the cutting component $j_{\parallel}$ for the larger values of $\chi$ are presented in Fig. \[Figure\_4\_14\], that corresponds to the case $\chi^{2}=7.5$. Again, owing to the complexity of the structure, we introduce the notation cT$^{(1)}$, cT$^{(2)}$ and cT$^{(3)}$, that is explained below. Let us first recall that the corner appears when the [*partial penetration*]{} regime cT$^{(3)}$ extinguishes and the full sample ($0<z<d/2$) satisfies the conditions $J_{\perp}=J_{c\perp}$ and $J_{\parallel}=J_{c\parallel}$ (i.e., CT). This property is clearly seen in the right pane of this figure. Thus, the cT$^{(1)}$ regime is characterized by a T region in the inner part of the sample ($J_{\perp}=J_{c\perp}$ and $J_{\parallel}=0$), that abruptly becomes CT at a point that progressively penetrates towards the center (T-CT structure). At a certain instant, the profile becomes T-CT-T because the outermost layers develop a [*subcritical*]{} $J_{\parallel}$. This is called cT$^{(2)}$. Then, the central CT band grows towards both ends. In first instance, the inner T region becomes CT, giving a global CT-T structure, that we call cT$^{(3)}$. In a final step, the surface T layer shrinks again to a null width and the full profile is a CT region. This instant establishes the appearance of the corner in the magnetization curves.
### \[ch-4-5\] *4.5 Smooth critical states in “3D” configurations* {#ch-4-5-4.5-smooth-critical-states-in-3d-configurations .unnumbered}
As stated before, our smooth double critical state theory (SDCST) allows to specify almost any critical state law by means a simple mathematical statement that includes an index $n$ accounting for the *smoothness* of the $J_{\parallel}(J_{\perp})$ relation, and a certain *bandwidth* characterizing the magnetic anisotropy ratio $\chi=J_{c\parallel}/J_{c\perp}$ \[see Eq. (\[Eq.2.17\])\]. The systematic consideration of the influence of these parameters is of remarkable importance as it allows a straightforward elucidation of the relation between diverse physical processes and the actual material law. Most of the experimental evidences reflecting accurate observations for the influence of the cutting effects onto the macroscopic measurements of magnetic moment for anisotropic superconducting samples, without transport current, may be summarized along two remarkable facts:
(i) The occurrence of magnetization peaks which are mainly evident in paramagnetic configurations [@P2-Voloshin_1997; @P2-Fisher_2000; @P2-Voloshin_2001; @P2-Fisher_1997; @P2-Voloshin_2010].
(ii) The collapse of the magnetization curves towards an ostensible isotropic response [@P2-Fisher_1997; @P2-Voloshin_2010; @P2-Fisher_2000_b].
Nowadays, it is well known that the DCSM and its precursors (the T-state and Isotropic models) are not able to achieve a fair understanding of the above effects in a wide number of configurations, or at least these models do not handle environments with high magnetic fields [@P2-Ruiz_PRB_2009]. This fact has lead to consider alternative models such as the two-velocity electrodynamic model [@P2-2hydrodynamic_model], or the helical electrodynamic model [@P2-helical_model], both lacking a solid physical basis for the mechanisms underlying the motion of vortices. In fact, being the threshold values for the cutting current component $(J_{c\parallel})$ and the depinning current component $(J_{c\perp})$ the main physical observables determining the magnetic anisotropy of a superconductor [@P2-Clem_2011_SUST; @P2-Campbell_2011_SUST; @P2-Karasik_1970], within these models, other parameters have to be included ad-hoc.
Moreover, it is worth anticipating the following chapter, by mentioning that neither of the above mentioned models allow a correct explanation for the experimental remarks when the superconductor is also carrying a longitudinal transport current. Thus, the purpose of this section is vindicating the physical mechanisms of cutting and pinning, through a comprehensive study of the magnetic anisotropy of type II superconductors, by means the modification of the [*conventional*]{} DCSM which leads to the establishment of the SDCST. Such modifications can be justified as corrections to the simplifying ideas that flux depinning is only related to $J_{\perp}$ and the flux cutting is only related to $J_{\parallel}$. We emphasize that in a in a general scenario, one should consider the dependencies $J_{c\perp}=J_{c\perp}(J_{\parallel})$ and $J_{c\parallel}=J_{c\parallel}(J_{\perp})$ [@P2-Ruiz_PRB_2009; @P2-Ruiz_SUST_2010; @P2-Brandt_2007; @P2-Clem_2011_SUST; @P2-Campbell_2011_SUST].
![\[Figure\_4\_15\] The magnetic moments $M_{x}$ and $M_{y}$ of the slab per unit area as a function of $h_{y_{0}}$ in the diamagnetic (top) and paramagnetic (bottom) cases with $h_{x0}=1.1$ and both for, a moderate perpendicular field $h_{z0}$=1.5 (left pane), and a low perpendicular field $h_{z0}=0.1$ (right pane). The “infinite band” or T-states model (blue solid lines), the DCSM or “rectangular regions” (other solid lines), the SDCST’s models with $n=4$ “superelliptical regions” (dashed-lines), and $n=1$ “elliptical regions” (dotted-lines), are shown for several values of the ratio $\chi^{2}\equiv j_{c\parallel}$ for a given $J_{c\perp}=1$. ](Figure_4_15.pdf){height="11cm" width="13cm"}
Recalling that, mathematically, the effect of [*smoothing the corners*]{} for the rectangular DCSM region may be represented by a [*one-parameter*]{} family of superelliptical functions with the generic form given in Eq. (\[Eq.2.17\]), i.e., $$\begin{aligned}
\left(\frac{J_{\parallel}}{J_{c\parallel}}\right)^{2n}+
\left(\frac{J_{\perp}}{J_{c\perp}}\right)^{2n}\leq 1 \, , \nonumber\end{aligned}$$ such kind of curves cover the whole range of interest just by allowing $n$ to take values over the positive integers. As the reader can easily verify for $\chi^{2}>1$, the index $n=1$ corresponds to the standard ellipse and $n\geq4$ is basically a rectangle with faintly rounded corners (see Fig. \[Figure\_4\_1\]).
In order to illustrate the effect of smoothing the material law $\Delta_{\textbf{r}}(\textbf{J}_{\parallel},\textbf{J}_{\perp})$ for different bandwidths $\chi$, below we will show the magnetization curves that are obtained for the diamagnetic and paramagnetic configurations considered before (Fig. \[Figure\_4\_2\]). The main results of our analysis are depicted in Figs. \[Figure\_4\_15\] & \[Figure\_4\_16\].
Figure \[Figure\_4\_15\] shows the behavior of the magnetization curves $M_x$ and $M_y$ for an external perpendicular field of either moderate intensity $h_{z0}=1.5$ (left pane) or a lower intensity $h_{z0}=0.1$ (right pane). In order to simplify their interpretation, at this stage we will only compare the prediction for the *smoothing* index $n=4$ (*superelliptic region*) with the limiting cases $n=1$ (*elliptic region*) and $n\rightarrow\infty$ (*a DCSM or rectangular region*), under consideration of different bandwidths $\chi$.
![\[Figure\_4\_16\] Current density vector $\textbf{J}$ in the planar representation \[$J_{\perp}$,$J_{\parallel}$\] for three different material law models, corresponding to the following $\Delta_{\textbf{r}}$ regions: rectangular (red), superelliptical (green), and elliptical (blue). Here, the diamagnetic case for both a moderate field $h_{z0}=1.5$ and the lower field $h_{z}=0.1$ are shown. Several vectors for several values of the ratio $\chi=J_{c\parallel}/J_{c\perp}$ and the applied field $h_{y0}$ are shown and labeled on each arrow. The scales on the horizontal axes that have been re-sized for visual purposes. ](Figure_4_16.pdf){width="100.00000%"}
Firstly, for the moderate perpendicular field region (left pane of Fig. \[Figure\_4\_15\]), we observe that the overall effect of reducing the value of $\chi\equiv J_{c\parallel}/J_{c\perp}$ is the same for the three material laws or $\Delta_{\textbf{r}}$ regions. The smaller the value of $\chi$, the higher reduction respect to the T-state ($\chi\to\infty$) master curve for the magnetic moment components. On the other hand, as regards the particular details for each model, we recall: (i) as expected the smooth models lead to smooth variations, i.e.: the corner is not present, (ii) the breakdown of the T-state behavior occurs before (at higher values of $\chi$ or lower values of $h_{y0}$) for the smoother models. Strictly speaking, the concept of T-state is only valid for the rectangular region, but it is asymptotically generated as the superelliptic parameter $n$ grows. Finally, (iii) the isotropic CS limit, given by the circular region $n=1$ and $\chi=1$ produces the expected results [@P2-Badia_PRL_2001]: $M_x$ collapses to zero, and $M_y$ develops a [*one dimensional*]{} critical state behavior.
![\[Figure\_4\_17\] Same as Fig. \[Figure\_4\_16\], but here the $J$ vectors corresponds to the paramagnetic case. ](Figure_4_17.pdf){width="100.00000%"}
Secondly, for the low perpendicular field region (right pane of Fig. \[Figure\_4\_15\], one can notice: (i) on the one hand, the rectangular and superelliptical models produce very similar results for the diamagnetic case, both for $M_x$ and for $M_y$, noticeably differing from the elliptical region predictions, that still show a practical collapse of $M_x$ and a saturation in $M_y$ as stated before. (ii) On the other hand, the paramagnetic case involves a higher complexity. Thus, we recall that the already mentioned feature of a “*flat*” behavior in $M_{x}$ for small values of $\chi$ within the rectangular region model, is no longer observed upon smoothing of the restriction region. On the contrary, the smooth models involve an initial negative slope and a minimum, resembling the behavior of $M_x$ for the rectangular model, but in moderate $h_{y0}$. As concerns $M_y$, important differences among the three models are also to be recalled.
In order to provide a physical interpretation of the behaviors reported in the above paragraphs for moderate and low perpendicular fields $h_{z0}$, a comparative plot of the current density vectors for each case is given in Figs. \[Figure\_4\_16\] & \[Figure\_4\_17\] respectively. For clarity, we restrict to the representation of the vector ${\bf J}$ at the surface of the sample ($z=a$) for a selected number of values of $h_{y0}$. Just at a first glance, one can relate the best coincidence in predicted magnetization to the more similar critical current density structures (superelliptical and rectangular regions for the diamagnetic case with $h_{z0}=0.1$). Recall that, in that case, the rectangular region produces a CT-state structure ($J_{\parallel}=J_{c\parallel}$ and $J_{\perp}=J_{c\perp}$) that is represented by a ${\bf J}$ vector, pinned in the corner. On the other hand, the vector ${\bf
J}$ related to the superelliptic model does not pin at any point, because such a singular point does not exist. However, it is basically oriented in the same fashion and this relates to the good agreement in ${\bf M}$. We emphasize that the cases in which strong differences occur for the magnetic moment are also related to important changes in the behavior of ${\bf J}$. Thus, if one considers the paramagnetic case at small values of $h_{z0}$ and $h_{y0}$, the significant differences in magnetization relate to an opposite behavior in ${\bf J}$. Moreover, the rectangular model predicts a transition towards a C-state ($J_{\parallel}=J_{c\parallel}$ and $J_{\perp}\approx 0$), while the smooth versions produce a tendency towards the T-state (see left bottom panel of Fig. \[Figure\_4\_17\]).
### \[appendix-ch-4\]*Appendix I Critical angle gradient in “3D” configurations* {#appendix-ch-4appendix-i-critical-angle-gradient-in-3d-configurations .unnumbered}
On the basis of minimum complexity, in this appendix the flux cutting criterion for 3D configurations will be revised under the assumption of a critical angle threshold instead of a superelliptical relation. Below let me present some results related to the concept of the critical angle gradient in 3D systems.
First recall that the limitation on $J_{\parallel}$ appears as related to the energy reduction by the cutting of neighboring flux lines when they are at an angle beyond some critical value [@P2-Brandt_1979; @P2-Clem_1980]. This concept has been largely exploited in the 2D slab geometry for fields applied parallel to the surface [@P2-Clem_DCSM], and it is introduced by the local relation $$\begin{aligned}
\label{Eq.4.12}
\left|\frac{d\alpha}{dz}\right|=\left|\frac{J_{\parallel}}{H}\right|\leq
K_{c} \, ,\end{aligned}$$ that establishes a critical angle gradient. Here, $\alpha$ stands for the angle between the flux lines and a given reference within the $XY$-plane (i.e.: an azimuthal angle). However, for the 3D cases under consideration, the relative misorientation between flux lines may also have a polar angle contribution, i.e.: ${\bf H}$ does not necessarily lie within the $XY$-plane or any other given plane.
As sketched in Fig. \[Figure\_2\_1\] (pag. ), one has to introduce the angle $\gamma$ within the plane defined by the pair of flux lines under consideration. After some mathematical manipulations, it can be shown that, for the infinite slab geometry, with a three dimensional magnetic field one has $$\label{Eq.4.13}
\frac{d\gamma}{dz}=\sqrt{\frac{J_{\parallel}^{2}}{H^{2}}+\frac{H_{z}^{2}J^{2}}{
H^ { 4}}}
=\frac{1}{H}\sqrt{J_{\parallel}^{2}+\frac{H_{z}^{2}}{H^{2}}\left(J_{\parallel}^{
2}+J_{\perp}^{2}\right)}\, ,$$ where the third component is also introduced. Actually, the above result is just a particular case of the relation $$\begin{aligned}
\label{Eq.4.14}
\nabla \times \left( B \hat {\bf B}\right)=\left[\left(\nabla B\right) \times
\hat {\bf B}\right] +
\left[ B\;\left(\nabla\times\hat {\bf B}\right)\right]
\equiv \left[{\bf J}_{\perp ,1}\right]+\left[{\bf J}_{\perp ,2}+{\bf
J}_{\parallel}\right]\, ,
$$ showing that, in general, both ${\bf J}_{\parallel}$ and ${\bf J}_{\perp}$ can contribute to the spatial variation of the direction $\hat{\bf B}$.
Below, we display the effects of using the cutting limitation $$\label{Eq.4.15}
\left|\frac{d\gamma}{dz}\right|\leq \kappa_{c} \, ,$$ instead of assuming a constant value for the parallel critical current. Fig. \[Figure\_4\_17\] contains the main results. The calculations have been performed for the same diamagnetic and paramagnetic initial configurations displayed in Fig. \[Figure\_4\_2\].
![\[Figure\_4\_18\] The magnetic moments per unit area $M_{x}$ (solid lines) and $M_{y}$ (dotted lines) of the slab as a function of $h_{y_{0}}$ for the critical angle gradient model \[Eq.(\[Eq.4.15\])\]. The unrestricted case ($\kappa_{c}^{2}\to\infty$) is shown for comparison with several cases with a restricted angle gradient: $\kappa_{c}^{2}=$0.20, 0.30 and 0.40 (dimensionless units are defined by $\kappa_{c}\equiv K_{c}a$). Shown are the diamagnetic (left pane) an paramagnetic (right pane) cases for $h_{x_{0}}=1.1$ and $h_{z_{0}}$=1.5. The insets detail the evolution of the angle gradient profiles for $\kappa_{c}^{2}\to\infty$. ](Figure_4_18.pdf){height="7cm" width="13cm"}
In general, one can see that the smaller values for the cutting threshold in whatever form produce the smaller magnetic moments (compare Figs. \[Figure\_4\_10\] & \[Figure\_4\_11\] with Fig. \[Figure\_4\_18\]). However, some important differences are to be quoted. On the one hand, the critical angle criterion $|{\gamma}'|\leq\kappa_{c}$ produces a smooth variation, by contrast to the corner structure induced by the critical current one $J_{\parallel}\leq J_{c\parallel}$. On the other hand, the effect of changing the value of $\kappa_{c}$ is much less noticeable, especially for the diamagnetic case, in which the full range of physically meaning values of $\kappa_{c}$ produce a negligible variation. Moreover, we call the readers’ attention that the above mentioned range for $\kappa_{c}$ is established by the application of Eq. (\[Eq.4.13\]) to the initial state of the sample. Thus, if one takes $J_{\parallel}=0,\, H_{z0}=1.5,\, H_{x0}=1.1$, the squared angle gradient takes the value ${\gamma}'^2 = 0.19$ and one has to use $\kappa^2 >
0.19$ in order to be consistent with the initial critical state assumed.
\[ch-5\] **The Longitudinal Transport Problem**
===============================================
It is well known that various striking phenomena may occur when a type-II superconductor with intrinsic magnetic anisotropy is under the action of a transport current and a longitudinal magnetic field [@P2-Ruiz_PRB_2009; @P2-Ruiz_SUST_2010; @P2-Voloshin_2001; @P2-Clem_1980; @P2-Ruiz_PRB_2011; @P2-Ruiz_SUST_2011; @P2-Sanchez_2010; @P2-Blamire_2003; @P2-LeBlanc_2003; @P2-LeBlanc_2002; @P2-Matsushita_1984; @P2-Matsushita_1998; @P2-Matsushita_2012; @P2-LeBlanc_1993; @P2-LeBlanc_1991; @P2-Voloshin_1991; @P2-Cave_1978; @P2-Walmsley_1977; @P2-Esaki_1976; @P2-Walmsley_1972; @P2-London_1968; @P2-LeBlanc_1966; @P2-Watanabe_1992; @P2-Blamire_1986; @P2-Boyer_1980; @P2-Gauthier_1974; @P2-Karasik_1970; @P2-Sugahara_1970; @P2-Taylor_1967; @P2-Sekula_1963; @P2-Clem_2011_PRB; @P2-Nakayama_1972]. In particular, a remarkable enhancement of the critical current density by means of its *compression* towards the center of the superconducting sample has been observed in a wide number of conventional and high temperature superconducting systems within a certain set of experimental conditions [@P2-Clem_1980; @P2-Blamire_2003; @P2-Sanchez_2010; @P2-Watanabe_1992; @P2-Blamire_1986; @P2-Boyer_1980; @P2-Gauthier_1974; @P2-Karasik_1970; @P2-Sugahara_1970; @P2-Taylor_1967; @P2-Sekula_1963]. This property, together with other intriguing phenomena, such as the observation of paramagnetic moments, and outstandingly, the experimental observation of a counter intuitive phenomenon of negative resistance by the action of a parallel magnetic field, have been reported in the course of intense experimental and theoretical activities [@P2-Voloshin_2001; @P2-LeBlanc_2003; @P2-LeBlanc_2002; @P2-Matsushita_1984; @P2-Matsushita_1998; @P2-Matsushita_2012; @P2-LeBlanc_1993; @P2-LeBlanc_1991; @P2-Voloshin_1991; @P2-Cave_1978; @P2-Walmsley_1977; @P2-Esaki_1976; @P2-Walmsley_1972; @P2-London_1968; @P2-LeBlanc_1966]. Most of these works were primarily concerned with the arrangement of the macroscopic current density ${\bf J}$ along the so-called nearly [*force free*]{} trajectories [@P2-Bergeron_1972]. Recall that if ${\bf J}$ is [*nearly parallel*]{} to the magnetic induction ${\bf B}$, moderate or weak pinning forces are needed for avoiding the detrimental flux-flow losses related to the drift of flux tubes driven by the magnetostatic force (${\bf J}\times{\bf B}$ per unit volume). More specifically, negative voltages have been observed by different groups [@P2-LeBlanc_2003; @P2-LeBlanc_2002; @P2-Matsushita_1984; @P2-Matsushita_1998; @P2-Matsushita_2012; @P2-LeBlanc_1993; @P2-LeBlanc_1991; @P2-Voloshin_1991; @P2-Matsushita_1984; @P2-Cave_1978; @P2-Walmsley_1977] when recording the current-voltage characteristics at specific locations on the surface of the sample (central region).
![\[Figure\_5\_1\] (a) Schematic representation of the helical model for the longitudinal transport problem in an infinite superconducting cylinder. An array of parallel helical fluxoids without flux cutting is assumed. (b) Pictorial illustration of the real situation where the flux cutting events appear. The cylindrical and slab symmetries are shown. ](Figure_5_1.pdf){height="5.2cm" width="13cm"}
In a first approach, such resistive structure has been intuitively understood in terms of helical domains, closely connected to the force-free current parallel to the flux-lines [@P2-helical_model; @P2-Matsushita_1984; @P2-Matsushita_1998; @P2-Matsushita_2012; @P2-LeBlanc_1966; @P2-Gauthier_1974; @P2-Esaki_1976]. The basic idea of this model relies in the fact that the averaged direction of the flux flow in a superconducting cylinder subjected to a longitudinal magnetic field and transport current, is the same as the direction of the Poynting’s vector $(\textbf{E}\times\textbf{H})$ onto the external surface, suggesting the occurrence of a continuous helical flux flow without flux cutting \[see Fig. \[Figure\_5\_1\] (a)\]. Also, in order to achieve a concordance with the experimental evidences [@P2-LeBlanc_1966; @P2-Gauthier_1974; @P2-Esaki_1976], the resulting helical flux over the cylindrical surface must be subdivided in two domains for which the Poynting’s vector is directed in two concomitant directions: inwards (allowing the compressing of **J** towards the center of the specimen), and outwards (allowing the occurrence of surface regions with **J** flowing in counter direction to the flux of transport current). Nevertheless, despite the seeming simplicity of the helical model and its intuitive explanation for the increasing of the current density and the simultaneous occurrence of surface negative currents, the helical symmetry shown in Fig. \[Figure\_5\_1\](a) does not exist in any other symmetry different to the infinite cylinder, and furthermore, it does not include the flux cutting mechanism which causes many derived effects \[see Fig. \[Figure\_5\_1\](b)\]. Actually, regardless of the symmetry considered, there are some remarkable effects which can not be explained under this scenario. On the one hand, it has been stated that the direction of the helical structure and that of the magnetic field at the surface are really different [@P2-Matsushita_1984; @P2-Matsushita_1998]. On the other hand, by increasing the magnetic field a continuous torsion of the helical domain should be expected, such that the vanishing of the negative current within a finite interval of the applied magnetic fields can not be conceived. Furthermore, it does neither explain the bounded increase of the transport current (i.e., the occurrence of a maximal peak on the longitudinal current density) as one raises the magnitude of the applied magnetic field. Finally, as a detail of fine structure, local paramagnetic domains cannot either be predicted within the above scenario. Therefore, a most accurate description of the diverse effects underlying to the longitudinal transport problem, have to include the physics behind the flux cutting mechanism.
Relying on our theoretical approach for the superconducting critical state problem in 3D magnetic field configurations and the aforementioned scenario, below we present an exhaustive analysis of the electrodynamic response for the so-called longitudinal transport problem of type-II superconductors in the slab geometry. Remarkable numerical and conceptual difficulties related to the implementation of the magnetic anisotropy and the relation between the flux-line cutting (crossing and recombination) and the flux-line depinning mechanisms, will be overcome by means simplified analytical models for extremal cases and the further comparison with the most general solution of the smooth double critical state theory (SDCST) for analogous material laws (subchapter 5.1). Then, supported by numerical simulations that cover an extensive set of experimental conditions, we put forward a much more complete physical scenario which is based upon a set of superelliptical material laws. Thus, subchapter 5.2 is devoted to show how the striking existence of negative flow domains, local and global paramagnetic structures, emergence of peak-like structures in both the critical current density and the longitudinal magnetic moment, as well as the compression of the transport current in type-II superconductors under parallel magnetic fields, are all predicted by our general critical state theory. In addition, we shall introduce some ideas that could be applied for the determination of the flux cutting threshold from local measurements of the current density flowing along specific layers of the superconducting sample, as correlated to the behavior of the magnetic moment components.
### \[ch-5-1\] *5.1 Simplified analytical models and beyond* {#ch-5-1-5.1-simplified-analytical-models-and-beyond .unnumbered}
In this subchapter, we call the reader’s attention to the fact that two analytical approaches for the slab geometry in extreme situations may be found in the literature. The first one was introduced by Brandt and Mikitik in Ref. [@P2-Brandt_2007] for the regime of strong pinning with very weak longitudinal current conditions, i.e., $h_{z0}$ must be very high as compared to the in-plane applied field $h_{xy}(a)$ (then $J_{\parallel}\ll J_{c\perp}$).On the other hand, the opposite limit ($h_{z}\to 0$) was recently developed in our group (Ref. [@P2-Ruiz_PRB_2011]). Thus, in a first stage let us show how the physical properties of the longitudinal transport problem may be understood within our simplified analytical model, and then we will move onto a general description of the problem in terms of the SDCST.
***5.1.1 The simplest analytical model***
\[ch-5-1-1\]
First, recall that the Ampère’s law takes the following form for the infinite slab geometry considered in the previous chapter: $$\label{Eq.5.1}
-\frac{dh_{y}}{dz}=j_{x}
\quad ; \quad
\frac{dh_{x}}{dz}=j_{y} \, .
$$ Also notice that, in the particular case $h_{z0}=0$ \[i.e., $\theta=\pi/2$ in Fig. \[Figure\_4\_1\] (pag. )\], the material law or region $\Delta_{p}$ becomes a rectangle with axis defined by the in-plane directions parallel and perpendicular to $\textbf{h}$. Thus, recalling the statements issued in chapter 4.1, one can show that such expressions may be transformed into the polar form $$\label{Eq.5.2}
-h\frac{d\alpha}{dz}=j_{\parallel}^{\rm p}
\quad ; \quad
\frac{dh}{dz}=j_{\perp}
$$ with $h=\sqrt{h_{x}^{2}+h_{y}^{2}}$ the modulus of the magnetic field vector, and $\alpha={\rm atan}(h_y/h_x)$ the angle between such vector and the $x$-axis.
Now, the thresholds of flux depinning and cutting imply the in-plane conditions $$\label{Eq.5.3}
|j_{\parallel}^{\rm p}|\leq j_{c\parallel}^{\rm p}(\theta =\pi
/2)=j_{c\parallel}
\quad ; \quad
|j_{\perp}^{\rm p}|\leq 1 \, .
$$ Notice that, in general, Eq. (\[Eq.5.2\]) and the critical constraints defined in Eq. (\[Eq.5.3\]) would not straightforwardly lead to the solution of the problem. Thus, one should also use Faraday’s law, either by explicit introduction of the related electric fields (as in Refs. [@P2-Clem_2011_SUST; @P2-Campbell_2011_SUST]), or by our variational statement. However, as in this case $\theta
=\pi/2$ and consequently $j_{c\parallel}^{\rm p}=j_{c\parallel}$, the resolution noticeably simplifies. In fact, for the considered situation, we will have a combination of the cases $j_{\parallel}^{\rm p}=\{0$ or $\pm
j_{c\parallel}\}$, and $j_{\perp}^{\rm p}=\{0$ or $1\}$, and then integration of Eq. (\[Eq.5.2\]) is straightforward. For further mathematical ease, we will also consider $j_{c\parallel}$ and $j_{c\perp}$ to be field independent constants.
Following the notation introduced in chapter 4.4 we will refer to different zones within the sample that are basically related to macroscopic regions where well defined dissipation mechanisms occur. In brief, we will speak about T-zones, where only flux depinning (transport) occurs (${j}_{\parallel}=0\, ,{j}_{\perp}=\pm 1$), C zones, where only flux cutting occurs (${j}_{\parallel}=\pm\chi\, ,{j}_{\perp}=0$), CT zones where both transport and cutting occur (${j}_{\parallel}=\pm\chi\, , {j}_{\perp}=\pm 1$), and O-zones where neither flux transport nor cutting take place (${j}_{\parallel}=0\, , {j}_{\perp}=0$). Introducing the above set of possibilities in Eqs. (\[Eq.5.2\]) & (\[Eq.5.3\]) one gets the following cases for the incremental behavior of the magnetic field in polar components $$\begin{aligned}
\label{Eq.5.4}
dh=\left\{
\begin{array}{rr}
0\qquad \rm{(O,C)}&
\\
\pm\, dz\;\; \rm{(T,CT)}&
\end{array}
\right.
\!; \;
d\alpha=\left\{
\begin{array}{rr}
0\qquad\qquad \rm{(O,T)}&
\\
\pm\,({\chi}/{h})\,dz\;\; \rm{(C,CT)}&
\end{array}
\right. \, ,
$$ and all that remains for obtaining the penetration profiles is to solve successively (integrate) for $h$ and $\alpha$ with the corresponding boundary conditions (evolutionary surface values $h_{0},\alpha_{0}$). The case selection has to be made according to Lenz’s law. We note, in passing, that further specification related to the sign is usually included in the notation. Thus, a T$_{+}$ zone will exactly mean $dh = +dz$.
![\[Figure\_5\_2\] *Left pane:* Schematic representation of the simplest experimental configuration of the longitudinal transport problem which is solved by analytical methods. *Right pane* Penetration profiles of the magnetic field components and rotation angle in the longitudinal transport experiment ($h_{z0}=0$) for a superconducting slab of thickness $2a$, as calculated from Eq. (\[Eq.5.1\]). The zone structure induced by increasing the field $h_{y0}$ is marked upon some of the curves. The dashed line corresponds to the transition regime between the states $O/T_{+}/C_{-}T_{+}$ to the unstable regime of states $C_{+}T_{+}/T_{+}/C_{-}T_{+}$ (see text). ](Figure_5_2.pdf){width="100.00000%"}
In detail, our experimental process starts with the application of the transport current along the y-axis (see left pane of Fig. \[Figure\_5\_2\]) which produce a T$_{+}$ zone, i.e., the starting profile can be depicted as follows: $$\begin{aligned}
\label{Eq.5.5}
dh= dz \quad ; \quad d\alpha = 0
\nonumber\\
{\Downarrow}\qquad\qquad
\\
h=h_{x0}+{\tt z}-1\quad ; \quad \alpha=0 \, ,
\nonumber\end{aligned}$$ that penetrates from the surface until the point where $h$ equals $0$, i.e.: ${\tt z}_{p0}=1-I_{tr}$. In our units, ${\tt z}_{p0}=0.5$ for $I_{tr}=h_{x0}=0.5$.
On the one hand, an O zone appears in the inner region $0<{\tt z}<{\tt z}_{p0}$ as far as $I_{tr}<1$. On the other hand, unless an external source of magnetic field is switched on, the above situation remains valid. Thus, upon increasing the external field $h_{y0}$, a flux line rotation starts on the surface and the perturbation propagates towards the center in the form of a C$_{-}$T$_{+}$ zone defined by $$\begin{aligned}
\label{Eq.5.6}
dh= dz \quad ; \quad d\alpha = -\frac{\chi}{h}\, dz\qquad\qquad
\nonumber\\
{\Downarrow}\qquad\qquad\qquad\qquad\qquad
\\
h=h_{x0}+{\tt z}-1\; ; \; \alpha=\alpha_{0}+\chi{\rm ln}\left[1+\frac{{\tt
z}-1}{h_{0}}\right] \,
,
\nonumber\end{aligned}$$ which covers the range ${\tt z}_{c}^{-}<{\tt z}<1$, defined by $\alpha=0$ $\Rightarrow$ ${\tt z}_{c}^{-}$ $=$ $1+h_{0}$ $[{\rm exp}(-\alpha_{0}/\chi)-1]$ (see Fig. \[Figure\_5\_2\]). The former T$_{+}$ zone is pushed towards the center and occupies the interval ${\tt z}_{p}^{-}<{\tt z}<{\tt z}_{c}^{-}$ with ${\tt z}_{p}^{-}=1-h_{0}$. Finally, an O-zone fills the core $0<{\tt z}<{\tt
z}_{p}^{-}$ until the condition ${\tt
z}_{p}^{-}=0\Leftrightarrow h_{0}=1$ is reached. Then, with further increase of the local component $h_{y0}$, an instability towards the center of the sample consisting of a transient structure that becomes $C_{+}T_{+}/C_{-}T_{+}$ is induced.
In physical terms, flux vortices penetrate from the surface with some orientation given by the components of the vector $(h_{x},h_{y})$. Owing to the critical condition for the penetration of the field $dh/dz = 1$, as soon as the modulus reaches the center, flux rotation must take place there. This is needed for accommodating the vector to the condition ${\bf h}({\tt z}=0)=(0,h_{y}({\tt
z}=0))$. On the other hand, as the angle variation is determined by the value of $J_{c\parallel}$, a jump is induced at the center, i.e.: $\alpha({\tt
z}=0)\to\pi /2$, and the related instability may be visualized by a critical $C_{+}T_{+}/T_{+}/C_{-}T_{+}$ profile (dashed lines at the right pane of Fig. \[Figure\_5\_2\]) in which the field angle decreases from its surface value $\alpha_{0}$ to $0$ in the $C_{-}T_{+}$ region, then keeps null within the $T_{+}$ zone, and suddenly increases to the value $\pi/2$ in the inner $C_{+}T_{+}$ band defined by $$\begin{aligned}
\label{Eq.5.7}
dh= dz \quad ; \quad d\alpha = \frac{\chi}{h}\, dz\qquad\qquad\quad
\nonumber\\
{\Downarrow}\qquad\qquad\qquad\qquad\qquad
\\
h=h_{x0}+{\tt z}-1\; ; \; \alpha=\pi/2-\chi{\rm ln}\left[1+\frac{{\tt
z}}{h_{0}-1}\right]\, .
\nonumber\end{aligned}$$ In fact, the $C_{+}T_{+}/C_{-}T_{+}$ structure is stabilized with the intersection between regions at the point \[$\alpha^{{+},{+}}({\tt z}_{\rm
v})=\alpha^{{-},{+}}({\tt z}_{\rm v})$\] given by $$\begin{aligned}
\label{Eq.5.8}
{\tt z}_{{\rm v}}=1-h_{0}+\sqrt{h_{0}(h_{0}-1)}~{\rm
exp}\left[\frac{\pi/2-\alpha_{0}}{\chi}\right] \, .\end{aligned}$$ Finally, note that upon further increasing $h_{y0}$ the point ${\tt z}_{\rm v}$ follows the rule ${\tt z}_{\rm v}(h_{y0}\to\infty)\to (1+h_{x0}/\chi)/2$
In brief, our simple analytical model allows to identify the following physical phenomena as the longitudinal component of the magnetic field $h_{y0}$ (parallel to the flow direction of the transport current) is increased:
1. The appearance of a surface layer with negative transport current density (mind the slope of $h_{x}$ in Fig. \[Figure\_5\_2\] in view of Eq. (\[Eq.5.1\])).
2. The applied magnetic field [*re-entry*]{} as related to the inner $C_{+}T_{+}$ zone, predicting the occurrence of local paramagnetic states near of the center of the superconducting sample.
These features will be confirmed along the forthcoming section, where the SDCST statements are thoroughly presented and a further comparison with the analytical model of Ref. [@P2-Brandt_2007] will be displayed.
***5.1.2 The SDCST statement and the BM’s approach***
\[ch-5-1-2\]
Below, let us introduce an experimental configuration rather to the opposite side of the previous situation, and that may be used for comparison to the work in Ref. [@P2-Brandt_2007]. To be specific, based upon our well-known 3D physical scenario for superconductors with magnetic anisotropy, we shall consider the time evolution of magnetic profiles $\textbf{h}_{l+1}(z)$ within an infinite superconducting slab of thickness $2a$ (see Fig. \[Figure\_5\_3\]), cooled under the assumption of an initial state defined by an uniform vortex lattice perpendicular to the external surfaces, i.e., a constant magnetic field $h_{z0}$. In terms of the geometrical interpretation for the material law, the above starting point or initial state corresponds to choose $\theta=0$ in Fig. \[Figure\_4\_1\] (pag. ). Then, a transport current is injected along the superconducting slab in the direction of the $y-$axis inducing a rotation of the critical current region $\Delta_{\textbf{r}}$ on the plane of currents $xy$ by means the induced magnetic field component $h_{x}$ (i.e., $\theta\neq0$ in Fig. \[Figure\_4\_1\]). Finally, a third source of magnetic field is switched on along the $y-$axis, which induces a new rotation of the current density vector by means the introduced local component of magnetic field $h_{y}$ (i.e., $\alpha\neq0$ in Fig. \[Figure\_4\_1\]).
![\[Figure\_5\_3\] *Left pane:* Schematic representation of the 3D experimental configuration of the longitudinal transport problem in the slab geometry. *Central and right panes:* Profiles of the magnetic field components $h_x(z)$ and $h_y(z)$ for the longitudinal problem corresponding to a transport current along the $y$ axis of value $I_{\rm
transport}\equiv J_{c\perp}a/2=0.5$ and at several increasing values of the magnetic field $h_{y0}$ as labeled in the curves. Here, we have assumed an uniform perpendicular field $h_{z0}=20$ (central pane) and then $h_{z0}=200$ (right pane). The plot shows the comparison of the full range numerical solution in the infinite widthband model or so-called T-state model (continuous lines) to the analytical approximation in Eq. (\[Eq.5.15\]) (dashed). The insets show the initial flux penetration profiles for both components of the magnetic field. ](Figure_5_3.pdf){height="7cm" width="13cm"}
It is worth mentioning that, by symmetry, the current density is confined to the $xy$-plane, such that the distribution of the current can be displayed in a proper set of circuits naturally defined by a collection of in-plane current layers located at the heights $z_{i}$, and each one carrying a current density given by $[J_{x}(z_i),J_{y}(z_i)]$ which must satisfy a certain set of physical constraints. In particular, this means that in practice one should impose the restriction that ${\bf J}$ belongs to the projection of the critical current region onto the plane (${\bf J}\in\Delta_p$) and that $\nabla\cdot\textbf{J}\equiv 0$. As a main fact it will be established that, when building the parallel configuration, the response of the superconductor depends on the limitations for the current density established by the depinning threshold $J_{c\perp}$ on the orientation of the local magnetic field, and eventually on the threshold value for the cutting component or $J_{c\parallel}$. This is easily understood at a qualitative level just by glancing once more the left-side of Fig. \[Figure\_4\_1\] (pag. ). Within this picture, with transport current density flowing along the $xy$-plane, notice that for moderate values of the angle $\theta$ between the local magnetic field and the $z$-axis, the critical current restriction or material law $\Delta_{\rm p}$ becomes an ellipse of semi-axes $J_{c\perp}$ and $J_{c\parallel}^{\rm p}$ with $$\begin{aligned}
\label{Eq.5.9}
J_{c\parallel}^{\rm
p}={J_{c\perp}}/{\cos{\theta}}=J_{c\perp}{\sqrt{H_{x}^{2}+H_{y}^{2}+H_{z}^{2}}}/
{H_{z}} \, .\end{aligned}$$ An increase of the in-plane magnetic field component will result in a tilt of the cylinder by an increase of the angle $\theta$. In particular notice that, initially the maximum value of the in-plane parallel current density, $J_{c\parallel}^{\rm p}$, grows with the angle $\theta$, independent of $J_{c\parallel}$ (which is, thus, absent from the theory) until the maximum value $\sqrt{J_{c\perp}^{2}+J_{c\parallel}^{2}}$ is reached. Then, the ellipse is truncated and eventually would be practically a rectangle of size $2J_{c\parallel}\times 2J_{c\perp}$ when $\gamma \to\pi/2$. Outstandingly, for large values of $\chi\equiv J_{c\parallel}/J_{c\perp}$ (long cylinders), the critical current along the parallel axis $J_{c\parallel}^{\rm p}$ increases more and more as the weight of $H_{z0}$ decreases, and furthermore, this quantity is always beyond the individual values $J_{c\perp}$ and $J_{c\parallel}$.
Recall that the variational statement for three dimensional configurations on the slab geometry has been thoroughly discussed in chapter \[ch-4-1\].1. Thus, based on the numerical resolution for discretized layers, and within the mutual inductance formulation \[Eq. (\[Eq.3.3\])\], the above described longitudinal problem takes the following form: $$\begin{aligned}
\label{Eq.5.10}
{\cal
F}[I_{l+1}]=&&~\frac{1}{2}\sum_{i,j}I_{i,l+1}^{x}{\rm
M}_{ij}^{x}I_{j,l+1}^{x}-\sum_{i,j}
{\rm I}_{i,l}^{x}{\rm M}_{ij}^{x}I_{j,l+1}^{x}\nonumber \\
&&+\frac{1}{2}\sum_{i,j}I_{i,l+1}^{y}{\rm M}_{ij}^{y}I_{j,l+1}^{y}
-\sum_{i,j} {\rm I}_{i,l}^{y}{\rm M}_{ij}^{y}I_{j,l+1}^{y}\nonumber \\
&&+\sum_{i}I_{i,l+1}^{y}(i-1/2)({\rm h}_{0,l+1}^{y}-{\rm h}_{0,l}^{y}) \, .\end{aligned}$$
Moreover, we already know that the local components of the magnetic fields have to be evaluated according to equations of the kind (\[Eq.4.1\]) & (\[Eq.4.2\]), and the parallel and perpendicular projections of the sheet current components are given by $$\begin{aligned}
\label{Eq.5.11}
I_{\perp}^{2}=\left(1-(h_{i}^{x})^{2}\right)
(I_{i}^{x})^{2}+\left(1-(h_{i}^{y})^{2}\right)(I_{i}^{y})^{2}-2h_{i}^{x}h_{i}^{y
} I_{i}^{x}I_{i}^{y} \, ,\end{aligned}$$ and $$\begin{aligned}
\label{Eq.5.12}
I_{\parallel}^{2}=(h_{i}^{x} I_{i}^{x})^{2} + (h_{i}^{y} I_{i}^{y})^{2}+
2h_{i}^{x}h_{i}^{y}I_{i}^{x} I_{i}^{y} \, . \end{aligned}$$ Technically, the main differences with the problems considered in the above chapter are the inclusion of an additional constraint for the transport current, and new expressions for the inductance matrices.
On the one hand, for our problem with transport current, one has to consider for each temporal step the external constraint $$\begin{aligned}
\label{Eq.5.13}
\sum_{i}I_{i}^{y}(t)=I_{tr}(t) \, .\end{aligned}$$ On the other hand, as related to the symmetry properties for the transport configuration \[$I_{i}^{y}(z)=I_{i}^{y}(-z)$ as opposed to the antisymmetry for the case of shielding currents\], here one has to use the mutual inductance expressions $$\begin{aligned}
\label{Eq.5.14}
M_{i,j}^{x}\equiv 1+2\left[min\{i,j\} \right]\,\qquad\, , \qquad
M_{i,i}^{x}\equiv 2\left(\frac{1}{4}+i-1 \right)\nonumber \, , \\ \, \\
M_{i,j}^{y}\equiv 1+2\left[N_{s}-max\{i,j\} \right]\,\qquad\, , \qquad
M_{i,i}^{y}\equiv 2\left(\frac{1}{4}+N_{s}-i\right) \, \nonumber.\end{aligned}$$ with $N_{s}$ the full number of layers in the discretized slab.
Finally, within the framework of the SDCST, it is to be recalled that the local components of the current density have to be constrained according to one of the models contained by our generalized material law depicted in the right pane of Fig. \[Figure\_4\_1\] (pag. ) \[Mathematically see Eq. (\[Eq.2.17\]), pag. \]. Thus, with the aim of achieving a comparison between our numerical results and the analytical approach of Ref. [@P2-Brandt_2007], in Fig. \[Figure\_5\_3\] we show some of the results for the T-states assumption under consideration of a strong perpendicular field $h_{z0}=200$, and the arising discrepancies when a perpendicular field of moderate intensity (e.g., $h_{z0}=20$) is considered.
Firstly, let us emphasize that within our SDCST there is no restriction for the ratios $\chi^{-1}\equiv
J_{c\perp}/J_{c\parallel}$ and $\varsigma\equiv J_{c\perp}a/H_{z0}$, which become small parameters within the analytical approach of Ref. [@P2-Brandt_2007]. Notice that the smallness of $\chi^{-1}$ means that the arising critical state is approximated by the unbounded band region $|j_{\perp}|=1, 0 < |j_{\parallel}|
<\infty$ described in the chapter 4.6 (T-states). The smallness of $\varsigma$ was meant to indicate a small deviation of the full magnetic field respect to the $z-axis$. Then, moderate values of $j_{\parallel}$ are expected. Notoriously, the above hypotheses of Ref. [@P2-Brandt_2007] allow to state the problem by a set of approximate analytic formulas for the electromagnetic quantities which in turn allow to bypass the numerical solution of the differential equations. However, as it will be shown below, the range of application is narrower than expected.
Figure \[Figure\_5\_3\] shows the comparison between the penetration profiles for the local magnetic field components $h_{x}$ and $h_{y}$ obtained from our SDCST with $\chi\rightarrow\infty$ and $j_{c\perp}=1$, and from the analytic expressions in Ref. [@P2-Brandt_2007], i.e., $$\begin{aligned}
\label{Eq.5.15}
h_{x}&=&\frac{\alpha}{{\rm cos}\,\theta}{\rm
arcsinh}\left(\frac{z}{\alpha}\right)
\nonumber\\
h_{y}&=&h_{y0}-\alpha\left(\sqrt{1+\frac{1}{\alpha^{2}}}-\sqrt{1+\frac{z^2}{
\alpha^2}}\right)
\nonumber\\
{\rm cos}\,\theta &=& 2\alpha\,{\rm arcsinh}\left(\frac{1}{\alpha}\right) \, .
$$ Here, $\alpha$ has to be obtained for each value of the applied field from the condition $cos\theta=h_{z0}/\sqrt{h_{z0}^{2}+h_{y0}^{2}}$. One can notice that the agreement is rather good for the higher value of the perpendicular field $h_{z0}=200$ (right pane of Fig. \[Figure\_5\_3\]), whereas remarkable differences appear for a moderate field $h_{z0}=20$ as $h_{y0}$ increases (central pane of Fig. \[Figure\_5\_3\]). Our interpretation of the above facts is as follows.
On the one hand, as regards to the establishment of the full penetration profile, we have straightforwardly obtained this condition through the step-by-step integration starting from the state $h_{y0}=0$ (the evolution is shown in the insets of the Fig. \[Figure\_5\_3\]). Whereas the value 0.796 is estimated for the penetration field $h_{y0}^{p}$ within the analytical approach of Ref. [@P2-Brandt_2007], by the straightforward method described above we get $h_{y0}^{p}=0.845$. Remarkably, in spite of some small differences for the low field profiles $h_{x}$ and $h_{y}$, at moderate values of the transverse field ($h_{y0}<h_{z0}$) the curves always coincide. On the other hand, the failure of the analytical approximation for the higher values of $h_{y0}$ is readily explained by the observation of the plot. Thus, increasing $h_{y0}$ compresses the transport current towards the center of the sample (as indicated by the slope of $h_{x}(z)$). For the case of $h_{z0}=20$ (central pane of Fig. \[Figure\_5\_3\]), one gets $j_{y,max}\approx 5$ when $h_{y}\approx 100$ and $j_{y,max}\approx 50$ when $h_{y}\approx 1000$, then a considerable value of $j_{\parallel}$ is obtained. This leads to a not so good approximation from the analytic condition in the approximation of Ref. [@P2-Brandt_2007], which one is only valid for small values of this quantity. However, when comparison is made for $h_{z0}=200$, one gets $j_{y,max}\approx 1$ when $h_{y}\approx 100$ and $j_{y,max}\approx 5$ when $h_{y}\approx 1000$. Then, a much better performance is obtained for the analytical limit even for very high applied fields $h_{y0}$.
In brief, from the above discussion we may conclude that our SDCST overcomes previous limitations related to the [*weak longitudinal current*]{} conditions. By contrast, in the following section we will show a wide set of numerical calculations for material laws even most complicated than the simplest T-state model here considered, allowing to display the corrections needed in the general critical states.
### \[ch-5-2\] *5.2 Magnetic anisotropy and the uncommon effects* {#ch-5-2-5.2-magnetic-anisotropy-and-the-uncommon-effects .unnumbered}
This section will be devoted to unveil the features of longitudinal transport problems under general critical state conditions, and to identify the influence of a number of physical parameters along the different stages of the magnetization process. Here, for the experimental configuration depicted in the left pane of Fig. \[Figure\_5\_3\] and based on the numerical resolution of the variational statement, a complete tour along the whole set of values for the perpendicular field will be presented. We shall concentrate on the effect of the flux cutting boundary ($j_{c ||}$) considering several conditions for the material law introduced by our SDCST (see left pane at Fig. \[Figure\_4\_1\], pag. ). Firstly, the extreme case $\chi\to\infty$ or infinite bandwidth model (T-states model) will be considered (Subchapter 5.2.1). Secondly, several anisotropic models characterized by the *superelliptic* relation \[Eq. (\[Eq.2.17\]), pag. \] $$\begin{aligned}
\label{Eq.5.16}
j_{\perp}^{2n}+(j_{\parallel}/\chi)^{2n}\leq1\end{aligned}$$ will be thoroughly analyzed (Subchapter 5.2.2). In particular, we will set of material laws defined by the parameters $\chi=1$, 2, 3, and 4 with the *smoothing* index n=4.
Remarkably, our procedure will reveal the fingerprints of the cutting and depinning mechanisms, thus being a theoretical pathway for the reconstruction of the material law, represented by the proper region $\Delta_{\textbf{r}}$.
***5.2.1 Extremal case: The T-states model***
\[ch-5-2-1\] The T-state model for three dimensional configurations of the applied magnetic field has been exhaustively studied in chapter 4.3. Here, although the depicted scenario is similar, the assumption of a longitudinal transport current allow a straightforward understanding of the physical scopes of this model beyond the analytical approximations. First, we will analyze the properties of the local field \[$\textbf{h}(z)$\] and the current density profiles \[$\textbf{j}(z)$\] for a longitudinal configuration built in the fashion described in the left pane of Fig. \[Figure\_5\_3\] (a third component of the magnetic field $h_{z0}$ is incorporated).
![\[Figure\_5\_4\] Profiles of the local magnetic field components $h_{x}[z,h_{y0}]$ and $h_{y}[z,h_{y0}]$, and the corresponding current-density profiles $j_{y}[z,h_{y0}]$ and $j_{x}[z,h_{y0}]$ for the *T*-state model and perpendicular magnetic field components $h_{z0}=10$ (top), $h_{z0}=2$ (middle) and $h_{z0}=0.5$ (bottom). The different curves correspond to the following sets of values for the applied longitudinal field at the surface: (i) top row: $h_{y0}=0.005, 0.050, 0.170, 0.340, 0.500, 0.680,
0.845,1.0, 40.0,80.0, 150.0, 300.0, 500.0, 750.0$, $1000$, (ii) middle row: $h_{y}(a)=0.0050.050,0.170,0.340,0.500,0.680,0.845,1.0,5.0,10.0$, $20.0,30.0,40.0,50.0,60.0$, (iii) bottom row: $h_{y}(a)=0.001,0.050,
0.170,0.340,0.500$, $0.680, 0.845, 1.0, 2.0, 3.0,5.0,7.0, 8.0, 9.0, 10.0$. Insets to the middle pane correspond to the $j_{y}[z,h_{y0}]$ profiles closer to the center of the sample. Analogously, insets to the right correspond to the $j_{x}[z,h_{y0}]$ profiles. Finally, inner insets at right pane correspond to a specific profile of $h_{y}[z,h_{y0}]$ so as to highlight the occurrence of magnetic field reentry at low values of $h_{z0}$. ](Figure_5_4.pdf){height="8.59cm" width="13cm"}
Fig. \[Figure\_5\_4\] shows the behavior of the magnetic field profiles and the induced currents subsequent to the application of the transport current for three different initial conditions: $h_{z0}=10$, $h_{z0}=2$, and $h_{z0}=0.5$, all of them under assumption of the T-state model. The initial state for the transport current condition ($I_{tr}=J_{c \perp}a/2$) establishes the initial transport profile $j_{y}\{0\leq z<a/2\}=0$ and $j_{y}\{a/2\leq z\leq
a\}=1$. As the transport current is no longer modified, the condition $h_{x}(a)=0.5$ can be applied in what follows. On the other hand, by symmetry, one has the condition $h_{x}(0)=0$ at the center of the slab.
In detail, when the external magnetic field $[h_{y0}=h_{y}(a)]$ (the applied parallel field) is linearly increased from $h_{y}(a)=0$, a current density $j_{x}$ is induced from the superconducting surface as an effect of Faraday’s law. Simultaneously, the local component of the magnetic field $h_{x}(z)$ increases monotonically following two continuous stages fulfilling the aforementioned boundary conditions. First, the superconducting sample is fully penetrated by the transport current when $h_{y}^{\star}(a)=0.845\pm0.003$ and eventually, the condition $j_{y}(0)=1$ is reached as soon as $h_{y}(a)\rightarrow 0.860$. We notice that the value of $h_{y}^{\star}(a)$ for the full penetration profile is basically independent of $h_{z0}$ (at least to the numerical precision of our numerical calculations), in agreement with our analytical solution [@P2-Ruiz_PRB_2011] introduced in the previous section (Chapter 5.1.1). Second, a remarkable enhancement of the transport current density occurs around the center of the slab as $h_{y}(a)$ increases beyond $h_{y}^{\star}(a)$. Furthermore, an eventual negative current density appears shielding the positive transport current around the center of the slab. It is to be noticed that the appearance of negative current flow is enhanced when the magnetic component $h_{z0}$ is decreased (Fig. \[Figure\_5\_5\]).
It is noteworthy that for the range of values $h_{z}(0)<h_{y}^{\star}(a)$ negative surface current appears even for the partial penetration regime, e.g., for $h_{z0}=0.5$ one has $j_{y}(a)<0$ for $h_{y}(a)>0.722$. Further, another outstanding property is the occurrence of profiles with magnetic field reentry (paramagnetism in the component $h_{y}$ around the center of the slab) for $h_{z0}\lesssim 1$ and under relatively low applied magnetic fields $h_{y}(a)$ (see Fig. \[Figure\_5\_4\]). In fact, we call the readers’ attention that the above mentioned effects, local paramagnetism and negative current zones, have both been shown analytically in the limiting case $h_{z0}=0$ (Chapter 5.1.1). Along this line, as a general rule, we can conclude that the smaller the value of $h_{z0}$, the sooner the surface of negative transport current and even paramagnetic local effects appear.
![\[Figure\_5\_5\] Dynamics of the local current density as a function of the applied longitudinal magnetic field $h_{y0}=h_{y}(a)$ along the central and external sheets of the slab. The results are shown for the T-state model ($J_{c||}\to\infty ~\&~ J_{c\perp}=1.0$). Top: the components $j_{y}$ and $j_{x}$ at $(z=0)$ and $(z=a)$. Middle: details of the above behavior. Bottom: dynamics of the parallel and perpendicular components of $\bf j$ in the same conditions as above. The different curves correspond to the values of the perpendicular magnetic field given by $h_{z0}= 0.5, 1, 2, 5, 10, 20, 50, 100,
200$. Several scales have been displayed to avoid information loss with all plots running over the same color scale. ](Figure_5_5.pdf){height="10.5cm" width="12cm"}
On the other hand, figure \[Figure\_5\_5\] displays the evolution of the current density vector focusing us on the specific values at the superconducting surface ($z=a$) and at the center of the superconducting slab ($z=0$) as the longitudinal magnetic field $h_{y}(a)$ is increased. Outstandingly, the involved physical features are straightforwardly explained by the polar decomposition of the current density introduced in chapter 4.1 \[See Eqs. (\[Eq.4.6\]) - (\[Eq.4.8\]), pag. \]. In brief, notice that the unbounded parallel current density allows unconstrained rotations for the flux lines as the applied magnetic field increases. In particular, this leads to negative values of $j_{y}(a)$ (slope of $h_{x}(a)$), simultaneous to high $j_{y}(0)$ (slope of $h_{x}(0)$). Moreover, it should be noticed that negative values of the transport current are favored by smaller and smaller values of the field component perpendicular to the surface of the sample $h_{z0}$. Also, notice that at the center of slab the flux line dynamics is mainly governed by the longitudinal transport current density $j_{y}(0)$. The basic idea is that for moderate values of $h_z$, when $h_{y}$ increases $j_{y}$ practically becomes $j_{\parallel}$. As this component is unconstrained, it grows indefinitely at the center.
For a closer connection with real experiments, below let’s concentrate on the magnetostatic properties by means of the *global* sample’s magnetization curve **M**(**H**). Thus, we have calculated **M** as a function of the longitudinal magnetic field $h_{y}(a)$.
![\[Figure\_5\_6\] The magnetic moment components $(M_{x},M_{y})$ of the slab as a function of the applied magnetic field component $h_{y}(a)$ for the T-state model. The curves are labeled according to the perpendicular magnetic field $h_{z0}= 0.5, 1, 2, 5, 10, 20, 50, 100, 200$. The insets show different zooms of the magnetic moment components for low values of the applied longitudinal field. The same color scale applies to all the plots. ](Figure_5_6.pdf){height="8cm" width="10cm"}
Fig. \[Figure\_5\_6\] displays the magnetic moment components $M_{x}(h_{y0})$ and $M_{y}(h_{y0})$ in units of $J_{c\perp}a^{2}$. First, notice that within the partial penetration regime ($h_{y}(a)\leqslant
h_{y}^{\star}(a)$) the magnetic moment components are almost independent of the perpendicular magnetic field $h_{z0}$ (at least for non small values of this quantity). On the contrary, when $h_{z0}<1$ and the patterns of negative current even occur before the fully penetrated state, the magnetization slightly increases. This is accompanied by faint field [*reentry*]{} effects that are also shown within the figure. Furthermore, as the threshold cutting current density $j_{c\parallel}$ is unbounded for the T-state model, the magnetic moment $M_{x}$ always increases as related to the diverging behavior of $j_{y}(0)$.
Let us emphasize that, as the unbounded behavior for the parallel current density assumed above leads to the prediction of arbitrarily high values of the transport current density, consequently the T-state model must be physically reconsidered. For example, on the one hand, the trend of the magnetic moment $M_{x}$ and also the unbounded longitudinal current density $j_{y}$ disagree with the experimental evidences recollected in Refs. [@P2-Voloshin_2001; @P2-LeBlanc_2003; @P2-LeBlanc_2002; @P2-Esaki_1976; @P2-LeBlanc_1966; @P2-Gauthier_1974; @P2-Karasik_1970; @P2-Sugahara_1970; @P2-Taylor_1967; @P2-Sekula_1963]. On the other hand, in Fig. \[Figure\_5\_5\] one can observe that, as soon as the flow of negative current along the superconducting surface is reached, it never disappears notwithstanding the longitudinal magnetic field remains increasing. By contrast, the disappearance of the patterns of superficial negative current were detected in Refs. [@P2-Matsushita_1984; @P2-Matsushita_1998; @P2-Matsushita_2012; @P2-Voloshin_1991; @P2-Esaki_1976; @P2-LeBlanc_1966]. These observations have lead to consider $j_{c\parallel}$ [*bounded*]{} descriptions as satisfactory solutions of the peculiar phenomena involved on the longitudinal transport current problem [@P2-LeBlanc_2003; @P2-LeBlanc_2002; @P2-Matsushita_1984; @P2-Matsushita_1998; @P2-Matsushita_2012]. Rather recent experimental data on high temperature superconductors [@P2-Clem_2011_SUST; @P2-Campbell_2011_SUST] also indicate that physical bounds are to be considered for both components of the critical current.
Thus, as will be described below in section 5.2.2, our generalized SDCST suits the necessity of dealing with a physically acceptable description of both local and global issues about the electromagnetic quantities involved in the longitudinal transport current problem [@P2-Ruiz_PRB_2009; @P2-Ruiz_PRB_2011; @P2-Ruiz_SUST_2011].
***5.2.2 Material laws with magnetic anisotropy: CT$\chi$ – models***
\[ch-5-2-2\]
More realistic models for the material law are presented below. Henceforth, we shall use the simplified notation T or CT$\chi$ as regards to the infinite bandwidth model (T by transport) or the *superelliptical* critical state models with anisotropy $\chi=|J_{c ||}/J_{c \perp}|$ (CT by cutting and transport, and $\chi$ the index controlling the magnetic anisotropy of the model). Recall that neither the isotropic model (n=1) nor the rectangular models ($n\rightarrow\infty$) are able to achieve a full explanation of the involved physical events. Thus, hereafter a value $n=4$ will be chosen for the smoothing index, corresponding to an intermediate regime between the above mentioned material laws. As an additional advantage, we want to mention that, technically, the use of smooth models produces stable and faster numerical convergence.
In order to obtain continuity with the T-state results obtained above, the electrodynamic quantities of interest have been obtained under the same experimental configuration shown at the left pane of Fig. \[Figure\_5\_3\], and a similar analysis scheme to that developed in the previous section will be tackled here. Thus, with the aim of getting a detailed physical interpretation on how the longitudinal and transverse magnetic fields affect the dynamics of the transport current problem, we will show the magnetic penetration profiles for low and high perpendicular fields, i.e., $h_{z0}=0.5$ (Figs. \[Figure\_5\_7\] & \[Figure\_5\_8\]) and $h_{z0}=10$ (Figs. \[Figure\_5\_9\] & \[Figure\_5\_10\]), for different $CT\chi$ conditions. Then, for completeness, the set of initial conditions $h_{z0}$ is extended in Figs. \[Figure\_5\_11\] & \[Figure\_5\_12\], by means a thorough analysis of the dynamics of the current density over the central layer $[j_{y}(z=0)]$ and external layer $[j_{y}(z=a)]$ of the slab. Finally, it will be shown that the fingerprint of the different CT$\chi$ models is identified as a peak effect in the curves for the magnetic moment components (Fig. \[Figure\_5\_13\] & \[Figure\_5\_14\]) caused by the maximal enhancement of the transport current density along the central layer.
##### ( *A.* ) *Penetration Profiles and current density behavior*\
On the one hand, let us recall that under the SDCST the square condition is given by $\chi=1$ (CT1) assuming the customary condition $J_{c\perp}=1$ (i.e., $j_{c\parallel}=1$ as $\chi\equiv
J_{c\parallel}/J_{c\perp}$). This can be considered as a lower bound for such quantity because the experimental values reported in the literature are typically above unity. On the other hand, as we have observed an intricate behavior for the local dynamics of the electromagnetic quantities under the CT$\chi$ conditions, henceforward, we will split the experimental process in three successive stages as the longitudinal magnetic field component $h_{y}(a)$ is increased. We mean:
![\[Figure\_5\_7\] Profiles of the magnetic field components $h_{x}[z,h_{y0}]$ and $h_{y}[z,h_{y0}]$ for a perpendicular field component $h_{z0}=0.5$. Also included are the corresponding current-density profiles $j_{y}[z,h_{y0}]$ and $j_{x}[z,h_{y0}]$ for the CT1 model. For clarity, the longitudinal magnetic field is applied in three stages: Top, $h_{y}(a)=$0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 1.1, 1.3; Middle, $h_{y}(a)=$1.3, 1.6, 1.9, 2.2, 3.0, 4.0, 5.0, 6.0, and Bottom $h_{y}(a)=10,
20, 40, 80, 100, 125, 150, 1000$. Insets at top pane show a zoom of the current density profiles close the external sheet of the superconducting slab. Insets at either middle or bottom panes show the shape of one of the profiles $h_{y}[z,h_{y}(a)]$ for the corresponding stage. Dashed line arrows are included to supply a trace method from the initial profile until the last one, in correspondence to the aforementioned stages. ](Figure_5_7.pdf){height="9.4cm" width="13cm"}
- The current density at the center $j_{y}(0)$ increases until a maximum value is obtained. The set of profiles for the physical quantities \[$h_{x}(z)$,$j_{y}(z)$\] and \[$h_{y}(z)$,$j_{x}(z)$\] in the partial penetration regime are also included within this stage (see e.g. top of Figs. \[Figure\_5\_7\] - \[Figure\_5\_10\]). The occurrence of possible negative values for the longitudinal current density along the superconducting surface $j_{y}(a)$ is also focused for low values of the perpendicular magnetic field $h_{z0}$.
- The longitudinal current density profiles $j_{y}(z)$ show a *bow tie pattern*, whose evolution is shown until the minimum value for the longitudinal current density along the superconducting surface, $j_{y}(a)$, is reached (see e.g., the middle row of Figs. \[Figure\_5\_7\] - \[Figure\_5\_10\]).
- Eventually, the longitudinal current density $j_{y}(a)$ grows up by increasing the longitudinal applied field $h_{y}(a)$, and further it stabilizes around a certain value (e.g., $j_{y}(a)\approx0.5$ for CT1 case as it is shown in the bottom of Figs. \[Figure\_5\_7\] - \[Figure\_5\_9\]).
![\[Figure\_5\_8\] Profiles of $h_{x}[z,h_{y0}]$ and $j_{y}[z,h_{y0}]$ for the 1st (top) and 2nd (middle) stages of the magnetic dynamics described in the cases CT2 (left pane), CT3 (central pane) and CT4 (right pane), all under a field $h_{z0}=0.5$. The 3rd stage is only defined for the CT2 case (left pane - bottom), as we have not noticed remarkable difference between the 2nd and 3rd stages for cases CT3 and CT4, at least for values of the longitudinal applied field not beyond of $h_{y}(a)=1000$. So, in return, the lower panes in cases CT3 and CT4 show the profiles for $j_x[z,h_{y}(a)]$ for the 1st and 2nd stages. $h_{y}(a)$ is as follows. CT2: (*1st stage*) $h_{y}(a)=$0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 1.1, 1.3, 1.6, 1.9, 2.2; (*2nd stage*) $h_{y}(a)=$3, 4, 5, 6, 8, 10, 20, 40; and (*3rd stage*) $h_{y}(a)=$ 80, 120, 160, 200, 300, 400, 600, 800, 1000. CT3: (*1st stage*) $h_{y}(a)=$0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 1.5, 1.8, 2.2, 2.6, 3.0; and (*2nd stage*) $h_{y}(a)=$4, 5, 6, 8, 15, 20, 40, 70, 100, 150, 200, 300, 400, 600, 1000. CT4: (*1st stage*) $h_{y}(a)=$0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 1.5, 1.8, 2.2, 2.6, 3.0, 4.0; and (*2nd stage*) $h_{y}(a)=$5, 6, 8, 15, 20, 40, 100, 200, 400, 600, 1000. ](Figure_5_8.pdf){height="8.1cm" width="13cm"}
As far as concerns to the field and current density penetration profiles, it is to be noticed that the trend of the profiles for the partial penetration regime is fairly independent of the perpendicular magnetic field $h_{z0}$ (Figs. \[Figure\_5\_7\] - \[Figure\_5\_10\]). Moreover, the partial penetration regime in which the transport current zone progressively penetrates the sample is nearly independent of the magnetic anisotropy of the critical state (compare the above profiles to their respective ones for the T-state condition in Fig. \[Figure\_5\_4\]).
![\[Figure\_5\_9\] Same as Fig.\[Figure\_5\_7\], but for $h_{z0}=10$ and the values of the longitudinal field: (Top) $h_{y}(a)=$ 0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 4.0, 6.0, 8.0, 11, (Middle) $h_{y}(a)=$ 13, 15, 17, 20, 25, 35, 50, 65, (Bottom) $h_{y}(a)=$ 80, 100, 125, 150, 200, 300, 400, 1000. ](Figure_5_9.pdf){height="9.4cm" width="13cm"}
We call readers’ attention to the fact that, in the aforementioned first stage, the negative current patterns are also found under the low applied magnetic fields $h_{z0}<0.5$. However, by contrast to the results within the previous section, it is to be recalled that for the T-state model the condition $j_{c ||}\rightarrow\infty$ allows unbounded values for the longitudinal current $j_{y}$ at the center of the sample. By contrast, for the bounded cases CT$\chi$, the magnetic anisotropy of the material law $\Delta_{r}$ defines the maximal current density for the critical state regime. In other words, the maximal length of the vector $\textbf{j}$ corresponds to the optimal orientation of the region $\Delta_{r}(\chi,n)$ in which the biggest distance into the superelliptical condition is reached, i.e., such situation corresponds to the maximal transport current allowed in the superconductor, and can be calculated by the following analytical expression $$\begin{aligned}
\label{Eq.5.17}
{\rm Max}\{j_{c\parallel}(\Delta_{\textbf{r}})\}=j_{y}^{max}=
\left( 1+\chi^{2n/(n-1)} \right)^{(n-1)/2n} \, . \end{aligned}$$
![\[Figure\_5\_10\] Similar to Fig. \[Figure\_5\_8\], but for $h_{z0}=10$. The curves are labeled as follows. CT2 (left pane): 1st stage (top) $h_{y}(a)=$ 0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 5.0$, $10, 15, 20, 30, and 2nd stage (middle) $h_{y}(a)=$ 40, 60, 80, 120, 160, 200, 300, 400, 600, 800, 1000. CT3 (middle pane): 1st stage (top) $h_{y}(a)=$ 0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 5.0, 10, 15, 20, 25, 30, 40, and 2nd stage (middle) $h_{y}(a)=$60, 80, 120, 160, 200, 300, 400, 600, 800, 1000. CT4 (right pane): 1st stage (top) $h_{y}(a)=$0.005, 0.050, 0.170, 0.340, 0.500, 0.680, 0.845, 1.0, 5.0, 10, 15, 20, 35, 30, 40, 60, and 2nd stage (middle) $h_{y}(a)=$ 80, 120, 160, 200, 300, 400, 600, 800, 1000. In the bottom of panes CT2, CT3 and CT4, the corresponding profiles of $j_x[z,h_{y}(a)]$ are shown. ](Figure_5_10.pdf){height="9.4cm" width="13cm"}
##### ( *B.* ) *Central and superficial patterns of the current density*\
Note that Eq. \[Eq.5.17\] allows us to obtain the maximum value expected for $j_{y}$ in terms of the actual critical state model in use. Thus, as we have assumed n=4 for the different CT$\chi$ models considered here, one gets $j_{y}^{max}=1.2968$ for the CT1 case, $j_{y}^{max}=2.1127$ for the CT2 case, $j_{y}^{max}=3.0591$ for the CT3 case, and $j_{y}^{max}=4.0369$ for the CT4 case. These values may be checked by means our numerical results in Figs. \[Figure\_5\_11\] & \[Figure\_5\_12\] for the intensity of the transport current density in the central layer of the slab, i.e., the obtained patterns $j_{y}(0)$.
![\[Figure\_5\_11\] Evolution of the current density vector for the CT1 model. To the left, we show the patterns of transport current over the central layer ($j_{y}(z=0)$) and external layer ($j_{y}(z=a)$) of the slab. To the right, we show the variation of the perpendicular component $j_{x}(z=0)$ and $j_{x}(z=a)$. The curves are labeled according to the perpendicular magnetic field component $h_{z0}=$ 0.1, 0.5, 1, 2, 5, 10, 20, 50, 100, 200. ](Figure_5_11.pdf){height="11.5cm" width="13cm"}
In addition, note that by a thorough analysis of the current density components along the central ($z=0$) and external layers ($z=|a|$) of the superconducting slab, several remarkable physical properties can be straightforward depicted (Figs. \[Figure\_5\_11\] & \[Figure\_5\_12\]). On the one hand, notice that the full penetration regime can be clearly distinguished from the partial penetration regime, and remarkably the emergence of negative states for the transport current density close to the external surface of the superconducting sample it is more evident either when $h_{z0}$ is reduced and/or the current density anisotropy factor $\chi$ is increased. On the other hand, outstandingly the maximal value of the current density along the original direction of the transport current ($y-$axis) can be depicted in terms of $j_{y}(0)$, where its maximal value defined by Eq. (\[Eq.5.17\]) turns out to be independent of the perpendicular applied magnetic field at least as regards the existence of the *peak effect* in the transport current density. Thus, the enhancement of the transport current density can be either accelerated or decelerated with the tilt of the applied magnetic field, but in general terms, its maximum directly relates to the limitation introduced by the cutting mechanism. Physically, this means that the role played by the magnetic anisotropy of the material law may be characterized by the influence of the threshold cutting value on the enhancement of the critical current density.
![\[Figure\_5\_12\] Dynamics of the current density vector for the models CT2 (left pane), CT3 (middle pane) and CT4 (right pane). The first row of plots corresponds to the patterns of transport current along the surface layers of the slab \[$j_{y}(z=a)$\]. The second one, corresponds to the respective values for the parallel component of the current density \[$j_{||}(a)$\]. The dynamics of the transport current $j_{y}$ and the component $j_{||}$ at the central layer of the superconducting slab $(z=0)$ is shown at the third and fourth rows, respectively. The curves are labeled according to the perpendicular magnetic field component $h_{z0}=$1E-7, 0.5, 1, 2, 5, 10 with all plots having the same color scheme.](Figure_5_12.pdf){height="10cm" width="13cm"}
In order to confirm the above physical interpretation, in Fig. \[Figure\_5\_12\] we show the magnetic dynamics of the longitudinal current density $j_{y}$, and the cutting current component $j_{||}$ for the conditions CT2, CT3, and CT4. We have taken a wide set of values for the perpendicular field component ($h_{z0}$). On the one hand, as far as concerns the sample’s surface, we have observed that the longitudinal current density $j_{y}(a)$ does not display significant differences when one has $\chi\geq2$ (see also Fig. \[Figure\_5\_5\] for the T-state model with $\chi\rightarrow\infty$). Hence, the disappearance of the negative current flow along the external superconducting surface does not occur despite a very high applied magnetic field has been considered ($h_{y}(a)=1000$). On the other hand, it is important to notice that the patterns of the parallel current density along the superconducting surface ($j_{||}(a)$) are almost indistinguishable as soon as the condition $\chi\geq 2$ (CT2) is reached (upper half of Fig. \[Figure\_5\_12\]). This implies that for an accurate picture of the parallel critical current, surface properties do not provide a useful information.
![\[Figure\_5\_13\] Magnetic moment components $M_{x}$ and $M_{y}$ of the slab as a function of the applied magnetic field $h_{y}(a)$ for the CT1 model. The curves are labeled according to the perpendicular magnetic field component $h_{z0}$ for each.](Figure_5_13.pdf){height="12cm" width="13cm"}
However, Fig. \[Figure\_5\_12\] shows that the threshold value for the cutting current density can be estimated from the experimental measurement of the transport current density along the central sheet of the superconducting sample. Moreover, regardless to the experimental conditions ($h_{z0}$, $h_{y}(a)$) and also for different bandwidths $\chi$ no significant change occurs in the parallel current density around the central sheet of the sample (lower half of Fig. \[Figure\_5\_12\]).
##### ( *C.* ) *Features on the magnetic moment*\
Outstandingly, it is to be noticed that the limitation introduced by the flux cutting mechanism imposes a maximal compression of the current density within the sample. Thus, the peak effects both for the transport current density $j_{y}$ (Figs. \[Figure\_5\_11\] & \[Figure\_5\_12\]) and for the magnetic moment component $M_{x}$ (Figs. \[Figure\_5\_13\] & \[Figure\_5\_14\]) are defined by the instant at which the maximal transport current density occurs. Additionally, upon further increasing the longitudinal applied magnetic field component $h_{y}(a)$, the profile $h_{x}(z)$ will be forced to decrease from the central sheet ($z=0$) towards the external surface ($z=a$). This reversal generates a local distortion of the longitudinal current density $j_{y}$ in a bow tie shape (see the middle row of Figs. \[Figure\_5\_7\] & \[Figure\_5\_10\]). Likewise, as soon as the profile $j_{||}(0)=j_{c||}$ is reached, the magnetic moment $M_{x}$ starts decreasing as one can see by comparison of Figs. \[Figure\_5\_11\] & \[Figure\_5\_13\] and Figs \[Figure\_5\_12\] & \[Figure\_5\_14\].
![\[Figure\_5\_14\] Same as Fig. \[Figure\_5\_13\], but here the curves are corresponding to the CT2 (left pane), CT3 (central pane), and CT4 (right pane) critical state models. ](Figure_5_14.pdf){height="12cm" width="13cm"}
Finally, one additional feature is to be noted: the interval between the instant at which the maximal transport current condition is reached ($j_{y}(0)=j_{y}^{max}$), and the instant at which the slope of the magnetic moment $M_{x}$ changes sign, could be assumed as transient or stabilization period required for an accurate determination of the value $j_{c ||}$ when measurements are performed in terms of the applied longitudinal field $h_{y0}=h_{y}(a)$. Apparently, this transient increases with the value of the perpendicular field $h_{z0}$. From this point on, $j_{y}(0)$ may be basically identified with $j_{c\parallel}$.
\[ch-6\] **Electromagnetism For Superconducting Wires**
=======================================================
Type-II superconducting wires are deemed promising elements for large-scale technological applications such as power transmission cables, magnet systems for large particle accelerators, and magnetic-based medical techniques such as MRI. The usefulness of these kind of technology is straightforwardly linked to the local electromagnetic response of the superconductor under variations of the ambient magnetic field and the customary condition of transport-current. Special interest relies on determining the value of the maximum dissipation-free current and characterizing the mechanisms of reduction of electric power dissipation due to alternating fields and/or alternating current flow (commonly called AC losses).
Despite the electric power hysteretic losses associated to a superconducting material are somewhat smaller than the range of power dissipated in normal metals, from the practical point of view, the superconducting technology is still not so attractive to replace the power technology based in copper wires, because in order to keep the temperature of the superconductor below $T_{c}$, heat removal requires a sophisticated and costly cryogenic system. Thus, in order to make the superconducting devices more attractive and competitive with respect to other technologies, it is of utter importance to understand, predict, and eventually, reduce the AC losses of superconducting wires under practical configurations.
Major features of the macroscopic electromagnetic behavior of type-II superconducting wires have been captured in Bean’s model of the critical state [@P2-Bean_1964]. In this framework, magnetization currents of density $\bf J$ are induced within the superconductor during variations of the magnetic flux which accordingly redistribute themselves to screen the penetrating flux within the sample. Their magnitude adopts the critical value $J_c$ at a given temperature and specified field. Although simple for idealized configurations, the electrodynamics underlying Bean’s model becomes cumbersome when realistic configurations are addressed. Thus, penetration of magnetic flux must be typically obtained by sophisticated numerical methods.
For example, in order to comprehend the distributions of the field and current in two- and three-dimensional samples subject to time-varying electromagnetic fields a sort of free-boundary problem should be solved. In a superconducting slab or a large cylinder (radius much smaller than their length) exposed either to a parallel magnetic field or a longitudinal transport current, the magnetic field has only one component, and analysis of the field and current distribution is simple [@P2-Bean_1964]. Nevertheless, if the situation is such that the local magnetic field has two components that are functions of two spatial variables, then solving the free-boundary problem for arbitrary relations of the external excitations becomes more and more complicated. In fact, for real applications of superconducting wires, the scenario is such that a simultaneous field and transport current condition must be satisfied. Then, in addition to the ambient field one has the magnetic field generated by the transport current itself [@P2-Ruiz_APL_2012]. This is the situation, for example, in superconductor windings where each wire is subjected to the magnetic field of its neighboring wires. Thus, as this is the configuration very often met in practice, this chapter is devoted to study round superconducting wires under different configurations of transverse magnetic field and transport current flow.
Subchapter 6.1 is devoted to introduce the theoretical framework of the problem with special attention to the numerical procedure of our variational approach. It is to be noticed that the theoretical statements developed along this section are not only valid for round wires but rather for long superconductors with any topology of the transverse section, e.g., it may include strips of rectangular cross section, wires of elliptical or circular cross section, also any intermediate shape between them, and even multifilament structures as the observed in practical superconducting wires.
Then, in order to introduce our numerical results and to get a fairly clear understanding of the phenomena involved in the dynamics of the electromagnetic quantities for superconducting wires with simultaneous transport current and applied magnetic field, subchapter 6.2 deals with the simpler cases where the superconducting wire is only subjected to a transverse magnetic field in absence of transport current, or on the contrary, assume that the superconducting wire is only subjected to a transport current condition. For a closer connection with the experimental quantities, we have calculated the wire’s magnetic moment ($\textbf{M}$) and the hysteretic losses ($L$) as a function of AC external excitations, and their comparison with classical analytical approaches are featured. On the other hand, we present a thorough discussion about the observed patterns for the local dynamics of the electromagnetic quantities such as the inner distribution of current density $\textbf{J}$, the components of the magnetic flux density $\textbf{B}$ and the isolevels of the vector potential $\textbf{A}$ (i.e., the lines of magnetic field), as well as the local distribution of the density of power dissipation $\textbf{E}\cdot\textbf{J}$, whose profiles are displayed at the end of the chapter in the corresponding section of supplementary material.
In subchapter 6.3, numerical simulations of filamentary type II superconducting wires under simultaneous AC transport current and oscillating transverse magnetic field are performed. A wide number of configurations have been considered, with special attention to the aforementioned experimental quantities, $\textbf{M}$ and $L$. On the one hand, according to the temporal evolution of the AC sources, exotic magnetization loops are envisaged. On the other hand, remarkable numerical corrections to several widely used approximate formulae for the hysteretic losses are identified. Also, in view of the differences found between the different cases studied, we present a comprehensive study of the local dynamics of the electromagnetic quantities in analogy to the previous section, which allows to reveal some of the main “*control*” factors affecting the losses into the superconducting wires. The full set of profiles describing the temporal dynamics for the electromagnetic quantities $\textbf{J}$, $\textbf{B}$, and $\textbf{E}\cdot\textbf{J}$ within the superconducting wire are shown as function of the external excitations at the end of this chapter. In more detail, subchapter 6.3.1 addresses the study of cases with synchronous oscillating excitations, where significant differences between the obtained AC losses and those predicted by regular approximation formulas are reported. Furthermore, noticeable non-homogeneous dissipation and field distortions are displayed, as well as an outstanding [*low pass filtering*]{} effect intrinsic to the magnetic response of the system is described. Then, premagnetized superconducting wires are also considered, i.e.: a magnetic moment is induced previous to switching on the synchronous oscillating excitation. Thus, the above envisioned results are straightforwardly generalized. Finally, subchapter 6.3.2 deals with asynchronous oscillating sources, focusing on the calculations of the double frequency effects provided by one or the other source. This closes our discussion about the main parameters controlling the hysteretic losses in superconducting wires, and how this knowledge can help in the design of new kinds of applications.
### \[ch-6-1\] *6.1 Theoretical framework and general considerations* {#ch-6-1-6.1-theoretical-framework-and-general-considerations .unnumbered}
From the theoretical point of view and as has been stated in the previous chapters, the major features of the macroscopic electromagnetic behavior of type-II superconductors have been captured by Bean in the phenomenological model of the critical state [@P2-Bean_1964], and its ideas have been extended in our generalized critical state theory or also called smooth double critical state theory (SDCST) [@P2-Ruiz_PRB_2009; @P2-Ruiz_SUST_2010]. It is worth mentioning that despite our SDCST allows to include any experimental dependence of the critical current density on the local properties of the superconducting specimen (e.g., $J_{c}(H,T)$), the simple Bean’s statement $(J\leq
J_{c})$ allows to achieve a clearest interpretation of the physical phenomena involved in the electromagnetic dynamics of type II superconductors. Moreover, it allows to establish the limiting values expected for macroscopic quantities such as magnetization and energy losses. Thus, in what follows we will assume that the critical current density does not depend on the applied magnetic field (at least for the intensities here considered) neither on the temperature.
Being more specific, the technical problem in the critical state theory consists of solving a free boundary problem for the distribution of penetrating current (or magnetic flux) for a given time of the external electromagnetic excitation. From the analytical point of view, in simplified configurations such as infinite slabs and cylinders, either considering transport current or parallel magnetic field, the inner flux-free region can be straightforwardly depicted by a planar front of flux or radial flux fronts centered in the symmetry axis of the superconductor respectively [@P2-London_1963; @P2-Hancox_1966]. Thus, once a method is found to obtain the actual size of the flux free region, all the magnetic properties such as the magnetic field lines, magnetization, and AC losses can be deduced from the knowledge of the penetrating current profiles, also called flux fronts. However, for a superconducting strip or cylinder exposed to a transverse magnetic field the flux fronts are not radially symmetric, and the free boundary problem is not so easy to be tackled. Nevertheless, already in early calculations as those performed in 1970s and 1980s various analytical simplifications have often been used, as for example sinusoidal or elliptical ansatz for the flux-free region [@P2-Carr_1975; @P2-Zenkevitch_1980; @P2-Minervini_1989], these studies being summarized in more detail in the books by Carr [@P2-Carr_2001] and Gurevich et al. [@P2-Gurevich_1997]. Even for extremal cases such as infinitely thin strips, exact analytical solutions can be achieved [@P2-Swan_1968; @P2-Halse_1970; @P2-Norris-1970; @P2-Brandt_1993; @P2-Zeldov_1994]. Most recently, analytical expressions for the magnetic field and current distributions within the CS model for hollow superconducting tubes of thickness much smaller than their external radius were reported by Mawatari [@P2-Mawatari_2011]. On the other hand, either for the case of a strip with finite thickness or a bulk superconducting cylinder exposed to a transverse magnetic field, exact analytical solutions have not been obtained. Moreover, when the problem is such that the superconductor is simultaneously subjected to a transport current, asymmetric deformations of the flux free region can appear, and analytical approaches are even less conceivable. Thus, implementation of numerical procedures becomes mandatory when handling intricate configurations where simultaneous alternating transport current and transverse applied field occur.
First, a consistent implementation of a variational approach allowed Ashkin [@P2-Ashkin_1979] to trace out the true structure of the partly flux-penetrated state of a superconducting wire subject to a transverse magnetic field, whose results were confirmed in the meantime by various numerical and analytical calculations [@P2-Pang_1981; @P2-Telschow_1994; @P2-Kuzovlev_1995; @P2-Bhagwat_2001; @P2-Gomory_2002; @P2-Haken_2002]. For example, Telschow and Koo [@P2-Telschow_1994] suggested an integral-equation approach for determining the flux-front profile, thus reducing the problem (for the case of a constant critical current density $J_{c}$) to solving a single Fredholm integral equation of the first kind which may be performed by several algorithms; the method applying generally to a sphere or long cylinders exposed to axisymmetric external fields. In a similar approach, Kuzovlev has established the integral equation for the flux-free zone and an exact value of the full penetration field ($B_{p}$) for a three-dimensional superconductor of an arbitrary axisymmetric form [@P2-Kuzovlev_1995]. Bhagwat and Karmakar developed a method allowing for determination of the flux-front form in cylinder superconductors of different cross-sections and field orientations by solving a (formally infinite, in fact: large) system of nonlinear ordinary differential equations for coefficients determining the front [@P2-Bhagwat_2001]. On the other hand, in a similar fashion to the numerical solution for the ellipsoid geometry developed in Ref. [@P2-Navarro_1991], minimization procedures were used in Refs. [@P2-Gomory_2002] & [@P2-Haken_2002] to optimize the trial boundary of the flux-free region avoiding assuming an *a priori* shape for the flux fronts as was customarily done in the precedent works.
At this point, it is worth of mention that currently the most popular trend in the analysis of magnetic flux dynamics in superconductors are the numerical simulations implementing finite-element methods in conjunction with nonlinear power-law voltage-current characteristics [@P2-Amemiya_2001; @P2-Nibbio_2001; @P2-Stavrev_2002; @P2-Hong_2008]. Also of mention is the development of new formulations of the critical state model [@P2-Prigozhin_2004; @P2-Barrett_2010; @P2-Campbell_2007; @P2-Ruiz_PRB_2009; @P2-Ruiz_SUST_2010], or new algorithms to approach the CS in commercial finite element codes [@P2-Farinon_2010]. For example, on the one hand, Campbell et al. [@P2-Campbell_2007] have suggested that the critical state model could be made amenable in COMSOL-multiphysics by modifying the so called material law by an explicit function of the vector potential, and Farinon et al. [@P2-Farinon_2010] have proposed a special algorithm to be implemented in commercial ANSYS-code approaching the CS by an iterative adjustment of the material resistivity. On the other hand, a dual formulation approach to the free boundary problem was developed by Prigozhin [@P2-Prigozhin_2004] which allows consideration of a wide class of variational problems, particularly the treatment of the critical states in superconductors of complicated shapes without assuming a priory specific shape of the flux free zone. Likewise, in a way similar to the spirit of Prigozhin’s work, our above introduced variational statement for the most general SDCST [@P2-Ruiz_PRB_2009; @P2-Ruiz_SUST_2010] allowing for arbitrary mutual orientation of the external field and transport current and implementing finite-element methods is able to tackle in a very efficient way the so called *front tracking* problem for superconducting wires [@P2-Ruiz_APL_2012]. In the present case of interest (a superconductor of thickness much smaller than its length, subjected to time dependent transverse magnetic field and a simultaneous transport current condition), the cumbersome analysis of the intrinsic anisotropy effects may be straightforwardly avoided, as the streamlines preserve only one direction (perpendicular to the applied magnetic field), which means a significant reduction of the computational time.
##### ( *A.* ) *Statements For The Variational Approach*\
Following the same methodology introduced in the previous chapters, here the whole superconducting region is involved in the calculation and the free boundary is obtained as a part of the solution of the minimization procedure. In this sense, the shape of the superconducting sample may be arbitrary and it is related to the mesh design (see Fig. \[Figure\_3\_1\]). Going into detail, in our case of interest we are enabled to discretize the samples according to their cross-section area ($\Omega$) through a collection of points ($\textbf{r}_{i}$) depicting the straight infinite elementary filaments fulfilling the condition $r_{i}\in\Omega$. Thus, for a sufficiently large mesh, a uniform current density can be assumed within each elementary wire such that $I_{i}=J_{i}s_{i}$ with $s_{i}$ the cross sectional area of the filament. Then, the problem can be straightforwardly written in terms of local contributions of the vector potential $A_{i}(r_{i})$ accordingly to cylindrical filaments of section $s_{i}=\pi a^{2}$ with $a\ll R$, being $R$ the maximal distance from the geometrical center of the superconducting sample to its external surface (in our case, $R$ defines the radius of the circular section $\Omega$). Therefore, the vector potential of each filament ($A_{i}$) splits up into two expressions, one within the filaments of radius $a$: $$\begin{aligned}
\label{Eq.6.1}
A_{i}(r_{i}\in s_{i})=\frac{\mu_{0}}{4\pi} \left[2\pi
a^{2}J_{i}\ln{\left(a\right)}-\pi (a^{2}-{r}_{i}^{2})J_{i}\right]+
C_{1} \,~\,~,~\,
\forall ~\, ~\, {r}_{i}<a \, ,\end{aligned}$$ and one outside the filament: $$\begin{aligned}
\label{Eq.6.2}
A_{i}(r_{j}\notin s_{i})=-\frac{\mu_{0}}{4\pi} \left[ J_{i}
\ln(r_{ij}^{2}/a^{2})\right] + C_{2} \,~\,~,~\, \forall ~\,
~\, r_{ij} \, .\end{aligned}$$ Here, $r_{ij}$ denotes the distance between the centers of filaments $i$ and $j$, and $C_{1}$ and $C_{2}$ are arbitrary integration constants, one of them determined by continuity at $r_{i}=a$ and the other one can be absorbed in a global constant for the whole section $\Omega$ (*C in what follows*). In fact, as it was established in chapter 2.1 any arbitrary constant may be added to the vector potential $A_{i}$ without altering the magnetic field produced by the wires, and therefore one can choose $C\equiv0$ to solve the critical state problem according to the minimization functional in Eq. (\[Eq.2.11\]). However, some care must be taken when electric fields related to flux variations are calculated. Let us be more specific. In general, an electrostatic term $\nabla\varPhi$ enters the definition of the electric field ($\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi$). In the long wire geometry, $\nabla\phi$ may be argued to be spatially constant ($C_{t}$) by symmetry reasons. Then, as related to gauge invariance, the vector potential $\textbf{A}$ may be recalibrated in the form $A\rightarrow \tilde{A}+\nabla\varPhi$, and therefore, arbitrary constants may be induced so as to fit the physical condition $E=0$ for those regions with absence of magnetic flux variations. In fact, if the minimization functional has not to be constrained by a transport current condition, i.e. $I_{tr}(t)=0$, $C_{t}$ disappears in the optimization process.
We recall that in quasi-steady regime (excellent approximation for the large scale application frequencies) the discrete form of Faraday’s law ($\delta\textbf{B}_{i} =- \nabla\times\textbf{E}_{i}(\textbf{J}_{i})\,\delta t$) in a mesh of circuits that carry the macroscopic electric current density $J_{i}$, is obtained in general terms by minimizing the action of an averaged field Lagrangian ( of density ${\cal L}=[{\bf
B}({t+\delta t})-{\bf B}({t})]^{2}/2$ ) coupling successive time layers. Thus, by using this procedure, and introducing the magnetic vector potential, the quantity to be minimized transforms to the so called objective function \[viz., Eqs. (\[Eq.3.2\]) & (\[Eq.3.3\])\] $$\begin{aligned}
\label{Eq.6.3}
\frac{1}{2}\sum_{i,j}I_{i,l+1}{\rm M}_{ij}I_{j,l+1}-\sum_{i,j}I_{i,l}{\rm M}_{
ij } I_ { j , l+1 }+\sum_{i}I_{i,l+1}\Delta A_{0} \, ,\end{aligned}$$ with $\{I_{i,l+1}\}$ the set of filaments with unknown current for the time steps $l+1$, $A_{0}$ the vector potential related to *non-local* sources, and $M_{ij}$ the mutual inductance matrix between filaments $i$ and $j$, which accordingly to Eqs. (\[Eq.6.1\]) & (\[Eq.6.2\]), for filaments of cylindrical cross section $s_{i}$ centered at the positions $r_{i}\in\Omega$ and subject to uniform distributions of current density $J_{i}\in s_{i}$ may be defined as: $$\begin{aligned}
\label{Eq.6.4}
{\rm M}_{ij}=
\left\{
\begin{array}{ll}
\dfrac{\mu_{0}}{8\pi} \,~\,~ & ,~\, \forall ~\,~\, r_{i}=r_{j} \in
\Omega \,
\\ \, \\
-\dfrac{\mu_{0}}{4\pi}\ln(r_{ij}/a)\,~\,~ & ,~\, \forall ~\, ~\,
r_{i}\neq r_{j}
\in \Omega \, ~ \, ~ \, .
\end{array}
\right. \end{aligned}$$ For our cases of interest, $A_{0}$ corresponds to the magnitude of the vector potential produced by a uniform transverse magnetic field $\textbf{B}_{0}$, which can be calculated from the components of the vectorial expression $$\begin{aligned}
\label{Eq.6.5}
\textbf{A}_{0}(\textbf{r}_{i})=\textbf{B}_{0}\times \textbf{r}_{i} \, .\end{aligned}$$ Furthermore, when required, optimization must to be performed under the restriction of applied transport current, i.e., $$\begin{aligned}
\label{Eq.6.6}
\sum_{i\in\Omega}I_{i}=I_{\rm tr} \, .\end{aligned}$$ and the physically admissible solutions have to be constrained by the CS material law for the current density, that in this case reads $|J_{i}|\leq J_{c}$.
Minimization is done under prescribed sources $(B_{0},I_{tr})$ for the time step $l$ and the above CS material law, and as result of the optimization procedure one gets the distribution of current filaments along the cross section of the superconducting sample at the time step $l+1$. Eventually, the vector potential can be evaluated in the whole space by superposition of Eqs. (\[Eq.6.1\]), (\[Eq.6.2\]), and (\[Eq.6.5\]). Then, one may plot the magnetic flux lines as the isolevels of the total vector potential $\textbf{A}$, and the components of the magnetic flux density can be evaluated according to its definition $\textbf{B}=\nabla\times\textbf{A}$.
Furthermore, in order to achieve a closer connection with experiments the sample’s magnetic moment per unit length ($l$) has been calculated by means the vectorial expression $$\begin{aligned}
\label{Eq.6.7}
\textbf{M}=\frac{l}{2}\int_{\Omega}\textbf{r}\times\textbf{J} d\Omega\, ,\end{aligned}$$ and the hysteretic AC losses per unit time and volume ($\Phi$) for cyclic excitations of frequency $\omega$ can be calculated by integration of the local density of power dissipation ($\textbf{E}\cdot\textbf{J}$) as follows $$\begin{aligned}
\label{Eq.6.8}
L=\omega\oint_{f.c.}dt\int_{\Phi}\textbf{E}\cdot\textbf{J} d\Phi \, .\end{aligned}$$ Here, $f.c.$ denotes a full cycle of the time-varying electromagnetic sources.
Applications of the above statements are developed along the following subchapters, in a systematic study of infinite cylindrical wires under a wide variety of different experimental conditions.
##### ( *B.* ) *Numerical procedure*\
Some technical details are worth of mentioning as far as concerns the numerical procedure.
On the one hand, the mesh utilized for the whole set of calculations presented along this chapter is defined in terms of a rectangular grid with filaments equally distanced under the prescribed condition $r_{ij}\geq
2a$ to satisfy Eq. (\[Eq.6.4\]). The number of filaments which have been considered to fill out the cross section of the superconducting cylinder is 3908. However, owing to the planar symmetry of the problem one is allowed to reduce the number of variables to 1954 (i.e., 977 filaments per quadrant), which is still a large number because the objective function to be minimized is highly nonlinear. To be more specific, the number of quadratic terms in Eq. (\[Eq.6.3\]) involve minimizing the action of 1910035 elements, i.e., the sum of elements produced by the mutual inductance terms between filaments (977\*977\*2), and the self inductance terms (977).
On the other hand, contrary to the common choice of a sinusoidal oscillation process we have run the simulations for triangular oscillating processes, which indeed do not change the electromagnetic response of the superconductor if resistive currents are neglected. In fact, an instantaneous response takes place in the absence of resistance. Also, under the critical state framework, Joule heat release may be calculated by $\dot{L}=J_{c}E$ as overcritical flow ($J>J_{c}$) is neglected because instantaneous response is assumed \[Fig. \[Figure\_1\_1\] (pag. )\].
### \[ch-6-2\] *6.2 SC wires subjected to isolated external sources* {#ch-6-2-6.2-sc-wires-subjected-to-isolated-external-sources .unnumbered}
***6.2.1 Wires with an injected AC transport current***
\[ch-6-2-1\]
As it is well known, the magnetic flux penetrates a superconducting material first entering from the surface towards the center while is shielded by screening currents flowing at the critical value $J_{c}$. Penetration occurs to a depth known as the flux front boundary, where the magnetic flux density drops to zero. Trivially, for long cylindrical wires (length much higher than its radius $R$) with transport current \[see Fig. \[Figure\_6\_1\](a)\], the flux front profile may be defined in terms of a set of circular front boundaries tracking the time evolution of the injected transport current. For example, and mainly to illustrate how the patterns of the main electromagnetic quantities evolve along the cyclic process depicted in the Fig. \[Figure\_6\_1\](b), our numerical results for the local profiles of current $I_{i}$, the components of the magnetic flux density $\textbf{B}$ and the corresponding isolevels of the vector potential $\textbf{A}$ (i.e., the flux lines of $\textbf{B}$), as well as the local distribution of the density of power dissipation $\textbf{E}\cdot\textbf{J}$ are shown in the section of supplementary material \[Figs. S1 & S2 (pags. , )\]. These figures will be of much help for eventual discussion about the AC hysteretic losses when simultaneous oscillating excitations must to be considered. However, already for this simple case, several aspects have to be noticed.
![\[Figure\_6\_1\] For the experimental configuration displayed in subplot (a), and for an injected AC transport current of amplitude $I_{\tt{a}}$ according to the temporal process depicted in subplot (b), we show the calculated components of the magnetic moment for half cross section of the superconducting wire, where (c1) corresponds to $\Omega^{+}$, i.e.: $(x,y)\in\Omega~|~x>0$, and (c2) to $\Omega^{-}$, i.e.: $(x,y)\in\Omega~|~x<0$. Notice that both sections cancel each other to satisfy $\textbf{M}(\Omega)=0$. In subplot (d) we show the calculated hysteretic AC losses per excitation cycle (since time-step 2 to 10). Red solid line corresponds to the exact analytical expression of Eq. (\[Eq.6.11\]), and solid diamonds corresponds to our numerical results. Inset shows the same results in linear scale. Here and henceforth, units are $\pi
R^{2}J_{c}\equiv{I_c}$ for $I_{\rm tr}$, $(\mu_{0}/4\pi)J_{c}R$ for B, $J_{c}R^{3}$ for M, and $(\mu_{0}/4\pi)\omega R^{2}J_{c}^{2}$ for the hysteretic losses per cycles of frequency $\omega$. ](Figure_6_1.pdf){height="7cm" width="13cm"}
First of all, let us to focus on the circulating transport currents so as to visualize their contribution to the total magnetic moment of the superconducting sample. Recall that, according to Eq. (\[Eq.6.7\]), those cases where only transport current is applied, the total magnetization of the superconducting sample becomes zero and the screening currents can be simply called “*injected current lines*”. They distribute accomplishing the critical state condition $j_{i}=j_{c}$ in the regions where $B(r_{i})\neq0$. In fact, although these currents produce local patterns of magnetization as it is shown in Fig. \[Figure\_6\_1\](c), by symmetry, the sum over the whole sample is fully compensated. Nevertheless, for this case, **M**$(\Omega)=0 $ does not imply the absence of hysteretic losses (see Fig. \[Figure\_6\_1\](d)) which are actually produced by the Joule heat release in those regions where flux transport occurs. On the other hand, as one can see in Fig. S1 (pag. ), the local density of power dissipation ($\textbf{E}\cdot \textbf{J}$) is not uniform despite the material law assumes the critical state condition $j_{i}=j_{c}$. This means that, locally one cannot assume a unique value for the electric field or $E=E_{c}$. In fact, for this case the maximal power dissipation always occurs over the superconducting surface decreasing to zero beyond the flux front.
Eventually, it is worth of mention that for this simple case the current distribution can be reliably calculated by following the boundary flux front defined through axisymmetric circumferences of radius $\tilde{r}=R\sqrt{1-I_{tr}/I_{c}}$, whose section $\tilde{r}<r<R$ produces the intensity of magnetic flux (in polar coordinates): $$\begin{aligned}
\label{Eq.6.9}
B_{\phi}=\frac{\mu_{0}I_{c}}{2\pi r}
\left[\frac{I_{tr}}{I_{c}}+\left(\frac{r^{2}}{R^{2}}-1\right)\right] \, ~ \, ~
\,
\forall \, ~ \, ~ \, \tilde{r}<r<R \, .\end{aligned}$$ Notice that here, $I_{tr}$ stands for the applied transport current at a given time, and $I_{c}\equiv J_{c}\pi R^{2}$. Thus, the electric field can be analytically calculated by the one dimensional Maxwell equation $\partial_{r}E_{z}=\partial_{t}B_{\phi}$ satisfying the condition $E_{z}(r<\tilde{r})=0$, i.e., $$\begin{aligned}
\label{Eq.6.10}
E_{z}=\frac{\mu_{0}}{2\pi}\ln{\left[\frac{r}{R}\left(1-\frac{I_{tr}}{I_{c}}
\right)^{
-1/2 }\right]} \dot{I}_{tr}
\, ~ \, ~ \,
\forall \, ~ \, ~ \, \tilde{r}<r<R \, .\end{aligned}$$ In fact, if the temporal evolution of the injected transport current is monotonic the average specific hysteretic loss rate per unit length can be calculated by integration of the local density of power dissipation ($\textbf{E}\cdot \textbf{J}$), $$\begin{aligned}
\dot{L}_{m}(I_{tr}(t))=
\frac{1}{\pi R^{2}}\int_{0}^{2\pi}\int_{\hat{r}}^{R}E_{z}J_{c}~r d\phi dr=
-\frac{\mu_{0}}{4\pi^{2}R^{2}}
\left[I_{tr}+I_{c}\ln\left(1-\frac{I_{tr}}{I_{c}}\right)\right]\dot{I}_{tr} \,
,\nonumber\end{aligned}$$ and integrating it with respect to time, the monotonic hysteretic losses is then $$\begin{aligned}
\label{Eq.6.11}
{L}_{m}(I_{tr}(t))=
\frac{\mu_{0}}{4\pi^{2}R^{2}}
\left[I_{tr} I_{c} \left(1-\frac{I_{tr}}{2I_{c}}\right)
+I_{c}^{2}\left(1-\frac{I_{tr}}{I_{c}}\right)
\ln\left(1-\frac{I_{tr}}{I_{c}}\right)\right]\end{aligned}$$ Moreover, if $I_{tr}(t)$ is periodic, the dependence of the hysteretic loss density per period on $I_{tr}=I_{\tt{a}}$ ($\tt{a}$ for the amplitude of the oscillating source) remains the same, and therefore the magnitude of the loss per cycle may be straightforwardly obtained from the monotonic first branch as, $L_{f.c.}=4 L_{m}$.
***6.2.2 Wires under an external AC magnetic flux***
\[ch-6-2-2\]
With the aim of providing a clear picture of the effects related to the occurrence of local *magnetization currents* (screening currents produced by external magnetic fields), the second case under consideration will correspond to a superconducting wire under zero transport current condition ($I_{tr}(t)=0$) and subjected to an external magnetic field perpendicular to its surface. Here, we must call readers’ attention to the fact that for this seemingly simple case an exact analytical solution for the dynamics of the flux front profiles has not been reported, although remarkable efforts have been done along the last five decades to implement diverse analytical and numerical approaches [@P2-Ashkin_1979; @P2-Carr_1975; @P2-Zenkevitch_1980; @P2-Minervini_1989; @P2-Carr_2001; @P2-Gurevich_1997; @P2-Gomory_2002]. In fact, for an arbitrary relation between the amplitude of the applied field ($H_{a}$) and the full penetration value $H_{p}$, the cyclic hysteretic losses can only be found numerically. Nevertheless, for cylindrical superconducting wires subjected to a monotonic source $H_{0}(t)$, the losses may be approached by the so-called Gurevich’s relation [@P2-Gurevich_1997]: $$\begin{aligned}
\label{Eq.6.12}
L_{m}(B_{0}(t))=\dfrac{2B_{p}^{2}}{3\mu_{0}}
\left\{
\begin{array}{ll}
\left(\dfrac{B_{0}}{B_{p}}\right)^{3} \left(1-\dfrac{B_{0}}{2B_{p}}\right)
\,~\,~ & ,~\, \forall ~\, ~\, B_{0}<B_{p} \,
\\ \, \\
\dfrac{B_{0}}{B_{p}}-\dfrac{1}{2}\,~\,~ & ,~\, \forall ~\, ~\, B_{0}\geq B_{p}
\, ~ \, ~ \, .
\end{array}
\right. \end{aligned}$$ Here, the customary relation for superconducting materials $\textbf{B}=\mu_{0}\textbf{H}$ has been assumed, and $B_{p}$ is given by: $$\begin{aligned}
\label{Eq.6.13}
B_{p}=\frac{2}{\pi}\mu_{0}J_{c}R \, .\end{aligned}$$ Just as in the previous case (only transport current), if $B_{0}(t)$ is periodic, the dependence of the hysteretic loss density per period remains the same, and the magnitude of the cyclic losses is higher by a factor of four regarding the monotonic branch.
![\[Figure\_6\_2\] For the experimental configuration represented in subplot (a), and before applying an oscillating magnetic flux with amplitude $B_{\tt{a}}=B_{0,y}(t'')-B_{0,y}(t')$, three different initial states $(B_{0,y}(t'))$ have been considered. Firstly, a non magnetized wire has been assumed accordingly to the magnetic temporal process displayed in subplot (b1). Subplots (b2) and (b3) show the evolution of the dimensionless magnetic moment $M_{y}/M_{p}$ as a function of the temporal process and the applied magnetic field $B_{0,y}$ respectively. Secondly, a premagnetized sample with $B(t')<B_{p}$ has been considered (subplot c1), and the corresponding $M_{y}/M_{p}$ curves are displayed (subplot c2). Thirdly, in (d1) the initial state has been assumed to be $B(t')=B_{p}$, and the corresponding $M_{y}/M_{p}$ curves are displayed in (d2). For all plots, curves are labeled according to $B_{\tt{a}}$. In subplots (b1) and (b2) an additional temporal scale in red has been incorporated to allow an straightforward interpretation for the sequence of profiles displayed in the section of supplementary material, Figs. S3 (pag. ) & S4 (pag. ). Recall that units for B are $(\mu_{0}/4\pi)J_{c}R$, and $J_{c}R^{3}$ for M. ](Figure_6_2.pdf){height="8cm" width="13cm"}
##### ( *A.* ) *Features on the magnetic response*\
For a major understanding of the electromagnetic quantities involved, we have considered the experimental configuration depicted in Fig. \[Figure\_6\_2\](a) under different amplitudes for the applied magnetic flux $B_{0,y}$. Just for completeness, three different magnetization pre-conditions have been considered as relates the first branch of the temporal processes $t<t'$ (see top pane in Fig. \[Figure\_6\_2\]). Thus, subplots (b1) to (b3) correspond to a non magnetized wire for which one gets magnetization loops centered at the coordinates (0,0) of the plane ($M_{y},B_{0,y}$). However, in those cases where $B_{0,y}(t')\neq0$ \[Figs. \[Figure\_6\_2\] (c1) and (d1)\] a different behavior reveals. An initial magnetization branch is present when the external AC magnetic flux is switched on ($t=t'$) \[see dashed lines in Figs. \[Figure\_6\_2\] (c2) and (d2)\]. Then, for $t>t'$ the SC wire is subjected to an oscillating magnetic flux of amplitude $B_{\tt{a}}$, such that the first quarter of the subsequent periodic excitation occurs along the interval $t'<t<t''$, and consequently the AC losses are calculated relative to cycles departing from $t''$. For the oscillating process in the above figure, the alternating magnetic field is applied along the same direction of the previous magnetization under the flux conditions $B_{0,y}(t')<B_{p}$ and $B_{0,y}\geq B_{p}$ respectively. The following facts have to be noticed:
(*i*) Concerning to the numerical accuracy, it has to be noticed that the mutual inductance matrices used in this fore, corresponds to the exact analytical solution for filaments of cylindrical cross section. Thus, some discrepancies between the analytical and numerical quantities can be expected. In fact, in the classical results of Ref. [@P2-Ashkin_1979] it may be noticed that the magnetization curve saturates to a flat value ($M_{p}$) before reaching the analytical limit $B_{p}=8$, which is in agreement with our numerical predictions in Fig. \[Figure\_6\_2\] ($B_{p}\simeq
7\pm5\times10^{-3}$). Accordingly to our numerical method, discrepancies with the analytical solution are explained in geometrical terms. Notice that even for a highly refined mesh, by assuming cylindrical filaments, the entire superconducting area cannot be filled, and therefore some deviation should be expected. We emphasize that other choices for the mesh elements filling the superconducting area could be done, but the complication seems unnecessary if one considers that the accuracy in the obtained physical quantities is already very high, and furthermore this mesh has been recognized to fit well to experimental evidences in the same geometry [@P2-Ashkin_1986].
(*ii*) Recalling Eq. (\[Eq.6.7\]), the sum of the local magnetic moments associated to each of the filaments over the entire superconducting sample increases as $B_{0,y}$ grows monotonically until the condition $B_{0,y}=B_{p}$ is met. Then, the magnetization of the sample saturates to the analytical value $M_{p}=2J_{c}R^{3}/3$ (in our dimensionless units $M_{p}=2/3\approx0.6667$). Regarding to our numerical results we have obtained $M_{p}\approx0.655\pm5\times10^{-3}$.
(*iii*) Once the cyclic process starts ($t>t''$), and regarding the magnetization loops characterized by AC cycles of external magnetic flux with amplitudes higher than $B_{p}$, the magnetic moment saturates at different values given by the dimensionless relation $$\begin{aligned}
\label{Eq.6.14}
B_{p^{\dag}}=\mp \left( 2 B_{p}- B_{\tt{a}} \mp B_{0}(t')-1/2\right) \, ,\end{aligned}$$ where the signs choice is made simultaneously, it for consider the time derivative of the cyclic excitation.
For example, in Fig. \[Figure\_6\_2\](b3) if $B_{0}(t')=0$ and the amplitude of the external magnetic flux is $B_{\tt{a}}=8$, into the cyclic process $(t>t'')$ the magnetization of the superconducting wire saturates at $B_{0,y}=\mp5.5$. On the other hand, if the sample has been previously premagnetized, the center of the magnetization loop is displaced in the axis $B_{0,y}$ of the plane $(M_{y},B_{0,y})$ by the amount $B_{0,y}(t')$ \[see Fig. \[Figure\_6\_2\] (c2,d2)\], and therefore the sample saturates at two different values of the applied magnetic field (e.g., for $B_{\tt{a}}=8$, $M_{y}$ is equals to $M_{p}$ at $B_{0,y}=-1.5$, and $M_{y}$ is equals to $-M_{p}$ at $B_{0,y}=9.5$).
(*iv*) Remarkably, the set of magnetization loops displayed in Fig. \[Figure\_6\_2\] serves as a map for any magnetization loop and any arbitrary relation between the experimental parameters $B_{0,y}(t')$ and $B_{\tt{a}}$. In fact, it can be done by the simple interpolation of the known shape of the magnetization loops where the first corner of $M_{y}$ (corresponding to the higher excitation peak for the first half of the excitation cycle) always falls over the magnetization branch for $t<t''$. Moreover, if $B_{\tt{a}}+B_{t'}>B_{p}$ the position of the corners is straightforwardly given by the saturation values $B_{p^{\dag}}$.
##### ( *B.* ) *Flux penetration and local power density*\
For a clear understanding of the different terms affecting the calculation of the hysteretic losses in superconducting wires, it is advisable to get familiar with the local dynamics of the electromagnetic quantities in the same way as it was done in the previous section. Therefore, in the section of supplementary material, readers will find out some of our numerical results for one of the experimental processes depicted in Fig. \[Figure\_6\_2\]. In particular, in Fig. S3 (pag. ) we show the flux penetration profiles for an external AC magnetic flux of amplitude $B_{\tt{a}}=6$ at intervals of $\Delta B_{0,y}=3$, assuming an initially non magnetized wire \[see by reference the temporal process depicted in Fig. \[Figure\_6\_2\](b)\]. Also shown are corresponding patterns for the local density of power dissipation across the section of the superconducting wire. The direction of the magnetic field can be tracked from the dynamics of the Cartesian components $B_{x}$ and $B_{y}$ both displayed in Fig. S4 (pag. ).
It becomes clear that the distribution of magnetization currents across the section of the superconducting wire preserves some symmetry respect to both Cartesian axes, although rotational invariance characteristic for transport problems \[Fig. S1 (pag. )\] is not fulfilled. Actually, we can argue that for cases with no rotating transverse magnetic field the numerical problem may be reduced to considering only two of the four Cartesian quadrants according to the following symmetry condition: (*i*) $I_{i}(r_{i}(y^{+})=I_{i}(r_{i}(y^{-})$, being “$y$” the axis of the applied magnetic field $B_{0}$. Moreover, for our current case also the symmetry condition (*ii*) $I_{i}(r_{i}(x^{+})=-I_{i}(r_{i}(x^{-})$, being $x$ the axis perpendicular to the direction of $B_{0}$ may be called. However, the latter can only be fulfilled as long as the transport current condition $I_{tr}=0$ occurs.
As it may be observed in Fig. S3 (pag. ), once the external magnetic flux $B_{0,y}$ is switched on, a set of screening currents symmetric along the $y$-axis but antisymmetric by sign along the $x-$axis appears so as to expell the magnetic field from the inner sample \[see also Fig. S4 (pag. )\]. Actually, for transverse magnetic fields, it is the Faraday’s law which produces the simultaneous occurrence of positive and negative screening currents distributed along the positive and negative semi-axis, but both orthonormal to the direction of the applied magnetic field. Thus, if the rate $\Delta
B_{0,y}(t)$ is monotonic, the shape of the flux free region approaches to ellipses with their foci along the $y-$axis, but with acute nodes in the boundaries of their major axis. Moreover, the associated lengths to the major and minor semi-axis of the ellipse also change responding to the variation of the external magnetic flux in the different points where the screening currents are allocated, which is in concordance with Refs. [@P2-Ashkin_1979; @P2-Telschow_1994; @P2-Kuzovlev_1995; @P2-Bhagwat_2001; @P2-Navarro_1991; @P2-Gomory_2002; @P2-Brandt_1996].
As regards the local power loss density \[right pane in Fig. S3 (pag. )\], we have observed some fine structure details which are worth of mention. First, if the rate of magnetic flux $\Delta B_{0,y}$ is monotonic (without change of sign), it is to be noticed that the specific local power density preserves the same kind of pattern observed for the dynamics of the local profiles of the magnetic flux density component $B_{y,i}$. However, the role played by the orthonormal component $B_{x,i}$ is rather different and geometry dependent \[see subplots (1) and (2) in Figs. S3 (pag. ) & S4 (pag. )\]. Then, as soon as $\Delta B_{0,y}$ changes sign, the local distribution of power density is much more complex as the core enclosed by the flux fronts satisfying the condition $E=0$ must be shielded \[see the sequence of subplots (3) to (10) in Figs. S3 (pag. ) & S4 (pag. )\].
![\[Figure\_6\_3\] Hysteretic AC losses per cycle for an oscillating magnetic field of amplitude $B_{\tt{a}}$. Red solid line corresponds to the analytical approach of Eq. (\[Eq.6.12\]), and diamonds correspond to our numerical calculations. Inset shows the same results in linear scale. Units are $(\mu_{0}/4\pi)J_{c}R$ for $B_{0,y}$, and $(\mu_{0}/4\pi)\omega R^{2}J_{c}^{2}$ for the hysteretic losses per cycles of frequency $\omega$. ](Figure_6_3.pdf){height="8cm" width="10cm"}
In general, power losses appear for the whole section where the magnetization currents are displayed, but their maximum contribution always occurs over the external surface of the superconducting sample. On the other hand, for the hysteretic losses per cycle the total amount of heat release can be straightforwardly calculated based on energy transport over the external surface of the superconducting sample as explained below.
***6.2.3 Ultimate considerations on the AC losses***
\[ch-6-2-3\]
At this point and after gaining some experience by the analysis of wires under transport or applied magnetic field, we are in the position of casting Eq. (\[Eq.6.8\]) in different forms for convenient interpretation of the AC losses. Thus, taking advantage of the knowledge generated in the above sections, below we reformulate the argument $\dot{L}$ of Eq. (\[Eq.6.8\]) in terms of the definitions for $\textbf{E}$ or $\textbf{J}$. This will allow a physical interpretation of the mechanism responsible for AC losses in complex configurations.
##### \[B-oriented\]( *A.* ) *Conventional approach \[B-oriented\]*\
First of all, recalling that the electric field may be defined for discretized time layers as $\textbf{E}=-\Delta \textbf{A} / \Delta t$, and furthermore, that it can be computed from the sum of the external vector potential $\textbf{A}_{0}(\textbf{B}_{0})$ \[see Eq. (\[Eq.6.5\])\] and the vector potential induced by the screening currents $\textbf{A}_{ind}(\textbf{J}(r_{i}))$ \[see Eqs. (\[Eq.6.1\]) & (\[Eq.6.2\])\], the losses of power density $\dot{L}\equiv \Delta L/\Delta t$ may be rewritten as $$\begin{aligned}
\label{Eq.6.15}
\Delta L=-\int_{\Omega} \Delta\textbf{A}_{0}\cdot \textbf{J} ~ dr
-\int_{\Omega} \Delta\textbf{A}_{ind}\cdot \textbf{J} ~ dr \, .\end{aligned}$$ Thus, recalling Eq. (\[Eq.6.7\]), the first term on the right-hand side of Eq. (\[Eq.6.15\]) becomes $$\begin{aligned}
\label{Eq.6.16}
\Delta L_{0}(\Delta B_{0,y})=-\int_{\Omega} \Delta \textbf{A}_{0}\cdot
J ~ dr \equiv \int_{\Omega}M_{y} ~ dB_{0,y} \, ,\end{aligned}$$ which corresponds to the losses produced by an external excitation of magnetic flux density $B_{0}(t)$.
In fact, since the work by Ashkin (Ref. [@P2-Ashkin_1979]), it is well known that for screening currents produced by external variations of an applied magnetic field of flux density $B_{0,y}$, the so called magnetization currents, the average AC losses can be straightforwardly calculated as the enclosed area by the magnetization loop $M_{y}$ as a function of the excitation $B_{0,y}$ (Fig. \[Figure\_6\_2\]). Thus, as long as the experimental configuration is such that $I_{tr}(t)=0$, physically the AC hysteretic losses may be understood as surface losses due to remagnetization.
Nevertheless, as in most applications of superconducting wires, the system is subjected to a simultaneous oscillating transport current, a most careful analysis of the second term at the right-hand side of Eq. (\[Eq.6.15\]) is needed.
First recall that in the CS theory the MQS approach allows us to define Ampere’s law as the spatial condition $\mu_{0}\nabla\times\textbf{B}=\textbf{J}$, where $\textbf{B}_{i}$ is the local density of magnetic flux produced by the induced screening currents of density $\textbf{J}(\textbf{r}_{i})$ plus the density of magnetic flux produced by the external excitation $\textbf{B}_{0}(\textbf{r}_{i})$, i.e., $\textbf{B}_{i}=\textbf{B}_{0}(\textbf{r}_{i})+\textbf{B}_{ind}(\textbf{r}_{ij}
)$, so that the second term at the right-hand side of Eq. (\[Eq.6.15\]) may be rewritten as $$\begin{aligned}
\label{Eq.6.17}
\Delta L_{ind}=-\mu_{0}\int_{\Omega} \Delta\textbf{A}_{ind}\cdot
\nabla\times\left(\textbf{B}_{0}+\textbf{B}_{ind}\right) ~ dr \, .\end{aligned}$$
On the other hand, accordingly to the CS statement, $\Delta{J_{i}}=\pm
J_{c}\neq 0$ accomplishes for the flux front profile, ergo $\Delta
\textbf{A}_{ind}\equiv {\rm M}_{ij}\Delta J_{i} \neq0$. Then, for systems only subjected to external magnetic fields, the distribution of screening currents over the cross section of the superconducting sample is such that the induced magnetic field over flux front is rotationally invariant with respect to the direction of the applied magnetic field. For example, considering the first monotonic branch in cases of Fig. \[Figure\_6\_2\], the flux front profile is defined by $B_{ind}(t)=-B_{0,y}(t)$. Said in other words, the induced magnetic field flows only in opposite direction to $B_{0}$ so that $\Delta L_{ind}=0$ as long as $I_{tr}(t)\equiv 0$, and therefore the hysteretic losses may be straightforwardly computed as $\Delta L\equiv \Delta L_{0}$ in the fashion of Eq. (\[Eq.6.16\]). Nevertheless, if $I_{tr}(t)\neq 0$ the local density of magnetic flux is not rotationally invariant with respect to the components of the induced magnetic field. For example, for the cases described in the previous subchapter (Fig. \[Figure\_6\_1\]), where $B_{0,y}(t)\equiv0$ (absence of magnetization currents), the magnetic flux density produced by the screening currents $J_{i}$, so called there *injected current lines*, shows two components $B_{x,i}$ and $B_{y,i}$ so that $\nabla\times\textbf{B}_{i}\neq0$, and thence that $\Delta L\equiv\Delta L_{ind} \neq 0$ despite $\int_{\Omega}M(\textbf{r}_{i})\equiv0$.
In order to understand the underlying physics behind the the concept of injected current lines into the critical state theory, at least for the 2D configurations studied along this chapter, it becomes useful to further analyze Eq. (\[Eq.6.17\]) by the vectorial definition $\textbf{A}\cdot\nabla\times\textbf{B}=$ $\nabla\cdot\textbf{B}\times\textbf{A}+$ $\textbf{B}\cdot\nabla\times\textbf{A}$, such that we can write $$\begin{aligned}
\label{Eq.6.18}
\Delta L_{ind}=-\mu_{0}\int_{\Omega}
\nabla\cdot(\textbf{B}\times \Delta \textbf{A}_{ind}) ~ dr -
\mu_{0}\int_{\Omega}\frac{\textbf{B}\cdot\Delta \textbf{B}}{2} ~ dr
\, . \end{aligned}$$ Into this framework the first integral turns to a surface integral over the curved walls of the cylinder, which does not contribute to the average losses because the contribution to the surface integral from the end planes vanishes as $\textbf{B}\times\textbf{A}_{ind}\cdot \hat{\textbf{z}}=0$, while the integral over the lateral surface turns zero in a closed cycle. Therefore, the contribution of the injected current lines may be understood by the simple relation $$\begin{aligned}
\label{Eq.6.19}
\Delta L_{ind}=-\mu_{0}\int_{\Omega}\frac{\textbf{B}\cdot\Delta
\textbf{B}}{2} ~ dr
\, ,\end{aligned}$$ where it is pointed out that significant reductions of the hysteretic losses may be achieved by reducing the magnitude of the local inductive magnetic field.
##### \[S-oriented\]( *B.* ) *Alternative approach \[S-oriented\]*\
In order to justify that the hysteretic losses may be calculated by the knowledge of the electromagnetic quantities over the external surface of a superconducting material, it is interesting to transform Eq. (\[Eq.6.8\]) in terms of a second formulation based upon the definition of $\textbf{J}$ instead of $\textbf{E}$. Indeed, assuredly the specific losses can be evaluated by the conservation energy principle defining $\textbf{J}=\nabla\times\textbf{H}-\partial_{t}\textbf{D}$ and by using the divergence theorem as follows:
$$\begin{aligned}
\label{Eq.6.20}
\dot{L}=\int_{\Phi}\textbf{E}\cdot\left(\nabla\times\textbf{H}-\partial_{t}
\textbf{D}\right) dr =
-\int_{\Phi}(\textbf{E}\cdot\partial_{t}\textbf{D}+\textbf{H}
\cdot\partial_{t}\textbf{B}) ~ dr -
\oint_{s}\textbf{S}\cdot\hat{\textbf{n}} ~ ds \, , \nonumber \\\end{aligned}$$
where $\Phi$ is introduced to distinguish the volume integral over the entire superconducting sample from the surface integral over the flux fronts defined by the Poynting’s vector $\textbf{S}=\textbf{E}\times\textbf{H}$, and $\hat{\textbf{n}}$ the unit vector normal to its surface element $(d\hat{\textbf{s}}_{i}=ds\hat{\textbf{n}})$.
The first term on the right-hand side of Eq. (\[Eq.6.20\]) represents the total electromagnetic energy stored within the superconductor volume, and the second one corresponds to the energy flow produced by the local variations of magnetic field as a consequence of the occurrence of screening currents. Then, if one is only interested on the hysteretic losses per closed cycles, one is entitled to evaluate the AC losses between two well-defined stationary regimes, $$\begin{aligned}
\label{Eq.6.21}
L= - \omega \int_{peak~a}^{peak~b} dt
\oint_{s}\textbf{S}\cdot\hat{\textbf{n}} ~ ds
= -2 \omega \int_{h.f} dt
\oint_{s}\textbf{S}\cdot\hat{\textbf{n}} ~ ds
\, , \end{aligned}$$ because the total electromagnetic energy is a conserved quantity between two consecutive peaks defined by the wavelength of the oscillating source. In fact, the calculation can be simplified following the same argument to consider only half cycle ($h.f$) of the excitation process. Remarkably, this fact can be straightforwardly observed by comparison between the local profiles $\textbf{E}\cdot \textbf{J}$ for the excitations of Figs. \[Figure\_6\_1\](a) or \[Figure\_6\_2\](b1), accordingly to the steps (6) and (10) in Figs. S1 (pag. ) and S3 (pag. ), respectively.
Finally, taking advantage of the two dimensional symmetry of our problem (as we have assumed wires of infinite length), Eq. (\[Eq.6.18\]) may be transformed to a path integral for the flux of energy over the external surface of the superconducting wire, which for cylindrical wires in polar coordinates is equivalent to say that the hysteretic losses per unit length can be reliably calculated by the following expression:
$$\begin{aligned}
\label{Eq.6.22}
L = -2 \omega \int_{h.f} dt \oint_{l}\textbf{S}\cdot
d\hat{\textbf{\textit{l}}} = \left. -2 \omega R \int_{h.f} dt
\oint_{l}\textbf{S}\cdot \hat{\textbf{r}} ~ d\phi ~ \right|_{r=R}
\, .\end{aligned}$$
### \[ch-6-3\] *6.3 SC wires under simultaneous AC excitations $(\textbf{B}_{0}$ , $I_{tr})$* {#ch-6-3-6.3-sc-wires-under-simultaneous-ac-excitations-textbfb_0-i_tr .unnumbered}
As we have mentioned before, the implementation of superconducting wire technology straightforwardly depends on the demonstration of their reliability, competitive advantages in terms of improved efficiency and reduced operating costs, with capital costs comparable to those of conventional devices. Thus, for the development of competitive devices for the industry, it is important to precisely understand the AC loss properties when realistic non-trivial AC excitations have to be considered. In fact, almost in all the conceived applications for superconducting wires, is well known that each one of the wires holds an AC transport current and experiences an additional AC magnetic field due to the neighboring wires. This situation is found, for example, in superconductor windings for AC magnets, generators, transformers and motors, where each turn feels the magnetic field of all the others [@P2-Larbalestier_2001; @P2-Oomen_2003; @P2-Hull_2003; @P2-Paul_2005; @P2-Yunis_1995; @P2-Takeuchi_1998].
The first conceptualization of the problem of superconducting wires under configurations of simultaneous alternating current and applied magnetic field was provided in 1966 by Hancox [@P2-Hancox_1966], who studied the AC losses through simplified analytical methods for determining the flux front profile in an infinite slab subjected to a field applied parallel to the direction of the injected transport current. This work went almost unnoticed for over a decade, until a similar approach was proposed in 1979 by Carr [@P2-Carr_1979]. Then, the same kind of experimental configurations but for monotonic rates of the experimental sources $(B_{0},I_{tr})$ has been studied since the 1990’s, under the assumptions of very thin superconducting strips to allow different analytical considerations [@P2-Brandt_1993; @P2-Zeldov_1994; @P2-Schonborg_2001]. However, in more realistic situations, where the cross section of the superconducting sample cannot be reduced one dimension, the use of exact analytical methods is not feasible. Thus, the use of numerical methods as the described in previous sections becomes in the more attainable procedure for the forecast and understanding of the electromagnetic observables such as the magnetization curves and the AC power density losses.
It is worth mentioning that despite the fact that there is a significant number of works assuming isolated superconducting wires of diverse geometries, mainly strips subjected to synchronous excitations [@P2-Carr_1979; @P2-Amemiya_1998; @P2-Yazawa_1998; @P2-Yazawa_1999; @P2-Amemiya_2001; @P2-Schonborg_2001; @P2-Zannella_2001; @P2-Stavrev_2002; @P2-Tebano_2003; @P2-Tonsho_2003; @P2-Ogawa_2003; @P2-Enomoto_2004; @P2-Stavrev_2005; @P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Vojenciak_2006; @P2-Pardo_2007; @P2-Pardo_2005; @P2-Pi_2010; @P2-Thakur-2011a; @P2-Thakur-2011b], a thoroughly study of cylindrical superconducting wires was still absent, and therefore some outstanding predictions had not been reported before [@P2-Ruiz_APL_2012].
In the present subchapter, we show a comprehensive study of the physical features associated to the local electrodynamics of superconducting wires under the combined action of AC current and AC magnetic field, which continues our previous discussion and constitutes a step forward in the understanding of the electromagnetic observables and the local effects associated to the AC losses. Section 6.3.1 is restricted to the situation of a *synchronous* AC excitation $(B_{0},I_{tr})$, corresponding to uniform AC magnetic field $B_{0,y}$ in phase with the injected transport current $I_{tr}$, both with the same oscillating frequency (see Fig. \[Figure\_6\_4\]). Also, synchronous excitations are considered in situations where the superconducting wire has been premagnetized (see Fig. \[Figure\_6\_9\], pag. ). On the other hand, section 6.3.2 addresses the effects related to the consideration of *asynchronous* excitations, in which, both sources may be out of phase and apply at different frequencies (see Fig. \[Figure\_6\_12\], pag. ). Premagnetized wires subjected to synchronous or asynchronous sources, may be found in superconducting multicoils for the production of high magnetic fields [@P2-Iwasa_2009], accelerator magnet technologies [@P2-Iwasa_2009; @P2-Wanderer_2006; @P2-Kashikhin_2006], and superconducting magnetic energy storage systems [@P2-He_2010].
![\[Figure\_6\_4\] Sketch of some of the experimental processes analyzed along this chapter. Here, cylindrical SC wires subjected to synchronous oscillating excitations $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}$ and $I_{\tt{a}}$ have been considered, ](Figure_6_4.pdf){height="8cm" width="13cm"}
***6.3.1 Synchronous excitations***
\[ch-6-3-1\] Accordingly to the cases explored in previous sections and holding the aim of achieving a clearest understanding of the electromagnetic quantities involved in the actual use of superconducting wires, is continued by our discussion considering the experimental scenario displayed in Fig. \[Figure\_6\_4\].
##### ( *A.* ) *Flux penetration profiles*\
Certainly, the flux front profile in the initial stage penetrates from the surface as the intensity of the external excitations ($B_{0,y}$,$I_{tr}$) increases (see Figs. \[Figure\_6\_5\] & \[Figure\_6\_6\]). Recall that screening currents produced by the external excitations have been conveniently introduced in terms of two different groups depending on the nature of the source. On the one hand, we speak about *magnetization currents* produced by the excitation magnetic flux density $B_{0,y}$, and on the other hand, we refer to the *injected current lines* which must accomplish the additional global constraint $\sum_{i}{I}_{i}(t)\equiv{I}_{tr}(t)$ \[Eq. (\[Eq.6.6\])\]. When the action of isolated sources is conceived, the distribution of screening currents preserves a well defined symmetry. However, for simultaneous application of both sources (Fig. \[Figure\_6\_4\]), the consumption of the magnetization currents by the injected current lines distorts the axisymmetric orientation of the flux-front, by displacing the current free core to the left during the monotonic branch (Figs. \[Figure\_6\_5\] & \[Figure\_6\_6\]).
![\[Figure\_6\_5\] Evolution of the magnetic flux lines and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=2$ (*low field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\]. For the branches corresponding to the synchronous cyclic excitation see Fig. S5 (pag. ). ](Figure_6_5_a.pdf "fig:"){height="5.0cm" width="3cm"} ![\[Figure\_6\_5\] Evolution of the magnetic flux lines and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=2$ (*low field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\]. For the branches corresponding to the synchronous cyclic excitation see Fig. S5 (pag. ). ](Figure_6_5_b.pdf "fig:"){height="5.0cm" width="3cm"} ![\[Figure\_6\_5\] Evolution of the magnetic flux lines and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=2$ (*low field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\]. For the branches corresponding to the synchronous cyclic excitation see Fig. S5 (pag. ). ](Figure_6_5_c.pdf "fig:"){height="5.0cm" width="3cm"} ![\[Figure\_6\_5\] Evolution of the magnetic flux lines and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=2$ (*low field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\]. For the branches corresponding to the synchronous cyclic excitation see Fig. S5 (pag. ). ](Figure_6_5_d.pdf "fig:"){height="5.0cm" width="3cm"}
For low magnetic fields (Fig. \[Figure\_6\_5\]), the profiles of current density are rather similar to those obtained for $B_{0,y}=0$. The basic difference is that the center of the current free core moves towards the semiaxis $x$-negative. In fact, if the intensity of the transport current is high enough, the flux front becomes nearly circular (*current-like*). Then, the distribution of screening currents may be understood as the straightforward overlapping of the profiles of current density for isolated sources $I_{tr}$ \[Fig. S1 (pag. )\] and $B_{0,y}$ \[Fig. S3 (pag. )\], and additionally displacing the center of the core devoid of electric current and magnetic flux (green zone) by the respective difference between the known flux fronts. It should be mentioned that such assumption has been made in Refs. [@P2-Brandt_1993] & [@P2-Zeldov_1994] for calculating the current profiles for thin strips with synchronous excitations $(B_{0},I_{tr})$. Nevertheless, recently it has been proved that even in this simple configuration, the overlapping principle for the flux front tracking may be only fulfilled for high current and low applied field [@P2-Pardo_2007].
![\[Figure\_6\_6\] Evolution of the magnetic flux lines (projected isolevels of the vector potential) and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\], i.e., label (1) identifies the time-step corresponding to half of the first branch, and (2) the first excitation peak. For visualizing the electromagnetic response in the following branches (cyclic response) reader is advised to see Fig. S8 (pag. ). ](Figure_6_6_a.pdf "fig:"){height="5.0cm" width="3cm"} ![\[Figure\_6\_6\] Evolution of the magnetic flux lines (projected isolevels of the vector potential) and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\], i.e., label (1) identifies the time-step corresponding to half of the first branch, and (2) the first excitation peak. For visualizing the electromagnetic response in the following branches (cyclic response) reader is advised to see Fig. S8 (pag. ). ](Figure_6_6_b.pdf "fig:"){height="5.0cm" width="3cm"} ![\[Figure\_6\_6\] Evolution of the magnetic flux lines (projected isolevels of the vector potential) and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\], i.e., label (1) identifies the time-step corresponding to half of the first branch, and (2) the first excitation peak. For visualizing the electromagnetic response in the following branches (cyclic response) reader is advised to see Fig. S8 (pag. ). ](Figure_6_6_c.pdf "fig:"){height="5.0cm" width="3cm"} ![\[Figure\_6\_6\] Evolution of the magnetic flux lines (projected isolevels of the vector potential) and their corresponding profiles of current with simultaneous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, *1st column*: $I_{\tt{a}}=0.25$, *2nd column*: $I_{\tt{a}}=0.5$, *3rd column*: $I_{\tt{a}}=0.75$, and *4th column*: $I_{\tt{a}}=1$. Subplots are labeled according to the monotonic branch of the experimental processes depicted in Fig. \[Figure\_6\_4\], i.e., label (1) identifies the time-step corresponding to half of the first branch, and (2) the first excitation peak. For visualizing the electromagnetic response in the following branches (cyclic response) reader is advised to see Fig. S8 (pag. ). ](Figure_6_6_d.pdf "fig:"){height="5.0cm" width="3cm"}
Likewise, if the applied magnetic field is intense enough as compared to the transport current (see e.g., left side in Fig. \[Figure\_6\_6\]) the distribution of screening currents is *field-like*. Nevertheless, the inherent existence of injected current lines makes it impossible to discern which filaments correspond to the so called magnetization currents, and which are the injected current lines. Certainly, for the monotonic branch of the cyclic excitation and before attaining a full penetration state by the screening currents, the “active” zone (blue) where $I_{i}$ takes the value $-I_{c}$ straightforwardly corresponds to the so called magnetization currents. However, the remaining “active” zone (red) defined by screening currents $I_{i}$ taking the value $I_{c}$ is not spatially symmetric as regards the direction of the applied magnetic field, which means that a certain amount of the magnetization currents are contributing in the same direction as the transport current, whilst another part has been consumed by the injected current lines. As it will be shown below, a parallel effect is that the density of magnetic flux increases in the “active” zone, where the patterns of injected current lines dominate.
For the cyclic processes displayed in Fig. \[Figure\_6\_4\] tracking the flux front for synchronous excitation with low magnetic field is intuitive \[Fig. S5 (pag. )\], although following up the components of the magnetic flux density $B_{x}$ \[Fig. S6 (pag. )\] and $B_{y}$ \[Fig. S7 (pag. )\] is not. For high magnetic fields, ascertaining the distribution of screening currents in the cyclic stage is much more elaborated, as long as the electromagnetic history is not erased by the maximal condition for the amplitude of the AC transport current $I_{\tt{a}}=I_{c}$. Actually, if $I_{\tt{a}}<I_{c}$ the flux fronts do not overlap to a unique contour line defined by the filaments with current alternating between $I_{c}$ and $-I_{c}$ \[Fig. S8 (pag. )\]. Likewise, describing the evolution of the magnetic flux density \[Figs. S9-S10 (pags. -)\] is also complicated if one compares them with the simplest cases in which isolated sources are assumed \[Fig. S2 (pag. ) & Fig. S4 (pag. )\].
Although the analysis of the magnetic flux density $B$ is complicated, one of the most outstanding observations for considering synchronous excitations as shown in Fig. \[Figure\_6\_4\], is that the local distribution of magnetic field preserves the same kind of pattern along the cyclic stage, independently of the intensity of the external sources. Thus, one can notice that the maximal density of magnetic flux occurs always to the right side of the superconducting wire, which corresponds to the “active” zone where the injected current lines are dominating the system. Concomitantly, substantial distortions of the magnetic flux density outside the wire appear. These are particularly marked when $B_{0,y}$ and $I_{tr}$ tend to zero during the excitation.
![\[Figure\_6\_7\] Local density of power dissipation E$\cdotp$J at the time frame of full cycle (step number 10 in Fig. \[Figure\_6\_4\]) for synchronous oscillating sources of amplitudes: (*1st pane*) $B_{\tt{a}}=8$ and $I_{\tt{a}}=0.25$, (*2nd pane*) $B_{\tt{a}}=8$ and $I_{\tt{a}}=1.0$, (*3rd pane*) $B_{\tt{a}}=2$ and $I_{\tt{a}}=0.25$, (*4th pane*) $B_{\tt{a}}=2$ and $I_{\tt{a}}=1.0$. The local dynamics for the aforementioned quantities in the full cyclic process including the initial monotonic branch, can be inferred from the supplementary figures for the low-field regime \[Fig. S14 (pag. )\], and the high-field regime \[Fig. S15 (pag. )\]. ](Figure_6_7.pdf){height="4.1cm" width="13cm"}
Remarkably, the strong localization of the inner density of magnetic flux density produces a significant change in the local distribution of density of power dissipation $\textbf{E}\cdot \textbf{J}$, which rises from low-value parts (blue) to high-value parts (red) \[see, Fig. \[Figure\_6\_7\]\], in such manner that the heat release from the superconducting wire is highly localized too. In fact, this asymmetric distribution of power losses regarding the perpendicular direction to the orientation of $\textbf{B}$, remains along the entire cyclic process as long as both excitations evolve synchronous \[see e.g., Figs. S14-S16 (pags. -)\]. Therefore, its pronounced bias unfolding across the wire could increase the probability of quench.
![\[Figure\_6\_8\] The dimensionless magnetic moment $M_{y}/M_{p}$ for the synchronous AC excitations displayed in Fig. \[Figure\_6\_4\]. Curves are shown as function of the injected transport current $I_{tr}$ (*left pane*), the applied magnetic field $B_{0,y}$ *(central pane)*, or either by its temporal evolution *(right pane)*. In this figure, the amplitudes for the electromagnetic AC sources can be extracted either from color comparison with curves in Fig. \[Figure\_6\_4\], or from the respective limits along the abscissas in left and right panes. ](Figure_6_8.pdf){height="10cm" width="12cm"}
##### ( *B.* ) *Magnetic response*\
The above described behavior for the local flux distributions gives way to the following features on the magnetic moment response.
For the set of cases displayed in Fig. \[Figure\_6\_4\], we have analyzed the dynamics of the magnetic moment component $M_{y}$ as a function of the amplitudes of the electromagnetic sources $B_{\tt{a}}$ and $I_{\tt{a}}$ (Fig. \[Figure\_6\_8\]). Results are shown accordingly to the temporal dependence with the synchronous AC excitations (right pane), and also by their dependence with each one of the electromagnetic sources, say $I_{tr}$ (left pane) and $B_{0,y}$ (middle pane). We realize that only for small values of the amplitude of the ac transport current, almost Bean-like loops of $M_{y}$ obtain. However, as $I_{\tt{a}}$ grows we notice a progressive disappearance of the flat saturation behavior for values of $B_{\tt{a}}$ higher than $B_{p}$. Actually, the notorious change of sign for the slope of the magnetic moment curve along a monotonic branch of the synchronous AC excitation, allows an unambiguous glimpse of the consumption of magnetization lines by effect of injected current lines. Remarkably, this phenomenon ends up with a symmetrization of the loops, both as functions of $B_{y}$ and $I_{tr}$, into characteristic lenticular shapes. As a consequence of this process, a distinct low-pass filtering effect comes to the fore which, in the case of the triangular input excitations considered here, yields a nearly perfect sinusoidal (first-harmonic) output signal $M_{y}(t)$.
![\[Figure\_6\_9\] Dimensionless magnetic moment $M_{y}/M_{p}$ as a function of the applied magnetic field $B_{0,y}$ for cycles of simultaneous ac excitations $B_{0,y}$ and $I_{tr}$ of amplitudes $(B_{\tt{a}},I_{\tt{a}})$. Although the excitation peak to peak of both sources assumes synchronous and with equal frequency, several premagnetized samples have been considered according to: $B(t')=2$, $B(t')=4$ and $B(t')=8$ (see 1st row), for the time instant when the ac current $I_{tr}(t')$ is switched on. Regarding to the cyclic process (i.e, from $t''$ to 1), several cases are shown accordingly to the amplitudes $I_{tr}(t'')=0.25$ (see 2nd row), $I_{tr}(t'')=0.5$ (see 3rd row), and $I_{tr}(t'')=1$ (see 4th row), as well as to $B_{\tt{a}}=$2 (dotted lines), $B_{\tt{a}}=$4 (dashed lines) and $B_{\tt{a}}=$8 (straight lines) respectively.](Figure_6_9.pdf){height="10cm" width="12cm"}
Interestingly, from the cycles of $M_{y}$, it furthermore appears that a proper determination of the “active” zones depends on the history of the virgin branch, thus bearing witness to the system’s memory. For example, a positive slope in the synchronous excitation $B_{y}$ and $I_{tr}$ produces a higher power dissipation in the positive $x$-direction perpendicular to the wire. On the other side, if the superconducting wire has been premagnetized before switching on the synchronous AC excitation \[$t=t'$ at Fig. \[Figure\_6\_9\]\], the magnetic moment curve is displaced in such a manner that the center of the magnetization loop lies over the master curve for the isolated excitation $B_{0,y}$ at $M_{y}(t')$ (see Fig. \[Figure\_6\_2\]), and the nodes move towards the boundaries $B_{0,y}(t')\pm B_{\tt{a}}$.
##### ( *C.* ) *AC Losses*\
Regarding the power density losses attained along the premagnetization process, it does not seem to play any role in the calculation of the AC losses (Fig. \[Figure\_6\_10\]). At least, this was observed to the precision of our calculations. However, the definition of the flux front profile becomes much more tangled, because multiple domains enclosed by contour lines defined by the screening currents alternating from $I_{c}$ to $-I_{c}$ arise \[see Fig. S11 (pag. )\].
Notwithstanding, the Bean-like magnetic moment curves as the described above, and the ostensible explanation for the distribution of screening currents in terms of the consumption of magnetization lines by the injected current lines, is actually insufficient for the proper interpretation of the actual AC losses produced by synchronous excitations. In fact, despite the collection of reliable experimental data is quite laborious (because there are many pitfalls in the measurement procedures), there is an extended consensus that the heat release by the superconducting wire may come from the electromagnetic sources $B_{0}$ and $I_{tr}$ in independent manners [@P2-Vojenciak_2006; @P2-Zannella_2001; @P2-Tonsho_2003; @P2-Ogawa_2003]. Moreover, many works dealing with this issue argue that the transport loss and the magnetization loss can be separately determined by electromagnetic measurements at least for low values of the magnetic field and high currents or vice versa [@P2-Ogawa_2011; @P2-Inada_2005; @P2-Jiang_2004; @P2-Rabbers_2001; @P2-Reuver_1985]. However, as we have discussed before, for general cases, the competence between the magnetization currents and the injected current lines involving axisymmetric distributions of the screening currents, generates a strong localization of the local density of magnetic flux, as well as of the local density of power losses, which makes it difficult discriminating the role of the AC losses introduced by the inductive terms \[Eq. (\[Eq.6.19\])\]. Therefore, from the theoretical point of view, it is advisable using experimental methods based upon the S-oriented approach (pag. ), such as calorimetric methods which directly make a measurement of the release flux of energy over the superconducting surface [@P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Ramos_2010; @P2-Ashworth_2000; @P2-Magnusson_2000].
As we will show below, the feasibility of approaching the total AC loss by overlapping the isolated contributions, strongly depends on the relative magnitudes of the AC field $B_{0}$, and the AC transport current $I_{tr}$. Accordingly to the numerical experiments shown in Figs. \[Figure\_6\_4\] & \[Figure\_6\_9\], eventually, we will present a percentage analysis of the actual AC loss for synchronous excitations, $L(B_{\tt{a}},I_{\tt{a}})$, in comparison with the most celebrated approaches. Our numerical results for $L(B_{\tt{a}},I_{\tt{a}})$ are displayed in Fig. \[Figure\_6\_10\] both in logarithmic and linear -scales with the aim of remarking the actual differences at low and high magnetic fields.
![\[Figure\_6\_10\] Hysteretic ac losses per cycle for synchronous AC magnetic flux density and oscillating transport current of amplitudes $(B_{\tt{a}},I_{\tt{a}})$ accordingly to the Figs. \[Figure\_6\_4\] & \[Figure\_6\_9\]. Results of this work are shown as color solid lines with markers. Comparisons with results from conventional approaches are shown for, *(i) Left pane*: separate excitations $L(B_{\tt{a}})$ (black solid line) and $L(I_{\tt{a}})$ (straight color lines), as well as their linear superposition $L(B_{\tt{a}})+L(I_{\tt{a}})$ (color dashed lines); *(ii) Central pane*: an ac magnetic field together with a dc transport current of intensity $I_{\rm tr}^{\rm dc}=I_{\tt{a}}$ , $L(B_{\tt{a}},I_{\rm tr}^{dc})$; *(ii) Right pane*: the whole set of results is also plotted in linear scale. Units for losses are $(\mu_{0}/4\pi)\omega
R^{2}
J_{c}^{2}$.](Figure_6_10.pdf){height="7cm" width="13cm"}
First, let us recall that, according to Eq. (\[Eq.6.11\]) for isolated sources, the AC transport loss $L(I_{\tt{a}})$ may be calculated according to $$\begin{aligned}
\label{Eq.6.23}
L(i_{\tt{a}})\equiv
\frac{\mu_{0} I_{c}^{2}}{\pi^{2}R^{2}}
\left[
i_{\tt{a}}\left(1-\frac{i_{\tt{a}}}{2}\right)
+(1-i_{\tt{a}})\ln(1-i_{\tt{a}})
\right]
\,~\,~\, \forall ~\, ~\, 0< i_{\tt{a}} \leq 1\, \, ,\end{aligned}$$ where the dimensionless parameter $i_{\tt{a}}=I_{\tt{a}}/I_{c}$ has been introduced.
On the other hand, the hysteretic loss produced by magnetization effects may be estimated from Eq. (\[Eq.6.12\]), in such manner that $L(B_{\tt{a}})$ is calculated from $$\begin{aligned}
\label{Eq.6.24}
L(b_{\tt{a}})\equiv
\dfrac{8B_{p}^{2}}{3\mu_{0}}
\left\{
\begin{array}{ll}
b_{\tt{a}}^{3} ~ \left(1-\dfrac{1}{2}b_{\tt{a}}\right)
\,~\,~ & ,~\, \forall ~\, ~\, 0 < b_{\tt{a}} \leq 1 \,
\\ \, \\
b_{\tt{a}}-\dfrac{1}{2}
\,~\,~ & ,~\, \forall ~\, ~\, b_{\tt{a}} \geq 1
\, ~ \, ~ \, .
\end{array}
\right. \end{aligned}$$
with the dimensionless parameter $b_{\tt{a}}=B_{\tt{a}}/B_{p}$. Recall that full penetration is given by $b_{\tt{a}}\geq1$ (or $i_{\tt{a}}=1$).
The simplest approach for determining the AC losses of cylindrical superconducting wires subjected to synchronous AC excitations relies in the linear superposition of the separate contributions, $L(B_{\tt{a}})+L(I_{\tt{a}})$. Another possibility is to assume that the actual AC losses for synchronous excitations do not strongly differ of the hysteretic losses for superconducting samples carrying a constant transport current $I_{tr}^{dc}$ and a simultaneous oscillating magnetic field of amplitude $B_{\tt{a}}$ [@P2-Gurevich_1997]. This idea was applied in the analytical approach by Zenkevitch et al. in Ref. [@P2_Zenkevitch_1983]. In such a framework, the hysteretic loss for a period is approximated by: $$\begin{aligned}
\label{Eq.6.25}
L(B_{\tt{a}},I_{tr}^{dc})\equiv
\dfrac{8 B_{p}^{2}}{3\mu_{0}}
\left\{
\begin{array}{ll}
b_{\tt{a}}^{3} \left(1-\dfrac{1}{2}b_{\tt{a}}\right)
\,~\,~ & ,~\, \forall ~\, ~\, b_{\tt{a}} < i_{\tt{a}}^{\dag} \,
\\ \, \\
i_{\tt{a}}^{\dag^{3}} \left(1-\dfrac{1}{2}i_{\tt{a}}^{\dag}\right)+
\left(1+i_{\tt{a}}^{2}\right)
\left(b_{\tt{a}}-i_{\tt{a}}^{\dag}\right)
\,~\,~ & ,~\, \forall ~\, ~\, b_{\tt{a}} \geq i_{\tt{a}}^{\dag}
\, ~ ,
\end{array}
\right. \end{aligned}$$
where we have introduced the dimensionless parameter $i_{\tt{a}}^{\dag}\equiv1-i_{\tt{a}}^{2/3}$ and the condition $b_{\tt{a}} \geq
i_{\tt{a}}^{\dag}$. Here, $I_{\tt{a}}\equiv I_{tr}^{dc}$. Thus, $b_{\tt{a}}\geq1$ or $i_{\tt{a}}^{\dag}\equiv0$, both define a full penetrated sample.
Fig. \[Figure\_6\_10\] shows our numerical results for the variation of the actual AC losses of cylindrical SC wires in terms of the amplitude of the synchronous oscillating sources, $L(B_{\tt{a}},I_{\tt{a}})$ (*solid-diamond lines*), compared to the customary approaches $L(B_{\tt{a}})+L(I_{\tt{a}})$ (*dashed lines at the left pane*) and $L(B_{\tt{a}},I_{tr}^{dc})$ (*dash-dotted lines at the central pane*), for four different amplitudes of the AC/DC transport current $I_{\tt{a}}$, and a set of amplitudes of the AC density of magnetic flux $B_{\tt{a}}$. The whole set of results is also plotted in linear scale at the right pane of this figure.
Comparison reveals the important fact that a linear superposition of contributions due to either type of excitation may be only appropriate for high amplitudes of the magnetic field ($B_{\tt{a}}\geq B_{p}$) and low currents ($I_{\tt{a}}<I_{c}/4$), or the converse limit, very low magnetic fields ($B_{\tt{a}}\leq 1$) and extremely high currents $I_{\tt{a}}\lessapprox I_{c}$; a finding which adds to previous work dealing with a rectangular geometry [@P2-Pardo_2007] and sheds new light on the validity of approximate formulae at the same time. Actually, notice that the actual AC loss $L(B_{\tt{a}},I_{\tt{a}})$ is always higher than the instinctive approach $L(B_{\tt{a}})+L(I_{\tt{a}})$, whilst the deviation respect to $L(B_{\tt{a}},I_{tr}^{dc})$ strongly depends on the intensity of the electromagnetic excitations. Consequently, approximations such as $
L\left(B_{\tt{a}} \right) + L\left( I_{\tt{a}} \right)$ and $ L\left(
B_{\tt{a}}, I_{tr}^{dc} \right)$ can drastically under- or overestimate the true losses.
![\[Figure\_6\_11\] Percent change between the actual AC loss $L(B_{\tt{a}},I_{\tt{a}})$ numerically calculated and the intuitive approaches $L(B_{\tt{a}})+L(I_{\tt{a}})$ (left pane) and $L(B_{\tt{a}})+L(I_{tr}^{dc})$ (right pane). ](Figure_6_11_a.pdf "fig:"){height="6cm" width="6.6cm"} ![\[Figure\_6\_11\] Percent change between the actual AC loss $L(B_{\tt{a}},I_{\tt{a}})$ numerically calculated and the intuitive approaches $L(B_{\tt{a}})+L(I_{\tt{a}})$ (left pane) and $L(B_{\tt{a}})+L(I_{tr}^{dc})$ (right pane). ](Figure_6_11_b.pdf "fig:"){height="5.8cm" width="6.2cm"}
For further understanding of the above behavior, Fig. \[Figure\_6\_11\] shows the percentage relation between the actual AC loss, $L(B_{\tt{a}},I_{\tt{a}})$, and the intuitive approaches, $ L\left(B_{\tt{a}}\right) + L\left(I_{\tt{a}}
\right)$ (*left pane*) and $ L\left(B_{\tt{a}},I_{tr}^{dc} \right)$ (*right pane*), stacked according to the values of $I_{\tt{a}}$. On the one side, we note that for the approach $L\left(B_{\tt{a}}\right) +
L\left(I_{\tt{a}}\right)$, and for small values of $I_{\tt{a}}$ (e.g. $I_{\tt{a}}=0.25$), the deviation is gradually reduced as one increases the amplitude of the magnetic field $B_{\tt{a}}$. However, as $I_{\tt{a}}$ increases deviations may either reduce (for low values of $B_{\tt{a}}$, e.g., $B_{\tt{a}}=1$) or increase (for high values of $B_{\tt{a}}$, e.g., $B_{\tt{a}}=8$). Moreover, for moderate fields (e.g., $2\leq B_{\tt{a}}\leq6$) percentage deviations first grow as a function of $I_{a}$ until $I_{\tt{a}}=0.5I_{c}$, and then decrease as $I_{\tt{a}}$ approaches the current limit $I_{c}$. On the other side, the approach $L\left(B_{\tt{a}},I_{tr}^{dc}\right)$ is not even comparable with the actual AC losses $L(B_{\tt{a}},I_{\tt{a}})$ for the regime of low magnetic fields (particularly for $b_{\tt{a}}<i_{\tt{a}}^{\dag}$), a range in which the approximation conceals the effect of $I_{a}$. However, for moderate and high magnetic fields, the percentage ratio between the actual AC loss and the AC loss predicted by the latter approach decrease as $I_{\tt{a}}$ increases even reaching negative values. Hence, $L\left(B_{\tt{a}},I_{tr}^{dc}\right)$ may either overestimate or underestimate the actual AC loss for wires subjected to synchronous oscillating sources. Remarkably, for high amplitudes of the oscillating magnetic flux density $B_{\tt{a}}$, we note that a synchronous oscillating transport current of amplitude $I_{\tt{a}}$ produces a lower amount of hysteretic losses per period, than those predicted when the superconducting sample is carrying a constant transport current $I_{tr}^{dc}$.
***6.3.2 Asynchronous excitations***
\[ch-6-3-2\]
In many of the large-scale power applications for superconducting wires, such as windings of motors, transformers, generators and power three-phase transmission lines, the SC wire is subjected to diverse configurations of electromagnetic excitations, where the AC transport current flowing through and the magnetic field to which the wire is exposed could not fulfill the synchronous conditions referred above (same phase and frequency). Moreover, remarkable experimental differences between the AC loss measured for superconducting wires or tapes with synchronous and asynchronous oscillating transport current and perpendicular magnetic field have been already reported by several authors [@P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Vojenciak_2006].
##### ( *A.* ) *General considerations on the “asynchronous” AC losses*\
Our analysis of the AC loss formulae for cylindrical superconducting wires subjected to simultaneous oscillating transport current $(I_{tr})$ and perpendicular magnetic flux density ($B_{0,y}$), has revealed that the total AC loss may be controlled by reducing the inductive magnetic flux density produced by the superposition between the external magnetic field and the contribution by the whole set of screening currents \[recalling Eq. (\[Eq.6.9\])\]. This can be achieved just by a time shift respect to one of the AC electromagnetic sources (either $B$ or $I$) respect to the other, so that the occurrence of the peaks of excitation for each one of the electromagnetic sources is no longer synchronous with the other. Thus, some eccentric branches with opposite temporal derivatives appear between two consecutive peaks of the dominant excitation (i.e., the excitation with lower frequency this is the case), which may counterbalance the local variation of the magnetic flux density produced by the other one in the zone of maximum heat release.
Evidently, by competition between the magnetization currents and the so-called injected current lines, the magnitude of the local density of magnetic flux ($B_{i}$) may be reduced in half section of the superconducting cylinder shifting the relative phase between the electromagnetic excitations. Thus, as long as the electromagnetic excitations have the same oscillating period, reductions of the actual AC loss could be announced for shifts in the relative phase measured between the synchronous case and a temporal displacement of half period \[i.e., $\Delta\phi=\pi$ if both sources accomplish the generic relation $f=f_{0}cos(\omega t+\phi)$\], as it has been observed in Refs. [@P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Vojenciak_2006]. Recall that, we have shown that the total AC loss decreases ensuring minimal variation of $\Delta B_{i}$ along the whole sample. In fact, if the relative phase shift equals half a period, it means that the occurrence of the excitation peaks for the electromagnetic sources are likewise synchronous but are pointing in opposite directions. Thus, as the total AC loss may be calculated by integration of the excitations peak to peak, for this case $\Delta B_{i}$ is maximum and therefore also the actual AC loss. On the other hand, the forecast of the minimal variation of the set of integrands in Eq. (\[Eq.6.9\]) and consequently the total AC loss, may be done by considering a temporal displacement of a quarter of period \[i.e., $\phi=\pi/2$ for circular excitation functions\], so that the local competition between the magnetic flux densities $\mid B_{0,i} \mid$ and $\mid B_{ind,i} \mid$ minimizes as $B_{0,y}(\Omega)=0$ when $\mid I_{tr} \mid =I_{a}$. Then, under this simplified scenario, and at least for cases where the local distribution of screening currents is current-like (very low magnetic field, $b_{a}\approx 0$, and high transport current, $i_{a}\approx1$), or field-like (high magnetic field, $b_{a}\gtrsim1$, and very low transport current, $i_{a}\approx0$), the minimal AC loss is envisioned to appears for a relative phase difference of a quarter of period. Likewise, the maximal AC loss may be predicted when both sources are fully synchronous or when there is a relative phase shift of half period. Latter facts agree with the analytical approaches for the slab [@P2_Takacs_2007] and strip [@P2_Mawatari_2006] geometries. Also, in further agreement with the experimental evidences of Refs. [@P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Vojenciak_2006], within our statement we predict that as long as a phase shifting occurs, at least a minimal reduction of the AC loss should be observed.
![\[Figure\_6\_12\] Sketch of some of the experimental processes analyzed along this chapter. Here, a cylindrical SC wire subjected to asynchronous oscillating excitations in the configuration shown in pane (a) are considered according to the temporal processes depicted in panes (b) and (c). ](Figure_6_12.pdf){height="8cm" width="13cm"}
When the distribution of screening currents is “*nothing-like*”, i.e., it shows a strong deformation when compared to the obtained profiles for the isolated excitations, it is not obvious to deduce a general rule for the position of the maximum and minimum of the total AC loss, as the nonhomogeneous interplay between the injected current lines and the magnetization currents affects the total AC loss. In fact, the situation may be very much complicated for the actual applications of superconducting transformers and three-phase transmission lines [@P2_Hamajima_2006; @P2_Hamajima_2007; @P2_Hamajima_2008; @P2_Gouge_2005], by the fact that the self induced magnetic field and the external magnetic field may differ considerably in phase. Especially, one can foretell complicated behaviors when effects of phase transposition and, frequency shifts appear. Here, we will show how the effects of double frequency which may be occasionally found in the power supply networks, can drastically alter the efficiency of the superconducting machines.
Fig. \[Figure\_6\_12\], shows the configuration analyzed below. Notice that, in what follows, we consider the effect of introducing one of the excitations with an oscillating frequency twice as big as the other. Thus, calculation of the AC loss, i.e., integration of the local density of power dissipation $\textbf{E}\cdot\textbf{J}$ is made along the smaller frequency excitation. In detail, we have considered the following cases:
I. The injected transport current is the source within the double frequency regime \[see Fig. \[Figure\_6\_12\] (b)\], and
II. The temporal dynamics of the magnetic flux density associated to the external source of magnetic field shows a double frequency behavior \[see Fig. \[Figure\_6\_12\] (c)\].
Most remarkable features for the flux dynamics, magnetic response, and AC losses for the above mentioned configurations are detailed below.
![\[Figure\_6\_13\] Evolution of the magnetic flux lines and their corresponding profiles of current with asynchronous oscillating sources $B_{0,y}$ and $I_{tr}$ of amplitudes $B_{\tt{a}}=4$ and $I_{\tt{a}}=0.5$ (left side into each pane), accordingly to the temporal processes displayed into Fig. \[Figure\_6\_12\](b) “left pane herein” and Fig. \[Figure\_6\_12\](c) “right pane herein”. Also the corresponding profiles for the local density of power dissipation **E**$\cdotp$**J** are shown (right side into each pane). In particular, in this figure we show the set of results for the last branch of the dominant excitation according to the time-steps marked with the labels (6), (8), and (10) in Fig. \[Figure\_6\_12\]. More details to follow up the electromagnetic quantities along the cyclic process are found in the section of supplementary material, pages -. ](Figure_6_13_a.pdf "fig:"){height="7.5cm" width="6.4cm"} ![\[Figure\_6\_13\] Evolution of the magnetic flux lines and their corresponding profiles of current with asynchronous oscillating sources $B_{0,y}$ and $I_{tr}$ of amplitudes $B_{\tt{a}}=4$ and $I_{\tt{a}}=0.5$ (left side into each pane), accordingly to the temporal processes displayed into Fig. \[Figure\_6\_12\](b) “left pane herein” and Fig. \[Figure\_6\_12\](c) “right pane herein”. Also the corresponding profiles for the local density of power dissipation **E**$\cdotp$**J** are shown (right side into each pane). In particular, in this figure we show the set of results for the last branch of the dominant excitation according to the time-steps marked with the labels (6), (8), and (10) in Fig. \[Figure\_6\_12\]. More details to follow up the electromagnetic quantities along the cyclic process are found in the section of supplementary material, pages -. ](Figure_6_13_b.pdf "fig:"){height="7.5cm" width="6.4cm"}
##### ( *B.* ) *Flux dynamics*\
Before discussing the results obtained for the total AC loss in the set of experiments displayed in Fig \[Figure\_6\_12\], some outstanding facts related to the rich phenomenology found for the local dynamics of the electromagnetic quantities are worth of mention. For example, when the applied magnetic field and the transport current are synchronous, and their associated amplitudes $(B_{\tt{a}},I_{\tt{a}})$ are weak enough such that a flux-free core remains along the AC cycles (i.e., as long as $i_{\tt{a}}^{*}\neq1$ and $b_{\tt{a}}^{*}<1$), it is more or less simple to identify the active zones in the AC cycles via the previous knowledge of the virgin branch (see Figs. \[Figure\_6\_5\] & \[Figure\_6\_6\], pags. -), and therefore, explaining and obtaining the AC loss may be achieved if the distribution of screening currents is well known for the first half of the AC period. However, when the temporal dynamics of the isolated excitations shows an asynchronous response, this is not longer valid. The reason is, that the hysteretic losses produced along the virgin branch are not monotonic concerning the temporal evolution of both electromagnetic excitations, such that the accruing hysteretic losses for the lower limit in the first integral of Eq. (\[Eq.6.21\]) are different for the first and second half of the cyclic period. In other words, the distribution of screening currents in the first peak of the excitation with smaller frequency or below so called dominant excitation \[time step 2 in Fig. \[Figure\_6\_12\]\] may drastically differ from those conceived in the second and third peaks \[time steps 6 and 10\].Hence, the distribution of screening currents for the first half period of the AC cycle cannot be fetched through their distribution in the virgin branch. Therefore, we want to call reader’s attention to the fact that for making use of Eq. (\[Eq.6.21\]) for the calculation of AC losses, the steady regimes for the limits of the time-integral have to be defined for the excitation peaks defining the second half period of the dominant excitation. Thus, a proper description of the profiles of current density in an asynchronous AC regime must be done at least for this temporal branch (see Fig. \[Figure\_6\_13\]).
![\[Figure\_6\_14\] The dimensionless magnetic moment $M_{y}/M_{p}$ for the AC asynchronous excitations displayed in Fig. \[Figure\_6\_12\](b) where the applied magnetic field have the role of dominant excitation. Curves are shown as function of the injected transport current $I_{tr}$ in units of their amplitude $I_{\tt{a}}$ (*left column*), the applied magnetic field $B_{0,y}$ *(central column)*, or either by its temporal evolution *(right column)*. Same color scheme to point out the amplitude of the AC magnetic field ($B_{\tt{a}}$) has been used in all subplots. ](Figure_6_14.pdf){height="10cm" width="13cm"}
Analyzing the distribution of current density profiles in Fig. \[Figure\_6\_13\] and their corresponding profiles for the local density of power dissipation $\textbf{E}\cdot\textbf{J}$, we found at least two interesting facts which are worth of mention. On one side, whether it is the magnetic field or the transport current, the excitation which leads the role of dominant, multiple domains or active zones connected between them may appear, which makes it impossible to find out a feasible analytical solution for the flux front boundary in the infinite spectra of combinations between the amplitudes $B_{\tt{a}}$ and $I_{\tt{a}}$, specially if the pattern of current density is far away of the approaches for profiles of the kind *current-like* or *field-like*. On the other side, a most striking fact revealed in Fig. \[Figure\_6\_13\] is that contrary to the behavior displayed for the local profiles of density of power dissipation $\textbf{E}\cdot\textbf{J}$, when both electromagnetic excitations are synchronous (see e.g. Fig. \[Figure\_6\_7\], pag. ), in asynchronous cases the zone of heat release is no longer localized in one side of the superconducting sample. Thus, the idea of focusing heat release in some part of the superconductor requires a special attention in the synchronization of sources.
##### ( *C.* ) *Magnetic response*\
Another interesting feature which derives from the study of asynchronous excitations is the actual possibility of finding “exotic” magnetization loops as a function of the AC sources (see Figs. \[Figure\_6\_14\] & \[Figure\_6\_15\]), where the straightforward competition between the magnetization currents (by consumption) and the injected current lines, may be visualized in terms of a non local macroscopic measurement.
![\[Figure\_6\_15\] The dimensionless magnetic moment $M_{y}/M_{p}$ for the AC asynchronous excitations displayed in Fig. \[Figure\_6\_12\](c) where the transport current has the role of dominant excitation. Curves are shown as function of the injected transport current $I_{tr}$ in units of their amplitude $I_{\tt{a}}$ (*left column*), the applied magnetic field $B_{0,y}$ *(central column)*, or either by its temporal evolution *(right column)*. Same color scheme to point out the amplitude of the AC magnetic field ($B_{\tt{a}}$) has been used in all subplots. ](Figure_6_15.pdf){height="10cm" width="13cm"}
In Fig. \[Figure\_6\_14\], the component of magnetic moment $M_{y}$ is displayed for the AC process in Fig. \[Figure\_6\_12\](b), where the AC magnetic field dominates the cyclic period of excitation. The whole set of results for $M_{y}$ have been renormalized according to the maximal expected value for the magnetic moment when only applied magnetic field is considered ($M_{p}=2/3$). Thus, curves are shown as function of the isolated electromagnetic excitations, $I_{tr}$ (in units of their associated AC amplitude $I_{\tt{a}}$) at the left column, $B_{0,y}$ at the central column, as well as by the time defining the first steady-period (right column). Notice that, the virgin branch which does not play any role for the integration of the AC losses per cyclic periods is shown through dashed lines. Furthermore, results have been organized accordingly to the associated amplitudes for the applied density of magnetic flux $B_{\tt{a}}$, such that $B_{\tt{a}}=2$ corresponds to the green curves, $B_{\tt{a}}=4$ to the blue curves, and $B_{\tt{a}}=8$ to the red curves. Likewise, the set of curves shown for each row can be straightforwardly associated to a single value for the amplitude of the AC transport current, $I_{\tt{a}}=0.25$ (first row), $I_{\tt{a}}=0.5$ (second row), and $I_{\tt{a}}=1$ (third row). Analogously, the corresponding set of curves obtained for the component of magnetic moment $M_{y}$ in those cases where the AC transport current dominates the cyclic period of excitation, are shown in the same fashion above described in Fig. \[Figure\_6\_15\].
Outstandingly, whether $B_{0,y}$ (Fig. \[Figure\_6\_14\]) or $I_{tr}$ (Fig. \[Figure\_6\_15\]) is the dominant excitation, and for low values of $I_{\tt{a}}$, the magnetization loops as function the magnetic flux density $M_{y}(B_{0,y})$ show a Bean-like behavior. As the value of $I_{\tt{a}}$ increases, notorious deformations of the Bean-like structures for the magnetic moment appear. Nevertheless, the behavior is radically different comparing the double frequency effects provided by one or another excitation, as it is explained below.
##### ( *i.* ) *Transport current with double frequency*\
On the one hand, when it is the AC transport current, $I_{tr}$, that shows a double frequency, the magnetization curves $M_{y}(B_{0,y})$ display a symmetric behavior in the regions (left-right) of the periods $[B_{\tt{a}}\Rightarrow-B_{\tt{a}}]$ and $[-B_{\tt{a}}\Rightarrow B_{\tt{a}}]$ (see Fig. \[Figure\_6\_14\]). On the contrary, for $M_{y}(I_{tr})$, one can notice the existence of a symmetrization of the curves of magnetic moment regarding their positive and negative values (*up/down*). Evidently, the steady-states where the maximum consumption of the magnetization currents occurs, always arise when the asynchronous AC excitation $[B_{0,y},I_{tr}]$ reaches the values for the current’s peaks \[see Fig. \[Figure\_6\_12\](b)\], i.e., for the time-steps (3) $[B_{\tt{a}}/2,-I_{\tt{a}}]$, (5) $[-B_{\tt{a}}/2,I_{\tt{a}}]$, (7) $[-B_{\tt{a}}/2,-I_{\tt{a}}]$, and (9) $[B_{\tt{a}}/2,I_{\tt{a}}]$. Then, by increasing (decreasing) the value of $I_{\tt{a}}$, a progressive decreasing (increasing) of the magnetic moment at these points ends up in the simultaneous collapsing of the magnetization curves $(M_{y}\equiv0)$ for the half periods of the dominant excitation $B_{0,y}$, as long as $I_{\tt{a}}=\pm I_{c}$. Latter fact is followed by the symmetrization of the loops, either as functions of $B_{0,y}$ and $I_{tr}$, into characteristic lenticular shapes bounded by two non-connected magnetization curves, both defined by the elapsed periods in which the time derivative of $I_{tr}(t)$ is positive, i.e., for the temporal branches $(3\Rightarrow5)$ and $(7\Rightarrow9)$. Then, the connecting curves for the abovementioned magnetization branches shows characteristic lashing shapes when the time derivative of $I_{tr}(t)$ is negative. Remarkably, as a consequence of this process, the output signal $M_{y}(t)$ does not show the low-pass filtering effects conceived for synchronous excitations. In fact, given our study, we prove that the low-pass filtering effect for superconducting wires, may only be envisaged when the temporal evolution of the injected AC transport current and the perpendicular magnetic field is fully synchronous (in phase and frequency).
##### ( *ii.* ) *Applied magnetic field with double frequency*\
On the other hand, when it is the AC density of magnetic flux, $B_{0,y}$, the electromagnetic source disclosing the double frequency effect \[Fig. \[Figure\_6\_12\] (c)\], the calculated curves of magnetization $M_{y}(B_{0,y})$, are outstandingly different. Thus, there are no symmetry conditions which may be observed in this representation (2nd column in Fig. \[Figure\_6\_15\]). However, for the set of magnetization curves as a function of the transport current, $M_{y}(I_{tr})$, we have noticed a well-defined symmetrization of the magnetization loops regarding to the positive and negative values of $I_{tr}(t)$ (*left/right*). Notwithstanding, in terms of the magnetization curves $M_{y}(B_{y})$, there is also a further fact to be mentioned. Strikingly, by using this representation it is possible to note that the steady-states where the consumption of the magnetization currents becomes evident, are mainly present along the period in which the time derivative of $B_{0,y}(t)$ is negative, whilst the AC transport current evolves through the current’s peaks, i.e., for the temporal branches defined by the time-steps (1) $[B_{\tt{a}},I_{\tt{a}}/2]$ to (3) $[-B_{\tt{a}},I_{\tt{a}}/2]$, and (5) $[B_{\tt{a}},-I_{\tt{a}}/2]$ to (7) $[-B_{\tt{a}},-I_{\tt{a}}/2]$. Moreover, the magnetic response of the superconductor is not monotonic along these branches. For example, from the time step (5) until the time step (6) $[0,-I_{\tt{a}}]$, the electromagnetic excitations, $B_{0,y}$ and $I_{tr}$, have the same tendency (both decreasing). However, if $I_{\tt{a}}$ tends to the limiting value $I_{c}$, $M_{y}$ may increase and decrease within the same period. Then, from the time step (6) to the time step (7) both evolve in opposite directions, but however the magnetic moment always increases along this period. On the other hand, for the following analogous branch in the AC period of magnetic field, say the time-steps (9) to (11), the magnetic moment curve is the same, but the competition between the electromagnetic sources $B_{0,y}$ and $I_{tr}$ is opposite to the aforementioned evolution. Thus, depending on the intensities of the electromagnetic sources, both reduction or enhancement of the hysteretic AC losses may envisaged when there is a difference between the oscillating excitation frequencies. Recall that, in contrast, by assuming only a relative difference in phase, only reductions of the actual AC loss may be foretell.
![\[Figure\_6\_16\] Hysteretic AC losses per cycle for asynchronous sources accordingly to the excitations shown in Fig. \[Figure\_6\_12\](b) “Herein, $L_{asynch}^{(b)}$ : square-solid-lines”, and Fig. \[Figure\_6\_12\](c) “Herein, $L_{asynch}^{(c)}$ : circle-solid-lines”. The results are compared with the curve of losses for synchronous sources, $L_{synch}\equiv
L(B_{\tt{a}},I_{\tt{a}})$ predicted above (Fig. \[Figure\_6\_10\]), and the curves for isolated excitations $L(B_{\tt{a}})$ and $L(I_{\tt{a}})$. The whole set of results is also plotted in linear scale. Units for losses are $(\mu_{0}/4\pi)\omega
R^{2} J_{c}^{2}$.](Figure_6_16.pdf){height="6.5cm" width="13cm"}
##### ( *D.* ) *AC Losses in asynchronous systems*\
In Fig. \[Figure\_6\_16\], the hysteretic AC loss calculated for the experimental configurations conceived in Fig. \[Figure\_6\_12\] \[panes (b) and (c)\] are shown. To be specific, $L_{asynch}^{(b)}$ and $L_{asynch}^{(c)}$, are shown in terms of the amplitude of the applied density of magnetic flux $B_{\tt{a}}$, whilst the different values for $I_{\tt{a}}$ are pointed in terms of the sequence of colors for the DC loss curves depicted in Fig. \[Figure\_6\_10\] (pag. ). Likewise, results are compared with the corresponding curves for the actual AC loss when the synchronous electromagnetic excitations were considered, i.e., $L_{synch}\equiv L(B_{\tt{a}},I_{\tt{a}})$. Outstandingly, for both cases, remarkable variations of the AC loss occur. Thus, with the aim of providing a clearest understanding of the range of variations in the AC loss curve for the configurations abovementioned, and further help the readers in the visualization of the numerical data, in Fig. \[Figure\_6\_17\] we show the percentage relation between the calculated losses for synchronous excitations, $L_{synch}$, and the calculated losses for the asynchronous cases, $L_{asynch}^{(b)}$ and $L_{asynch}^{(c)}$.
![\[Figure\_6\_17\] Percent change between the AC loss for synchronous excitations, $L_{synch}$, and the losses $L_{asynch}^{(b)}$ (at the left-side) and $L_{asynch}^{(c)}$ (at the right-side), for combinations of three different amplitudes $B_{\tt{a}}$ and $I_{\tt{a}}$. ](Figure_6_17.pdf){width="70.00000%"}
Notice that, on the one hand, when the applied magnetic field provides the dominant oscillating period (the impressed AC transport current shows a relative double frequency, Fig. \[Figure\_6\_12\](b)), the resulting comparison between the calculated losses $L_{asynch}^{(b)}$ and $L_{synch}$ for high values of $B_{\tt{a}}$ \[Fig. \[Figure\_6\_17\]\], shows a small but sizable increase of the hysteretic loss as a consequence of the double frequency effect ($\sim4-10\%$ for $B_{\tt{a}}\equiv B_{p}=8$). Then, assuming that $I_{\tt{a}}\equiv I_{c}=1$, we have found that the AC loss reduces as $B_{\tt{a}}$ decreases. However, an outstanding fact is that for the lowest values of $B_{\tt{a}}$ and $I_{\tt{a}}$, a notorious increase of the actual AC losses appears. For example, for $B_{\tt{a}}=B_{p}/2=4$ and $I_{\tt{a}}=I_{c}/2=0.5$, deviation is about $36\%$. Likewise, for $B_{\tt{a}}=B_{p}/4=2$ and $I_{\tt{a}}=I_{c}/4=0.25$, deviation is about $29\%$
On the other hand, when the electromagnetic source with the double frequency is the applied magnetic field \[Fig. \[Figure\_6\_12\](c)\], the resulting AC loss ($L_{asynch}^{(c)}$) for high values of $B_{\tt{a}}$ shows a significant reduction as compared to the predicted losses for synchronous configurations ($\sim95\%$ for $B_{\tt{a}}\equiv B_{p}=8$ and $I_{\tt{a}}\equiv I_{c}/4=0.25$). However, by reducing $B_{\tt{a}}$ a notorious increase of the AC loss may be revealed depending on the value of $I_{\tt{a}}$. In fact, we call readers’ attention about the relative increase of the AC losses as compared to those of the synchronous cases: $\sim14\%$ for ($B_{\tt{a}}=4$ , $I_{\tt{a}}=1$), and $\sim18\%$ for ($B_{\tt{a}}=2$ , $I_{\tt{a}}=1$). Moreover, for those cases with the lower values of $B_{\tt{a}}$, i.e., $B_{\tt{a}}=2$, increases of the AC loss are also found for $I_{\tt{a}}=0.25$ ($\sim29\%$), whilst for the intermediate case \[$B_{\tt{a}}=2$,$I_{\tt{a}}=0.5$\], it shows a reduction of the AC loss of less than 1%. The remarkable point here, is that for asynchronous excitations of $B_{0}$ and $I_{tr}$, reductions of the AC losses can be only asserted if both sources evolve with the same frequency, i.e.: if one is restricted to shifts in phase.
{#section-4 .unnumbered}
{#section-5 .unnumbered}
In this part, we have shown that our general critical state theory for the magnetic response of type-II superconductors in the framework of optimal control variational theory and computational methods for large scale applications may be applied in an extensive number of configurations.
In order to summarize the main physical features extracted from our numerical experiments, the conclusions are presented according to the previous sequence of chapters as follows:
##### ***Chapter 4\
***
General critical state problems have been solved for a wide number of examples within the infinite slab geometry. All of them share a three dimensional configuration for the magnetic field, i.e., ${\bf H}=(H_{x},H_{y},H_{z})$, under various magnetic processes, and different models for the critical current restriction or material law. Thus, we have considered several physical scenarios classified by the ansatz for the flux depinning and cutting processes (basically affecting the critical current thresholds $J_{c\perp}$ and $J_{c\parallel}$), their relative importance (given by $\chi\equiv J_{c\parallel}/J_{c\perp}$), and a *coupling* index $n$ which controls the smoothness of the material law. In summary, the following scenarios have been analyzed:
1. Isotropic solutions, in which the limiting case with $\chi^{2}=1$ and $n=1$ produces states under the 1D constraint $J=J_{c}$.
2. T-state solutions, in which the approximation $~\chi\gg 1$ produces the result $J_{\perp}=J_{c\perp}$, and $J_{\parallel}$ may be arbitrarily high. Our predictions show an excellent agreement with previous analytical results in the literature, and extend the theory to the full range of applied magnetic fields.
3. CT-state solutions in which $\chi\geq 1$ for several cases within the [*rectangular region*]{} given by the threshold conditions $J_{\perp}\leq
J_{c\perp}$ and $J_{\parallel}\leq J_{c\parallel}$ are analyzed. Outstandingly, the appearance of the flux cutting limitation takes place as a sudden corner in the magnetic moment curves in many cases. The corner establishes a criterion for the range of application of T-state models.
4. SDCST solutions, in which the possible coupling between the flux depinning and cutting limitations has been studied through the solution of [*smoothed*]{} DCSM cases. In particular, we consider the effect of rounding the corners of the rectangular region $J_{\perp}\leq J_{c\perp}$ and $J_{\parallel}\leq
J_{c\parallel}$, by the [*superelliptic*]{} region criterion $(J_{\parallel}/
J_{c\parallel})^{2n}+(J_{\perp}/ J_{c\perp})^{2n}\leq 1$ with $1\leq n <
\infty$. It is shown that, under specific conditions (paramagnetic initial state and low perpendicular fields), important differences in the predictions of the magnetic moment behavior are to be expected. The differences in ${\bf M}$ have been related to the behavior of the critical current vector ${\bf J}_c$ around the corner of the rectangular region.
Remarkably, the whole set of physical features linked to the different material laws may be depicted in terms of the magnetization curves for the aforementioned experimental configurations \[see Fig. \[Figure\_4\_2\], pag. \]. The main findings are synthesized in figure .
![\[Figure\_Conc\_4\_1\] Figure II-1: The magnetic moment components of the slab $M_{x}$ (top) and $M_{y}$ (bottom) per unit area as a function of the applied magnetic field $h_{y_{0}}$ for the experimental configurations depicted in Fig. \[Figure\_4\_2\]. By comparison, results for several models in the diamagnetic (left panes) and paramagnetic (right panes) configurations are shown. In terms of our SDCST we display the magnetization curves for: the infinite bandwidth model or so called model of T-states ($\chi^{2}\rightarrow\infty$, $J_{c\perp}\neq0$), the conventional DCSM ($\chi^{2}=1$, $n\rightarrow\infty$), several material laws defined by the superelliptical regions $\chi^{2}=1$ and $n=2$, 3, 4, 5, 6 and 10, and finally the isotropic model or superelliptical region with $\chi^{2}=1$ and n=1. ](Figure_Conc_4_1.pdf){width="100.00000%"}
Firstly, we have noticed a pronounced peak effect in both components of the magnetic moment. We emphasize that whatever region is considered \[excepting the limiting cases “$\chi^{2}=1$, $n=1$” (isotropic model), and “$\chi^{2}\to\infty$” (T- or infinite bandwidth- model)\], the peak effect in the paramagnetic case is predicted for both components of the magnetization. Thus, we argue that the peak effect cannot be interpreted as a direct evidence of an elliptical material law. Instead of this, it is a universal signal of the anisotropy effects involved in a general description of the material law. The evolution of the peak effect as a function of $\chi^{2}$ has been shown in Fig. \[Figure\_4\_15\] (pag. ). There, we note that an increase of the bandwidth $\chi^2$ produces a stretched magnetic peak. Consequently, paramagnetic effects are visible over a wider range as the cutting threshold value $J_{c \parallel}$ increases. We also emphasize that the overall effect of increasing the value $\chi^{2}=(J_{c \parallel}/J_{c \perp})^{2}$ is that the components of ${\bf M}$ get closer to the [*master*]{} curves defined by $\chi\to\infty$.
Secondly, some additional and distinctive signals for the different models have been also observed. On the one hand, for the isotropic model, the collapse of the magnetization is achieved while $J_{\parallel}$ is monotonically reduced \[Figs. \[Figure\_4\_3\] – \[Figure\_4\_5\], pags. – \]. When the material law is the infinite bandwidth model ($\chi^{2}=\infty$, or so called T-state model) the magnetization collapse does not occur, and there is no restriction on the longitudinal component of the current that increases arbitrarily towards the center of the sample \[Figs. \[Figure\_4\_6\] – \[Figure\_4\_10\], pags. – \]. This corresponds to the absence of flux cutting, i.e.: $J_{\parallel}$ does not saturate by reaching a threshold value $J_{c \parallel}$. For rectangular or [*smooth rectangular*]{} regions \[Figs. \[Figure\_4\_11\] – \[Figure\_4\_15\], pags. – \], together with the absence of collapse, one also observes that $J_{\parallel}$ basically saturates to a value that depends on the smoothing parameter $n$ (exactly to $J_{c\parallel}$ for the very rectangular case $n\to\infty$). Remarkably, when a rectangular section is assumed, the sample globally reaches the CT state (corner of the rectangle). As a consequence of the sharp limitation for $J_{\parallel}$, a well-defined corner in the magnetic moment components $M_{x}$ and $M_{y}$ appears, both for the diamagnetic and paramagnetic cases (see e.g., Fig. ). This clear trace of the DCSM establishes the departure from the master curves defined by the T-state, and has been assigned to the instant at which the sample reaches the CT state.
Let us call the readers’ attention about a noticeable gap in Fig. , separating the isotropic model ($\chi^{2}=1,~n=1$) and the square model ($\chi^{2}=1,~n\to\infty$). In fact, if one compares Fig. and Fig. \[Figure\_4\_15\] (pag. ) one can realize that smooth models for a given ratio $\chi\equiv J_{c\parallel}/J_{c\perp}$ will fill the gap between the master limiting curves defined by the rectangular ($\chi,n\to\infty$) and elliptic ($\chi,n=1$) models, and their corresponding curves for different values of $\chi$ will intersect in a complicated fashion. In this sense, we argue that the magnetization curves by themselves do not provide unambiguous information on the material law which defines the critical state dynamics in type II superconductors. Moreover, notice that in the regime of low fields $H_{z}\sim h_{y}(a)$ the material law is indistinguishable and the magnetic moment may be reproduced even by the isotropic model. However, we have noticed that, although the dynamics of the profiles $H_{x}$, $J_{y}$, and $J_{x}$ is almost indistinguishable between the smooth and rectangular models a clear distinction arises by analyzing $J_{c\parallel}$. On the one hand, when the rectangular model is assumed $J_{\parallel}$ reaches the threshold value $J_{c\parallel}$, and the entire specimen verifies a CT-state as the applied magnetic field increases. On the other hand, when the rectangular region is smoothed by the index $n$, the parallel component of the current density eventually decreases to a value that depends on the values of $n$ and $\chi$. Thus, further research along this line is suggested [@P2-Clem_2011_PRB], i.e.: the design of some experimental routine that defines a well posed inverse problem for the determination of $\Delta_{\textbf{r}}$.
Finally, in Appendix-I the critical angle (between vortices) criterion that establishes the limitation on $J_{\parallel}$ has been modified for 3D problems. It is shown that, in general, the concept may involve both $J_{\parallel}$ and $J_{\perp}$ as one can see in Eqs. (\[Eq.4.12\]) & (\[Eq.4.15\]). Nevertheless, the influence of the local magnetic anisotropy and the underlying effects at the flux cutting mechanism are much less noticeable, especially for the diamagnetic case, in which the full range of physically meaning values of $\kappa_{c}$ produce a negligible variation.
##### ***Chapter 5\
***
Despite of extensive experimental and theoretical studies about the electrodynamic response of type-II superconductors in longitudinal geometries, much uncertainties remain about the interaction between flux depinning and cutting mechanisms, and their influence in such striking observations as the appearance of negative transport current flow, the enhancement of the critical transport current density, and the observation of peak effects on the magnetization curves. In this chapter, and based on the application of our SDCST, we have reproduced theoretically the existence of negative flow domains, local and global paramagnetic structures, emergence of peak-like structures in the longitudinal magnetic moment, as well as the compression of the transport current density for a wide number of experimental conditions.
Here, the longitudinal transport problem in superconducting slab geometry has been studied as follows: on the one hand, we have considered a superconducting slab lying at the $xy$ plane and subjected to a transport current density along the $y$ direction as it is shown in the left pane of Fig. \[Figure\_5\_3\] (pag. ). The slab is assumed to be penetrated by a uniform vortex array along the $z$ direction, so that the local current density along the thickness of the sample is entirely governed by the depinning component ($J_{\perp}$) perpendicular to the local magnetic field. Subsequently, a magnetic field source parallel to the transport current direction is switched on. Then, the experimental conditions have been changed through the value of the external magnetic field $H_{y0}$.
The dynamical behavior of the transport current density is shown to rely on the interaction between the cutting and depinning mechanisms. Moreover, the intensity of the inherent effects has been shown to depend on the perpendicular component $H_{z0}$, being more prominent as this quantity is reduced. In fact, for restricted situations (infinite slab geometry and only fields parallel to the surface, $H_{z0}=0$), we have shown that the prediction of the counterintuitive effect of negative current flow in type-II superconductors may be even predicted within a simplified analytical model (section 5.1.1). Then, the three-dimensional effects are straightforwardly incorporated by numerical methods when the third component of the local magnetic field ($H_{z}$) is considered.
By means of our SDCST that allows to modulate the influence of the different physical events, by using a *superelliptical* material law that depends on two parameters ($\chi\equiv J_{c\|}/J_{c\perp}$ and $n$) accounting respectively for the intrinsic material anisotropy and for the smoothness of the $J_{\parallel}(J_{\perp})$ law, we have quantitatively investigated the influence of the flux cutting mechanism and shown that the peak structures observed in the magnetization curves and the patterns of the transport current along the central section of a superconducting sample are both directly associated with the local structure of the vortex lattice. Such dependence may become more pronounced as the extrinsic pinning of the material is reduced, in favor of the flux cutting interactions. The same conclusion was pointed out from the experimental measurements of Blamire et.al. (Ref. [@P2-Blamire_2003; @P2-Blamire_1986]) for high critical temperature and conventional superconductors. It has been done by comparing the T-state model ($\chi\to\infty$), and the smooth double critical state conditions CT$\chi$ with $\chi=1$, 2, 3 and 4, all of them with the [*smoothing*]{} index $n=4$ and $J_{c\perp}=1$. Going into detail, when the cutting threshold is high ($J_{c\|}\gg J_{c\perp}$ or $\chi \gg 1$) the emergence of negative current patterns is ensured because unbounded parallel current density allows unconstrained rotations for the flux lines as the longitudinal magnetic field increases. Thus, under a range of conditions, the peak effects in the magnetic moment and a modulation of the negative surface currents have been predicted.
Only for completeness, it is to be mentioned that from our theoretical framework we have obtained that the isotropic model (circular region: $\chi=1$, $n=1$) does neither predict the appearance of negative current patterns nor the peak effects in the magnetic moment curves. However, as long as a clear distinction between the depinning and the cutting components of $\bf J$ is allowed (by letting $n > 1$), several remarkable facts can be explained.
In order to understand the different consequences and physical phenomena derived from the SDCST for the longitudinal transport problem, our numerical results may be summarized as follows:
1. For the magnetic process under consideration \[Fig. \[Figure\_5\_3\], pag. \], and concentrating on the local properties within the sample \[see e.g., Fig. \[Figure\_5\_4\], pag. \], a clear independence of the field and current density profiles relative to the anisotropy level of the material law has been obtained for the [*partial penetration regime*]{}, in which the flux free core progressively shrinks to zero \[top pane of Figs. \[Figure\_5\_7\] – \[Figure\_5\_10\], pags. – \].
2. Negative values for the transport current density $j_{y}$ are neither obtained for the T or CT$\chi$ states when $h_{z0}$ is high ($h_{z0}\gtrsim 50$) until extreme values of the longitudinal field ($h_{y0}\gtrsim 500$) are reached \[see e.g., $j_{y}(a)$ profiles in Figs. \[Figure\_5\_5\] (pag. ), \[Figure\_5\_11\] (pag. ) & \[Figure\_5\_12\] (pag. )\]. On the contrary, one can early find negative current flow for both cases when $h_{z0}\leq2$. Notice by Eq. (\[Eq.5.1\]), pag. , that the reduction of the perpendicular component of the magnetic field may be understood as an enhancement of the cutting current component. Thus, the negative values of ${j}_{y}(z)$ are obtained for smaller and smaller $h_{y0}$ as $h_{z0}$ also decreases. In fact, negative values can happen even for the partial penetration regime ($h_{y0}\lesssim 0.845$) when $h_{z0}$ tends to $0$, in accordance with the analytical model presented before (Section 5.1.1).
3. If $j_{\parallel}$ is unbounded (T states) the ${j}_{y}(z)$ structure becomes rather inhomogeneous as $h_{y0}$ increases and takes the form of a highly positive layer in the center [*shielded*]{} by a prominent negative region, i.e., the transport current is essentially *compressed* toward the center of the sample by the effect of the shielding currents. Also, it is worth of mention that, when simulating experiments in which the transport current is applied subsequent to the field, the SDCST does not predict negative flow values at all. On the contrary, in such cases, what one gets is a [*compression*]{} of the original field penetration profile, until the increasing transport current leads to dissipation.
4. When $j_{\parallel}$ is bounded (CT states) one observes a negative layer at the surface that eventually disappears when $h_{y0}$ increases more and more \[e.g., $h_{y0}>100$ for CT1 case - see bottom pane of Fig. \[Figure\_5\_11\] (pag. )\]. Thus, as a general rule, the smaller the value of $h_{z0}$, the sooner the negative transport current is found. In the CT cases, this also increases the range of longitudinal field for which negative values are observed.
5. The peaked structure of $j_{y}(z)$ for the T-states at $h_{z0}=0.5$ is accompanied by a similar behavior in $j_{x}(z)$ that relates to a subtle magnetic field reentry phenomenon in $h_{y}(z)$ \[see curves labeled $h_{y0}=10$ in the bottom pane of Fig. \[Figure\_5\_4\] (pag. )\]. For the corresponding CT$\chi$ cases, the occurrence of this phenomena is linked to the choice of widthbands larger than $\chi=1$ \[by comparison, see profiles for the labeled first stage at the bottom of Fig. \[Figure\_5\_8\] (pag. )\].
6. Unlimited growth of the global magnetic moment component $M_{x}$ as a function of the longitudinal magnetic field $h_{y0}$ occurs for the T-states in which $j_{\parallel}$ is unbounded. From the local point of view, this relates to an unlimited growth (*compression*) of the current density at the center of the slab \[$j_{y}(0)$\]. Remarkably, the appearance of a peak structure in $M_{x}(h_{y0})$ correlates with the maximum value of the transport current density at the center of the slab for the bounded CT states. For example, for the CT1 case (square with $\chi=1$ and a smoothed corner by $n=4$) the obtained maximum value $j_{y}^{max}(0)= 1.2968$ corresponds to the optimal orientation of the region $\Delta_{\textbf{r}}$ in which the biggest distance within the superelliptic hypothesis is reached. Such situation is sketched in Fig. \[Figure\_4\_1\] (pag. ) and one may check the numeric result from the expression $$\begin{aligned}
{\rm Max}\{j_{c\parallel}(\Delta_{\textbf{r}})\}=j_{y}^{max}=
\left( 1+\chi^{2n/(n-1)} \right)^{(n-1)/2n} \, .\nonumber\end{aligned}$$ Notice that, as a limiting case, it produces the expected value $2^{1/2}$ for the diagonal of a perfect square \[i.e., $n\to\infty$ in Eq. (\[Eq.2.17\]), pag. \].
{height="7.0cm" width="8.5cm"}
Additional physical considerations can be done so as to cover the full experimental scenario for the longitudinal transport problems. In particular, although our analysis has been performed within the infinite slab geometry, one can straightforwardly argue about the extrapolation to real experiments by means the inclusion of the third component of the magnetic field, $H_{z0}$, which qualitatively may be related to the importance of the [*finite size effects*]{} in real superconducting samples. Notice that our numerical calculations, imply that negative currents should be more prominent in those regions of the sample where the component of ${\bf H}$ perpendicular to the current density layers is less important. Thus, considering that a real sample in a longitudinal configuration will be typically a rod with field and transport current along the axis, the above idea is straightforwardly shown by plotting the penetration of an axial field in a finite cylinder (see Fig. ). Then, the aforementioned effect will occur at the central region of the sample, where end effects are minimal.
Just for visual purposes, Fig. shows the distortion of the magnetic field, shielded by the induced supercurrents in a finite superconducting cylinder, where the horizontal component of the magnetic field along the lateral surface layer has been outlined. It is apparent that the normal component of ${\bf H}$ will be enhanced close to the bases and tend to zero at the central region. Then, inhomogeneous surface current densities with negative flow at the mid part should be expected in agreement with the experimental evidences reported in Refs. [@P2-Voloshin_2001; @P2-LeBlanc_2003; @P2-LeBlanc_2002; @P2-Matsushita_1998; @P2-LeBlanc_1993; @P2-LeBlanc_1991; @P2-Voloshin_1991; @P2-Matsushita_1984; @P2-Cave_1978; @P2-Walmsley_1977; @P2-Esaki_1976; @P2-Walmsley_1972; @P2-London_1968; @P2-LeBlanc_1966].
##### ***Chapter 6\
***
In this chapter, we have presented a thorough study of the local and global electromagnetic response of a straight, infinite, cylindrical type-II superconducting wire subject to diverse AC-configurations of transverse magnetic flux density $B_{0}(t)$ and/or longitudinal transport current flow $I_{tr}(t)$. We have assumed that the superconductor follows the celebrated critical state model with a constant threshold for the critical current density $J_{c}$, such that $|I_{tr}|\leq |J_{c}| \pi R^{2}$. The problem is posed over a mesh of virtual filamentary wires each carrying a current $I_{i}$ across a surface $s_{i}$, filling up the whole cross section of the superconducting wire whose area is defined by $\Omega=\pi
R^{2}$.
After a brief theoretical review that concentrates on the physical nature of the different contributions to the AC response (Sec. \[ch-6-1\].1), we have performed extensive numerical calculations for several amplitudes of the impressed transport current, $I_{\tt{a}}$, as well as the amplitude of the magnetic flux density associated to the external excitation source, $B_{\tt{a}}$, for three different regimes of excitation:
(*i*) Isolated electromagnetic sources, Fig. \[Figure\_6\_1\] (pag. ) and Fig. \[Figure\_6\_2\] (pag. ).
(*ii*) Synchronous electromagnetic sources, Fig. \[Figure\_6\_4\] (pag. ) and Fig. \[Figure\_6\_9\] (pag. ).
(*iii*) Asynchronous electromagnetic sources, Fig. \[Figure\_6\_12\] (pag. ).
For each of the above cases, and in order to understand the influence of the electromagnetic excitations involved in the macroscopical physical processes found in this kind of systems, we have presented a detailed study of the local dynamics of the distribution of screening currents, $I_{i}=\pm J_{c} s_{i}$ or $0$, as well as the related local density of power dissipation $\textbf{E}\cdot\textbf{J}$ along a cyclic oscillating period. Likewise, for a wide number of experiments, we have presented the full behavior of the magnetic flux density vector $\textbf{B}$. Many of these results are summarized in the section of supplementary material at the end of this chapter. On the other hand, for a closer connection with the most common experimental observables, we have calculated the wire’s magnetic moment $\textbf{M}$ and the hysteretic losses $L$ as a function of AC external excitations, and their comparison with classical analytical approaches has been featured. Our main conclusions are summarized below.
##### ( *I.* ) *Isolated excitations.*\
Section \[ch-6-2\].2 is devoted to unveil the physics behind the simplest configurations, where the superconducting wire under zero field cooling is subjected to isolated external excitations. On the one hand, for cases with pure AC transport current, we have shown that our numerical method achieves an exact comparison with the well known analytical solution for the AC hysteretic loss (see Fig. \[Figure\_6\_1\], pag. ). Moreover, local magnetization effects have been revealed from integration of the magnetic moments of the screening currents in half cross section of the SC wire. Thus, although the global condition $\textbf{M}(\Omega)=0$ occurs, this does not imply the absence of hysteretic losses, which are in fact produced by the reassembling of the distribution of screening currents in the transient between two consecutive steady states, and therefore of local variation of the induced magnetic flux density in those active zones where the screening currents appear.
On the other hand, when the SC wire is under the action of an external magnetic flux density applied perpendicular to its surface, $B_{0,y}$, and $I_{tr}(t)=0$ (Fig. \[Figure\_6\_2\], pag. ), an exact analytical solution for the dynamics of the flux front profiles is not known. Nevertheless, we have shown that the analytical approach provided by Gurevich for monotonic losses fits well to our numerical calculations (Fig. \[Figure\_6\_3\], pag. ), if one assumes that the dependence of the hysteretic loss density for the periods $\pm B_{\tt{a}}\rightarrow0$ and $0\rightarrow \mp
B_{\tt{a}}$ is the same as the loss calculated for the first monotonic branch, i.e., the excitation branch before the cyclic peak to peak process. Then, the AC loss may be calculated by introducing a factor of four.
Regarding the magnetization curves, we have observed that for the first branch of the oscillating electromagnetic excitation $(0<B_{0,y}(t)\leq
B_{\tt{a}})$, or so called monotonic branch, the saturation point of the magnetic moment $M_{p}$ may be straightforwardly identified at the value of full penetration field $B_{p}$. On the contrary, within the cyclic stage, if $B_{\tt{a}}\geq B_{p}$, the magnetic moment saturates at different values of the magnetic field satisfying the dimensionless empirical relation $B_{p^{\dag}}=\mp(2 B_{p}- B_{\tt{a}} \mp B_{0}(t')-1/2)$. The simultaneous choice of both signs have to be made according to the sign’s rule in the time derivative of the cyclic excitation, i.e., for each half cycle of amplitude $B_{\tt{a}}$. Considerations of premagnetized samples are allowed by $B_{0}(t')\neq0$. Remarkably, the set of magnetization loops displayed in Fig. \[Figure\_6\_2\] (pag. ) serves as a map for drawing any magnetization loop for dealing with arbitrary relations between the experimental parameters $B_{0}(t')$ and $B_{\tt{a}}$
##### *Nature of the hysteretic losses, and their formulations.*\
As a conclusion of our analysis of the basic configurations where the cylindrical SC wire is only subjected to isolated electromagnetic excitations, $B_{0,y}$ or $I_{tr}$, we have compared our results to the hysteretic AC loss obtained by different methods. Thus, we have noticed the following remarkable aspects:
\(1) Despite the fact that an analytical solution for Eq. (\[Eq.6.8\]) can only be achieved for those cases with only transport current flow, the actual AC loss when the isolated electromagnetic excitation corresponds to an external magnetic field applied perpendicular to the superconducting surface, may be straightforwardly evaluated by calculating the enclosed area by the magnetization loop between the steady-peaks of the electromagnetic excitation, as a function of the density of magnetic flux provided by the external source \[Eq. (\[Eq.6.16\]), pag. \].
\(2) The expression of the AC losses provided by the inductive component of the vector potential may be conveniently rewritten when simultaneous occurrence of injected current lines or screening currents accomplishing transport current condition $I_{tr}(t)\neq0$, and the so-called magnetization currents, is given by \[Eq. (\[Eq.6.19\]), pag. \]. Thus, when the superconducting wire is subjected to both electromagnetic excitations, $B_{0}(t)$ and $I_{tr}(t)$, significant reductions (or increases) of the hysteretic losses may be envisaged by reducing (or increasing) the magnitude of the local density of magnetic flux $\textbf{B}=\textbf{B}_{0}+\textbf{B}_{ind}$.
\(3) Likewise, an alternative approach for calculating the AC hysteretic loss per closed cycles has been derived from the principle of conservation of energy and the definition of the Poynting’s vector, such that for cylindrical superconducting wires the AC hysteretic loss per unit length can be reliably calculated by \[Eq. (\[Eq.6.22\]), pag. \].
##### ( *II.* ) *Synchronous excitations.*\
In section 6.3.1 we have presented a detailed study of the physical features associated to the local electrodynamics of superconducting wires, subjected to the simultaneous action of oscillating synchronous excitations $B_{0,y}(t)$ and $I_{tr}(t)$, in the geometrical conditions above considered. Thus, a wide number of experiments based upon the combination of different amplitudes for the density of magnetic flux related to the external field source, $B_{\tt{a}}$, and the peak intensity of the impressed transport current, $I_{\tt{a}}$, have been analyzed for situations in which the wire is in the virgin state \[Fig. \[Figure\_6\_4\], pag. \], or it has been premagnetized \[Fig. \[Figure\_6\_9\], pag. \] before switching on the cyclic synchronous excitation $(B_{0,y},I_{tr})$. Our main remarks concerning the underlying physics of these systems are detailed below.
\(1) The local distribution of screening currents when the simultaneous action of the electromagnetic excitations $B_{0,y}$ and $I_{tr}$ is conceived, may be described as the consumption of the magnetization currents (screening currents induced by the external magnetic field) by effect of the occurrence of injected current lines (constrained by the local condition $\sum_{i}I_{i}=I_{tr}$). As a result of this, the flux front profile is displaced from the geometrical center of the wire towards one of the sides by a kind of “Lorentz force” effect on the injected current lines \[see e.g., Figs. \[Figure\_6\_5\] - \[Figure\_6\_6\] (pag. -), and their corresponding supplementary material, Figs. S5 (pag. ) and S8 (pag. )\].
\(2) For low magnetic fields ($B_{\tt{a}}\leq2$, in units of the penetration field), the distribution of screening currents is *current-like* as long as the peak intensity of the transport current is so high ($I_{\tt{a}}\geq0.5$, in units of the critical current) that the “magnetization lines” in counter direction to the injected current lines may be neglected (Fig. \[Figure\_6\_5\]). Likewise, for higher values of $B_{\tt{a}}$ and lower values of $I_{\tt{a}}$, *field-like* profiles may be envisaged (Fig. \[Figure\_6\_6\]). Nevertheless, for most of the possible combinations between $B_{\tt{a}}$ and $I_{\tt{a}}$, nonsymmetric distributions of the screening currents in a *nothing-like* fashion are predicted \[see also Figs. S5 (pag. ), S8 (pag. ), and S11 (pag. )\].
\(3) Remarkable distortions of the magnetic flux density outside of the superconducting wire are mainly observed at the instants when the synchronous excitation tends to zero in the AC excitation.
\(4) The maximal density of magnetic flux occurs in the side opposite to the location of the flux free core. Thus, recalling Eq. (\[Eq.6.19\]), the strongest localization of the density of magnetic flux in one side of the active zone of the material produces a remarkable localization of the local hysteretic losses in such manner that the heat release from the superconducting wire is highly localized too \[see e.g., Fig. \[Figure\_6\_7\] (pag. ) or Figs. S14-S16 (pags. -)\].
\(5) Having in mind that the local density of magnetic flux may involve the concomitant response of both the magnetization currents and the injected current lines, the *B-oriented* approach for the calculation of AC losses (pag. ) allows to understand why it is not possible to discriminate the magnetization loss from the transport loss by using electromagnetic measurements [@P2-Ogawa_2011; @P2-Inada_2005; @P2-Jiang_2004; @P2-Rabbers_2001; @P2-Reuver_1985]. On the other hand, the *S-oriented* approach (pag. ) justifies that calorimetric methods which directly measure the release flux of energy over the superconducting surface \[Eq. (\[Eq.6.21\])\] have been found crucial for the determining of the actual AC loss when simultaneous electromagnetic excitations act on the superconductor [@P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Ramos_2010; @P2-Ashworth_2000; @P2-Magnusson_2000].
\(6) Regarding the magnetic moment curves \[Fig. \[Figure\_6\_8\], pag. \], we argue that only for small values of $I_{\tt{a}}$, Bean-like loops are expected. However, as $I_{\tt{a}}$ increases, the derived effect by the consumption of the magnetization currents is most prominent, ending up with the symmetrization of the magnetization loop for cyclic periods as function of $B_{0}$ or $I_{tr}$ into striking lenticular shapes. As a consequence of this process, a distinct low-pass filtering effect comes to the fore which, in the case of the triangular input excitations with $I_{\tt{a}}=I_{c}$, yields a nearly perfect sinusoidal (first-harmonic) output signal $M_{y}(t)$.
\(7) If the superconducting wire has been magnetized before switching on the synchronous AC excitation ($\textbf{B}_{0},I_{tr}$), say at $t=t'$ \[see Fig. \[Figure\_6\_9\] (pag. )\], the center of the magnetization loop drifts from $\textbf{M}(0,0)$ towards $\textbf{M}(B_{0}(t'),0)$, such that the corners of the magnetization loop $\textbf{M}(\pm B_{\tt{a}},\pm I_{\tt{a}})$ lie on the excitation coordinates $(B_{0}(t') \pm B_{\tt{a}},\pm I_{\tt{a}})$. Thus, as the area enclosed by the magnetization loop remains the same, the power losses attained along the premagnetization process plays no role in the calculation of the hysteretic AC loss. Nevertheless, in these cases, the profiles drawn for the screening currents have revealed highly intricate patterns regarding to the coexistence of the magnetization currents and the injected current lines, as well as defining the flux front profile \[Fig. S11 (pag. )\]
\(8) Our straightforward calculation of the actual hysteretic losses by means the general definition $\delta L=\int_{\Phi}\textbf{E}\cdot\textbf{J}dr$, reveals important differences concerning the approximate formulae customarily used. In fact, we have shown that the actual AC losses are always higher than those envisaged by the linear superposition of contributions due to either type of AC excitation \[see Fig. \[Figure\_6\_10\], pag. \]. Comparisons reveal the important fact that the linear approach $L(B_{\tt{a}})+L(I_{\tt{a}})$ is only appropriate for high strengths of the magnetic flux density and low transport currents (screening currents with distribution *field-like*), as well as for low magnetic field and high transport current (screening currents with distribution *current like*). Likewise, for high amplitudes of the magnetic flux density ($B_{\tt{a}}\geq B_{p}$) and for any value of the transport current, significant reductions of the actual hysteretic loss may be envisaged if one compares it with the predicted losses for a wire carrying a DC current instead to the AC case \[Fig. \[Figure\_6\_11\], pag. \].
##### ( *III.* ) *Asynchronous excitations.*\
Our detailed analysis of the underlying physics behind the local and global electromagnetic response of superconducting wires subjected to isolated electromagnetic excitations $B_{0}(t)$ or $I_{tr}(t)$ in oscillating regimes, and then of the synchronous action of both, have revealed that the total AC loss may be controlled by locally reducing the total magnetic flux density resulting from the addition of the external magnetic field and that induced by the concomitant occurrence of magnetization currents and the injected current lines. Thus, either by displacing in time the electromagnetic excitations (phase shift), by introducing changes in frequency, or simply by considering excitation branches with time derivatives in counter directions, one can help to counterbalance the local density of magnetic flux in the zone of maximum heat release, which is further translated to the reduction of the hysteretic losses.
Thus, as long as both excitations evolve with the same oscillating frequency, it is possible to assert that at least a minimal reduction of the AC loss should appear for relative phase changes between the electromagnetic excitations. This is in agreement with the experimental and numerical evidences reported in Refs. [@P2-Nguyen_2005a; @P2-Nguyen_2005b; @P2-Vojenciak_2006]. For example, by the simple overlapping between the excitation curves of the electromagnetic sources, it is evident that the maximal losses are envisaged for the synchronous cases, as well as for those cases with a relative change in phase of half excitation period. Likewise, minimal losses are attained around a phase shifting of a quarter of period, although its exact position as a function of the electromagnetic excitations straightforwardly depends on the entangled competition between the external magnetic field and the induced fields by the injected current lines and the so called magnetization currents. Thus, unless the distribution of screening currents show patterns of the kind *current-like* or *field-like*, is not possible assert that the lower AC loss is given for a change of phase of a quarter of period.
Within the above scenario, we have argued that predictions are not straightforward for non-trivial time dependencies of the simultaneous electromagnetic excitations and, higher hysteretic AC losses could be also expected. Along this line, we have introduced a thorough study of the so-called double frequency effects, which arise when one of the isolated electromagnetic sources, $B_{0}(t)$ or $I_{tr}(t)$, is connected to a power supply with a double oscillating frequency \[see Fig. \[Figure\_6\_12\], pag. \]. The most outstanding observations about the local and global behavior of the involved electromagnetic quantities are detailed below:
\(1) For asynchronous excitations, the distribution of screening currents in the first peak of the dominant excitation (i.e., that with a longer period), may be strongly different as compared to the attained distributions for the subsequent excitation peaks \[Fig. S17, pag. \]. Thus, the first peak of excitation cannot be considered as a steady-state for integrating the AC loss when the integral is reduced to half period of excitation. In other words, the knowledge of the distribution of screening currents in the time elapsed for the second half-period of the dominant excitation is relevant. The latter fact has been experimentally recognized in Ref. [@P2-Nguyen_2005a].
\(2) Whatever electromagnetic excitation carries the double frequency, we have found that, in the AC regime, complex arrays of domains connected by boundary lines with currents switching between $I_{c}$ and $-I_{c}$ appear \[see Fig. \[Figure\_6\_13\] (pag. ) or Fig. S17 (pag. )\].
\(3) Contrary to the strong localization of the local density of power losses observed in cases with synchronous excitations, for asynchronous excitations the active zone with higher heat release is no longer focused at only one side of the superconducting wire \[see Fig. \[Figure\_6\_13\] (pag. ) or Fig. S20 (pag. )\].
\(4) As far as concerns to the magnetic response of the superconducting wire when double frequency effects are incorporated, *exotic* magnetization loops are predicted \[see Fig. \[Figure\_6\_14\] (pag. ) and Fig. \[Figure\_6\_15\] (pag. )\]. Outstandingly, for low amplitudes of the transport current, $I_{\tt{a}}$, Bean-like loops may be observed for any of the above mentioned cases. However, as $I_{\tt{a}}$ increases, strongest differences between the magnetization loops arise, and the global behavior is radically different to the synchronous cases.
\(5) A comprehensive analysis of the magnetization curves as a function of the temporal evolution of the electromagnetic excitations has been carried out. Thus, several symmetry conditions for the magnetization loop, when the electromagnetic excitation with double frequency is either $I_{tr}$ or $B_{0}$, concerning to the amplitudes $I_{\tt{a}}$ and $B_{\tt{a}}$ have been illustrated.
\(6) In order to attain the low-pass filtering effect in the temporal evolution of the magnetization curve when $I_{\tt{a}}\rightarrow I_{c}$, it becomes absolutely necessary to assure that both oscillating excitations evolve synchronous in time.
\(7) Accordingly to the different setups in panes (b) and (c) of Fig. \[Figure\_6\_12\], we have calculated the hysteretic AC loss for several values of $B_{\tt{a}}$ and $I_{\tt{a}}$, when the dominant excitation is either the applied magnetic field, $L_{asynch}^{(b)}$, or the impressed transport current, $L_{asynch}^{(c)}$ \[Fig. \[Figure\_6\_16\], pag. \]. By comparing them with the hysteretic AC loss predicted for the synchronous cases \[see also Fig. \[Figure\_6\_17\], pag. \], we have shown the AC loss may either increase or decrease by double frequency effects when asynchronous excitations are involved. The relative intensities of the excitations $B_{\tt{a}}$ and $I_{\tt{a}}$ play an additional role on this. Then, the big number of possible combinations makes it imperative to have a previous knowledge of the operational environment of the superconducting wire, for attaining valid predictions of the actual AC losses.
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{#section-6 .unnumbered}
### \[ch-6-s\] {#ch-6-s .unnumbered}
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![\[Figure\_S\_1\] Figure S1: For an oscillating transport current of amplitude $I_{\tt{a}}=I_{c}$, and the temporal steps defined in Fig. \[Figure\_6\_1\](b) (pag. ), we show the dynamics of the magnetic flux lines (projected isolevels of the vector potential), together with the profiles of current $I_{i}$ across the superconducting wire, in the 1st and 2nd column. Next, 3rd and 4th column show the corresponding evolution of the local density of power dissipation. The profiles of current are displayed according to: red ($+I_{i}$), blue ($-I_{i}$), and green (zero). The plotting interval is $\Delta I_{tr}(t)=I_{\tt{a}}/2$, with t=0 defining the virgin state (i.e., $I_{tr}=0$). Units are $\pi R^{2}J_{c}\equiv{I_c}$ for $I_{\rm tr}$, and $(\mu_{0}/4\pi)J_{c}^{2}R^{2}\delta t^{-1}$ for E$\cdotp$J. ](Figure_S_1_a.pdf "fig:"){height="13cm" width="5.8cm"} ![\[Figure\_S\_1\] Figure S1: For an oscillating transport current of amplitude $I_{\tt{a}}=I_{c}$, and the temporal steps defined in Fig. \[Figure\_6\_1\](b) (pag. ), we show the dynamics of the magnetic flux lines (projected isolevels of the vector potential), together with the profiles of current $I_{i}$ across the superconducting wire, in the 1st and 2nd column. Next, 3rd and 4th column show the corresponding evolution of the local density of power dissipation. The profiles of current are displayed according to: red ($+I_{i}$), blue ($-I_{i}$), and green (zero). The plotting interval is $\Delta I_{tr}(t)=I_{\tt{a}}/2$, with t=0 defining the virgin state (i.e., $I_{tr}=0$). Units are $\pi R^{2}J_{c}\equiv{I_c}$ for $I_{\rm tr}$, and $(\mu_{0}/4\pi)J_{c}^{2}R^{2}\delta t^{-1}$ for E$\cdotp$J. ](Figure_S_1_b.pdf "fig:"){height="13cm" width="6.1cm"}
![\[Figure\_S\_2\] Figure S2: Colormaps for the intensity of the components of magnetic flux density $B_{x}$ (left pane) and $B_{y}$ (right pane) for the square section of area $4R^{2}$ enclosing the cylindrical wire of radius $R=1$. The profiles are plotted in according to Fig. S1, and the superconducting surface is depicted by black dashed lines. Units are $(\mu_{0}/4\pi)J_{c}R$ for B. ](Figure_S_2.pdf){height="14cm" width="13cm"}
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![\[Figure\_S\_3\] Figure S3: For an external AC magnetic flux applied along the $y-axis$, with amplitude $B_{\tt{a}}=6$, and for the temporal steps defined in Fig. \[Figure\_6\_2\](b) (pag. ); the 1st and 2nd column show the dynamics of the magnetic flux lines and their corresponding profiles of current $I_{i}$. Next, 3rd and 4th column show the corresponding dynamics of the local density of power dissipation. The profiles of current are displayed according to: red ($+I_{i}$), blue ($-I_{i}$), and green (zero). The plotting interval is $\Delta B_{0,y}=3$, with t=0 defining the virgin state (i.e., $B_{0}=0$). Units are $(\mu_{0}/4\pi)J_{c}R$ for B, and $(\mu_{0}/4\pi)J_{c}^{2}R^{2}\delta t^{-1}$ for E$\cdotp$J. ](Figure_S_3_a.pdf "fig:"){height="13cm" width="5.8cm"} ![\[Figure\_S\_3\] Figure S3: For an external AC magnetic flux applied along the $y-axis$, with amplitude $B_{\tt{a}}=6$, and for the temporal steps defined in Fig. \[Figure\_6\_2\](b) (pag. ); the 1st and 2nd column show the dynamics of the magnetic flux lines and their corresponding profiles of current $I_{i}$. Next, 3rd and 4th column show the corresponding dynamics of the local density of power dissipation. The profiles of current are displayed according to: red ($+I_{i}$), blue ($-I_{i}$), and green (zero). The plotting interval is $\Delta B_{0,y}=3$, with t=0 defining the virgin state (i.e., $B_{0}=0$). Units are $(\mu_{0}/4\pi)J_{c}R$ for B, and $(\mu_{0}/4\pi)J_{c}^{2}R^{2}\delta t^{-1}$ for E$\cdotp$J. ](Figure_S_3_b.pdf "fig:"){height="13cm" width="6.1cm"}
![\[Figure\_S\_4\] Figure S4: Colormaps for the intensity of the components of magnetic flux density $B_{x}$ (left pane) and $B_{y}$ (right pane) for the square section of area $4R^{2}$ enclosing the cylindrical wire of radius $R=1$. The profiles are plotted in according to the Fig. S3. The superconducting surface is depicted by the black dashed lines. Recall that units for B are $(\mu_{0}/4\pi)J_{c}R$. ](Figure_S_4.pdf){height="14cm" width="13cm"}
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![\[Figure\_S\_5\] Figure S5: Evolution of the magnetic flux lines and profiles of current with simultaneous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=2$ (*low field*) and: $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time-step (1) defines the condition $(1,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. By symmetry, only the first half of the AC cycle is shown \[i.e., time-steps (2) to (6)\]. ](Figure_S_5_a.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_5\] Figure S5: Evolution of the magnetic flux lines and profiles of current with simultaneous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=2$ (*low field*) and: $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time-step (1) defines the condition $(1,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. By symmetry, only the first half of the AC cycle is shown \[i.e., time-steps (2) to (6)\]. ](Figure_S_5_b.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_5\] Figure S5: Evolution of the magnetic flux lines and profiles of current with simultaneous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=2$ (*low field*) and: $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time-step (1) defines the condition $(1,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. By symmetry, only the first half of the AC cycle is shown \[i.e., time-steps (2) to (6)\]. ](Figure_S_5_c.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_5\] Figure S5: Evolution of the magnetic flux lines and profiles of current with simultaneous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=2$ (*low field*) and: $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time-step (1) defines the condition $(1,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. By symmetry, only the first half of the AC cycle is shown \[i.e., time-steps (2) to (6)\]. ](Figure_S_5_d.pdf "fig:"){height="15cm" width="3cm"}
![\[Figure\_S\_6\] Figure S6: Colormaps for the evolution of the component of magnetic flux density $B_{x}$, corresponding to the current profiles displayed in Fig. S5. ](Figure_S_6.pdf){height="12.5cm" width="13cm"}
![\[Figure\_S\_7\] Figure S7: Colormaps for the evolution of the component of magnetic flux density $B_{y}$, corresponding to the current profiles displayed in Fig. S5. ](Figure_S_7.pdf){height="12.5cm" width="13cm"}
![\[Figure\_S\_8\] Figure S8: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time step (1) defines the condition $(8,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. ](Figure_S_8_a.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_8\] Figure S8: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time step (1) defines the condition $(8,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. ](Figure_S_8_b.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_8\] Figure S8: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time step (1) defines the condition $(8,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. ](Figure_S_8_c.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_8\] Figure S8: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $(B_{0,y},I_{tr})$ of amplitudes $B_{\tt{a}}=8$ (*high field*) and, $I_{\tt{a}}=0.25$ *(1st column)*, $I_{\tt{a}}=0.5$ *(2nd column)*, $I_{\tt{a}}=0.75$ *(3rd column)*, and $I_{\tt{a}}=1$ *(4th column)*. Plotting interval is $(B_{\tt{a}}/2,I_{\tt{a}}/2)$, where the time step (1) defines the condition $(8,I_{\tt{a}}/2)$ \[see also Fig. \[Figure\_6\_4\] (pag. )\]. ](Figure_S_8_d.pdf "fig:"){height="15cm" width="3cm"}
![\[Figure\_S\_9\] Figure S9: Colormaps for the evolution of the component of magnetic flux density $B_{x}$, corresponding to the current profiles displayed in Fig. S8. ](Figure_S_9.pdf){height="12.5cm" width="13cm"}
![\[Figure\_S\_10\] Figure S10: Colormaps for the evolution of the component of magnetic flux density $B_{y}$, corresponding to the current profiles displayed in Fig. S8. ](Figure_S_10.pdf){height="12.5cm" width="13cm"}
![\[Figure\_S\_11\] Figure S11: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $B_{0,y}$ and $I_{tr}$ of amplitudes $B_{\tt{a}}=4$ (*intermediate field*) and $I_{\tt{a}}=0.5$. Results for two premagnetized samples with $B(t')=2$ (left pane) and $B(t')=8$ (right pane) are shown. Numeric tags in the upper left corner of each subplot have been incorporated according to the following time-steps for the experimental process depicted in Fig. \[Figure\_6\_9\]: (1) corresponds to half of the time between $t=0$ and $t=t'$, then (2) at $t=t'$, and for (2) to (12) increases of $\Delta t\equiv 1/8$ per unit cycle have been considered. Thus, the full cycle peak-to-peak corresponds to the subplots (4) to (12), respectively. ](Figure_S_11_a.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_11\] Figure S11: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $B_{0,y}$ and $I_{tr}$ of amplitudes $B_{\tt{a}}=4$ (*intermediate field*) and $I_{\tt{a}}=0.5$. Results for two premagnetized samples with $B(t')=2$ (left pane) and $B(t')=8$ (right pane) are shown. Numeric tags in the upper left corner of each subplot have been incorporated according to the following time-steps for the experimental process depicted in Fig. \[Figure\_6\_9\]: (1) corresponds to half of the time between $t=0$ and $t=t'$, then (2) at $t=t'$, and for (2) to (12) increases of $\Delta t\equiv 1/8$ per unit cycle have been considered. Thus, the full cycle peak-to-peak corresponds to the subplots (4) to (12), respectively. ](Figure_S_11_b.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_11\] Figure S11: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $B_{0,y}$ and $I_{tr}$ of amplitudes $B_{\tt{a}}=4$ (*intermediate field*) and $I_{\tt{a}}=0.5$. Results for two premagnetized samples with $B(t')=2$ (left pane) and $B(t')=8$ (right pane) are shown. Numeric tags in the upper left corner of each subplot have been incorporated according to the following time-steps for the experimental process depicted in Fig. \[Figure\_6\_9\]: (1) corresponds to half of the time between $t=0$ and $t=t'$, then (2) at $t=t'$, and for (2) to (12) increases of $\Delta t\equiv 1/8$ per unit cycle have been considered. Thus, the full cycle peak-to-peak corresponds to the subplots (4) to (12), respectively. ](Figure_S_11_c.pdf "fig:"){height="15cm" width="3cm"} ![\[Figure\_S\_11\] Figure S11: Evolution of the magnetic flux lines and profiles of current with synchronous oscillating sources $B_{0,y}$ and $I_{tr}$ of amplitudes $B_{\tt{a}}=4$ (*intermediate field*) and $I_{\tt{a}}=0.5$. Results for two premagnetized samples with $B(t')=2$ (left pane) and $B(t')=8$ (right pane) are shown. Numeric tags in the upper left corner of each subplot have been incorporated according to the following time-steps for the experimental process depicted in Fig. \[Figure\_6\_9\]: (1) corresponds to half of the time between $t=0$ and $t=t'$, then (2) at $t=t'$, and for (2) to (12) increases of $\Delta t\equiv 1/8$ per unit cycle have been considered. Thus, the full cycle peak-to-peak corresponds to the subplots (4) to (12), respectively. ](Figure_S_11_d.pdf "fig:"){height="15cm" width="3cm"}
![\[Figure\_S\_12\] Figure S12: Colormaps for the evolution of the components of magnetix flux density $B_{x}$ (left) and $B_{y}$ (right) for the current density profiles displayed at the left pane of Fig. S11. For $B_{y}$, subplots (3) to (6), and (12), the colormap have to be renormalized to a linear scale of limits 10 and -10. ](Figure_S_12.pdf){height="16cm" width="13cm"}
![\[Figure\_S\_13\] Figure S13: Colormaps for the evolution of the components of magnetix flux density $B_{x}$ (left) and $B_{y}$ (right) for the current density profiles displayed at the rigth pane of Fig. S11. ](Figure_S_13.pdf){height="16cm" width="13cm"}
![\[Figure\_S\_14\] Figure S14: Evolution of the local density of power dissipation **E**$\cdotp$**J** for oscillating sources of amplitude $B_{\tt{a}}=2$ and: $I_{\tt{a}}=0.25$ *(left pane)*, and $I_{\tt{a}}=1$ *(right pane)*. Each step has been plotted according to the temporal process depicted in Fig. \[Figure\_6\_4\]. In the right pane, the colormap for subplots (2), (6), and (10) have to be renormalized by a factor of 5. Also, some of the corresponding flux profiles have been displayed in the left and right columns of Figs. S5- S7. ](Figure_S_14_a.pdf "fig:"){height="12cm" width="6cm"} ![\[Figure\_S\_14\] Figure S14: Evolution of the local density of power dissipation **E**$\cdotp$**J** for oscillating sources of amplitude $B_{\tt{a}}=2$ and: $I_{\tt{a}}=0.25$ *(left pane)*, and $I_{\tt{a}}=1$ *(right pane)*. Each step has been plotted according to the temporal process depicted in Fig. \[Figure\_6\_4\]. In the right pane, the colormap for subplots (2), (6), and (10) have to be renormalized by a factor of 5. Also, some of the corresponding flux profiles have been displayed in the left and right columns of Figs. S5- S7. ](Figure_S_14_b.pdf "fig:"){height="12cm" width="6cm"}
![\[Figure\_S\_15\] Figure S15: Evolution of the local density of power dissipation **E**$\cdotp$**J** for oscillating sources of amplitude $B_{\tt{a}}=8$ and, $I_{\tt{a}}=0.25$ *(left pane)*, and $I_{\tt{a}}=1$ *(right pane)*. Each step has been plotted according to the temporal process depicted in Fig. \[Figure\_6\_4\], and some of the corresponding flux profiles have been displayed in the left and right columns of Figs. S8- S10. In the right pane, the colormap for subplots (6) and (10) have to be renormalized by a factor of 2. ](Figure_S_15_a.pdf "fig:"){height="12cm" width="6cm"} ![\[Figure\_S\_15\] Figure S15: Evolution of the local density of power dissipation **E**$\cdotp$**J** for oscillating sources of amplitude $B_{\tt{a}}=8$ and, $I_{\tt{a}}=0.25$ *(left pane)*, and $I_{\tt{a}}=1$ *(right pane)*. Each step has been plotted according to the temporal process depicted in Fig. \[Figure\_6\_4\], and some of the corresponding flux profiles have been displayed in the left and right columns of Figs. S8- S10. In the right pane, the colormap for subplots (6) and (10) have to be renormalized by a factor of 2. ](Figure_S_15_b.pdf "fig:"){height="12cm" width="6cm"}
![\[Figure\_S\_16\] Figure S16: Evolution of the local density of power dissipation **E**$\cdotp$**J** for cases with synchronous sources $(B_{0,y},I_{tr})$ and premagnetized superconducting wires with: $B(t')=2$ (*left pane*) and $B(t')=8$ (*right pane*), in correspondence to the profiles of current density displayed in Fig. S11. ](Figure_S_16_a.pdf "fig:"){height="15cm" width="6cm"} ![\[Figure\_S\_16\] Figure S16: Evolution of the local density of power dissipation **E**$\cdotp$**J** for cases with synchronous sources $(B_{0,y},I_{tr})$ and premagnetized superconducting wires with: $B(t')=2$ (*left pane*) and $B(t')=8$ (*right pane*), in correspondence to the profiles of current density displayed in Fig. S11. ](Figure_S_16_b.pdf "fig:"){height="15cm" width="6cm"}
- ******
![\[Figure\_S\_17\] Figure S17: Evolution of the magnetic flux lines and profiles of current with asynchronous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=4$ and $I_{\tt{a}}=0.5$. In the left pane, some of the results for the temporal process displayed in pane (b) of Fig. \[Figure\_6\_12\] ($B_{0,y}$ has the lower frequency) are shown. Analogously, the results for the pane (c) of Fig. \[Figure\_6\_12\] ($I_{tr}$ has the lower frequency) are shown at the right pane. ](Figure_S_17_a.pdf "fig:"){height="13cm" width="3cm"} ![\[Figure\_S\_17\] Figure S17: Evolution of the magnetic flux lines and profiles of current with asynchronous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=4$ and $I_{\tt{a}}=0.5$. In the left pane, some of the results for the temporal process displayed in pane (b) of Fig. \[Figure\_6\_12\] ($B_{0,y}$ has the lower frequency) are shown. Analogously, the results for the pane (c) of Fig. \[Figure\_6\_12\] ($I_{tr}$ has the lower frequency) are shown at the right pane. ](Figure_S_17_b.pdf "fig:"){height="13cm" width="3cm"} ![\[Figure\_S\_17\] Figure S17: Evolution of the magnetic flux lines and profiles of current with asynchronous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=4$ and $I_{\tt{a}}=0.5$. In the left pane, some of the results for the temporal process displayed in pane (b) of Fig. \[Figure\_6\_12\] ($B_{0,y}$ has the lower frequency) are shown. Analogously, the results for the pane (c) of Fig. \[Figure\_6\_12\] ($I_{tr}$ has the lower frequency) are shown at the right pane. ](Figure_S_17_c.pdf "fig:"){height="13cm" width="3cm"} ![\[Figure\_S\_17\] Figure S17: Evolution of the magnetic flux lines and profiles of current with asynchronous oscillating sources $B_{0,y}$ and $I_{tr}$, of amplitudes $B_{\tt{a}}=4$ and $I_{\tt{a}}=0.5$. In the left pane, some of the results for the temporal process displayed in pane (b) of Fig. \[Figure\_6\_12\] ($B_{0,y}$ has the lower frequency) are shown. Analogously, the results for the pane (c) of Fig. \[Figure\_6\_12\] ($I_{tr}$ has the lower frequency) are shown at the right pane. ](Figure_S_17_d.pdf "fig:"){height="13cm" width="3cm"}
![\[Figure\_S\_18\] Figure S18: Colormaps for the evolution of the components of magnetix flux density $B_{x}$ (left) and $B_{y}$ (right) for the current density profiles displayed at the left pane of Fig. S17. ](Figure_S_18.pdf){height="13.5cm" width="13cm"}
![\[Figure\_S\_19\] Figure S19: Colormaps for the evolution of the components of magnetix flux density $B_{x}$ (left) and $B_{y}$ (right) for the current density profiles displayed at the right pane of Fig. S17. ](Figure_S_19.pdf){height="13.5cm" width="13cm"}
![\[Figure\_S\_20\] Figure S20: Evolution of the local density of power dissipation **E**$\cdotp$**J** for cases with asynchronous AC sources $B_{0,y}$ and $I_{tr}$, according to the current density profiles displayed in Fig. S17. ](Figure_S_20_a.pdf "fig:"){height="12.5cm" width="6cm"} ![\[Figure\_S\_20\] Figure S20: Evolution of the local density of power dissipation **E**$\cdotp$**J** for cases with asynchronous AC sources $B_{0,y}$ and $I_{tr}$, according to the current density profiles displayed in Fig. S17. ](Figure_S_20_b.pdf "fig:"){height="12.5cm" width="6cm"}
{#section-7 .unnumbered}
\[Intro-P2\] **Introduction** {#intro-p2-introduction-1 .unnumbered}
-----------------------------
After one century of the discovery of superconductivity we are still awaiting for a conclusive theory at least beyond normal metals, that are described within the framework of the BCS theory [@P3-BCS57]. However, this does not mean that we lack well-established theories to explain some of the experimental facts, despite many of the thermodynamic properties as the high superconducting transition temperature $(T_{c})$ can not be reproduced under a unique scheme. Interestingly, although that the theoretical background behind the understanding of the microscopical aspects involved in the creation of the superconducting state is rather complex, we have found that relatively simple numerical techniques may be used in order to withdraw relevant conclusions from specific experimental data [@P3-Ruiz2009; @P3-Ruiz2011_JS; @P3-Ruiz2011_CAP].
Our point of interest is the following. In conventional metals, the electron-phonon (E-Ph) coupling mode has long been recognized as the main mechanism involved in the superconducting properties, because the strength of this interaction essentially determines the value of $T_{c}$. However, in the HTSC the experimentally determined $d-$wave pairing in their layered crystal structure with one or more CuO$_{2}$ planes per unit cell [@P3-Bednorz86], introduces considerable theoretical complications even when other coupling modes are considered [@P3-Kulic04]. Fortunately, with the appearance of a new era of analyzers for Angle Resolved Photoemission Spectroscopies (ARPES) with improved resolution both in energy and momentum [@P3-Hufner07], the controversy on the influence of the anisotropic character of the superconducting gap in the electron properties of HTSC can be directly avoided just by analyzing preferential directions within the $CuO_{2}$ planes or the so-called nodal directions. These directions are basically characterized by a negligible contribution of the superconducting gap along the $(0,0)-(\pi,\pi)$ direction in the Brillouin zone, providing a smart solution if one is merely interested in identifying the energy modes of the quasiparticles involved in the superconducting pairs formation, i.e., in the origin of the coupling mode which binds two electrons (holes) in the formation of Cooper pairs.
In order to understand the relevance of the ARPES technique, we recall that the photoemission process results in both an excited photoelectron and a photohole in the final state. On the one hand, unlike other probes, the ARPES technique has the advantage of momentum resolving, which becomes a useful probe of the related scattering mechanisms contributing to the electrical transport in different materials. We want to note that the single-particle scattering rate measured in ARPES is not identical to the scattering rate measured in transport studies themselves. Nonetheless, direct proportionality between them has been established [@P3-Kulic00; @P3-Smith01]. On the other hand, one of the most telling manifestations of the E-Ph mode is a mass renormalization of the electronic dispersion at the energy scale associated with the phonons. This renormalization effect is directly observable in the ARPES measurements as a low-energy excitation band in the dispersion curves of photo-emitted electrons, known as *kink* [@P3-Lanzara01; @P3-Zhou03]. In other words, the kink effect can be understood as a well-defined slope change in the electronic energy-momentum dispersion in a similar energy scale ($E_{k}-E_{F}\sim40 -
80meV$). This feature, so far universal in the HTSC, has been regarded as a signature of the strength of the boson mechanism which causes the pair formation in the superconducting state. In fact, all the interactions of the electrons which are responsible for the unusual normal and superconducting properties of cuprates are believed to be represented in this anomaly [@P3-Zhou07]. This has prompted an intense debate about the nature of the coupling mode involved in the density of low-energy electronic excited states in the momentum-energy space, and its influence on the emergence of the superconducting state [@P3-Ruiz2009; @P3-Ruiz2011_JS; @P3-Ruiz2011_CAP; @P3-Lanzara01; @P3-Zhou03; @P3-Zhou07; @P3-Devereaux04; @P3-Zhou02; @P3-Zhou05; @P3-Kordyuk06; @P3-Xiao07; @P3-Takahashi07; @P3-Gweon04; @P3-Douglas07; @P3-Johnson01; @P3-Borisenko06; @P3-Zhang08; @P3-Graf08; @P3-Reznik08; @P3-Giustino08; @P3-Park08; @P3-Chang08].
In an effort to clarify the influence of the phonon coupling mode (either weak or strong), we have analyzed the influence of the E-Ph interaction on the electronic dispersion relations for several cuprate compounds. A full discussion of the ARPES technique, as well as a detailed description of the strong correlations theory which have been used to reproduce the nodal kink effect in HTSC is more appropriately reserved for specialized texts in photoemission spectroscopies and many body theories \[see e.g., Refs. [@P3-Doniach98; @P3-Gross86; @P3-Allen76; @P3-Allen82; @P3-Carbotte90; @P3-Ashcroft76]\]. However, we can, in a brief way, introduce the basic concepts of the E-Ph coupling theory for photoemission spectroscopies, and give a thorough interpretation of the influence of this boson coupling mode on the momentum distribution-curves (MDC) measured by ARPES (chapter \[ch-7\]), and finally argue about how strong can be considered the phonon coupling mode from the analysis of the predicted values for $T_{c}$, the ratio gap $2\Delta_{0}/k_{B}T_{c}$, and the zero temperature gap $\Delta_{0}$ (chapter \[ch-8\]).
On the one hand, in chapter \[ch-7\] our analysis shows a remarkable agreement between theory and experiment for different samples and at different doping levels. Universal effects such as the nodal kinks at low energies are theoretically reproduced, emphasizing the necessary distinction between the general electron mass-enhancement parameter $\lambda^{*}$and the conventional electron-phonon coupling parameter $\lambda$. On the other hand, in chapter \[ch-8\] a thorough analysis of the superconducting thermodynamic quantities and the Coulomb effects based on different approaches will reveal as, contrary to the predictions for LSCO samples, in more anisotropic materials as Bi2212 and Y123 families, it seems unavoidable to consider additional coupling modes in order to justify their high critical temperatures.
\[ch-7\] **E-Ph Theory And The Nodal Kink Effect In HTSC**
==========================================================
The photoemission process formally measures a complicated nonlinear response function. However, it is helpful to notice that the analysis of the optical excitation of the electron in the bulk greatly simplifies within the “sudden approximation” [@P3-Randeria95; @P3-Hedin02]. It means that the photoemission process is supposed to occur [*suddenly*]{}, with no post-collisional interaction between the photoelectron and the system left behind [@P3-Damascelli03]. In particular, it is assumed that the excited state of the sample (created by the ejection of the photo-electron) does not relax in the time it takes for the photo-electron to reach the detector [@P3-Zhou07]. It can be shown that within the sudden approximation using Fermi’s Golden Rule for the transition rate, the measured photo-current density is basically proportional to the spectral function of the occupied electronic states in the solid, i.e.: $J_{\bf k}\propto A_{\bf k}(E_{k})$. Eventually, and validated by whether or not the spectra can be understood in terms of well defined peaks representing poles in the spectral function, one may connect $A_{\bf k}(E_{k})$ to the quasiparticle Green’s function $G(\textbf{k},E)=1/(E_{k}-\Sigma_{k}(E_{k})-\varepsilon_{k})$, with $\Sigma_{k}(E_{k})$ defining the electronic self-energy and $\varepsilon_{k}$ the bare band dispersion. In fact, customarily the inversion method for the experimental data in ARPES is based upon the so called sudden approximation trough the relation $A(\textbf{k},\omega)=-(1/\pi)$Im$
G(\textbf{k},\omega+i0^{+})$. Beyond the sudden approximation, one would have to take into account the screening of the photoelectron by the rest of the system, and the photoemission process could be described by the generalized golden rule formula, i.e, a three-particle correlation function [@P3-Hedin02]. However, for our purposes, it is important mention the evidence that the sudden approximation is justified for the cuprate superconductors even at low photon energies [@P3-Randeria95; @P3-Koralek06]. In the end, the suitability of the approximations invoked, will be justified by the agreement between the theory and the experimental observations.
In the diagrammatic language, the above approach can be reduced to calculate the quasiparticle self energy $\Sigma_{k}$ within the framework of the Fermi-liquid theory, where electron-like quasiparticles populate bands in the energy-momentum space up to the cut-off at the Fermi energy. In the case of normal metals, this sophisticated description was firstly introduced by Migdal [@P3-Migdal58] who showed that the small parameter $N(0)\theta_{D}$ allows to consider the higher order corrections negligible, assuming that the density of states $N(\varepsilon)$ is approximately a constant $N(0)$ over the interval $(-\theta_{D},\theta_{D})$ around the Fermi level $\varepsilon_{F}$. Here, $\theta_{D}$ is the so-called Debye energy.
However, in the case of superconductors, the theory requires to incorporate the Cooper-pairs condensation through a bosonic coupling function assuming that both the electronic and the bosonic spectrum are possible to obtain from inelastic neutron scattering (INS) measurements, X-Ray scattering (XRS) experiments, tunneling experiments, or ab-initio calculations of the electronic band structure. This complex picture can be understood, in general terms, from the so-called Eliashberg theory [@P3-Eliashberg60], and the works by Nambu [@P3-Nambu60], Schrieffer [@P3-Schrieffer63], and Morel and Anderson [@P3-Morel62]. In this scenario, the boson spectrum is directly associated to the lattice vibration (phonons) as the binding mechanism for the Cooper-pairs formation. Nevertheless, this theory also has been often considered as the base of another possible mechanisms with a magnetic origin [@P3-Kulic00; @Muschler2010; @P3-Hwang07; @Schachinger06; @P3-Carbotte05; @P3-Dordevic05; @P3-Schachinger03; @P3-Schachinger00; @P3-Carbotte99; @P3-Dahm09]. As there is not any argument which allows to validate this assumption, onwards, we will refer only to phonons as the boson coupling mechanism.
In order to support our “phononic choice” we want to recall that the absence of the magnetic-resonance mode in LSCO ($La_{2-x}Sr_{x}CuO_{4}$) [@P3-Zhou05], Bi2212 ($Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$) over-doped (x=0.23) [@P3-Hwang04], and its appearance only below $T_{c}$ in some cuprates (e.g., Bi2201 [@P3-Sato03]) are not consistent with the universality of the kink effect. Moreover, recent studies on electron doped systems [@P3-Park08; @P3-Graf07; @P3-Wilson06; @P3-Zhao07] have shown that the intensity of the magnetic resonance mode is seemingly weak in comparison with the phonon mode to be considered as the cause of the strong electron energy dispersion measured by ARPES.
In the forthcoming paragraphs, and relying on a solid theoretical basis that is introduced within the supplementary material section for the reader’s sake, we present our analysis of available experimental data.
### \[ch-7-1\] *7.1 Basic statements for the E-Ph coupling* {#ch-7-1-7.1-basic-statements-for-the-e-ph-coupling .unnumbered}
As a manifestation of the electron-phonon coupling interaction one can introduce the mass renormalization of the electronic dispersion at the energy scale associated with the phonons. This may be technicallydefined from the real part of the electron self energy as a mass-enhancement parameter $\lambda^{*}$ [@P3-Grimvall81] given by, $$\begin{aligned}
\label{Eq-7.1}
\lambda_{k}^{*}\equiv\left.-\partial_{\omega}\Sigma_{1}\right|_{\omega=0} \, .\end{aligned}$$
Although related (as it will be later clarified), $\lambda^{*}$ must not be interpreted as the strength of the E-Ph interaction which is estimated from the so-called boson coupling parameter $\lambda$. This dimensionless parameter is commonly defined in terms of the electron-phonon spectral density as $$\label{Eq-7.2}
\lambda\equiv2\int_{0}^{\infty}d\nu\,\frac{\alpha^{2}F(\nu )}{\nu}\, ,$$ and it is customarily related to the superconducting transition temperature. In fact, as we have argued in Ref. [@P3-Ruiz2009], equality would just be warranted at very low temperatures, and ensuring that the spectral density measured from the experiments is fully satisfying the ME approach for the Feynman diagram of lowest order for the E-Ph interaction \[see Fig. SIII-I (pag. ), in the section of supplementary material III\].
Hence, taking into consideration the inherent existence of phonons in the HTSC, we have evaluated the electron spectral densities for $La_{2-x}Sr_{x}CuO_{4}$ (LSCO), $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$ (Bi2212), and $YBa_{2}Cu_{3}O_{6+x}$ (Y123), in the flat model of Ref. [@P3-Shiina90] as well as by solving the isotropic Eliashberg equations on the Matsubara frequencies. The structure of these spectral densities is restricted to the isotropic nodal direction under the assumption that $\alpha^{2}F(\nu)=G(\nu)\times C$, with C an adjustable constant and $G(\nu)$ the generalized phonon density of states extracted from the inelastic neutron scattering experiments [@P3-Shiina90]. Our results are shown in Fig. \[Figure\_7\_1\] (*Shiina’s* lines). In addition, other reproducible methods to calculate the E-Ph spectral density have been taken into consideration. Specifically, we mean the simple method by Islam & Islam [@P3-Islam00] (*Islam’s* lines), and the method by Gonnelli et al. [@P3-Gonnelli98], the last only referred for the Bi2212 family (*Gonnelli’s* line). The method of Refs. [@P3-Shiina90] & [@P3-Islam00] is based on the INS experimental data by Renker *et al.* [@P3-Renker87; @P3-Renker88; @P3-Renker89], and the method in Ref. [@P3-Gonnelli98] is based on the tunneling data reported in Refs. [@P3-Gonnelli97; @P3-Ummarino97].
![\[Figure\_7\_1\] E-Ph spectral density $\alpha^{2}\textit{F}(\nu)$ for (a) $La_{2-x}Sr_{x}CuO_{4}$ (LSCO), (b) $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$ (Bi2212), and (c) $YBa_{2}Cu_{3}O_{6+x}$ (Y123), determined by different methods. The solid lines correspond to the method in Ref. [@P3-Islam00], dashed lines to the method in Ref. [@P3-Shiina90], and the dotted line in (b) corresponds to the method of Ref. [@P3-Gonnelli98].](Figure_7_1.pdf "fig:"){width="100.00000%"}\
On the other hand, as it is shown in the supplementary material section, and after some simple mathematical manipulations, the E-Ph self energy ($\Sigma$) can be rewritten within the ME approach as a complex function of the form: $$\begin{aligned}
\label{Eq-7.3}
\Sigma(\textbf{k},i\omega)=\int_{0}^{\infty}d\nu\alpha^{2}F(\textbf{k},\nu)
\left\{-2\pi~i\left[N(\nu)+\frac{1}{2}\right]+
\Upsilon(\nu,\omega,T)
\right\}
\, ,\end{aligned}$$ where the function $\Upsilon(\nu,\omega,T)$ is defined in terms of the so-called digamma functions $\psi(z)$, $$\begin{aligned}
\label{Eq-7.4}
\Upsilon(\nu,\omega,T)=\psi\left(\frac{1}{2}+i\frac{\nu-\omega}{2\pi
T}\right)-\psi\left(\frac{1}{2}-i\frac{\nu+\omega}{2\pi T}\right)
\, .\end{aligned}$$ At this point, is useful recall that the bare band energy $\varepsilon_{\textbf{k}}$ is related to the dressed band energy $E_{\textbf{k}}$ by $$\begin{aligned}
\label{Eq-7.5}
E_{\textbf{k}}=\varepsilon_{\textbf{k}}+Re\Sigma(E_{k}) \, .\end{aligned}$$
Whereas the direct extraction of the self-energy from experiments appears to be troublesome because the underlying band structure of the bare electrons is *a priori* unknown, a theoretical determination of the bare band structure and its relation to the full energy renormalization effects observed in the experiments seems much more attractive. In this sense, the nodal ARPES spectra are of great importance to check the validity of the quasiparticle concept discussed above, and also for understanding the nature of the involved interactions.
### \[ch-7-2\] *7.2 E-Ph model for the nodal kink effect* {#ch-7-2-7.2-e-ph-model-for-the-nodal-kink-effect .unnumbered}
Let me make some final remarks before moving onto the application of the above ideas to the analysis of the nodal kink effect in the ARPES data. On the one hand, is to be mentioned, that the energy distribution- and momentum distribution-curves (EDC - MDC) are the two most popular ways for analyzing photoemission data. The dichotomy between the MDC- and EDC-derived bands from the same data raises critical questions about its origin and also about which one represents the intrinsic band structure. In this sense, we want to recall that at the larger bandwidth along the nodal direction, the MDC method can be reliably used to extract high quality data of dispersion in searching for fine structure. It has also been shown theoretically that this approach is reasonable in spite of the momentum-dependent coupling [@P3-Devereaux04]. In a typical Fermi liquid picture, the MDC- and EDC-derived dispersions are identical. Moreover, in an electron-boson coupling system the lower and higher energy regions of the MDC- and EDC-derived dispersions are still consistent, except right over the kink region [@P3-Zhang08]. On the other hand, recalling that the bare electron band energy $\varepsilon_k$ is not directly available from the experiments. Instead, the electron momentum dispersion curve $E_{k}(k-k_{F})$ may be measured. Below, we show that the good agreement between our theoretical data and the experimental facts supports this general picture.
As the kink effect is a common feature of the whole set of HTSC families i.e., it is observable regardless of the doping level or the temperature at which the measurement is performed, the only possible scenario seems to be the coupling of quasiparticle with phonons. Thus, we have noted that the relation between the dressed and bare energies is a central property as related to the kink structure. The key issue seems to be the existence of an energy scale (in the range of 40-80 meV).
We have proposed an integral equation formalism based on the combination of Eqs. \[Eq-7.2\] – \[Eq-7.5\] from which, one can consider that $\varepsilon_k$ implicitly depends on $E_k$ through the boson coupling parameter $\lambda$ [@P3-Ruiz2009]. In other words, Eq. \[Eq-7.2\] acts as an integral constraint, through the definition of $\lambda$. Thus, based on an interpolation scheme between the numerical behavior of the dressed energy band and the experimental data, one can introduce a universal dispersion relation which allows to reproduce in a quite general scheme the nodal dispersions close to the Fermi level [@P3-Ruiz2009; @P3-Ruiz2011_JS; @P3-Ruiz2011_CAP]. As a central result, we have encountered that the practical totality of data are accurately reproduced by a universal linear dispersion relation of the kind $$\begin{aligned}
\label{Eq-7.6}
{k}-{k}_{F}=\frac{\varepsilon_{k}}{v_{F_{<}}}\left(1-\delta\lambda\right)\, ,\end{aligned}$$ with $v_{F_{<}}$ the Fermi velocity at low-energies, and $\varpi$ as the (only) “*free*” parameter required for incorporating the specific renormalization for each superconductor. From the physical point of view, our empirical [*ansatz*]{} \[Eq.(\[Eq-7.6\])\] may interpreted as follows. Into the ME approach, the involved quantities are not far from their values at the Fermi level, and one can start with $\varepsilon_k$ replaced by $E_{k}-\Sigma_{1}(E_{k})$, i.e., $$\begin{aligned}
\label{Eq-7.7}
E_{k}-\Sigma_{1}(E_{k})=\left({k}-{k}_{F}\right)v_{F_{<}}
\left(1-\delta\lambda\right)^{-1} \, .\end{aligned}$$ Then, taking derivatives with respect to $E_{k}$ and evaluating them for $E_{k}\to 0$, one gets $$\begin{aligned}
\label{Eq-7.8}
1+\lambda^{*}=\left.\frac{\partial (k-k_{F})}{\partial
E_{k}}\right|_{E_{k}=0} v_{F<}(1-\delta\lambda)^{-1}\, ,\end{aligned}$$ and recalling that $v_{F<}$ is obtained as the slope of the lower part of the momentum dispersion curve, this equation leads to $1+\lambda^{*}=(1-\delta\lambda)^{-1}$. Thus, a physical interpretation of the fit parameter $\delta$ is straightforwardly obtained from the analytical relation $$\begin{aligned}
\label{Eq-7.9}
\delta =\frac{\lambda^{*}}{\lambda(1+\lambda^{*})} \, .\end{aligned}$$ To the lowest order, the dimensionless parameter $\delta$ is basically the ratio between the defined mass-enhancement and phonon-coupling parameters $\delta\approx\lambda^{*}/\lambda$. Outstandingly, this fact reassembles the differences obtained by tight-binding Hamiltonian models [@P3-Weber87; @P3-Weber88] and the “density-functional” band theories [@P3-Giustino08; @P3-Heid08] appealing to the differences between $\lambda$ and $\lambda^{*}$. To understand these differences, recall that, in principle, the density-functional theory [@P3-Kohn66] gives a correct ground-state energy, but the bands do not necessarily fit the quasi-particle band structure used to describe low-lying excitations.
### \[ch-7-3\] *7.3 Numerical procedure and results* {#ch-7-3-7.3-numerical-procedure-and-results .unnumbered}
In order to reproduce the experimental results, our numerical program is as follows:
1. $v_{F_{<}}$ is determined from the momentum dispersion curve in the ARPES measurements \[$\sim $ 1.4 eV$\cdot$Å - 2.2 eV$\cdot$Å\],
2. To determine $\delta$, we calculate the so called logarithmic frequency $\omega_{\rm log}$ as introduced by Allen and Dynes [@P3-Allen75], $$\begin{aligned}
\label{Eq-7.10}
\omega_{\rm
log}\equiv
exp\left\{\frac{2}{\lambda}\int_{0}^{\infty}\ln(\nu)[{\alpha^{2}F(\nu)}/{
\nu }]
d\nu\right\} \, .\end{aligned}$$
This constant frequency (properly defining the corresponding spectral densities $\alpha^{2}F(\nu)$ for each $\lambda$) is introduced only for scaling purposes $(\delta=\varpi/\omega_{\rm log})$. Thus, we achieve a reduction in the scattering of numerical values when dealing with different samples. In this sense, $\varpi$ is a mere mathematical instrument. For the spectral densities shown in Fig. \[Figure\_7\_1\] we get $\omega_{\rm
log}^{LSCO}\simeq
16.1455\, meV$, $\omega_{\rm
log}^{Y123}\simeq 35.5900\, meV$ and $\omega_{\rm
log}^{Bi2212}\simeq 33.8984\, meV$, respectively.
3. $\varepsilon_{k}$ is numerically determined from the relation $E_{k}=\varepsilon_{k}+Re\{\Sigma(E_{k})\}$, where the digamma functions are simply subroutines of our code.
4. Finally, correlation between theory and experiment is established by the application of Eq. (\[Eq-7.6\]). The best fit with the experimental curves has been obtained for $\delta=0.185$ in LSCO, $\delta=0.354$ in Bi2212, and $delta=0.365$ in Y123 families, respectively.
![\[Figure\_7\_2\] The renormalized electron quasiparticle energy dispersion $E_{k}$ as a function of the momentum $k-k_{F}$ for several samples of $\texttt{La}_{2-x}\texttt{Sr}_{x}\texttt{CuO}_{4}$, measured along the (0,0)-($\pi$,$\pi$) nodal direction at a temperature of 20 K. The doping level $x$ ranges between 0.03 (top left) up to 0.30 (bottom right). The experimental data (symbols) are taken from Ref. [@P3-Zhou03] and the theoretical curves have been obtained using the spectral densities of Islam & Islam \[Ref. [@P3-Islam00]\] (open squares), and Shiina & Nakamura (solid lines)[@P3-Shiina90].](Figure_7_2.pdf){height="11.2cm" width="13cm"}
In Fig. \[Figure\_7\_2\] we show the results found in LSCO covering the doping range ($0<x\leq0.3$). Remarkably, within this range, the hole concentration in the $CuO_{2}$ plane is well controlled by the $Sr$ content and a small oxygen non-stoichiometry, where the physical properties span over the insulating, superconducting, and overdoped non-superconducting metal behavior. Superconducting transition temperatures $T_{c}$ in the interval of 30-40K have been observed in Refs. [@P3-Bednorz86; @P3-Uchida87; @P3-Dietrich87]. For the application of Eq. (\[Eq-7.6\]), we have considered $v_{F_{<}}=2 eV\cdot$Å as related to the experimental results of Refs. [@P3-Lanzara01; @P3-Zhou03; @P3-Zhou07; @P3-Zhou05]. The best fit of the whole set of experimental data has been obtained for $\delta=0.185$, and the derived $\lambda^{*}$ values are shown in Table \[Table\_7\_1\]. For comparison, recall that values of “$\lambda=2-2.5$” in the range $0.2>x>0.1$ were reported in the Ref. [@P3-Weber87] by Weber. In that case, these values were obtained within the framework of the nonorthogonal-tight-binding theory of lattice dynamics, based on the energy band results of Mattheiss [@P3-Mattheiss87]. It must be emphasized that in both cases (Ref. [@P3-Weber87] & Refs [@P3-Ruiz2009; @P3-Ruiz2011_JS]), $\lambda$ was obtained in agreement with the observed $T_c$ values in LSCO. Moderate discrepancies between the predictions of our model and the analysis of Refs. [@P3-Weber87] & [@P3-Weber88] may be ascribed to some uncertainty in the experimental spectral densities.
On the other hand, we have taken advantage of the widespread availability of experimental information in LSCO [@P3-Lanzara01] and our numerical method, to describe the evolution of the E-Ph coupling parameter $\lambda$ as a function of the doping level. It can be fitted to the simple expression $$\begin{aligned}
\label{Eq-7.11}
\lambda=2\tilde{\omega}exp(-\frac{\tilde{\omega}}{\delta} x)+1 \, ,\end{aligned}$$ within a precision factor of $\sim95$% (Fig. \[Figure\_7\_3\]). Here, $\tilde{\omega}$ defines the ratio between the phonon characteristic energies introduced by McMillan [@P3-McMillan68], $$\begin{aligned}
\label{Eq-7.12}
\omega_{\rm 1}=(2/\lambda)\int_{0}^{\infty}\alpha^{2}F(\nu)d\nu\equiv
(2/\lambda)S \, , \end{aligned}$$ and the logarithmic frequency $\omega_{\rm log}$ \[Eq. (\[Eq-7.10\])\]. In this case, we get $\omega_{\rm log}^{LSCO}\simeq 16.1455\, meV$ and $\omega_{1}^{LSCO}\simeq 25.2627\, meV$.
$La_{2-x}Sr_{x}CuO_{4}$ (LSCO). $\delta=0.85$
------------------------- ------------------ ----------- --------------- ------------------------
$x$ Ref. $\lambda$ $\lambda^{*}$ $T_{c}(\lambda)$ \[K\]
0.03 3.30 0.61 –
0.05 2.90 0.54 –
0.063 2.80 0.52 –
0.075 2.70 0.50 –
0.10 2.20 0.41 42.10
0.12 2.10 0.39 40.77
0.15 1.90 0.35 37.83
0.18 1.80 0.33 36.19
0.22 1.50 0.28 30.47
0.30 1.30 0.24 –
0.1-0.2 [@P3-Weber87] 2-2.5 – 30–40
– [@P3-Shiina90] 1.78 – 40.6
0.15 [@P3-Giustino08] 1–1.32 0.14–0.22 –
0.22 [@P3-Giustino08] 0.75–0.99 0.14–0.20 –
: \[Table\_7\_1\] Values of the E-Ph coupling parameter $\lambda$ and the corresponding mass-enhancement parameter $\lambda^{*}$ obtained from the analysis of ARPES data at several doping levels ($x$) of $La_{2-x}Sr_{x}CuO_{4}$, $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$. $\lambda^{*}$ has been obtained by means of Eq. (\[Eq-7.6\]) to the lowest order $\lambda^{*}\approx\delta\lambda$. The predicted superconducting transition temperatures $T_{c}$ are also shown. Our results are presented in contrast with other models available in the literature.
![\[Figure\_7\_3\] Evolution of the E-Ph coupling parameter $\lambda$ as a function of the dopant content in LSCO. The $\lambda$-values have been obtained from our best fit with the nodal kink dispersion (black squares) shown in Fig. \[Figure\_7\_2\]. Correspondingly, the evolution of the area $S$ as a function of the dopant content is also shown (right scale).](Figure_7_3.pdf "fig:"){width="51.00000%"}\
$Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$ (Bi2212). $\delta=0.354$
------------------------------- ------------------ ------------- --------------- ------------------------
$x$ Ref. $\lambda$ $\lambda^{*}$ $T_{c}(\lambda)$ \[K\]
0.12 2.15 0.76 64.81
0.16 1.33 0.47 42.45
0.21 0.85 0.30 19.93
– [@P3-Shiina90] 3.28 – 85
– 3.28 1.16 81.66
– [@P3-Gonnelli98] 3.34 1.05 93
– 3.34 1.18 82.40
0.16 [@P3-Kordyuk06] $\sim 1.28$ $\sim0.43$ –
: \[Table\_7\_2\] Same as Table \[Table\_7\_1\] but for $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$.
We want to clarify that, although Eq. (\[Eq-7.11\]) can be considered as a useful relation between the physical and chemical properties of LSCO, the shaping of other HTSC families by similar expressions cannot be guaranteed.
In Table \[Table\_7\_2\] and Fig. \[Figure\_7\_4\], we display the results found for the available experimental data in Bi2212 samples. In this case, the theoretical curves for $E_{k}(k-k_{F})$ have been predicted by our interpolation method for the under-doped (UD70) “$x=0.12$, $T_{c}\approx 70K$”, optimally doped (0PD90) “$x=0.16$, $T_{c}\approx 90K$”, and over-doped (OVD58) “$x=0.21$, $T_{c}\approx 58K$” samples, having use of the spectral densities of Fig. \[Figure\_7\_1\]. The experimental data were taken from the work by Johnson *et al.* [@P3-Johnson01] with $v_{F_{<}}=1.6eV\cdot$[Å]{} as a value consistent with the experimental results of Refs. [@P3-Lanzara01; @P3-Zhou02; @P3-Kordyuk06; @P3-Johnson01]. The best fit with experimental data has been found for $\delta=0.354$. We show that, regardless the method used for obtaining the E-Ph spectral density, the same conclusions can be achieved.
$YBa_{2}Cu_{3}O_{6+x}$ (Y123). $\delta=0.365$
------------------------ ---------------- ------------- --------------- ------------------------
$x$ Ref. $\lambda$ $\lambda^{*}$ $T_{c}(\lambda)$ \[K\]
0.4 0.80 0.29 17.51
0.6 0.65 0.24 9.82
0.85 0.50 0.18 3.39
– [@P3-Shiina90] $\sim$3.4 – 91
– 3.45 1.26 84.19
– [@P3-Weber88] $\sim$0.5 – $\sim$3
– [@P3-Weber88] $\sim$1.3 – $\sim$30
– 1.30 0.47 36.43
– [@P3-Heid08] – 0.18 - 0.22 –
– 0.49 - 0.60 0.18 - 0.22 $\sim$3.0–6.6
: \[Table\_7\_3\] Same as Table \[Table\_7\_1\] but for $YBa_{2}Cu_{3}O_{6+x}$.
Finally, in Table \[Table\_7\_3\] and Fig. \[Figure\_7\_5\] we show the results found for Y123 with the following dopant levels: Under-Doped 35 “$x=0.4$, $T_{c}\approx 35K$” (UD35), Under-Doped 61 “$x=0.6$, $T_{c}\approx 61K$” (UD61), and Over-Doped 90 “$x=0.85$, $T_{c}\approx
90K$” (OVD90). The experimental data were taken from the work by Borisenko *et al.* [@P3-Borisenko06]. To our knowledge, no more experimental evidence of kinks in the nodal direction is available for this material. The value $v_{F_{<}}=1.63eV \cdot$Å has been used for consistency with the experimental results reported by those authors. The best fitting between the experimental data and our model has been found with the value $\delta=0.365$. It must be noted that the appearance of a second kink in the underdoped case may not be allocated with the simple assumptions of our model. Nevertheless, although this effect could be explained either by introducing high perturbation orders beyond the ME approach or introducing a renormalization factor for the electronic band structure, the range of energies for the E-Ph spectral density does not seems provide a physically admissible explanation.
![\[Figure\_7\_4\] Same as in Fig. \[Figure\_7\_2\] but in samples of Bi2212. The experimental data for the normal state (full diamond) and superconducting state (full circle) both have been taken from the Ref. [@P3-Johnson01]. The theoretical curves (lines -diamond, or -circles) have been obtained from our interpolation method \[Eq. (\[Eq-7.6\])\] according to the best fit values for the E-Ph coupling parameter $\lambda$, and the spectral densities of Islam & Islam [@P3-Islam00] (top), Shiina & Nakamura [@P3-Shiina90] (middle), and Gonnelli et al. [@P3-Gonnelli97] (bottom). The different plots correspond to the doping levels of Bi2212: under-doped “x=0.12” (left), optimally doped “x=0.16” (center), and over-doped “x=0.21” (right) respectively.](Figure_7_4.pdf "fig:"){width="80.00000%"}\
![\[Figure\_7\_5\] Same as in Fig. \[Figure\_7\_2\] but in samples of Y123 with (left to right): $x=0.4$ (underdoped), $x=0.6$ (underdoped), and $x=0.85$ (overdoped). The solid squares correspond to the experimental data of Ref. [@P3-Borisenko06]. The theoretical curves have been obtained using the spectral densities of Islam & Islam \[Ref. [@P3-Islam00]\] (open squares), and Shiina & Nakamura \[Ref. [@P3-Shiina90]\] (solid line). All curves have been obtained at 30K. ](Figure_7_5.pdf "fig:"){width="60.00000%"}\
\[ch-8\] **Is it necessary to go beyond the E-Ph mode?**
========================================================
According to the previous chapter, the results for optimally doped Bi2212 (OPD90, $x=0.16$) and over doped Bi2212 (OVD85, $x=0.21$) samples (Fig. \[Figure\_7\_4\]), with measured $T_{c}=90K$ and $85K$ respectively, have revealed that the influence of the E-Ph coupling mechanism is seemingly weak in spite of the kink effect is reproduced for the whole energy spectrum. On the other hand, the discrepancies become even larger when the thermodynamic properties for Y123 samples are analyzed.
*To address the question:*
In order to answer Is it necessary to go beyond the E-Ph coupling mode for the superconducting pairs formation?, we can *a priori* consider that some of the E-Ph spectral densities of Fig. \[Figure\_7\_1\] will allow explain the high $T_{c}$ values and the zero temperature gap $\Delta_{0}$ reported in the literature. From this point of view, several approaches can be argued for each one of the materials. For example, in Fig. \[Figure\_8\_1\] we show our results for $T_{c}$, the ratio gap $2\Delta_{0}/k_{B}T_{c}$ and the zero temperature gap $\Delta_{0}$ for the different HTSC families analyzed in the above chapter. The different curves have been obtained from the point of view of three different approaches
1. The celebrated McMillan’s equation [@P3-McMillan68], $$\begin{aligned}
\label{Eq-8.1}
T_{c}=\frac{\omega_{1}}{1.2}exp\left[-1.04\frac{1+\lambda}{\lambda-\mu^{*}
(1+0.62\lambda)}\right] \, ,\end{aligned}$$ with $\mu^{*}$ the so-called Coulomb pseudopotential.
2. The refined formula by Allen and Dynes [@P3-Allen75], which is obtained by replacing $\omega_{1}$ \[Eq. (\[Eq-7.12\])\] in Eq. (\[Eq-8.1\]) by the so called logarithmic frequency $\omega_{log}$ \[see Eq. (\[Eq-7.10\])\].
For the HTSC families considered along this section, and the different spectral densities of Fig \[Figure\_8\_1\], we get $\omega_{\rm log}^{LSCO}\simeq 16\,
meV$, $\omega_{1}^{LSCO}\simeq 25\, meV$, $\omega_{\rm log}^{Bi2212}\simeq 33\,
meV$, $\omega_{1}^{Bi2212}\simeq 39\, meV$, $\omega_{\rm log}^{Y123}\simeq 35\,
meV$, and $\omega_{1}^{Y123}\simeq 39\, meV$.
3. Finally, the less conventional Kresin’s formula [@P3-Kresin87], $$\begin{aligned}
T_{c}=0.25\; \varpi \; exp\left(\frac{2}{\lambda_{eff}}-1\right)^{-1/2}
\; ,\label{Eq-8.2}\end{aligned}$$ where $\varpi=[(2/\lambda)\int_{0}^{\infty}\nu\alpha^{2}F(\nu)d\nu]^{1/2}$ and $\lambda_{eff}=(\lambda-\mu^{*})[1+2\mu^{*} + (3/2) \lambda \mu^{*}
exp(-0.28\lambda)]$.
![\[Figure\_8\_1\] Plots of the critical temperatures $T_{c}$ (top), the ratio gap $2\Delta_{0}/k_{B}T_{c}$ (middle) and the gap $\Delta_{0}$ (bottom) for LSCO, Bi2212 and Y123, all of them represented as functions of the E-Ph coupling parameter $\lambda$ obtained from the E-Ph spectral densities shown in Figure \[Figure\_8\_1\]. We have used three different approaches: the McMillan’s formula [@P3-McMillan68], the Allen-Dynes formula [@P3-Allen75], and the Kresin’s formula [@P3-Kresin87]. ](Figure_8_1.pdf "fig:"){height="11.5cm" width="11cm"}\
In all calculations, the Coulomb’s pseudopotencial was given by a typical value, $\mu^{*}=0.13$ [@P3-Allen82]. We have to be mention that is essential to be aware that there is no a small parameter that enables a satisfactory perturbation theory to be constructed for the Coulomb interaction between electrons. Thus, Coulomb contributions to the electron self energy $\Sigma$ introduced in chapter \[ch-7\] cannot be reliably calculated [@P3-Allen75]. Fortunately this is not a serious problem in superconductivity because a reasonable assumption is to consider that the large Coulomb effects for the normal state are already included in the electronic bare band structure $\varepsilon_{k}$. The remaining off-diagonal terms of the superconducting components of the Coulomb self-energy turn out to have only a small effect on superconductivity, which is treated phenomenologically [@P3-Allen82].
Is to be noticed that the high values of the critical temperatures strongly depend on the approximation invoked, and (in some cases) on the inversion method for the boson coupling spectral density. Thus, we argue that the critical temperature $T_{c}$ should not be considered as a fit parameter for adjusting the theory, i.e.: one should not predict $\lambda$ from the approximate $T_{c}$ formulas and then use it for calculating the electron self-energy. As a proof of this, we recall that although attractive, this idea has led to underestimates the phonon contribution to the photoemission kink in HTSC [@P3-Giustino08; @P3-Heid08].
To our knowledge, the most suitable way for determining the influence of an interaction mechanism in the pair formation for HTSC could be (i) evaluate the strength of the boson coupling mode from the electron renormalization effects and then (ii) solve the Eliashberg equations for the superconducting $T_{c}$, or have use of semiempirical approaches as the introduced before. From such analysis, we conclude that the consideration of the E-Ph interaction in LSCO strongly suggests that the high $T_{c}$ values can be caused by the conventional coupling to phonons (see table \[Table\_7\_1\]). However, for the Bi2212 and Y123 samples the obtained results are not so encouraging as they are for the LSCO family (see tables \[Table\_7\_2\] & \[Table\_7\_3\]).
On the one hand, we have noticed that the obtained critical temperatures for Bi2212-UD70 in the framework of the spectral densities of Islam & Islam \[$T_{c}^{Kresin}(\lambda=2.15)=70.0 K$ and $T_{c}^{McMillan}(\lambda=2.15)=64.81 K$\], and Gonnelli et al. \[$T_{c}^{Kresin}(\lambda=2.15)=67.06 K$ and $T_{c}^{McMillan}(\lambda=2.15)=58.19 K$\], are in some sense, good estimations for the experimental values of $T_{c}$. However, from the method by Shiina & Nakamura [@P3-Shiina90], and the framework of the Allen & Dynes formula [@P3-Allen75], a strong reduction of $T_{c}$ ($\sim 40\%$) can be found. This can be interpreted as a first signal about the need of considering additional perturbation mechanisms into the matrix elements of the Eliashberg equations, or perhaps, and in an optimistic way for the phonon hypothesis, this fact could be revealing that the flat model used in Ref. [@P3-Shiina90] is not consistent with the experimental facts of this kind of material.
On the other hand, regarding the Bi2212-OPD90 and Bi2212-OVD85 samples, independently of the inversion method used for the E-Ph spectral density (Fig. \[Figure\_7\_1\]), the maximal values for $\lambda$ which are able to reproduce the kink structure ($\lambda \simeq 1.3$ and $\lambda \simeq 0.85$ respectively), both underestimate the experimental critical temperatures in about $50\%$. The disagreement can be even higher ($\sim80\%$) if we consider the results for Y123 samples (see Fig. \[Figure\_7\_5\] & Table \[Table\_7\_3\]). However, before moving on thinking in the necessity of additional boson coupling mechanisms, is necessary to revalidate the influence of the Coulomb effects along the framework of the different invoked approaches.
![\[Figure\_8\_2\] Plots of the critical temperatures $T_{c}$ (top), and the ratio gap $2\Delta_{0}/k_{B}T_{c}$ for Bi2212 and Y123 samples, as functions of the E-Ph coupling parameter $\lambda$ and the Coulomb pseudopotential $\mu^{*}$. $\lambda$ has been determined from the spectral densities of: (a) Islam & Islam [@P3-Islam00], (b) Shiina & Nakamura [@P3-Shiina90], and (c) Gonnelli et al. [@P3-Gonnelli98]. Three different approximations are shown: the Kresin’s formula [@P3-Kresin87], the McMillan’s formula [@P3-McMillan68], and the Allen-Dynes formula [@P3-Allen75]. ](Figure_8_2.pdf "fig:"){height="11cm" width="13cm"}\
Assuming that the Coulomb effects are almost negligible for the renormalized energy of the electronic quasiparticles participating in the superconducting pairs formation, a remarkable enhancement of the thermodynamic properties could be expected (Fig. \[Figure\_8\_2\]). For example, taking $\mu^{*}=0.001$ rather than the conventional $\mu^{*}=0.13$, and assuming the most favorable scenario for the phonon hypothesis, i.e.: (*i*) determine the E-Ph spectral density from the methods by Islam & Islam (Ref. [@P3-Islam00]), and/or Gonnelli et al. (Ref. [@P3-Gonnelli98]), (*ii*) use of the empirical Kresin’s formula to determine $T_{c}$ (see Fig. \[Figure\_8\_2\]), and (*iii*) check if it is possible to reproduce the renormalization effects of the scattered quasiparticles along the nodal direction by ARPES measurements; the maximal influence of the E-Ph mechanism for the families of Bi2212 and Y123 can be estimated. For Bi2212-OPD90, we have obtained $\lambda_{Islam}(T_{c}^{Kresin}=91K)\simeq1.82$ and $\lambda_{Gonnelli}(T_{c}^{Kresin}=91K)\simeq1.93$, with the renormalization parameters for the ARPES “bare” dispersion $\delta=0.264$ and $\delta=0.248$, respectively. Regarding to Bi2212-OVD58, the obtained E-Ph coupling parameters are: $\lambda_{Islam}(T_{c}^{Kresin}=58K)\simeq1.12$ and $\lambda_{Gonnelli}(T_{c}^{Kresin}=58K)\simeq1.17$, for the same $\delta$ values above considered. Along this line, we have observed a widening of the kink effect which can be only explained by the existence of at least one additional perturbation mechanism reducing the momentum of the dispersed quasiparticles (see Fig. \[Figure\_8\_3\]).
![\[Figure\_8\_3\] The renormalized energy $E_{k}$ as a function of the momentum $k-k_{F}$ under the assumption that the nodal dispersion is a consequence of the E-Ph coupling mode and the Coulomb effects are completely contained in the bare energies. Then, we assume the very weak Coulomb pseudopotential $\mu^*=0.001$ within the more favourable approximation for the calculus of $T_{c}$ and the phonon hypothesis, i.e. the Kresin Formula and the E-Ph spectral density $\alpha^2\emph{F}(\omega)$ obtained by the methods of Refs. [@P3-Islam00; @P3-Gonnelli97] in Bi2212 cases, and Refs. [@P3-Islam00; @P3-Shiina90] in Y123 cases. ](Figure_8_3.pdf "fig:"){height="9cm" width="13cm"}\
On the other hand, regarding the Y123 samples, the effect of reducing the Coulomb pseudopotential shows that the expected values for $T_{c}$ in Y123-UD35 can be explained under any of the aforementioned formulas. In detail, for $\lambda=0.80$ (see Fig. \[Figure\_7\_5\]) and the E-Ph spectral density extracted from the method by Islam & Islam (see Fig. \[Figure\_7\_1\]), we have obtained $T_{c}^{Kresin}\simeq37.69$, $T_{c}^{McMillan}\simeq40.04$, and $T_{c}^{Allen-Dynes}\simeq33.25$. And from the most rigorous method of Ref. [@P3-Shiina90], we have obtained: $T_{c}^{Kresin}\simeq36.86$, $T_{c}^{McMillan}\simeq35.01$, and $T_{c}^{Allen-Dynes}\simeq28.71$. Then, at least for Y123-UD35, is possible keep the idea that the E-Ph coupling is the the most relevant interaction mechanism for the superconducting pair formation. However, we can not argue the same for the whole set of Y123 samples, because the values of $T_{c}$ are still underestimated in about $62\%$ in Y123-UD61, and about $80\%$ in Y123-OVD90. Indeed, if once again we allow assume the most favorable scenario for the phonon hypothesis with the aim of reproduce the $T_{c}$ value for Y123-UD61 \[$\lambda_{Islam}(T_{c}^{Kresin}=61K)\simeq1.20$; $\lambda_{Shiina}(T_{c}^{Kresin}=61K)\simeq1.23$\], and Y123-OVD90 \[$\lambda_{Islam}(T_{c}^{Kresin}=90K)\simeq1.83$; $\lambda_{Shiina}(T_{c}^{Kresin}=90K)\simeq1.89$\], with the ARPES “bare” dispersion renormalized according to the experiments, a most notorious widening of the kink effect appears in disagreement with the experiments (see Fig. \[Figure\_8\_3\]).
Thus, despite the E-Ph mechanism by itself is able to explain many of the properties of low temperature superconductors and even some of the HTSC, it does not seem possible to avoid the idea of the existence of some additional perturbation mechanism contributing to the formation of pairs of Cooper. In fact, in Y123 samples, the importance of an additional mechanism seems to be more significant that in Bi2212, and it could be related to the appearance of a second kink in the under-doped phase. Recent experimental results on the oxygen isotope effect in Bi2212 also assert the need of consider additional coupling mechanism for the formation of pairs [@P3-Carbotte10].
Summarizing, we recall that there are several possibilities as candidates of additional perturbation mechanisms but any of them should modify the perturbation theory introduced along the last two chapters, i.e., by the superposition (adding) of different spectral functions emulating the boson coupling mechanism, we know that the thermodynamic properties could be reproduced. However, the calculation of the intrinsic effects in the renormalized energy band near of the Fermi Level might be compromised, and a satisfactory theory for both effects cannot achieved under the same scheme.
{#section-8 .unnumbered}
{#section-9 .unnumbered}
In summary, we have introduced a numerical model that allows to reproduce the appearance of the ubiquitous nodal kink for a wide set of ARPES experiments in cuprate superconductors. Our proposal is grounded on the Migdal-Eliashberg approach for the self-energy of quasi-particles within the electron-phonon coupling scenario. The main issue is the use of a linear dispersion relation for the bare band energy, i.e.: $\varepsilon_{k}=(k-k_{F})v_{F<}(1-\delta\lambda)^{-1}$. $\delta$, the only free parameter of the theory is a universal property for each family of cuprates, which has been interpreted as the relation between the mass-enhancement $\lambda^{*}$ and electron-phonon coupling $\lambda$ parameters.
On the one hand, for decades, a well-known controversy has arisen on the role of the parameter “$\lambda$” whose values noticeably scatter among different model calculations. As a central result, our proposal re-ensembles the “$\lambda$” values obtained from different models and, as a first approximation, it solves the controversy through the relation $\lambda^{*}\cong\delta\lambda$. We emphasize that the phenomenological parameter $\delta$ (obtained through the analysis of a wide collection of data) has allowed to go beyond the conventional Migdal-Eliashberg approach for restricted sets of experiments. Our model is directly supported by the “$\lambda$” values obtained in Refs. [@P3-Kordyuk06; @P3-Giustino08; @P3-Shiina90; @P3-Gonnelli98; @P3-Weber87; @P3-Weber88; @P3-Heid08]. Furthermore, an excellent agreement between the theory and the available collection of experiments is achieved.
On the other hand, our results support the idea that the strong renormalization of the band structure and the so-called universal nodal Fermi velocity, customarily related to the dressing of the electron with excitations, can be explained in terms of the conventional electron-phonon interaction. In fact, our results suggest that the electron-phonon interaction strongly influences the electron dynamics of the high-$T_{c}$ superconductors, and it is an important mechanism linked with the Fermi surface topology. Thus, we conclude that the electron-phonon interaction (strong or weak) must be included in any realistic microscopic theory of superconductivity, although its effect in the appearance of the superconducting state and the high critical temperatures is not clear yet.
In detail, we have studied the influence of the electron-phonon coupling mechanism through different doping levels in several families of HTSC-cuprates. On the one hand, we have evaluated different methods for obtaining the electron-phonon spectral densities and their influence on the electron bare band energy, and on the other hand, several approaches have been recalled to obtain the critical temperatures. Our results suggest that at least in the LSCO family, and in the so-called Bi2212-UD70 and Y123-UD35 superconductors, the electron-phonon interaction could be the most relevant mechanism involved in the formation of Cooper’s pairs. Our conclusion is supported by the experimental evidence of a mass renormalization of the electronic dispersion curves measured along the nodal direction in ARPES, and the reported $T_{c}$ values in good agreement with our theoretical predictions. In addition, we have evaluated the consequences of assuming an enhanced phonon mechanism, through the reduction of the Coulomb’s pseudopotential weight. When appropriate $T_c$ values are obtained by this method, a remarkable widening of the predicted kink effect arises. This fact, suggest that independently of the approximations invoked and even avoiding the influence of the *d*-wave superconducting gap through the nodal ARPES measurements, it doesn’t seem possible to elude the existence of additional mechanisms that reduce the momentum of the dispersed quasiparticles in comparison with the phonon mechanism. In this sense, despite the fact that in LSCO the influence of the magnetic mode seems not relevant, it is not possible to ignore its importance over the electron properties of other HTSC families.
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{#section-10 .unnumbered}
### \[PIII-s\] {#piii-s .unnumbered}
******
Lattice vibrations couple to electrons because the displacements of atoms from their equilibrium positions alter the band dispersions, either lowering or raising the total electron quasiparticle energy, where the phonon propagator or Green’s function for the phonon contribution can be defined as $$\begin{aligned}
\label{Eq-SIII-1}
D_{\alpha,\beta}(\textbf{Q},\tau)=-\left\langle
T_{\tau}\textbf{u}_{Q,\alpha}(\tau)_{H}\textbf{u}_{-Q,\beta}^{\dag}(\tau)_{H}
\right\rangle \, .
\nonumber\end{aligned}$$ The displacement operators $\textbf{u}_{Q}$ have been written in the Heisenberg picture, with the exception that the time *t* is replaced by an imaginary time $-i\tau$ with $\beta=T^{-1}$. $T_{\tau}$ is the Wick operator which reorders the operators in such a way that $\tau$ increases from right to left. In detail, the displacement operator $\textbf{u}_{Q}$ is a sum of phononic operators, where the operator $a_{Qi}\left(a_{Qi}^{\dag}\right)$ destroy (create) a phonon with energy $\omega_{Qi}$, wave vector $\textbf{Q}=Qi$, branch index i, and polarization vector $\hat{\epsilon}_{Qi}$, i.e., $$\begin{aligned}
\label{Eq-SIII-2}
\textbf{u}_{Q}=\sum_{i}\left(\frac{\hbar}{2NM\omega_{Qi}}\right)^{1/2}\hat{
\epsilon}_{Qi}\left(a_{Qi}+a_{Qi}^{\dag}\right) \, .
\nonumber\end{aligned}$$ Here, by simplicity, we have considered only one kind of ion-mass *M* for the displaced atoms. On the other hand, as we are interested in determine the Green’s function for the interacting system, is helpful to write the Dyson’s equations for electrons and phonons, i.e., $$\begin{aligned}
\label{Eq-SIII-3}
G^{-1}(\textbf{k},i\omega_{n})=G_{0}^{-1}(\textbf{k},i\omega_{n})-\Sigma(\textbf
{k},i\omega_{n}) \, ,
\nonumber\end{aligned}$$ $$\begin{aligned}
\label{Eq-SIII-4}
\left[D^{-1}(\textbf{Q},i\omega_{\nu})\right]_{ij}=&\left[D_{0}^{-1}(\textbf{Q},
i\omega_{\nu})\right]_{ij}-\Pi_{ij}(\textbf{Q},i\omega_{\nu}) \, ,
\nonumber\end{aligned}$$ in terms of their corresponding spectral representations, as follows: $$\begin{aligned}
\label{Eq-SIII-5}
G(\textbf{k},i\omega_{n})=\int_{-\infty}^{\infty}d\varepsilon
C(\textbf{k},\varepsilon)(i\omega_{n}-\varepsilon)^{-1} \, ,
\nonumber\end{aligned}$$ $$\begin{aligned}
\label{Eq-SIII-6}
D_{\alpha,\beta}(\textbf{Q},i\omega_{\nu})=\int_{-\infty}^{\infty}d\nu
B_{\alpha,\beta}(\textbf{Q},\nu)\times[(i\omega_{\nu}-\nu)^{-1}-(i\omega_{\nu}
+\nu)^{-1}] \, .
\nonumber\end{aligned}$$ Notice as, the last couple of equations are defined within the framework of the “Matsubara Frequencies” $(i\omega_{n}=i(2n+1)\pi T)$, where the Green’s function for non-interacting electrons is defined as follows: $$\begin{aligned}
\label{Eq-SIII-7}
G_{0}(\textbf{k},i\omega_{n})=\left(i\omega_{n}-\varepsilon_{k}\right)^{-1} \, ,
\nonumber\end{aligned}$$ and the phonon propagator in the harmonic approximation [@P3-Doniach98] can be defined as $$\begin{aligned}
\label{Eq-SIII-8}
D_{\alpha,\beta}^{0}(\textbf{Q},i\omega_{\nu})=\sum_{\textbf{Q}}
\left(\frac{\hbar}{2NM\omega_{\textbf{Q}}}\right)\epsilon_{Q\alpha}\epsilon_{
-Q\beta}
\left(\frac{1}{i\omega_{\nu}-\omega_{\textbf{Q}}}-\frac{1}{i\omega_{\nu}+\omega_
{\textbf{Q}}
}\right) \, .
\nonumber\end{aligned}$$ $C(\textbf{k},\varepsilon)$ and $B_{\alpha,\beta}(\textbf{Q},\nu)$ are the electron and phonon spectral functions, respectively, and $\Sigma(\textbf{k},i\omega_{n})$ and $\Pi(\textbf{Q},i\nu_{n})$ are the corresponding electron and phonon self-energies. On the one hand, the phonon self-energy $\Pi$ causes a large shift of the phonon energies which is very difficult to evaluate by numerical methods. Nevertheless, the spectral function $B_{\alpha,\beta}$ is directly measurable by INS or, more precisely, from a Born-Von Karman interpolation [@P3-Doniach98]. On the other hand, there is not possible get a direct measure of the spectral function *C*, and evaluate it is too complicated because in the electronic density of states $N(\varepsilon)$ there are some critical points in the Brillouin zone (Van-Hove singularities) where $\left|\nabla\varepsilon_{k}\right|$ becomes zero by symmetry reasons. Thus, is easier evaluate $\Sigma$ even though a proper theory implies the use of a Hamiltonian model that in the normal state corresponds to the Fröhlich Hamiltonian [@P3-Gross86]. Here, the key lies in resolving the Feynman diagram of figure SIII-I within the Migdal-Eliashberg (ME) approach.
{height="5.5cm" width="6.5cm"}
Strictly speaking, the Feynman diagram of Fig. SIII-I represents the lowest perturbation diagram for the self energy of electron quasiparticles $\Sigma(k,i\omega_{n})$, scattered from the state *k* to *k’* through the phonon propagator,
$$\begin{aligned}
\Sigma(k,i\omega_{n})=-\beta^{-1}\sum_{k',\nu} \langle
k|\nabla_{\alpha}V|k'\rangle
D_{\alpha,\beta}(k-k',i\omega_{\nu})\langle k'|\nabla_{\beta}V|k\rangle
G(k',i\omega_{n}-i\omega_{\nu}) \nonumber \, .
\nonumber\end{aligned}$$
Here, we have assumed that the electron correlations are responsible of the formation of quasiparticles which are well defined near the Fermi level. In addition, the so called vertex corrections can be a *priori* neglected because these can be shown to be reduced by the ratio between the phonon frequency ($0-100~meV$) and $\varepsilon_{F}$ ($\sim1-10~eV$). Thus, the presence of strong electron correlations mediated by the electron-phonon interaction is avoided, and the multi-phonon excitations are reduced to the single-loop approximation or ME approach. The advantage of this formulation is that the electron self energy can be defined in terms of an experimental spectral density which allow estimate the interaction between the electrons and the lattice vibrations for each one of the materials. In this sense, is useful define the E-Ph spectral density as $$\begin{aligned}
\label{Eq-SIII-9}
\alpha^{2}F(k,k',\nu)\equiv N(0)\langle k|\nabla_{\alpha}V|k'\rangle
B_{\alpha,\beta}(k-k',\nu)\langle k'|\nabla_{\beta}V|k\rangle\, ,
\nonumber\end{aligned}$$ where $N(0)=\sum_{k}\delta(\varepsilon_{k})$ represents the single-spin electronic density of states at the Fermi surface.
Is to be mentioned, that some of the photoemission experiments require consider the complexities of the $d-$band electron structure, and phonons from the “angular” and “energy” components of the phase space ($k$). Then, the $k$-space can be represented in terms of its harmonic components $(J,\varepsilon)$ [@P3-Allen76], and the electron self energy can be defined in terms of this set as,
$$\begin{aligned}
\label{Eq-SIII-10}
\Sigma_{J}(k,i\omega_{n})=T\sum_{J',\nu}\int_{0}^{\infty}d\varepsilon' &&
\frac{ N(\varepsilon')}{N(0)}\int_{0}^{\infty}d\nu
\alpha^{2}\emph{F}(J,J',\varepsilon,\varepsilon',\nu)\nonumber \\ &&
\left(\frac{2\nu}{\omega_{\nu}^{2}+\nu^{2}}\right)
G_{J'}(\varepsilon',i\omega_{n}-i\omega_{\nu}) \, .
\nonumber\end{aligned}$$
The electron-phonon spectral density $\alpha^{2}\emph{F}(J,J',\varepsilon,\varepsilon',\nu)$ represents a measure of the effectiveness of the phonons of frequency $\nu$ in the scattering of electrons from $k(J,\varepsilon)$ to $k'(J',\varepsilon')$. Here, only the harmonic approximation to the phonon propagator $D(k-k',\nu)=2\nu/(\omega_{\nu}^{2}+\nu^{2})$ has been considered. Nevertheless, the above equation for the electron self energy is still cumbersome, and requires an accurate determination of the electron-phonon spectral density from theoretical ab-initio calculations. Then, to avoid this tricky procedure our position is invoke the ME approach.
In order, the ME approach consists in simplify $\Sigma_{J}(k,i\omega_{n})$ by assuming that, it is possible to neglect the dependence on the energy surfaces ($\varepsilon,\varepsilon'$) of the $N(\varepsilon')\alpha^{2}F(J,J',\varepsilon,\varepsilon',\nu)$ function. This allow omit the processes violating the Born-Oppenheimer adiabatic theorem contained within the high order graphs. Formally, this means that the spectral function $\alpha^{2}F(JJ',\nu)$ is to be diagonal where its representation for the normal state will continue to hold in the superconducting state for the isotropic Cooper pairing or $s$-*wave* gap. However, other pairing schemes which break rotation symmetry are, in principle, possible [@P3-Allen82; @P3-Carbotte90].
Thus, taking advantage that the ARPES measurements at the nodal direction are not influenced by the anisotropy of the superconducting gap, we will refer to a (non-directional) isotropic quasiparticle spectral density, defined as the double average over the Fermi surface of the electron-phonon spectral density $\alpha^{2}F(\textbf{k},\textbf{k}',\nu)$; i.e., $$\begin{aligned}
\label{Eq-SIII-11}
\alpha^{2}F(\nu)=\frac{1}{N(0)}\sum_{\textbf{kk'},j} |
g_{\textbf{kk'}}^{j}|^{2}\delta(\nu-\nu_{\textbf{k}-\textbf{k'}}^{j})
\delta(\varepsilon_{k})\delta(\varepsilon_{k'}) \, ,
\nonumber\end{aligned}$$ where, $g_{\textbf{k}\textbf{k}'}^{j}=[\hbar/2M\nu^{j}_{\textbf{k}'\textbf{k}}]^{1/2}
\langle
\textbf{k}|\hat{\epsilon}^{\; j}_{\textbf{k}'\textbf{k}}\cdot\nabla
V|\textbf{k}'\rangle$ defines the matrix elements of the E-Ph interaction for electron scattering from $\textbf{k}$ to $\textbf{k'}$ with a phonon of frequency $\nu_{\textbf{k}-\textbf{k'}}^{j}$ ($j$ is a branch index). $V$ stands for the crystal potential, $\hat{\epsilon}^{\;
j}_{\textbf{k}'\textbf{k}}$ is the polarization vector, and $\delta(x)$ denotes the Dirac’s delta function evaluated at $x'=0$.
Notice that, $|g_{\textbf{k}\textbf{k}'}^{j}|^{2}$ is inversely proportional to the number of charge carriers contributed by each atom of the crystal to the bosonic coupling mode. Therefore, an increase in the doping level, which causes an increment in the hole concentration of the $CuO_{2}$ plane must be reflected in the coupling parameters as we will show in chapter \[ch-8\]. Moreover, recalling the outstanding feature of the theory of metals, that $|g_{\textbf{k}\textbf{k}'}^{j}|^{2}$ vanishes linearly with $|\textbf{k}-\textbf{k}'|$ when $|\textbf{k}-\textbf{k}'| \ll
{k}_{F}$ [@P3-Ashcroft76], one would expect a [*linear*]{} disappearance of the coupling effect that gives rise to the nodal kink in the vicinity of the Fermi surface. Thus, inspired by recent results on the “universality” of the nodal Fermi velocity $v_{F<}$ (at low energies) in the HTSC, a prominent role of this quantity is also expected.
Finally, in photoelectron scattering experiments the relevant dynamical information is contained in the analytic continuation $G(k,\omega+i0^{+})$ to points just above the real frequency axis, known as the “retarded” Green’s function [@P3-Doniach98]. One is therefore led to continue the electronic self-energy $\Sigma(k,i\omega_{n})$ analytically by $\Sigma(k,\omega+i0^{+})\equiv
\Sigma_{1}(k,\omega)+i\Sigma_{2}(k,\omega)$, where the bare electron band energy is determined by the poles of the Green’s function $G(k,\omega+i0^{+})$ or the zeros of $G^{-1}(k,\omega+i0^{+})$.
Assuming that a pole occurs near $\omega=0$, one gets $$\begin{aligned}
\label{Eq-SIII-12}
G^{-1}(k,\omega+i0^{+})
&& =\omega-\varepsilon_{k}-\Sigma_{1}(k,\omega)-i\Sigma_{2 } (k,\omega)
\nonumber \\ &&
\simeq\omega\left(1-\left.\frac{\partial\Sigma_{1}(k,\omega)}{\partial\omega}
\right|_{\omega=0}\right)-\left[
\varepsilon_{k}+\Sigma_{1}(k,0)\right]-i\Sigma_{2}(k,\omega) \nonumber \, .\nonumber\end{aligned}$$ Then, the pole of $G$ occurs at a frequency $\omega_{0}$ given by $\omega_{0}=E_{k}-i/2\tau_{k}$, with the quasiparticle scattering time defined by $\tau_{k}^{-1}=-2\left(1-\partial_{\omega}\Sigma_{1}\right)^{-1}\Sigma_{2}(k,E_{
k})$, and the electron dressed band energy $E_{k}$ by $$\begin{aligned}
\label{Eq-SIII-13}
E_{k}=(1-\partial_{\omega}\Sigma_{1})^{-1}\left[\varepsilon_{k}+\Sigma_{1}(k,
0)\right] \, .
\nonumber\end{aligned}$$
{#section-11 .unnumbered}
\[Glossary\]**Glossary** {#glossaryglossary .unnumbered}
------------------------
In order to provide an easiest reading of this book, below we introduce a list of the most used abbreviations in text. Greek symbols are either incorporated by their phonetic translation into Latin.
- **A** A. Magnetic vector potential. AC. Alternate Cycle. ARPES. Angle resolved photoemission spectroscopies.
- **B** B. Magnetic induction field (Bold-facing means vector). $B_{\tt{a}}$. Peak amplitude for the AC excitation $B_{0}$. $B_{0}$. Applied magnetic flux density. $B_{p}$. Penetration field. $B_{ind}$. Self (induced) magnetic flux density. Bi2212. $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$. Bi2201. $Bi_{2}Sr_{1.65}La_{0.35}CuO_{6+x}$. Bi-2221. $(Bi,Pb)_{2}Sr_{2}Ca_{2}Cu_{3}O_{x}$ BM. Brandt-Mikitik
- **C** $\chi$. Bandwidth of the critical state material law incorporated by the SDCST, i.e., $J_{c\parallel}/J_{c\perp}$. CS. Critical state.
- **D** $\Delta$. Variation (increment) of ... $\Delta_{0}$. The zero temperature superconducting gap (Only used within the third part of this dissertation). $\Delta_{\textbf{r}}$. Material law for the critical state problems. DC. Direct current. DCSM. Double critical state model.
- **E** E. Induced transient electric field (In bold means vector). $E_{c}$. Critical electrical field. $E_{F}$. Electron energy at the Fermi level. $E_{k}$. Electron dressed band energy. $\varepsilon_{k}$. Electron bare band energy. E-Ph. Electron-Phonon. EDC. Energy Distribution Curves. EDCSM. Elliptical double critical state models. Eq(s). Equation(s).
- **F** ${\cal F}$. Minimization functional or so-called Objective function. Fig(s). Figure(s)
- **H** H. Magnetic field (In bold means vector). HTSC. High-temperature superconducting copper oxides.
- **I** $I_{\tt{a}}$. Peak amplitude for the AC excitation $I_{tr}$. $I_{c}$. Critical current. $I_{tr}$. Transport current. $I_{\parallel}$. Parallel current. $I_{\perp}$. Perpendicular current. ICSM. Isotropic critical state model. INS. Inelastic Neutron Scattering.
- **J** J. Electrical current density (In bold means vector). $J_{c}$. Critical current density. $J_{c\parallel}$. Parallel component of J. $J_{c\perp}$. Perpendicular component of J.
- **K** $k_{B}$. Boltzmann’s constant. $k_{F}$. Electrom momentum at the Fermi level
- **L** ${\cal L}$. Lagrange density. $L$. Hysteretic AC loss. $\lambda$. Electron-phonon coupling parameter. $\lambda^{*}$. Mass-enhancement parameter. LANCELOT. A FORTRAN package for large-scale nonlinear optimization. LEED. Low electronic energy diffraction. LSCO. $La_{2-x}Sr_{x}CuO_{4}$.
- **M** $\mu^{*}$. Coulomb psudopotential. $\mu_{0}$. Permeability of the free space. $\mu_{r}$. Relative permeability associated to a ferromagnetic. material. M. Magnetization (In bold means vector). $M_{ij}$. Mutual/Self inductance matrix. MDC. Momentum distribution-curves. ME. Migdal-Eliashberg. MRI. Magnetic resonance imaging.
- **N** n. Smoothing index.
- **O** $\omega$. Electromagnetic oscillating frequency. $\Omega$. Superconducting volume. OPD. Optimally doped. OVD. Over doped.
- **P** $\Phi$. SC volume. $\varPhi$. Electric scalar potential. $\textbf{p}$. Lagrange multiplier. Pag(s). Page(s).
- **R** $R$. Radius of the cylinder.
- **S** **S**. Poynting’s vector. SC. Superconductor. SDCST. Smooth double critical state theory. SIF. Standard Input Format. STM. Scanning tunneling microscopy.
- **T** $T_{c}$. Superconducting critical temperature. TCSM. T critical state model.
- **U** UD. Under doped.
- **X** XRS. X-ray scattering.
- **Y** Y123. $YBa_{2}Cu_{3}O_{6+x}$.
{#section-12 .unnumbered}
**Publications** {#publications .unnumbered}
----------------
Some of the results presented in this dissertation have been published in the following scientific communications:
1. , A. Badía-Majós, Yu. A. Genenko, S.V. Yampolskii and H. Rauh.\
\
Superconducting wires under simultaneous oscillating sources: Magnetic response, dissipation of energy and low pass filtering.
2. and A. Badía-Majós.\
\
Strength of the phonon-coupling mode in $La_{2-x}Sr_{x}CuO_{4}$,\
$Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$ and $YBa_{2}Cu_{3}O_{6+x}$ composites along the nodal direction.
3. , A. Badía-Majós and C. López.\
\
Material laws and related uncommon phenomena in the electromagnetic response of type II superconductors in longitudinal geometry.
4. , C. López and A. Badía-Majós.\
\
Inversion mechanism for the transport current in type II superconductors.\
*Selected for the Virtual Journal of Applications of Superconductivity, Vol 29, Issue 9 (May 1st, 2011), Properties Important for Applications.*
5. and A. Badía-Majós.\
\
Relevance of the Phonon-Coupling Mode on the Superconducting Pairing Interaction of $La_{2-x}Sr_{x}CuO_{4}$.
6. and A. Badía-Majós and.\
\
Smooth double critical state theory for type II superconductors.
7. A. Badía-Majós, C. López and .\
\
General critical states in type II superconductors.\
*Selected for the Virtual Journal of Applications of Superconductivity, Vol 17, Issue 8 (October 15th, 2009), Properties Important for Applications.*
8. and A. Badía-Majós.\
\
Nature of the nodal kink in angle-resolved spectra of cuprate superconductors.\
*Selected for the Virtual Journal of Applications of Superconductivity, Vol 16, Issue 5 (March 1st, 2009), Materials Important for Applications.*
{#section-13 .unnumbered}
**Acknowledgments** {#acknowledgments .unnumbered}
-------------------
I am especially thankful for the great confidence and support provided by my advisor Dr. Antonio Badía-Majós.
I would like also to express my gratitude to Dr. Luis Alberto Angurel and Dr. Rafael Navarro for their undeniable support along my different stages at University of Zaragoza. I also thank to Dr. Yuri Genenko, Dr. Sergey Yampolskii, and Dr. Hermann Rauh, for the interesting discussions we held in Darmstadt University of Technology, as well as for their generous hospitality.
Thanks to the Spanish National Research Council (CSIC), the Institute of Materials Science of Arágon (ICMA), and the Department of Condensed Matter Physics of the University of Zaragoza by their academic and economical support.
Finally but not less important, a special thankful to my girlfriend Edna Corredor, who shares the same passion for physics than me. Without her inexhaustible patience and support, as well as that from my family, this would have not been possible.
It is to be mentioned that, funding of the research within this dissertation has been sponsored by the JAE program of the Spanish National Research Council (CSIC). Also, the attendance to events and dissemination of results was supported by the Spanish CICyT and FEDER program (projects No. MAT2008-05983-C03-01, MTM2006-10531, and MAT2005-06279-C03-01), the DGA project PI049/08, and DGA grant T12/2011.
*I authorize to the University of Zaragoza for the distribution and reproduction of this thesis in the digital repositories of ZAGUAN and TESEO, as well as any other one by direct consent of the University of Zaragoza, the Institute of Materials Science of Arágon (ICMA), or the Spanish National Research Council (CSIC).*
<span style="font-variant:small-caps;">Harold Steven Ruiz Rondan</span>
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---
author:
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[by]{}\
Izu Vaisman
title: A construction of Courant algebroids on foliated manifolds
---
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
A[BSTRACT. For any transversal-Courant algebroid $E$ on a foliated manifold $(M,\mathcal{F})$, and for any choice of a decomposition $TM=T\mathcal{F}\oplus Q$, we construct a Courant algebroid structure on $T\mathcal{F}\oplus T^*\mathcal{F}\oplus E$.]{}
Preliminaries
=============
General Courant algebroids were studied first in a paper by Liu, Weinstein and Xu [@LWX], which appeared in 1997 and became the object of an intensive research since then. Courant algebroids provide the framework for Dirac structures and generalized Hamiltonian formalisms. In [@V1] we have introduced the notions of transversal-Courant and foliated Courant algebroid, thereby extending the framework to bases that are a space of leaves of a foliation rather than a manifold. In the present note we show that a transversal-Courant algebroid over a foliated manifold can be extended to a foliated Courant algebroid. A similar construction for Lie algebroids (which is a simpler case) was given in [@V1]. We assume that the reader has access to the paper [@V1], from which we also take the notation, and he will consult [@V1] for the various definitions and results that we use here. In this paper we assume that all the manifolds, foliations, mappings, bundles, etc., are $C^\infty$-differentiable.
A Courant algebroid over the manifold $M$ is a vector bundle $E\rightarrow M$ endowed with a symmetric, non degenerate, inner product $g_E\in\Gamma\odot^2E^*$, with a bundle morphism $\sharp_E:E\rightarrow TM$ called the [*anchor*]{} and a skew-symmetric bracket $[\,,\,]_E:\Gamma E\times\Gamma
E\rightarrow\Gamma E$, such that the following conditions (axioms) are satisfied:
1\) $\sharp_E[e_1,e_2]_E=[\sharp_Ee_1,\sharp_Ee_2]$,
2\) $im(\sharp_{g_E}\circ^t\sharp_E)\subseteq ker\,\sharp_E$,
3\) $\sum_{Cycl}[[e_1,e_2]_E,e_3]_E=(1/3)\partial_E\sum_{Cycl}g_E([e_1,e_2]_E,e_3)$, $\partial_E=(1/2)\sharp_{g_E}\circ\,^t\sharp_E:T^*M\rightarrow E$, $\partial_E f=\partial_E(df)$,
4\) $[e_1,fe_2]_E=f[e_1,e_2]_E+(\sharp_Ee_1(f))e_2-g(e_1,e_2)\partial_E
f $,
5\) $(\sharp_Ee)(g_E(e_1,e_2))=g_E([e,e_1]_E+\partial_E g(e,e_1)
,e_2)+g_E(e_1,[e,e_2]_E+\partial_E g(e,e_2)).$\
In these conditions, $e,e_1,e_2,e_3\in\Gamma E$, $f\in C^\infty(M)$ and $t$ denotes transposition. Notice also that the definition of $\partial_E$ is equivalent with the formula $$\label{partialcug}
g_E(e,\partial_Ef)=\frac{1}{2}\sharp_Ee(f).$$
The index $E$ will be omitted if no confusion is possible.
The basic example of a Courant algebroid was studied in [@C] and it consists of the [*big tangent bundle*]{} $T^{big}M=TM\oplus
T^*M$, with the anchor $\sharp(X\oplus\alpha)=X$ and with $$\label{gC}
g(X_1\oplus\alpha_1,X_2\oplus\alpha_2)=\frac{1}{2}(\alpha_1(X_2)+\alpha_2(X_1)),$$ $$\label{crosetCou}
[X_1\oplus\alpha_1,X_2\oplus\alpha_2]=[X_1,X_2]\oplus
(L_{X_1}\alpha_2- L_{X_2}\alpha_1+\frac{1}{2}d(\alpha_1(X_2)
-\alpha_2(X_1))).$$ (The notation $X\oplus \alpha$ instead of the accurate $X+\alpha$ or $(X,\alpha)$ has the advantage of showing the place of the terms while avoiding some of the parentheses. The unindexed bracket of vector fields is the usual Lie bracket.)
Furthermore, let $\mathcal{F}$ be a foliation of the manifold $M$. We denote the tangent bundle $T\mathcal{F}$ by $F$ and define the transversal bundle $\nu\mathcal{F}$ by the exact sequence $$0\rightarrow F
\stackrel{\iota}{\rightarrow}TM\stackrel{\psi}{\rightarrow}\nu\mathcal{F}
\rightarrow0,$$ where $\iota$ is the inclusion and $\psi$ is the natural projection. We also fix a decomposition $$\label{descTM} TM=F\oplus
Q,\;Q=im(\varphi:\nu\mathcal{F}
\rightarrow TM),\,\psi\circ\varphi=id.,$$ which implies $$\label{descT*M} T^*M=Q^*\oplus
F^*,\;Q^*=ann\,F,\,F^*=ann\,Q\approx T^*M/ann\,F,$$ where the last isomorphism is induced by the transposed mapping $^t\iota$. The decompositions (\[descTM\]), (\[descT\*M\]) produce a bigrading $(p,q)$ of the Grassmann algebra bundles of multivector fields and exterior forms where $p$ is the $Q$-degree and $q$ is the $F$-degree [@V0].
The vector bundle $T^{big}\mathcal{F}=F\oplus (T^*M/ann\,F)$ is the big tangent bundle of the manifold $M^{\mathcal{F}}$, which is the set $M$ endowed with the differentiable structure of the sum of the leaves of $\mathcal{F}$. Hence, $T^{big}\mathcal{F}$ has the corresponding Courant structure (\[gC\]), (\[crosetCou\]). A cross section of $T^{big}\mathcal{F}$ may be represented as $Y\oplus\bar\alpha$ $(Y\in\chi(M^{\mathcal{F}}),\alpha\in
\Omega^1(M^{\mathcal{F}}))$, where the bar denotes the equivalence class of $\alpha$ modulo $ann\,F$ (this bar-notation is always used hereafter); generally, these cross sections are differentiable on the sum of leaves. If we consider $Y_{l}\oplus\bar\alpha_{l}$ ($l=1,2$) such that $Y_{l}\in\chi(M)$ and $\alpha_{l}\in\Omega^1(M)$ are differentiable with respect to the initial differentiable structure of $M$ we get the inner product and Courant bracket $$\label{gF} g_F(Y_1\oplus\bar\alpha_1,
Y_2\oplus\bar\alpha_2)=\frac{1}{2}(\alpha_1(Y_2)+\alpha_2(Y_1)),$$ $$\label{crosetF}
[Y_1\oplus\bar\alpha_1,Y_2\oplus\bar\alpha_2]
=([Y_1,Y_2]\oplus\overline{(L_{Y_1}\alpha_2-
L_{Y_2}\alpha_1+\frac{1}{2}d(\alpha_1(Y_2)-\alpha_2(Y_1))}),$$ where the results remain unchanged if $\alpha_{l}\mapsto
\alpha_{l}+\gamma_{l}$ with $\gamma_{l}\in ann\,F$. Formulas (\[gF\]), (\[crosetF\]) show that $T^{big}\mathcal{F}\rightarrow
M$, where $M$ has its initial differentiable structure, is a Courant algebroid with the anchor given by projection on the first term. Alternatively, we can prove the same result by starting with (\[gF\]), (\[crosetF\]) as definition formulas and by checking the axioms of a Courant algebroid by computation.
We will transfer the Courant structure of $T^{big}\mathcal{F}$ by the isomorphism $$\Phi=id\oplus\hspace{1pt}^t\hspace{-1pt}\iota:
F\oplus (T^*M/ann\,F)\rightarrow F\oplus ann\,Q,$$ i.e., $$\Phi(Y\oplus\bar\alpha)=Y\oplus\alpha_{0,1},\;\;(Y\in F,\alpha=
\alpha_{1,0}+\alpha_{0,1}\in T^*M).$$ This makes $F\oplus ann\,Q$ into a Courant algebroid, which we shall denote by $\mathcal{Q}=T^{big}_Q\mathcal{F}$, with the anchor equal to the projection on $F$, the metric given by (\[gF\]) and the bracket $$[Y_1\oplus\bar\alpha_1, Y_2\oplus\bar\alpha_2]_{\mathcal{Q}}
=[Y_1,Y_2]\oplus pr_{ann\,Q}(L_{Y_1}\alpha_2-
L_{Y_2}\alpha_1+\frac{1}{2}d(\alpha_1(Y_2)-\alpha_2(Y_1))$$ $\alpha_1,\alpha_2\in ann\,Q$. Using the formula $L_Y=i(Y)d+di(Y)$ and the well known decomposition $d=d'_{1,0}+d''_{0,1}+\partial_{2,-1}$ [@V0], the expression of the previous bracket becomes $$[Y_1\oplus\bar\alpha_1,
Y_2\oplus\bar\alpha_2]_{\mathcal{Q}}
=([Y_1,Y_2]\oplus(i(Y_1)d''\alpha_2-i(Y_2)d''\alpha_1$$ $$+\frac{1}{2}d''(i(Y_1)\alpha_2-i(Y_2)\alpha_1))\;\;(\alpha_1,\alpha_2\in
T^*_{0,1}M).$$
The extension theorem
=====================
Let $(M,\mathcal{F})$ be a foliated manifold. If the definition of a Courant algebroid is modified by asking the anchor to be a morphism $E\rightarrow\nu\mathcal{F}$, by asking $E,g,\sharp_E$ to be foliated, by asking only for a bracket $[\,,\,]_E:\Gamma_{fol}E\times\Gamma_{fol}E\rightarrow\Gamma_{fol}E$ and by asking the axioms to hold for foliated cross sections and functions, then, we get the notion of a [*transversal-Courant algebroid*]{} $(E,g_E,\sharp_E,[\,,\,]_E)$ over $(M,\mathcal{F})$ [@V1]. (The index $fol$ denotes foliated objects, i.e., objects that either project to or are a lift of a corresponding object of the space of leaves.)
On the other hand, a subbundle $B$ of a Courant algebroid $A$ over $(M,\mathcal{F})$ is a [*foliation*]{} of $A$ if: i) $B$ is $g_A$-isotropic and $\Gamma B$ is closed by $A$-brackets, ii) $\sharp_A(B)=T\mathcal{F}$, iii) if $C=B^{\perp_{g_A}}$, then the $A$-Courant structure induces the structure of a transversal-Courant algebroid on the vector bundle $C/B$; then, the pair $(A,B)$ is called a [*foliated Courant algebroid*]{} (see [@V1] for details).
In this section we prove the announced result:\
[**Theorem.**]{} [*Let $E$ be a transversal-Courant algebroid over the foliated manifold $(M,\mathcal{F})$ and let $Q$ be a complementary bundle of $F$ in $TM$. Then $E$ has a natural extension to a foliated Courant algebroid $A$ with a foliation $B$ isomorphic to $F$.*]{}
The proof of this theorem requires a lot of technical calculations. We will only sketch the path to be followed, leaving the actual calculations to the interested reader. We shall denote the natural extension that we wish to construct, and its operations, by the index $0$. Take $A_0=T^{big}_Q\mathcal{F}\oplus
E=\mathcal{Q}\oplus E$ with the metric $g_0=g_F\oplus g_E$ and the anchor $\sharp_0=pr_F\oplus\rho$, where $\rho=\varphi\circ\sharp_E$ with $\varphi$ defined by (\[descTM\]), therefore, $\psi\circ\rho=\sharp_E$. Notice that this implies $$\label{partial0}\partial_0\lambda=(0,\lambda|_F)+\frac{1}{2}
\sharp_{g_E}(\lambda\circ\rho)=(0,\lambda|_F)+\partial_E(\hspace{1pt}
^t\hspace{-1pt}\varphi\lambda)\;\;(\lambda\in T^*M)$$ and, in particular, $$\partial_0f =\partial_{\mathcal{Q}}(d''f)
\oplus\partial_E(d'f)=(0,d''f) \oplus\partial_E(d'f)\;\;(f\in
C^\infty(M)).$$
Then, inspired by the case $T^{big}M=
\mathcal{Q}\oplus\nu\mathcal{F}$ where the formulas below hold, we define the bracket of generating cross sections $Y\oplus\alpha\in\Gamma \mathcal{Q}$, $e\in\Gamma_{fol}E$ by $$\label{cr0} \begin{array}{l}
[Y_1\oplus\alpha_1,Y_2,\oplus\alpha_2]_0=
[Y_1\oplus\alpha_1,Y_2,\oplus\alpha_2]_{\mathcal{Q}}\vspace{2mm}\\
\hspace*{1cm}\oplus\frac{1}{2}\sharp_{g_E}((L_{Y_1}\alpha_2-
L_{Y_2}\alpha_1+\frac{1}{2}d(\alpha_1(Y_2)-\alpha_2(Y_1))\circ\rho)\vspace{2mm}\\
=([Y_1,Y_2]\oplus0)+\partial_0(L_{Y_1}\alpha_2-
L_{Y_2}\alpha_1+\frac{1}{2}d(\alpha_1(Y_2)-\alpha_2(Y_1)),\vspace{2mm}\\
[e,Y\oplus\alpha]_0= ([\rho e,Y]\oplus (L_{\rho e}\alpha)|_F)
\oplus\frac{1}{2}\sharp_{g_E}((L_{\rho e}\alpha)\circ\rho)\vspace{2mm}\\
\hspace*{1cm}=([\rho e,Y]\oplus0)+\partial_0L_{\rho e}\alpha),\vspace{2mm}\\
[e_1,e_2]_0=(([\rho e_1,\rho e_2]-\rho [e_1,e_2]_E)\oplus0) \oplus
[e_1,e_2]_E.\end{array}$$ The first term of the right hand side of the second formula belongs to $\Gamma \mathcal{Q}$ since $e\in\Gamma_{fol}E$ implies $[\rho e,Y]\in\Gamma F$. The first term of the right hand side of the third formula belongs to $\Gamma
\mathcal{Q}$ since we have $$\psi([\rho e_1,\rho e_2]-\rho [e_1,e_2]_E)=\psi([\rho e_1,\rho e_2])-
\sharp_E [e_1,e_2]_E$$ $$=\psi([\rho e_1,\rho e_2])-
[\sharp_Ee_1,\sharp_Ee_2]_{\nu\mathcal{F}}=0.$$
Furthermore, we extend the bracket (\[cr0\]) to arbitrary cross sections in agreement with the axiom 4) of Courant algebroids, i.e., for any functions $f,f_1,f_2\in C^\infty(M)$, we define $$\label{f} \begin{array}{l} [Y\oplus\alpha,fe]_0=
f[Y\oplus\alpha,e]_0\oplus(Yf)e,\vspace{2mm}\\
[f_1e_1,f_2e_2]_0=f_1f_2[e_1,e_2]_0+ f_1(\rho e_1(f_2))e_2
-f_2(\rho e_2(f_1))e_1\vspace{2mm}\\
\hspace*{2cm}-g_E(e_1,e_2)(f_1\partial_0f_2-f_2\partial_0f_1)\end{array}$$ ($Y\in\Gamma F,\alpha\in ann\,Q, e,e_1,e_2\in\Gamma_{fol}E$). It follows easily that formulas (\[cr0\]) and (\[f\]) give the same result if $f\in C^\infty_{fol}(M,\mathcal{F})$.
We have to check that the bracket defined by (\[cr0\]), (\[f\]) satisfies the axioms of a Courant algebroid and it is enough to do that for every possible combination of arguments of the form $Y\oplus\alpha\in
\mathcal{Q}$ and $fe$, $e\in\Gamma_{fol}E$, $f\in
C^\infty(M)$.
To check axiom 1), apply the anchor $\sharp_0=pr_F+\rho$ to each of the five formulas (\[cr0\]), (\[f\]) and use the transversal-Courant algebroid axioms satisfied by $E$. To check axiom 2), use formula (\[partial0\]). The required results follow straightforwardly. It is also easy to check axiom 4) from (\[f\]) and from axiom 4) for $\mathcal{Q}$ and $E$.
Furthermore, technical (lengthy) calculations show that if we have a bracket such that axioms 1), 2), 4) hold, then, if 5) holds for a triple of arguments, 5) also holds if the same arguments are multiplied by arbitrary functions. Therefore, in our case it suffices to check axiom 5) for the following six triples: (i) $(e,e_1,e_2)$, (ii) $(Y\oplus\alpha,e_1,e_2)$, (iii) $(e,Y\oplus\alpha,e')$, (iv) $(Y\oplus\alpha,Y'\oplus\alpha',e)$, (v) $(e,Y_1\oplus\alpha_1,Y_2\oplus\alpha_2)$, (vi) $(Y\oplus\alpha,Y_1\oplus\alpha_1,Y_2\oplus\alpha_2)$, where all $Y\oplus\alpha\in\Gamma\mathcal{Q}$ and all $e\in\Gamma_{fol}E$. In cases (i), (vi) the result follows from axiom 5) satisfied by $E,\mathcal{Q}$, respectively. In the other cases computations involving evaluations of Lie derivatives will do the job.
Finally, we have to check axiom 3). If we consider any vector bundle $E$ with an anchor and a bracket that satisfy axioms 1), 2), 4), 5), then, by applying axiom 5) to the triple $(e,
e_1=\partial f,e_2)$ $(f\in C^\infty(M))$ we get $$\label{crosetpt5} [e,\partial
f]_E=\frac{1}{2}\partial(\sharp_Ee(f)),$$ whence (using local coordinates, for instance) the following general formula follows $$\label{gen-partial} [e,\partial_E\alpha]_E=
\partial_E(L_{\sharp_Ee}\alpha-\frac{1}{2}d(\alpha(\sharp_Ee))).$$
Furthermore, assuming again that axioms 1), 2), 4), 5) hold and using (\[partialcug\]) and (\[crosetpt5\]) a lengthy but technical calculation shows that, if axiom 3) holds for a triple $(e_1,e_2,e_3)$, it also holds for $(e_1,e_2,fe_3)$ $(f\in
C^\infty(M))$ provided that $$\label{Erond} \begin{array}{c} \mathcal{E}:=
g([e_1,e_2],e_3)+\frac{1}{2}\sharp e_2(g(e_1,e_3))-
\frac{1}{2}\sharp
e_1(g(e_2,e_3))\vspace{2mm}\\=\frac{1}{3}\sum_{Cycl}g([e_1,e_2],e_3)\end{array}$$ ($:=$ denotes a definition). But, if the last two terms in $\mathcal{E}$ are expressed by axiom 5) for $E$ followed by (\[crosetpt5\]), and after we repeat the same procedure one more time, we get $$\mathcal{E}=\frac{1}{4}\sum_{Cycl}g([e_1,e_2],e_3)
+\frac{1}{4}\mathcal{E},$$ whence we see that (\[Erond\]) holds for any triple $(e_1,e_2,e_3)$.
Hence, it suffices to check axiom 3) for the following cases: (i) $(e_1,e_2,e_3)$, (ii) $(e_1,e_2,Y\oplus\alpha)$, (iii) $(e,Y_1\oplus\alpha_1,Y_2\oplus\alpha_2)$, (iv) $(Y_1\oplus\alpha_1,Y_2\oplus\alpha_2,Y_3\oplus\alpha_3)$, where all $Y\oplus\alpha\in\Gamma\mathcal{Q}$ and all $e\in\Gamma_{fol}E$. In case (i), using the second and third formula (\[cr0\]), we get $$[[e_1,e_2]_0,e_3]_0=([[\rho e_1,\rho e_2],\rho e_3]-
\rho[[e_1,e_2]_E,e]_E,0)\oplus[[e_1,e_2]_E,e]_E$$ and the required result follows in view of the Jacobi identity for vector fields and of axiom 3) for $E$ (in this case, the right hand side of axiom 3) for $A_0$ reduces to the one for $E$).
To check the result in the other cases simpler, we decompose $$Y\oplus\alpha=(Y\oplus0)+(0\oplus\alpha)$$ and check axiom 3) for each case induced by this decomposition.
For a triple $(e_1,e_2,Y\oplus0)$ the right hand side of axiom 3) is zero and the left-hand side is $$([[\rho e_1,\rho
e_2],Y]+[[\rho e_2,Y],\rho e_1]+[[Y,\rho e_1],\rho
e_2])\oplus0=0$$ by the Jacobi identity for vector fields.
For a triple $(e_1,e_2,0\oplus\alpha)$, after cancelations, the right hand side of axiom 3) becomes $(1/2)\alpha([\rho e_1,\rho
e_2])$. The same result is obtained for the left hand side if we use the second form of the first two brackets defined by (\[cr0\]) and formula (\[gen-partial\]).
For a triple $(e,Y_1\oplus0,Y_2\oplus0)$ the two sides of axiom 3) vanish (the left hand side reduces to the Jacobi identity for the vector fields $(\rho e,Y_1,Y_2)$), hence the axiom holds.
For a triple $(e,0\oplus\alpha_1,0\oplus\alpha_2)$, using the second form of the second bracket (\[cr0\]) and formula (\[gen-partial\]), axiom 3) reduces to $0=0$, i.e., the axiom holds.
For a triple $(e,Y_1\oplus0,0\oplus\alpha)$, if we notice that $0\oplus\alpha=\partial_0\alpha$ (see (\[partial0\])) and use (\[gen-partial\]), we see that the two sides of the equality required by axiom 3) are equal to $(1/2)\partial_0(\alpha([\rho
e,Y])-(1/2)\rho e(\alpha(Y)))$, hence the axiom holds.
The case $(Y_1\oplus0,Y_2\oplus0,Y_3\oplus0)$ is trivial. In the case $(Y_1\oplus0,Y_2\oplus0,0\oplus\alpha=\partial_0\alpha)$ similar computations give the value $(1/4)\partial_0(\alpha([Y_1,Y_2])-d\alpha(Y_1,Y_2))$ for the two sides of the corresponding expression of axiom 3). Finally, in the cases $(Y\oplus0,\partial_0\alpha_1,\partial_0\alpha_2)$ and $(\partial_0\alpha_1,\partial_0\alpha_2,\partial_0\alpha_3)$ the two sides of the required equality are $0$ since the image of $\partial_0$ is isotropic and the restriction of the bracket to this image is zero (use axiom 2) and formula (\[gen-partial\])).
[xx]{} T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 361-661. Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Diff. Geom., 45 (1997), 547-574. I. Vaisman, Cohomology and Differential Forms, M. Dekker, Inc., New York, 1973. I. Vaisman, Foliated Lie and Courant Algebroids, Mediterranean J. of Math. (to appear) arXiv:0902.1296\[math.DG\].
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abstract: 'User-generated item lists are popular on many platforms. Examples include video-based playlists on YouTube, image-based lists (or “boards”) on Pinterest, book-based lists on Goodreads, and answer-based lists on question-answer forums like Zhihu. As users create these lists, a common challenge is in identifying what items to curate next. Some lists are organized around particular genres or topics, while others are seemingly incoherent, reflecting individual preferences for what items belong together. Furthermore, this heterogeneity in item consistency may vary from platform to platform, and from sub-community to sub-community. Hence, this paper proposes a generalizable approach for user-generated item list continuation. Complementary to methods that exploit specific content patterns (e.g., as in song-based playlists that rely on audio features), the proposed approach models the consistency of item lists based on human curation patterns, and so can be deployed across a wide range of varying item types (e.g., videos, images, books). A key contribution is in intelligently combining two preference models via a novel consistency-aware gating network – a general user preference model that captures a user’s overall interests, and a current preference priority model that captures a user’s current (as of the most recent item) interests. In this way, the proposed consistency-aware recommender can dynamically adapt as user preferences evolve. Evaluation over four datasets (of songs, books, and answers) confirms these observations and demonstrates the effectiveness of the proposed model versus state-of-the-art alternatives. Further, all code and data are available at https://github.com/heyunh2015/ListContinuation\_WSDM2020.'
author:
- Yun He
- Yin Zhang
- Weiwen Liu
- James Caverlee
bibliography:
- 'sample-bibliography.bib'
title: 'Consistency-Aware Recommendation for User-Generated Item List Continuation'
---
<ccs2012> <concept> <concept\_id>10002951.10003317.10003347.10003350</concept\_id> <concept\_desc>Information systems Recommender systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
This work is supported in part by NSF (\#IIS-$1841138$).
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---
abstract: 'In this paper, we present a parallel computing method for the coupled-cluster singles and doubles (CCSD) in periodic systems. The CCSD in periodic systems solves simultaneous equations for single-excitation and double-excitation amplitudes. In the simultaneous equations for double-excitation amplitudes, each equations are characterized by four spin orbitals and three independent momentums of electrons. One of key ideas of our method is to use process numbers in parallel computing to identify two indices which represent momentum of an electron. When momentum of an electron takes $N_ {\bm{k}}$ values, $N_ {\bm{k}}^2$ processes are prepared in our method. Such parallel distribution processing and way of distribution of known quantities in the simultaneous equations reduces orders of computational cost and required memory scales by $N_ {\bm{k}}^2$ compared with a sequential method. In implementation of out method, communication between processes in parallel computing appears in the outmost loop in a nested loop but does not appear inner nested loop.'
author:
- Takumi Yamashita
- Taichi Kosugi
- 'Yu-ichiro Matsushita'
- Tetsuya Sakurai
title: 'A Parallel Computing Method for the Coupled-Cluster Singles and Doubles'
---
= 10pt
Introduction {#Introduction}
============
The development of high-precision first-principles calculation methodologies stays an important research subject for the theoretical condensed-matter physics. Reflecting the recent growth of computational power, the wavefunction theories are getting attention not only in the quantum chemistry communities but also in the physics ones because they provide typically systematic ways to improve the accuracy, compared to the development of exchange correlation functionals for DFT calculations[@HK64; @KS65]. For example, the density-matrix-renormalization group (DMRG)[@White92] has been applied to isolated molecular systems such as H$_2$O [@WM99]. The transcorrelated method[@BH69] was reported to be applied to uniform electron gases[@UTOSC05] and semiconductors[@OAT17]. Self-energy functional theory[@Potthoff03; @Potthoff14] has recently been applied to isolated transition metal atoms[@KNFM18]. The coupled-cluster singles-and-doubles (CCSD)[@HJO00] has been applied to electron gases[@MLWMRLBC16] and various periodic systems[@MSCB17]. Furthermore, the one-particle spectra from the Green’s functions (GFs) in CCSD method (GFCCSD) have also been reported for realistic systems[@KNFM18; @FKNM18; @NKFM18; @PK18]. The relation between the equation of motion (EOM)-CCSD, a part of the GFCCSD procedure, and the $GW$ method[@Hedin65], which is already known as a high-precision calculation method in condensed-matter physics, has been examined in detail, suggesting the high potential of CCSD-based methods compared with the $GW$ method[@LB18].
However, while the CCSD method in periodic systems is expected as a high-precision calculation, its large computational cost and large required memory capacity are an obstacle for its practical applications. Actually, the computational cost is $N_{\rm band}^6 N_{\bm{k}}^{4}$, with $N_{\bm{k}}$ being the number of $k$ points and $N_{\rm band}$ the number of bands. For the required memory regions, the bottlenecks are the antisymmetrized two-electron integrals and double-excitation amplitudes: Required memory space scales as $N_{\rm band}^4N_k^3$. In this context, an interpolation method has been developed to reduce the computational costs. In fact, a Wannier interpolation technique which interpolates $k$ points for self-energy has been developed [@KM19]. On the other hand, a straightforward approach to the problem of the required large memory space is large-scale memory distribution using supercomputers. Actually, the usage of supercomputers is effective for shortening the calculation time as well. Thus, the algorithm development of minimizing the communication time in accordance with large-scale memory distribution is a serious problem.
In this study, we developed a new method for the CCSD method in periodic systems. It enables efficient memory distribution in large-scale parallel computing suppressing the communication time compared with a naïve parallel computing implementation. Actually, required memory space in each process can be suppressed to $N_{\rm band}^4 N_{\bm{k}}$ by using this new method. Note that this new method uses key ideas and some techniques for a parallel computing method [@YS19] for the Higher Order Tensor Renormalization Group (HOTRG) [@XCQZYX12]. This method for the HOTRG has successfully been applied [@AKYY19] to investigation of phase transition in the four-dimensional Ising model.
This paper is organized as follows. In Section \[BasicEqs\], basic equation as a start point of the presented method is explained. In Section \[OutlineBasicEq\], an outline of the presented method is described and rearranged simultaneous equations for the presented method are given. In Section \[ImplementMethod\], implementation of the presented method is described. Section \[Conclusion\] is devoted to conclusion.
Basic equations in the CCSD in a periodic system {#BasicEqs}
================================================
In the CCSD in a periodic system, basic equations are simultaneous equations. Unknown quantities are as follows:
- single-excitation amplitudes $t_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}$
- double-excitation amplitudes $t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$.
Known quantities are as follows:
- the matrix elements of the Fock operator $f_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}$, $f_{p \bar{\bm{k}} q \bar{\bm{k}}}$ and $f^{g \bar{\bm{k}} h \bar{\bm{k}}}$
- antisymmetrized two-electron integrals $\bra{w \bm{k}_w x \bm{k}_x} \ket{y \bm{k}_y z \bm{k}_z}_{\phi_w \phi_x \phi_y \phi_z}$
A lower case letter represents a spin orbital. If it is a subscript (superscript), then the spin orbital is occupied (virtual). For antisymmetrized two-electron integrals, if a symbol $\phi_u$ $( u: w, x, y, z )$ is “$o$” (“$v$”), then the spin orbital $u$ is occupied (virtual). A symbol $\bm{k}$ with an index represents momentum of an electron in the spin orbital which is represented by the index. Single-excitation amplitudes and the matrix elements of the Fock operator consider two spin orbitals. They may have a nonzero value only when momentums of an electron in the two spin orbitals have the same value because of conservation law of momentum. Then, a symbol $\bar{\bm{k}}$ represents the common momentum of an electron in these two spin orbitals.
Introducing momentum to the basic equation of CCSD given in [@GS95], we have simultaneous equations similarly to [@HPTB04] for single-excitation amplitudes as $$\begin{aligned}
0 = & f_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}
+ \sum_c \tilde{\mathcal{F}}^{g \bar{\bm{k}} c \bar{\bm{k}}} t_{p \bar{\bm{k}}}^{c \bar{\bm{k}}}
- \sum_r t_{r \bar{\bm{k}}}^{g \bar{\bm{k}}} \tilde{\mathcal{F}}_{r \bar{\bm{k}} p \bar{\bm{k}}}
+ \sum_{n, f, \hat{\bm{k}}} \tilde{\mathcal{F}}_{n \hat{\bm{k}}}^{f \hat{\bm{k}}} t_{p \bar{\bm{k}} n \hat{\bm{k}}}^{g \bar{\bm{k}} f \hat{\bm{k}}} \notag \\
&+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{g \bar{\bm{k}} r \hat{\bm{k}}} \ket{p \bar{\bm{k}} c \hat{\bm{k}}}_{voov} \notag \\
&- \frac{1}{2} \sum_{m, \bm{k}_m, n, \bm{k}_n, f} t_{m \bm{k}_m n \bm{k}_n}^{g \bar{\bm{k}} f \bm{k}_f} \bra{m \bm{k}_m n \bm{k}_n} \ket{p \bar{\bm{k}} f \bm{k}_f}_{ooov} \notag \\
&+ \frac{1}{2} \sum_{n, \bm{k}_n, e, \bm{k}_e, f} t_{p \bar{\bm{k}} n \bm{k}_n}^{e \bm{k}_e f \bm{k}_f} \bra{g \bar{\bm{k}} n \bm{k}_n} \ket{e \bm{k}_e f \bm{k}_f}_{vovv}, \label{ori_EqSingle}
\end{aligned}$$ and for double-excitation amplitudes as $$\begin{aligned}
0 = & \bra{a \bm{k}_a b \bm{k}_b} \ket{i \bm{k}_i j \bm{k}_j}_{vvoo} \notag \\
&+ P_- (ab) \sum_f t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a f \bm{k}_b} \left( \tilde{\mathcal{F}}^{b \bm{k}_b f \bm{k}_b} - \frac{1}{2} \sum_r t_{r \bm{k}_b}^{b \bm{k}_b} \tilde{\mathcal{F}}_{r \bm{k}_b}^{f \bm{k}_b} \right) \notag \\
&- P_- (ij) \sum_n t_{i \bm{k}_i n \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} \left( \tilde{\mathcal{F}}_{n \bm{k}_j j \bm{k}_j} + \frac{1}{2} \sum_c t_{j \bm{k}_j}^{c \bm{k}_j} \tilde{\mathcal{F}}_{n \bm{k}_j}^{c \bm{k}_j} \right) \notag \\
&+ \frac{1}{2} \sum_{m, \bm{k}_m, n} \tau_{m \bm{k}_m n \bm{k}_n}^{a \bm{k}_a b \bm{k}_b} \tilde{\mathcal{W}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}
+ \frac{1}{2} \sum_{e, \bm{k}_e, f} \tilde{\mathcal{W}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f} \tau_{i \bm{k}_i j \bm{k}_j}^{e \bm{k}_e f \bm{k}_f} \notag \\
&+ P_- (ab) P_- (ij) \left( \sum_{n, \bm{k}_n, f} t_{i \bm{k}_i n \bm{k}_n}^{a \bm{k}_a f \bm{k}_f} \tilde{\mathcal{W}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f} \right. \notag \\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \left. - \sum_{r, c} t_{i \bm{k}_i}^{c \bm{k}_i} t_{r \bm{k}_a}^{a \bm{k}_a} \bra{r \bm{k}_a b \bm{k}_b} \ket{c \bm{k}_i j \bm{k}_j}_{ovvo} \right) \notag \\
&+ P_- (ij) \sum_c \bra{a \bm{k}_a b \bm{k}_b} \ket{c \bm{k}_i j \bm{k}_j}_{vvvo} t_{i \bm{k}_i}^{c \bm{k}_i} \notag \\
&- P_- (ab) \sum_r t_{r \bm{k}_a}^{a \bm{k}_a} \bra{r \bm{k}_a b \bm{k}_b} \ket{c \bm{k}_i j \bm{k}_j}_{ovoo}, \label{ori_EqDouble}
\end{aligned}$$ where $P_- ()$ is an operator, $\tilde{\mathcal{W}}$ and $\tilde{\mathcal{F}}$ are intermediates and $\tau$ are effective double-excitation amplitudes. The operator $P_- ()$ acts on a quantity $Z$ as $$P_- (rs) Z(\cdots r \bm{k}_r s \bm{k}_s \cdots ) = Z(\cdots r \bm{k}_r s \bm{k}_s \cdots ) - Z(\cdots s \bm{k}_s r \bm{k}_r \cdots ).$$ The intermediates $\tilde{\mathcal{W}}$ are defined as $$\begin{aligned}
\tilde{\mathcal{W}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j} =& \bra{m\bm{k}_m n \bm{k}_n} \ket{i \bm{k}_i j \bm{k}_j}_{oooo} \notag \\
&+ P_- (ij) \sum_c t_{j \bm{k}_j}^{c \bm{k}_j} \bra{m\bm{k}_m n \bm{k}_n} \ket{i \bm{k}_i c \bm{k}_j}_{ooov} \notag \\
&+ \frac{1}{4} \sum_{e, \bm{k}_e, f} \tau_{i \bm{k}_i j \bm{k}_j}^{e \bm{k}_e f \bm{k}_f} \bra{m\bm{k}_m n \bm{k}_n} \ket{e \bm{k}_e f \bm{k}_f}_{oovv}, \\
\tilde{\mathcal{W}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f} =& \bra{a \bm{k}_a b \bm{k}_b} \ket{e \bm{k}_e f \bm{k}_f}_{vvvv} \notag \\
&- P_- (ab) \sum_r t_{r \bm{k}_b}^{b \bm{k}_b} \bra{a \bm{k}_a r \bm{k}_b} \ket{e \bm{k}_e f \bm{k}_f}_{vovv} \notag \\
&+ \frac{1}{4} \sum_{m, \bm{k}_m, n} \tau_{m\bm{k}_m n \bm{k}_n}^{a \bm{k}_a b \bm{k}_b} \bra{m\bm{k}_m n \bm{k}_n} \ket{e \bm{k}_e f \bm{k}_f}_{oovv}, \\
\tilde{\mathcal{W}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f} =& \bra{n \bm{k}_n b \bm{k}_b} \ket{f \bm{k}_f j \bm{k}_j}_{ovvo} \notag \\
&+ \sum_c t_{j \bm{k}_j}^{c \bm{k}_j} \bra{n \bm{k}_n b \bm{k}_b} \ket{f \bm{k}_f c \bm{k}_j}_{ovvv} \notag \\
&- \sum_r t_{r \bm{k}_b}^{b \bm{k}_b} \bra{n \bm{k}_n r \bm{k}_r} \ket{f \bm{k}_f j \bm{k}_j}_{oovo} \notag \\
&- \frac{1}{2} \sum_{x, \bm{k}_x, y} t_{j \bm{k}_j x \bm{k}_x}^{y \bm{k}_y b \bm{k}_b} \bra{n \bm{k}_n x \bm{k}_x} \ket{f \bm{k}_f y \bm{k}_y}_{oovv} \notag \\
&- \sum_c \sum_r t_{j \bm{k}_j}^{c \bm{k}_j} t_{r \bm{k}_b}^{b \bm{k}_b} \bra{n \bm{k}_n r \bm{k}_b} \ket{f \bm{k}_f c \bm{k}_j}_{oovv}. \label{Def_Woovv}
\end{aligned}$$ The intermediates $\tilde{\mathcal{F}}$ are defined as $$\begin{aligned}
\tilde{\mathcal{F}}_{p \bar{\bm{k}} q \bar{\bm{k}}} =& f_{p \bar{\bm{k}} q \bar{\bm{k}}}
+ \frac{1}{2} \sum_c f_{p \bar{\bm{k}}}^{c \bar{\bm{k}}} t_{q \bar{\bm{k}}}^{c \bar{\bm{k}}}
+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{p \bar{\bm{k}} r \hat{\bm{k}}} \ket{q \bar{\bm{k}} c \hat{\bm{k}}}_{ooov} \notag \\
&+ \frac{1}{2} \sum_{n, \bm{k}_n, e, \bm{k}_e, f} \tilde{\tau}_{q \bar{\bm{k}} n \bm{k}_n}^{e \bm{k}_e f \bm{k}_f} \bra{p \bar{\bm{k}} n \bm{k}_n} \ket{e \bm{k}_e f \bm{k}_f}_{oovv} , \label{ori_F_oo} \\
\tilde{\mathcal{F}}^{g \bar{\bm{k}} h \bar{\bm{k}}} =& f_{g \bar{\bm{k}} h \bar{\bm{k}}}
- \frac{1}{2} \sum_r f_{r \bar{\bm{k}}}^{h \bar{\bm{k}}} t_{r \bar{\bm{k}}}^{g \bar{\bm{k}}}
+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{g \bar{\bm{k}} r \hat{\bm{k}}} \ket{h \bar{\bm{k}} c \hat{\bm{k}}}_{ooov} \notag \\
&- \frac{1}{2} \sum_{m, \bm{k}_m, n, \bm{k}_n, f} \tilde{\tau}_{m \bm{k}_m n \bm{k}_n}^{g \bar{\bm{k}} f \bm{k}_f} \bra{m \bm{k}_m n \bm{k}_n} \ket{h \bar{\bm{k}} f \bm{k}_f}_{oovv} , \label{ori_F_vv} \\
\tilde{\mathcal{F}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}}} =& f_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}
+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{p \bar{\bm{k}} r \hat{\bm{k}}} \ket{g \bar{\bm{k}} c \hat{\bm{k}}}_{oovv}, \label{ori_F_ov}
\end{aligned}$$ where $\tilde{\tau}$ are effective double-excitation amplitudes $$\tilde{\tau}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} = t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} + \frac{1}{4} P_- (ab) P_- (ij) t_{i \bm{k}_i}^{a \bm{k}_a} t_{ j \bm{k}_j}^{b \bm{k}_b}.$$ The effective double excitation amplitudes are given as $$\tau_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} = t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} + \frac{1}{2} P_- (ab) P_- (ij) t_{i \bm{k}_i}^{a \bm{k}_a} t_{ j \bm{k}_j}^{b \bm{k}_b}.$$
The number of equations in the simultaneous equations for single-excitation amplitudes is $N_{\textrm{occ}} N_{\textrm{vir}} N_{\bm{k}}$, where $N_{\textrm{occ}}$, $N_{\textrm{vir}}$ and $N_{\bm{k}}$ are the number of occupied spin orbitals, that of virtual spin orbitals, and that of possible momenta of an electron, respectively. Double-excitation amplitudes and antisymmetrized two-electron integrals may have a nonzero value only when the four momentums which identify these quantities satisfy conservation law of momentum. Then, the number of equations in the simultaneous equations for double-excitation amplitudes is $N_{\textrm{occ}}^2 N_{\textrm{vir}}^2 N_{\bm{k}}^3$.
Outlines and rearranged simultaneous equations of the presented method {#OutlineBasicEq}
======================================================================
In this section, we describe outlines of the presented method and give its basic equations. The basic equations are obtained by rearranging the basic method shown in the previous section.
In Section \[MethodOutline\], outlines of the presented method is described. In Section \[MethodPreconditions\], preconditions in the presented method are given. In Section \[RearrangedEq\], rearranged simultaneous equations for the presented method are given.
Outlines of the presented method {#MethodOutline}
--------------------------------
In numerical computation of the CCSD in periodic systems, handling of quantities which have eight indices may become a problem. In a naïve parallel computing implementation, we will be suffered from a large amount of communication of such quantities between parallel computing processes. To avoid this problem, we use key ideas and some techniques for implementation developed for algorithm for the Higher Order Tensor Renormalization Group method (HOTRG) [@XCQZYX12] which is described in a paper in preparation by T. Y. and T. S. [@YS19] to be shown in another place. Akiyama, Kuramashi, T. Y. and Yoshimura [@AKYY19] argue phase transition of the four-dimensional Ising model using codes basically based on this algorithm. The key ideas for the HOTRG are to utilize process numbers in parallel computing to identify some indices of tensors and these indices should [*not be contracted*]{} during considering step in computation. The quantities which appear the basic equation of the CCSD in a periodic system can be regarded as tensor elements. Then, in the method we present in this paper, the process numbers are utilized to identify two of the eight indices. Slightly different from the case of the HOTRG, there are cases such that there exists only one index which is not contracted during considering step in computation. Then, a principal for choice of indices which are identified through a process number in the CCSD in periodic systems is that we should [*preferentially*]{} choose indices which [*are not contracted*]{} during considering step in computation. In addition to this principal, we consider conservation law of momentum since this consideration gives a good perspective for design of a method for communication between parallel computing processes. As a conclusion, two indices which represent momentum of an electron are identified though a process number. In the presented method, the $k$ points are represented by one of the numbers $0, 1, ..., N_{\bm{k}} - 1$. Then, our method prepares $N_{\bm{k}}^2$ processes. When the process number of a process is represented as $\alpha + \beta N_{\bm{k}}$ $( \alpha , \beta = 0, 1, ..., N_{\bm{k}} - 1 )$, the identified momentums $\bm{k}_1$ and $\bm{k}_2$ are $\bm{k}_1 = \alpha$ and $\bm{k}_2 = \beta$. Some techniques which are not used in the HOTRG are introduced in this paper.
We consider computational cost and required memory space. For simplicity, we consider a case such that the numbers of occupied and virtual bands are equal. Let us denote these numbers by $N_{\textrm{band}}$. In a naïve implementation, computational cost and required memory space are $O ( N_{\textrm{band}}^6 N_{\bm{k}}^4 )$ and $O ( N_{\textrm{band}}^4 N_{\bm{k}}^3 )$, respectively, because of conservation law of momentum. In the presented method, computational cost is $O ( N_{\textrm{band}}^6 N_{\bm{k}}^2 )$ and required memory space in each process is $O ( N_{\textrm{band}}^4 N_{\bm{k}} )$.
Preconditions {#MethodPreconditions}
-------------
In this section, preconditions in the presented method are described.
### Processes in parallel computing
As mentioned in outlines, $N_k^2$ processes are used in parallel computing. A process number can be represented by two integers as $\alpha + \beta N_{\bm{k}}$ $( \alpha , \beta = 0, 1, ..., N_{\bm{k}} - 1 )$. Each of $\alpha$ and $\beta$ identifies the momentum of an electron. An image shown in Fig. \[Fig\_Processes\] may help understanding of the presented method. Each box represents a process. Rows and columns correspond to $\alpha$ and $\beta$, respectively. A box at the intersection of each row and each column represents a process whose number is $\alpha + \beta N_{\bm{k}}$.
![Processes[]{data-label="Fig_Processes"}](PDF_Processes){width="30.00000%"}
During computing, there are cases such that we consider only processes whose process numbers are represented as $p + p N_{\bm{k}}$ $( p = 0, 1, ..., N_{\bm{k}} - 1 )$. See Fig. \[Fig\_Diag\_Process\]. Let us call these processes “the diagonal processes”.
![The diagonal processes[]{data-label="Fig_Diag_Process"}](PDF_Diag_Process){width="50.00000%"}
When we represent process numbers as $\alpha + \beta N_{\bm{k}}$, there are cases such that we want to handle processes which have the same $\alpha$ or $\beta$ as a group. For this purpose, let us introduce “direction” into the diagram shown in shown in Fig. \[Fig\_Processes\]. Horizontal and vertical directions are introduced as shown in Fig. \[Fig\_direction\].
![Directions[]{data-label="Fig_direction"}](PDF_direction){width="60.00000%"}
### Inputs and outputs {#IO}
Inputs are the known quantities and the initial values of the unknown quantities of simultaneous equations. The matrix elements of the Fock operator $f_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}$, $f_{p \bar{\bm{k}} q \bar{\bm{k}}}$ and $f^{g \bar{\bm{k}} h \bar{\bm{k}}}$ $( \bar{\bm{k}} = 0, 1, ..., N_{\bm{k}} - 1 )$ are stored in the diagonal process whose number is $\bar{\bm{k}} + \bar{\bm{k}} N_{\bm{k}}$. See Fig. \[Fig\_Diag\_Process\]. Antisymmetrized two-electron integrals are distributed to each process according to two of four indices which represent momentum of an electron. There are plural choices for such two indices and there is no necessity that one integral is distributed to only one process. The integrals are distributed to each process as shown in Table \[Table\_TEI\]. In the cases such that the integrals are distributed in plural ways, superscripts are attached to the integrals for identification. The symbol “$o$" (“$v$") means that corresponding index represents an occupied (a virtual) spin orbital.
Integrals Process number
-------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{oooo}$ $\bm{k}_{\gamma} + \bm{k}_{\delta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{ooov}^{[oo]}$ $\bm{k}_{\alpha} + \bm{k}_{\gamma} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{ooov}^{[ov]}$ $\bm{k}_{\gamma} + \bm{k}_{\delta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{oovv}^{[oo]}$ $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{oovv}^{[ov]}$ $\bm{k}_{\alpha} + \bm{k}_{\gamma} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{oovv}^{[vv]}$ $\bm{k}_{\gamma} + \bm{k}_{\delta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{ovvo}$ $\bm{k}_{\delta} + \bm{k}_{\beta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{ovvv}^{[ov]}$ $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{ovvv}^{[vv]}$ $\bm{k}_{\beta} + \bm{k}_{\delta} N_{\bm{k}}$
$\bra{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}} \ket{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}_{vvvv} $ $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$
: Distribution of antisymmetrized two-electron integrals to each process[]{data-label="Table_TEI"}
We can set the initial values of the unknown quantities of simultaneous equations, single-excitation and double-excitation amplitudes, to arbitrary values. For setting the initial values of single-excitation and double-excitation amplitudes in our implementation, we rewrite the equations to be solved to a form $t=f(t)$, where $t$ represents the amplitudes collectively. We set the initial amplitudes $t_0$ to the right-hand side with $t = 0$, that is, $t_0$ = $f(0)$. Outputs are these unknown quantities.
### Parallel computing of the right-hand sides of simultaneous equations {#RHS_abstract}
In the presented method, the basic equations are rearranged to another form of simultaneous equations described below. The rearranged simultaneous equations for single-excitation amplitudes (\[EqSingle\]) shown below can be conceptionally written as $$\label{Eq_R1}
0 = ( R_1 )_{g \bar{\bm{k}}}^{p \bar{\bm{k}}}.$$ Those for double-excitation amplitudes (\[EqDouble\]) shown below can be conceptionally written as $$\label{Eq_R2}
0 = ( R_2 )_{i {\bm{k}}_i j {\bm{k}}_j}^{a {\bm{k}}_a b {\bm{k}}_b}.$$ Intermediate results during computation and the final results in computation of $R_1$ are stored in the diagonal process whose process number is $\bar{\bm{k}} + \bar{\bm{k}} N_{\bm{k}}$. Those in computation of $R_2$ are stored in the process whose process number is $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. See Fig. \[Fig\_RHS\].
![Parallel computing of the right-hand sides of the simultaneous equations[]{data-label="Fig_RHS"}](PDF_RHS){width="80.00000%"}
### Distribution of single-excitation and double-excitation amplitudes to processes {#DistrbutionAmp}
Single-excitation amplitudes $t_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}$ are stored in all the processes. Those of double-excitation amplitudes $t_{i {\bm{k}}_i j {\bm{k}}_j}^{a {\bm{k}}_a b {\bm{k}}_b}$ are distributedly stored in a process according to the momentums ${\bm{k}}_i$ and ${\bm{k}}_a$. They are stored in a process whose process number is $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. See Fig. \[Fig\_t2\_stored\].
![Processes in which double-excitation amplitudes are stored[]{data-label="Fig_t2_stored"}](PDF_t2_stored){width="30.00000%"}
### Conservation law of momentum
Throughout the present algorithm, the momentum conservation in a periodic system has to be taken into account. Specifically, the equivalence of $\bm{k}_1 + \bm{k}_2$ and $\bm{k}_3 + \bm{k}_4$ means that there exists a reciprocal lattice vector $\bm{G}$ such that $\bm{k}_1 + \bm{k}_2 = \bm{k}_3 + \bm{k}_4 + \bm{G}$. If one adopt a regular mesh containing $\bm{k}= 0$ in the reciprocal space as usual, there will be no difficulty in finding the momentum from given three momenta.
Rearrangement of the basic equations in the CCSD in a periodic system {#RearrangedEq}
---------------------------------------------------------------------
In this section, we present simultaneous equations which are suitable for parallel computing in the presented method. These simultaneous equations are obtained by rearranging the basic equations and the intermediates shown in Section \[BasicEqs\]. The order of items in the right-hand sides in the simultaneous equations in Section \[BasicEqs\] for single-excitation and double-excitation amplitudes are changed. A few intermediates are newly introduced into the rearranged equations. Some of intermediates used in Section \[BasicEqs\] are decomposed, namely, are not used.
For antisymmetrized two-electron integrals, we apply the following relations $$\bra{p \bm{k}_p q \bm{k}_q} \ket{r \bm{k}_r s \bm{k}_s} = \bra{r \bm{k}_r s \bm{k}_s} \ket{p \bm{k}_p q \bm{k}_q}^*$$ and $$\begin{aligned}
\bra{p \bm{k}_p q \bm{k}_q} \ket{r \bm{k}_r s \bm{k}_s}
&= - \bra{q \bm{k}_q p \bm{k}_p} \ket{r \bm{k}_r s \bm{k}_s} \notag \\
&= - \bra{p \bm{k}_p q \bm{k}_q} \ket{s \bm{k}_s r \bm{k}_r}
= \bra{q \bm{k}_q p \bm{k}_p} \ket{s \bm{k}_s r \bm{k}_r}.
\end{aligned}$$
The rearrangement of the basic equations are described as follows. In Section \[introduce\_rho\], intermediates $\check{\rho}$ and $\hat{\rho}$ are newly introduced. In section \[arrange\_F\], rearrangement of the intermediates $\tilde{\mathcal{F}}$ is described. In Section \[Sec\_EqSingle\] and \[Sec\_EqSingle\], rearranged simultaneous equations for single-excitation and double-excitation amplitudes are given, respectively.
### Introduction of new intermediates $\check{\rho}$ and $\hat{\rho}$ {#introduce_rho}
Intermediates $\check{\rho}$ and $\hat{\rho}$ are newly introduced. In our parallel computing method, they depend on a process in which they are computed. When a process number is expressed as $\bm{k}_{\alpha} + N_{\bm{k}} \bm{k}_{\beta}$ $( \bm{k}_{\alpha}, \bm{k}_{\beta} = 0, 1, ..., N_{\bm{k}} - 1 )$, definitions of these intermediates are $$\begin{gathered}
\check{\rho}_p^g ( \bm{k}_{\alpha} ) = t_{p \bm{k}_{\alpha}}^{g \bm{k}_{\alpha}}, \label{def_hat_rho} \\
\hat{\rho}_p^g ( \bm{k}_{\beta} ) = t_{p\bm{k}_{\beta}}^{g \bm{k}_{\beta}}. \label{def_check_rho}
\end{gathered}$$
### Rearrangement of the intermediates $\tilde{\mathcal{F}}$ {#arrange_F}
The intermediates $\tilde{\mathcal{F}}$ are rearranged from (\[ori\_F\_oo\]), (\[ori\_F\_vv\]) and (\[ori\_F\_ov\]) as $$\begin{aligned}
\tilde{\mathcal{F}}_{p \bar{\bm{k}} q \bar{\bm{k}}}
=& f_{p \bar{\bm{k}} q \bar{\bm{k}}}
+ \frac{1}{2} \sum_c \check{\rho}_q^c ( \bar{\bm{k}} ) \left( f_{p \bar{\bm{k}}}^{c \bar{\bm{k}}} - \tilde{\mathcal{K}}_{p \bar{\bm{k}}}^{c \bar{\bm{k}}} \right) \notag \\
&+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{p \bar{\bm{k}} r \hat{\bm{k}}} \ket{q \bar{\bm{k}} c \hat{\bm{k}}}_{ooov}^{[oo]}
+ \sum_{\bm{k}_e} \tilde{\mathcal{Z}}_{p \bar{\bm{k}} q \bar{\bm{k}}}^{\bm{k}_e}, \label{def_F_oo} \\
\tilde{\mathcal{F}}^{g \bar{\bm{k}} h \bar{\bm{k}}}
=& f^{g \bar{\bm{k}} h \bar{\bm{k}}}
- \frac{1}{2} \sum_r \hat{\rho}_r^g ( \bar{\bm{k}} ) \left( f_{r \bar{\bm{k}}}^{h \bar{\bm{k}}} + \tilde{\mathcal{K}}_{r \bar{\bm{k}}}^{h \bar{\bm{k}}} \right) \notag \\
&+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{r \hat{\bm{k}} g \bar{\bm{k}}} \ket{c \hat{\bm{k}} h \bar{\bm{k}}}_{ovvv}^{[vv]}
- \sum_{\bm{k}_m} \tilde{\mathcal{Z}}_{\bm{k}_m}^{g \bar{\bm{k}} h \bar{\bm{k}}}, \label{def_F_vv} \\
\tilde{\mathcal{F}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}}} =& f_{p \bar{\bm{k}}}^{g \bar{\bm{k}}} + \tilde{\mathcal{K}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}, \label{def_F_ov}
\end{aligned}$$ where $ \tilde{\mathcal{Z}}$ and $ \tilde{\mathcal{K}}$ are newly introduced intermediates defined as $$\begin{gathered}
\tilde{\mathcal{Z}}_{p \bar{\bm{k}} q \bar{\bm{k}}}^{\bm{k}_e} = \frac{1}{2} \sum_e \sum_{n, \bm{k}_n, f} t_{q \bar{\bm{k}} n \bm{k}_n}^{e \bm{k}_e f \bm{k}_f} \bra{p \bar{\bm{k}} n \bm{k}_n} \ket{e \bm{k}_e f \bm{k}_f}_{oovv}^{[ov]} , \label{def_Z_oo} \\
\tilde{\mathcal{Z}}_{\bm{k}_m}^{g \bar{\bm{k}} h \bar{\bm{k}}} = \frac{1}{2} \sum_m \sum_{n, \bm{k}_n, f} t_{m \bm{k}_m n \bm{k}_n}^{g \bar{\bm{k}} f \bm{k}_f} \bra{m \bm{k}_m n \bm{k}_n} \ket{h \bar{\bm{k}} f \bm{k}_f}_{oovv}^{[ov]} , \label{def_Z_vv} \\
\tilde{\mathcal{K}}_{x \bar{\bm{k}}}^{y \bar{\bm{k}}} = \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{x \bar{\bm{k}} r \hat{\bm{k}}} \ket{y \bar{\bm{k}} c \hat{\bm{k}}}_{oovv}^{[ov]}. \label{def_K}
\end{gathered}$$
### The rearranged simultaneous equations for single-excitation amplitudes {#Sec_EqSingle}
The rearranged simultaneous equations for single-excitation amplitudes are obtained from (\[ori\_EqSingle\]) as $$\begin{aligned}
\label{EqSingle}
0 =& f_{p \bar{\bm{k}}}^{g \bar{\bm{k}}}
+ \sum_c \check{\rho}_p^c ( \bar{\bm{k}} ) \tilde{\mathcal{F}}^{g \bar{\bm{k}} c \bar{\bm{k}}}
- \sum_r \hat{\rho}_r^g ( \bar{\bm{k}} ) \tilde{\mathcal{F}}_{r \bar{\bm{k}} p \bar{\bm{k}}}
+ \sum_{n, f, \hat{\bm{k}}} t_{p \bar{\bm{k}} n \hat{\bm{k}}}^{g \bar{\bm{k}} f \hat{\bm{k}}} \tilde{\mathcal{F}}_{n \hat{\bm{k}}}^{f \hat{\bm{k}}} \notag \\
&+ \sum_{r, c, \hat{\bm{k}}} t_{r \hat{\bm{k}}}^{c \hat{\bm{k}}} \bra{r \hat{\bm{k}} g \bar{\bm{k}}} \ket{c \hat{\bm{k}} p \bar{\bm{k}}}_{ovvo}
- \sum_{\bm{k}_m} \tilde{\mathcal{L}}_{p \bar{\bm{k}} \bm{k}_m}^{g \bar{\bm{k}}}
+ \sum_{\bm{k}_e} \tilde{\mathcal{L}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}} \bm{k}_e}.
\end{aligned}$$ The intermediates $\tilde{\mathcal{L}}$ are newly introduced and defined as $$\begin{gathered}
\tilde{\mathcal{L}}_{p \bar{\bm{k}} \bm{k}_m}^{g \bar{\bm{k}}} = \frac{1}{2} \sum_m \sum_{n, \bm{k}_n, f} t_{m \bm{k}_m n \bm{k}_n}^{g \bar{\bm{k}} f \bm{k}_f} \bra{m \bm{k}_m n \bm{k}_n} \ket{p \bar{\bm{k}} f \bm{k}_f}_{ooov}^{[oo]}, \label{def_L_oo} \\
\tilde{\mathcal{L}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}} \bm{k}_e} = \frac{1}{2} \sum_e \sum_{n, \bm{k}_n, f} t_{p \bar{\bm{k}} n \bm{k}_n}^{e \bm{k}_e f \bm{k}_f} \bra{n \bm{k}_n g \bar{\bm{k}}} \ket{f \bm{k}_f e \bm{k}_e}_{ovvv}^{[vv]}. \label{def_L_vv}
\end{gathered}$$
### The rearranged simultaneous equations for double-excitation amplitudes {#Sec_EqDouble}
We rearrange the right-hand side of the rearranged simultaneous equations for double-excitation amplitudes (\[ori\_EqDouble\]). Numerical computation of the rearranged right-hand side is divided into four parts. Among each parts, ways of identification of momentums through a process number is different. Then, the rearranged simultaneous equations is expressed as $$\label{EqDouble}
0 = \bar{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
+ \check{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
+ \tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
+ \hat{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}.$$
For the first term $\bar{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$, the momentums $\bm{k}_i$ and $\bm{k}_a$ are identified through a process number. This term is given as $$\begin{aligned}
\label{A_1st}
\bar{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
=& P_-(ij) P_-(ab) \tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
+ \left( \bra{i \bm{k}_i j \bm{k}_j} \ket{a \bm{k}_a b \bm{k}_b}_{oovv} \right) ^* \notag \\
&+ \sum_{s, d} \left( t_{s \bm{k}_b}^{b \bm{k}_b} t_{j \bm{k}_j}^{d \bm{k}_j} \sum_{r, c} \tilde{\zeta}_{ri}^{ac} ( \bm{k}_i, \bm{k}_a ) \bra{r \bm{k}_a s \bm{k}_b} \ket{c \bm{k}_i d \bm{k}_j}_{oovv}^{[ov]} \right) .
\end{aligned}$$ Intermediates $\tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ and $\tilde{\zeta}$ are newly introduced. Similarly to the intermediates $\check{\rho}$ and $\hat{\rho}$, the intermediate $\tilde{\zeta}$ depends on a process. When a process number is expressed as $\bm{k}_{\alpha} + N_{\bm{k}} \bm{k}_{\beta}$ $( \bm{k}_{\alpha}, \bm{k}_{\beta} = 0, 1, ..., N_{\bm{k}} - 1 )$, the definition of $\tilde{\zeta}$ is $$\label{def_zeta}
\tilde{\zeta}_{p q}^{g h} ( \bm{k}_{\alpha}, \bm{k}_{\beta} ) = t_{p \bm{k}_{\alpha}}^{g \bm{k}_{\alpha}} t_{q \bm{k}_{\beta}}^{h \bm{k}_{\beta}}.$$ The intermediate $\tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ is given as $$\label{def_C}
\tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
= \sum_{n, \bm{k}_n, f} t_{i \bm{k}_i n \bm{k}_n}^{a \bm{k}_a f \bm{k}_f} \tilde{\mathcal{W}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f}
- \sum_{r, c} t_{r \bm{k}_b}^{b \bm{k}_b} t_{j \bm{k}_j}^{c \bm{k}_j} \bra{r \bm{k}_b a \bm{k}_a} \ket{c \bm{k}_j i \bm{k}_i}_{ovvo}.$$ The intermediate $\tilde{\mathcal{W}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f}$ is rearranged from the original definition in (\[Def\_Woovv\]) and given as $$\begin{aligned}
\label{rearranged_W}
\tilde{\mathcal{W}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f}
=&- \frac{1}{2} \sum_{x, y, \bm{k}_y} t_{j \bm{k}_j y \bm{k}_y}^{x \bm{k}_x b \bm{k}_b} \bra{n \bm{k}_n y \bm{k}_y} \ket{f \bm{k}_f x \bm{k}_x}_{oovv}^{[ov]} \notag \\
&+ \bra{n \bm{k}_n b \bm{k}_b} \ket{f \bm{k}_f j \bm{k}_j}_{ovvo}
+ \tilde{\mathcal{J}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f},
\end{aligned}$$ where $\tilde{\mathcal{J}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f}$ is a newly introduced intermediate given as $$\begin{aligned}
\label{def_J}
\tilde{\mathcal{J}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f}
=& \sum_c \hat{\rho}_j^c ( \bm{k}_j ) \bra{n \bm{k}_n b \bm{k}_b} \ket{f \bm{k}_f c \bm{k}_j}_{ovvv}^{[vv]} \notag \\
&- \sum_r \check{\rho}_r^b ( \bm{k}_b ) \bra{r \bm{k}_b n \bm{k}_n} \ket{j \bm{k}_j f \bm{k}_f}_{ooov}^{[oo]} \notag \\
&- \sum_{r, c} \tilde{\zeta}_{rj}^{bc} ( \bm{k}_b, \bm{k}_j ) \bra{r \bm{k}_b n \bm{k}_n} \ket{c \bm{k}_j f \bm{k}_f}_{oovv}^{[ov]} .
\end{aligned}$$
For the second term $\check{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$, the momentums $\bm{k}_i$ and $\bm{k}_j$ are identified through a process number. This term is given as $$\begin{aligned}
\label{A_2nd}
\check{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
=& \sum_{r, s} \tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b} \bra{r \bm{k}_a s \bm{k}_b} \ket{i \bm{k}_i j \bm{k}_j}_{oooo}
+ P_-(ij) \check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} \notag \\
&+ P_-(ab) \sum_f \left( t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a f \bm{k}_b} \left( \tilde{\mathcal{F}}^{b \bm{k}_b f \bm{k}_b} - \frac{1}{2}\sum_r t_{r \bm{k}_b}^{b \bm{k}_b} \tilde{\mathcal{F}}_{r \bm{k}_b}^{f \bm{k}_b} \right) \right) \notag \\
&+ \sum_{e, \bm{k}_e, f} t_{i \bm{k}_i j \bm{k}_j}^{e \bm{k}_e f \bm{k}_f} \tilde{\mathcal{Y}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f}.
\end{aligned}$$ Intermediates $\tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b}$, $\check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ and $\tilde{\mathcal{Y}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f}$ are newly introduced. The definition of the intermediate $\tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b}$ is $$\tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b} = t_{r \bm{k}_a}^{a \bm{k}_a} t_{s \bm{k}_b}^{b \bm{k}_b}.$$ That of the intermediate $\check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ is $$\begin{aligned}
\check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
=&\sum_c \Biggl( \hat{\rho}_j^c ( \bm{k}_j ) \biggl( \sum_{r, s} \tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b} \bra{r \bm{k}_a s \bm{k}_b} \ket{i \bm{k}_i c \bm{k}_j}_{ooov}^{[ov]} \biggr. \Biggr. \notag \\
&~~~~~~~~~~~~~~~~~~~~~~~~ \Biggl. \biggl. + \left( \bra{i \bm{k}_i c \bm{k}_j} \ket{a \bm{k}_a b \bm{k}_b}_{ovvv}^{[ov]} \right) ^* \biggr) \Biggr) .
\end{aligned}$$ That of the intermediate $\tilde{\mathcal{Y}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f}$ is $$\begin{aligned}
\label{def_Y_vv}
\tilde{\mathcal{Y}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f}
= \frac{1}{2} & \biggl( P_-(ab) \tilde{\mathcal{V}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f} - \bra{b \bm{k}_b a \bm{k}_a} \ket{e \bm{k}_e f \bm{k}_f}_{vvvv} \biggr. \notag \\
& ~~~~ \biggl. - \sum_{r, s} \tilde{\zeta}_{sr}^{ba} ( \bm{k}_b, \bm{k}_a ) \bra{s \bm{k}_b r \bm{k}_a} \ket{e \bm{k}_e f \bm{k}_f}_{oovv}^{[oo]} \biggr) ,
\end{aligned}$$ where, $\tilde{\mathcal{V}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f}$ is newly introduced intermediate defined as $$\tilde{\mathcal{V}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f} = \sum_r \check{\rho}_r^b ( \bm{k}_b ) \bra{r \bm{k}_b a \bm{k}_a} \ket{e \bm{k}_e f \bm{k}_f}_{ovvv}^{[ov]}.$$
The third term $\tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ is given as $$\label{A_3rd}
\tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
= \frac{1}{4} \sum_{m, \bm{k}_m, n} t_{m \bm{k}_m n \bm{k}_n}^{a \bm{k}_a b \bm{k}_b} \tilde{\mathcal{X}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$$ where newly introduced intermediate $\tilde{\mathcal{X}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$ is $$\label{def_X}
\tilde{\mathcal{X}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}
= \sum_{e, \bm{k}_e, f} t_{i \bm{k}_i j \bm{k}_j}^{e \bm{k}_e f \bm{k}_f} \bra{m \bm{k}_m n \bm{k}_n} \ket{e \bm{k}_e f \bm{k}_f}_{oovv}^{[oo]}.$$ In computation of the intermediate $\tilde{\mathcal{X}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$ in (\[def\_X\]), the momentums $\bm{k}_i$ and $\bm{k}_j$ are continuously identified through a process number. In computation of $\tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ in (\[A\_3rd\]), the momentums $\bm{k}_a$ and $\bm{k}_b$ are identified through a process number.
For the fourth term $\hat{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$, the momentums $\bm{k}_a$ and $\bm{k}_b$ are continuously identified through a process number. This term is given as $$\begin{aligned}
\label{A_4th}
\hat{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
=& \sum_{m, \bm{k}_m, n} t_{m \bm{k}_m n \bm{k}_n}^{a \bm{k}_a b \bm{k}_b} \tilde{\mathcal{Y}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j} \notag \\
&- P_-(ij) \sum_n \left( t_{i \bm{k}_i n \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} \left( \tilde{\mathcal{F}}_{n \bm{k}_j j \bm{k}_j} + \frac{1}{2} \sum_{c} t_{j \bm{k}_j}^{c \bm{k}_j} \tilde{\mathcal{F}}_{n \bm{k}_j}^{c \bm{k}_j} \right) \right) \notag \\
&- \sum_{c, d} \tilde{\eta}_{i j \bm{k}_i \bm{k}_j}^{c d} \bra{b \bm{k}_b a \bm{k}_a} \ket{c \bm{k}_i d \bm{k}_j}_{vvvv}
+ P_-(ab) \hat{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}.
\end{aligned}$$ Intermediates $\tilde{\eta}_{i j \bm{k}_i \bm{k}_j}^{c d}$, $\hat{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ and $\tilde{\mathcal{Y}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$ are newly introduced. The definition of the intermediate $\tilde{\eta}_{i j \bm{k}_i \bm{k}_j}^{c d}$ is $$\tilde{\eta}_{i j \bm{k}_i \bm{k}_j}^{c d} = t_{i \bm{k}_i}^{c \bm{k}_i} t_{j \bm{k}_j}^{d \bm{k}_j}.$$ That of the intermediate $\hat{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ is $$\begin{aligned}
\hat{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
=&\sum_r \Biggl( \check{\rho}_r^b ( \bm{k}_b ) \biggl( \sum_{c, d} \tilde{\eta}_{i j \bm{k}_i \bm{k}_j}^{c d} \bra{r \bm{k}_b a \bm{k}_a} \ket{c \bm{k}_i d \bm{k}_j}_{ovvv}^{[ov]} \biggr. \Biggr. \notag \\
&~~~~~~~~~~~~~~~~~~~~~~~ \Biggl. \biggl. + \left( \bra{i \bm{k}_i j \bm{k}_j} \ket{r \bm{k}_b a \bm{k}_a}_{ooov}^{[ov]} \right) ^* \biggr) \Biggr) .
\end{aligned}$$ That of the intermediate $\tilde{\mathcal{Y}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$ is $$\begin{aligned}
\label{def_Y_oo}
\tilde{\mathcal{Y}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}
= \frac{1}{2} & \biggl( P_-(ij) \tilde{\mathcal{V}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}
+ \bra{m \bm{k}_m n \bm{k}_n} \ket{i \bm{k}_i j \bm{k}_j}_{oooo} \notag \\
& ~~~~ \biggl. + \sum_{c, d} \tilde{\zeta}_{ij}^{cd} ( \bm{k}_i, \bm{k}_j ) \bra{m \bm{k}_m n \bm{k}_n} \ket{c \bm{k}_i d \bm{k}_j}_{oovv}^{[vv]} \biggr) ,
\end{aligned}$$ where $\tilde{\mathcal{V}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$is newly introduced intermediate defined as $$\tilde{\mathcal{V}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j} = \sum_c \hat{\rho}_j^c ( \bm{k}_j ) \bra{m \bm{k}_m n \bm{k}_n} \ket{i \bm{k}_i c \bm{k}_j}_{ooov}^{[ov]}.$$
Implementation of the presented method {#ImplementMethod}
======================================
In this section, a way of implementation of the presented method is described. In Section \[Sec\_overall\_flow\], overall flow of the presented implementation is described. In Section \[array\_RHS\], arrays used for the right-hand sides of simultaneous equations are explained. In Section \[common\_procedure\], procedures which are common to plural steps are described. In Section \[OurImplementaiton\], our implementation is given.
Overall flow {#Sec_overall_flow}
------------
Overall flow of the presented method is as follows:
The matrix elements of the Fock operator and antisymmetrized two-electron integrals are input as described in Section \[IO\]. The initial values of single-excitation and double-excitation amplitudes are set as described in Section \[IO\]. The right-hand sides of the simultaneous equations (\[EqSingle\]) and (\[EqDouble\]) are computed and single-excitation and double-excitation amplitudes are updated iteratively until these right-hand sides are sufficiently close to zero. Thus we obtain single-excitation and double-excitation amplitudes amplitudes that satisfy the CCSD equations as outputs.
Arrays for the right-hand sides of the simultaneous equations for single-excitation and double-excitation amplitudes {#array_RHS}
--------------------------------------------------------------------------------------------------------------------
We prepare arrays for the right-hand sides of the simultaneous equations (\[EqSingle\]) and (\[EqDouble\]) for single-excitation and double-excitation amplitudes. Intermediate results during computation and the final results are stored in these arrays. Let us denote these arrays for single-excitation and double-excitation amplitudes by $S_ 1$ and $S_2$, respectively. See Section \[RHS\_abstract\] for details.
Procedures which are common to plural steps in implementation {#common_procedure}
-------------------------------------------------------------
In the presented implementation, some procedures are common to plural steps. In this section, such procedures are described.
### Procedures executed only in the diagonal processes {#DiagProcedure}
There are procedures executed only in the diagonal processes whose numbers are $\bar{\bm{k}} + \bar{\bm{k}} N_{\bm{k}}$ $( \bar{\bm{k}} = 0, 1, ..., N_{\bm{k}} - 1 )$. Let us call such a procedure “Diagonal procedure”.
### Summations which are concerned with the intermediates $\tilde{\mathcal{L}}$ and $\tilde{\mathcal{Z}}$ {#Sum_L_Z}
In this section, we describe ways of computation of the summations $\sum_{\bm{k}_e} \tilde{\mathcal{Z}}_{p \bar{\bm{k}} q \bar{\bm{k}}}^{\bm{k}_e}$, $\sum_{\bm{k}_m} \tilde{\mathcal{Z}}_{\bm{k}_m}^{g \bar{\bm{k}} h \bar{\bm{k}}}$, $\sum_{\bm{k}_m} \tilde{\mathcal{L}}_{p \bar{\bm{k}} \bm{k}_m}^{g \bar{\bm{k}}}$ and $\sum_{\bm{k}_e} \tilde{\mathcal{L}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}} \bm{k}_e}$ in (\[EqSingle\]), (\[def\_F\_oo\]) and (\[def\_F\_vv\]). The intermediates $\tilde{\mathcal{L}}_{p \bar{\bm{k}} \bm{k}_m}^{g \bar{\bm{k}}}$ and $\tilde{\mathcal{Z}}_{\bm{k}_m}^{g \bar{\bm{k}} h \bar{\bm{k}}}$ are computed according to (\[def\_Z\_vv\]) and (\[def\_L\_oo\]), respectively, in the process whose process number is $\bm{k}_m + \bar{\bm{k}} N_{\bm{k}}$. The intermediatess $\tilde{\mathcal{L}}_{p \bar{\bm{k}}}^{g \bar{\bm{k}} \bm{k}_e}$ and $\tilde{\mathcal{Z}}_{p \bar{\bm{k}} q \bar{\bm{k}}}^{\bm{k}_e}$ are computed according to (\[def\_Z\_oo\]) and (\[def\_L\_vv\]), respectively, in the process whose process number is $\bar{\bm{k}} + \bm{k}_e N_{\bm{k}}$.
The ways of the summations $\sum_{\bm{k}_m}$ and $\sum_{\bm{k}_e}$ are described using Fig. \[Fig\_sum\_k\].
![Summation over the indices $\bm{k}_m$ and $\bm{k}_e$[]{data-label="Fig_sum_k"}](PDF_sum_k){width="70.00000%"}
The summations over the indeices $\bm{k}_m$ and $\bm{k}_e$ are taken over processes in vertical and horizontal directions, respectively. The results are stored in the diagonal processes whose numbers are $\bar{\bm{k}} + \bar{\bm{k}} N_{\bm{k}}$ according to $\bar{\bm{k}}$. When we use MPI (Message Passing Interface), this procedure can be executed by the subroutine MPI\_REDUCE in a communicator grouping processes in the vertical or horizontal direction.
### Change of an index to be identified through a process number {#IndexChange}
Let us consider a status that quantities $\Theta_{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta}}^{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}$ are distributed to each process and two momentums $\bm{k}_{\alpha}$ and $\bm{k}_{\gamma}$ are identified through a process number by representing it as $\bm{k}_{\alpha} + \bm{k}_{\gamma} N_{\bm{k}}$. In the presented method, redistribution of these quantities to the following situations is necessary.
- Two momentums $\bm{k}_{\beta}$ and $\bm{k}_{\gamma}$ are identified through a process number by representing it as $\bm{k}_{\beta} + \bm{k}_{\gamma} N_{\bm{k}}$.
- Two momentums $\bm{k}_{\gamma}$ and $\bm{k}_{\delta}$ are identified through a process number by representing it as $\bm{k}_{\delta} + \bm{k}_{\gamma} N_{\bm{k}}$.
See Figs. \[Fig\_ExcRow\] and \[Fig\_ExcCol\]. When we use MPI, the first (second) change can be achieved by the subroutine MPI\_ALLTOALL in a communicator grouping processes in the vertical (horizontal) direction.
![Change of index from $\bm{k}_{\alpha}$ to $\bm{k}_{\beta}$[]{data-label="Fig_ExcRow"}](PDF_ExcRow){width="70.00000%"}
![Change of index from $\bm{k}_{\gamma}$ to $\bm{k}_{\delta}$[]{data-label="Fig_ExcCol"}](PDF_ExcCol){width="70.00000%"}
### Summation between two quantities which have eight indices {#Sum_88}
In this section, we consider a summation represented as $$\Xi_{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta} \gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta}}
= \sum_{\varepsilon , \bm{k}_{\varepsilon}, \zeta} \Phi_{\alpha \bm{k}_{\alpha} \beta \bm{k}_{\beta} \varepsilon \bm{k}_{\varepsilon} \zeta \bm{k}_{\zeta}}
\Psi_{\gamma \bm{k}_{\gamma} \delta \bm{k}_{\delta} \varepsilon \bm{k}_{\varepsilon} \zeta \bm{k}_{\zeta}}.$$ In this section, indices do not represent kind of spin orbitals — occupied or virtual. Assume that the elements of $\Phi$ and $\Psi$ are stored in processes whose numbers are $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$ and $\bm{k}_{\gamma} + \bm{k}_{\delta} N_{\bm{k}}$, respectively, according to momentums $\bm{k}_{\alpha}$, $\bm{k}_{\beta}$, $\bm{k}_{\gamma}$ and $\bm{k}_{\delta}$. See Fig. \[Fig\_sum\_88\].
![Processes for $\Phi$ and $\Psi$[]{data-label="Fig_sum_88"}](PDF_sum_88){width="70.00000%"}
To compute $\Xi$, it is necessary to send $\Psi$ to an appropriate process with consideration of conservation law of momentum. For each process as a receiver of $\Psi$, a process as a sender of $\Psi$ is uniquely determined when the momentum $\bm{k}_{\gamma}$ is specified because of conservation law of momentum. For $\Delta = 0, 1, ..., N_{\bm{k}} - 1$, we repeat the following procedure. For $\bm{k}_{\alpha}$ particular to a process, $\bm{k}_{\gamma}$ is specified by $\bm{k}_{\gamma} = \bm{k}_{\alpha} + \Delta \mod N_{\bm{k}}$. Then, $\bm{k}_{\delta}$ is uniquely determined from conservation law of momentum. The elements of $\Psi$ in the process whose process number is $\bm{k}_{\gamma} + \bm{k}_{\delta} N_{\bm{k}}$ are sent to the process whose process number is $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$. When we use MPI, this procedure can be executed by the subroutines MPI\_SEND and MPI\_RECV. The elements of $\Xi$ are computed from $\Phi$ and the received $\Psi$.
### Exchange of the first and the second indices identified through a process number {#DiagSwap}
In our method, two momentums $\bm{k}_{\alpha}$ and $\bm{k}_{\beta}$ are identified through a process number by representing it as $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$. In some steps in our method, to make an equivalent status under a condition that the two momentums are identified by representing a process number as $\bm{k}_{\beta} + \bm{k}_{\alpha} N_{\bm{k}}$ is necessary for particular quantities. See Fig. \[Fig\_diag\_swap\]. This change can be achieved by transfer such quantities from a process whose process number is $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$ to the one whose process number is $\bm{k}_{\beta} + \bm{k}_{\alpha} N_{\bm{k}}$ $( \alpha \neq \beta )$. Such quantities in the diagonal processes are copied to another array. When we use MPI, this procedure can be executed by the subroutines MPI\_SEND and MPI\_RECV.
![Change of a way of representing a process number[]{data-label="Fig_diag_swap"}](PDF_diag_swap){width="70.00000%"}
Implementation {#OurImplementaiton}
--------------
In this section, we describe implementation of procedure after inputs and the initial values of single-excitation and double-excitation amplitudes are distributed to each process. This procedure is iterated until solution of the simultaneous equations (\[EqSingle\]) and (\[EqDouble\]) are obtained.
### Computation of the intermediates $\hat{\rho}$, $\check{\rho}$ and $\tilde{\zeta}$
Let us consider a case such that a process number is represented as $\bm{k}_{\alpha} + \bm{k}_{\beta} N_{\bm{k}}$. From single-excitation amplitudes, the intermediates $\hat{\rho}$ and $\check{\rho}$ are set according to (\[def\_hat\_rho\]) and (\[def\_check\_rho\]), and the intermediate $\tilde{\zeta}$ is computed through (\[def\_zeta\]).
### Computation of the intermediates $\tilde{\mathcal{F}}$
The summations $\sum_{\bm{k}_e} \tilde{\mathcal{Z}}_{p \bar{\bm{k}} q \bar{\bm{k}}}^{\bm{k}_e}$, and $\sum_{\bm{k}_m} \tilde{\mathcal{Z}}_{\bm{k}_m}^{g \bar{\bm{k}} h \bar{\bm{k}}}$ in (\[def\_F\_oo\]) and in (\[def\_F\_vv\]) are obtained from the way described in Section \[Sum\_L\_Z\] and are stored in the diagonal processes.
The intermediate $\tilde{\mathcal{K}}$ is computed according to (\[def\_K\]). It is Diagonal procedure described in Section \[DiagProcedure\].
The intermediates $\tilde{\mathcal{F}}$ are computed according to (\[def\_F\_oo\]), (\[def\_F\_vv\]) and (\[def\_F\_ov\]). It is Diagonal procedure described in Section \[DiagProcedure\].
Elements of the intermediates $\tilde{\mathcal{F}}$ are separately stored in the diagonal processes according to $\bar{\bm{k}}$ at this stage. It is necessary that each process stores all the elements of $\tilde{\mathcal{F}}$. We describe a procedure using Fig. \[Fig\_share\_F\]. The elements of $\tilde{\mathcal{F}}$ in the diagonal processes are broadcasted to the other processes in the vertical direction. When we use MPI, this procedure can be executed by the subroutine MPI\_BCAST in a communicator grouping processes in the vertical direction. After broadcasting, these results are gathered to each process in the horizontal direction. When we use MPI, this procedure can be executed by the subroutine MPI\_ALLGATHER in a communicator grouping processes in the horizontal direction. Thus, all the intermediates of $\tilde{\mathcal{F}}$ are stored in all the processes.
![Broadcasting and gathering of the intermediates $\tilde{\mathcal{F}}$[]{data-label="Fig_share_F"}](PDF_share_F){width="70.00000%"}
### Computation of the right-hand sides of simultaneous equations for single-excitation amplitudes
We compute the right-hand side of (\[EqSingle\]). The first item can directly be substituted to the array $S_1$. This is Diagonal procedure described in Section \[DiagProcedure\]. On the other items, when computation of each item is finished, addition to or subtraction from the array $S_1$ is done. This is also Diagonal procedure. Computation of from the second to the fifth items is Diagonal procedure and is straightforward. Computation of the sixth and the seventh items is as described in Section \[Sum\_L\_Z\].
Thus, the right-hand sides of simultaneous equations for single-excitation amplitudes are obtained in the diagonal processes.
### Computation of the intermediate $\tilde{\mathcal{W}}$
The intermediate $\tilde{\mathcal{W}}$ is computed according to (\[rearranged\_W\]).
We describe computation of the first term in the right-hand side. At the beginning of this computation, the indices $\bm{k}_j$ and $\bm{k}_x$ of double-excitation amplitudes $t_{j \bm{k}_j y \bm{k}_y}^{x \bm{k}_x b \bm{k}_b}$ are identified through a process number by representing it as $\bm{k}_j + \bm{k}_x N_{\bm{k}}$. It is desired that the indices $\bm{k}_j$ and $\bm{k}_b$ of double-excitation amplitudes are identified through a process number by representing it as $\bm{k}_j + \bm{k}_b N_{\bm{k}}$. For this purpose, the elements of double-excitation amplitudes are exchanged between processes by the procedure described in Section \[IndexChange\]. See also Fig. \[Fig\_ExcCol\]. After this exchange, the term is obtained using the procedure described in Section \[Sum\_88\].
Addition of the second term in the right-hand side is straightforward.
On the third term in the right-hand side, note that antisymmetrized two-electron integrals used in computation of the intermediate $\tilde{\mathcal{J}}$ is stored in a process whose process number is $\bm{k}_b + \bm{k}_j N_{\bm{k}}$. The elements $\tilde{\mathcal{J}}_{j \bm{k}_j n \bm{k}_n}^{b \bm{k}_b f \bm{k}_f}$ are computed according to (\[def\_J\]) in a process whose process number is $\bm{k}_b + \bm{k}_j N_{\bm{k}}$. After computation, the procedure described in Section \[DiagSwap\] is executed. Then, the third term is obtained in a process whose process number is $\bm{k}_j + \bm{k}_b N_{\bm{k}}$. This term is added.
Thus, the intermediate $\tilde{\mathcal{W}}$ is obtained.
### Computation of the intermediate $\tilde{\mathcal{C}}$ and operation by the operators $P_-(ij) P_-(ab)$ to it {#Compute_PPC}
The intermediate $\tilde{\mathcal{C}}$ is computed according to (\[def\_C\]). The first term in the right-hand side is computed by the procedure described in Section \[Sum\_88\]. The result is substituted to the array $S_2$. After computation of the second term, the result is subtracted from the array $S_2$. At this stage, it holds $S_2 = \tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$.
The result of operation by the operators $P_-(ij) P_-(ab)$ is concretely $$\label{concrete_PPC}
P_-(ij) P_-(ab) \tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
= \tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
- \tilde{\mathcal{C}}_{i \bm{k}_i j \bm{k}_j}^{b \bm{k}_b a \bm{k}_a}
- \tilde{\mathcal{C}}_{j \bm{k}_j i \bm{k}_i}^{a \bm{k}_a b \bm{k}_b}
+ \tilde{\mathcal{C}}_{j \bm{k}_j i \bm{k}_i}^{b \bm{k}_b a \bm{k}_a}.$$ When computation of $\tilde{\mathcal{C}}$ is finished, the elements of the first term in the right-hand side are distributed to each process according to the indices $\bm{k}_i$ and $\bm{k}_a$ by representing process numbers as $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. In computation of the other items, redistribution of the elements of $\tilde{\mathcal{C}}$ is necessary and we prepare another array different from $S_2$ for this purpose. The array $S_2$ is copied to this array before redistribution. To subtract the second (third) term, the elements of $\tilde{\mathcal{C}}$ should be redistributed according to the indices $\bm{k}_i$ and $\bm{k}_b$ ($\bm{k}_j$ and $\bm{k}_a$) by representing process numbers as $\bm{k}_i + \bm{k}_b N_{\bm{k}}$ ($\bm{k}_j + \bm{k}_a N_{\bm{k}}$). This redistribution of the elements can be achieved by the procedure described in Section \[IndexChange\]. To add the fourth term, the elements of $\tilde{\mathcal{C}}$ should be redistributed according to the indices $\bm{k}_j$ and $\bm{k}_b$ by representing process numbers as $\bm{k}_j + \bm{k}_b N_{\bm{k}}$. This redistribution of the elements can be achieved by the procedure described in Section \[IndexChange\] starting from the redistributed configuration of the elements of $\tilde{\mathcal{C}}$ for the subtraction in the second term. After each redistribution is finished, addition to or subtraction from the array $S_2$ is done.
### Computation of the term $\bar{\mathcal{A}}$
The term $\bar{\mathcal{A}}$ in (\[EqDouble\]) is computed according to he right-hand side of (\[A\_1st\]). The first term is obtained by the procedure described in Section \[Compute\_PPC\] and the result is stored in the array $S_2$. Addition of the second term to the array $S_2$ is straightforward. Note that the third term is computed in processes whose process numbers are represented not as $\bm{k}_i+ \bm{k}_a N_{\bm{k}}$ but as $\bm{k}_a+ \bm{k}_i N_{\bm{k}}$. After finish of this computation, the procedure described in Section \[DiagSwap\] is executed to transfer the results to the appropriate processes. Then, the transferred results are added to the array $S_2$. At this stage, it holds $S_2 = \bar{\mathcal{A}}$.
### Computation of the term $\check{\mathcal{A}}$ – Before redistribution of double-excitation amplitudes
The elements of the term $\check{\mathcal{A}}$ are computed in processes whose process numbers are $\bm{k}_i+ \bm{k}_j N_{\bm{k}}$. Intermediate results during computation and the final results are stored in an array $\check{S}_2$ which is different from $S_2$. Double-excitation amplitudes are redistributed according to the indices $\bm{k}_i$ and $\bm{k}_j$ by representing process numbers as $\bm{k}_i + \bm{k}_j N_{\bm{k}}$. The first and the second term in (\[A\_2nd\]) are computed before this redistribution. The first term and the intermediate $\check{\mathcal{G}}$ are computed. After this computation, it holds $$\label{S2_intermediate}
\check{S}_2 = \sum_{r, s} \tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b} \bra{r \bm{k}_a s \bm{k}_b} \ket{i \bm{k}_i j \bm{k}_j}_{oooo} + \check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}.$$ and the intermediate $\check{\mathcal{G}}$ is stored in another array. The procedure described in Section \[DiagSwap\] for operation of the operator $P_-(ij)$ is applied to the intermediate $\check{\mathcal{G}}$. The elements $\check{\mathcal{G}}_{j \bm{k}_j i \bm{k}_i}^{a \bm{k}_a b \bm{k}_b}$ stored in the processes whose process numbers are $\bm{k}_j + \bm{k}_i N_{\bm{k}}$ are subtracted from the array $\check{S}_2$ in (\[S2\_intermediate\]). After this procedure, it holds $$\check{S}_2 = \sum_{r, s} \tilde{\eta}_{r s}^{a b \bm{k}_a \bm{k}_b} \bra{r \bm{k}_a s \bm{k}_b} \ket{i \bm{k}_i j \bm{k}_j}_{oooo} + P_-(ij) \check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}.$$
### Redistribution of double-excitation amplitudes {#Redist_1st}
At the beginning of computation of $\check{\mathcal{A}}$, double-excitation amplitudes $t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ are distributed according to the indices $\bm{k}_i$ and $\bm{k}_a$ by representing process numbers as $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. The amplitudes should be redistributed according to the indices $\bm{k}_i$ and $\bm{k}_j$ by representing process numbers as $\bm{k}_i + \bm{k}_j N_{\bm{k}}$. For this purpose, elements of double-excitation amplitudes are exchanged between processes by the procedure described in Section \[IndexChange\]. See also Fig. \[Fig\_ExcCol\]. The original array in which the amplitudes are stored should be kept.
### Computation of the term $\check{\mathcal{A}}$ – After redistribution of double-excitation amplitudes
In computation of the third item in the right-hand side of (\[A\_2nd\]), no communication between processes occurs. When this computation is finished, it holds $$\check{S}_2 = \check{\mathcal{A}} - \sum_{e, \bm{k}_e, f} t_{i \bm{k}_i j \bm{k}_j}^{e \bm{k}_e f \bm{k}_f} \tilde{\mathcal{Y}}^{a \bm{k}_a b \bm{k}_b e \bm{k}_e f \bm{k}_f}.$$
Next, we consider computation of the fourth item in the right-hand side of (\[A\_2nd\]). Note that four arithmetic operations in computation of the intermediate $\tilde{\mathcal{Y}}$ given in (\[def\_Y\_vv\]) are done in processes whose process numbers are $\bm{k}_b + \bm{k}_a N_{\bm{k}}$. After computation of the intermediate $\tilde{\mathcal{V}}$, its elements are copied to the process whose process number is $\bm{k}_a + \bm{k}_b N_{\bm{k}}$ through the procedure described in Section \[DiagSwap\]. The copied elements are subtracted from the ones of the original $\tilde{\mathcal{V}}$. Thus, computation of the item $P_-(ab) \tilde{\mathcal{V}}$ is finished. After this step, the elements of the intermediate $\tilde{\mathcal{Y}}$ are obtained according to (\[def\_Y\_vv\]) without communication between processes. The summation $\sum_{e, \bm{k}_e, f}$ is executed through the procedure described in Section \[Sum\_88\]. The results are added to the array $\check{S}_2$. Now, it holds $\check{S}_2 = \check{\mathcal{A}}$.
### Addition of the term $\check{\mathcal{A}}$ to the array $S_2$
The elements of the term $\check{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ are stored in processes whose numbers are $\bm{k}_i + \bm{k}_j N_{\bm{k}}$. To add them to the array $S_2$, redistribution of them is necessary. The elements should be redistributed according to the indices $\bm{k}_i$ and $\bm{k}_a$ to processes whose numbers are $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. For this purpose, the elements of the array $\check{S}_2$ are exchanged between processes by the procedure described in Section \[IndexChange\]. See also Fig. \[Fig\_ExcCol\]. The redistributed elements are added to the array $S_2$. Now, it holds $S_2 = \bar{\mathcal{A}} + \check{\mathcal{A}}$.
### Redistribution of double-excitation amplitudes {#redistribution-of-double-excitation-amplitudes}
We redistribute double-excitation amplitudes $t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ again for computation of the terms $\tilde{\mathcal{A}}$ and $\hat{\mathcal{A}}$. These amplitudes are originally distributed according to the indices $\bm{k}_i$ and $\bm{k}_a$ by representing process numbers as $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. The amplitudes should be redistributed according to the indices $\bm{k}_b$ and $\bm{k}_a$ by representing process numbers as $\bm{k}_b + \bm{k}_a N_{\bm{k}}$. For this purpose, elements of double-excitation amplitudes are exchanged between processes by the procedure described in Section \[IndexChange\]. See also Fig. \[Fig\_ExcRow\]. The amplitudes which have been distributed in Section \[Redist\_1st\] for computation of $\check{\mathcal{A}}$ should be kept for computation of the intermediate $\tilde{\mathcal{X}}$ defined in (\[def\_X\]).
### Computation of the intermediate $\tilde{\mathcal{X}}$
The intermediate $\tilde{\mathcal{X}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$ defined in (\[def\_X\]) is obtained using the procedure described in Section \[Sum\_88\]. The elements of $\tilde{\mathcal{X}}$ are distributed according to the indices $\bm{k}_i$ and $\bm{k}_j$ by representing process numbers as $\bm{k}_i + \bm{k}_j N_{\bm{k}}$.
### Computation of the term $\tilde{\mathcal{A}}$
The elements of the terms $\tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ and $\hat{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$ are computed in processes whose process numbers are $\bm{k}_b + \bm{k}_a N_{\bm{k}}$. Intermediate results during computation and the final results of $\tilde{\mathcal{A}} + \hat{\mathcal{A}}$ are stored in an array $\hat{S}_2$ which is different from $S_2$ and $\check{S}_2$.
The term $\tilde{\mathcal{A}}$ defined in (\[A\_3rd\]) can be obtained using the procedure described in Section \[Sum\_88\]. The results are substituted to the array $\hat{S}_2$. Then, it holds $\hat{S}_2 = \tilde{\mathcal{A}}$.
### Computation of the term $\hat{\mathcal{A}}$ – The first term
We consider computation of the first item in the right-hand side of (\[A\_4th\]). Note that four arithmetic operations in computation of the intermediate $\tilde{\mathcal{Y}}$ given in (\[def\_Y\_oo\]) are done in processes whose process numbers are $\bm{k}_i + \bm{k}_j N_{\bm{k}}$. After computation of the intermediate $\tilde{\mathcal{V}}$, its elements are copied to the process whose process number is $\bm{k}_j + \bm{k}_i N_{\bm{k}}$ through the procedure described in Section \[DiagSwap\]. The copied elements are subtracted from the ones of the original $\tilde{\mathcal{V}}$. Thus, computation of the item $P_-(ij) \tilde{\mathcal{V}}$ is finished. After this step, the elements of the intermediate $\tilde{\mathcal{Y}}$ are obtained according to (\[def\_Y\_oo\]) without communication between processes. The summation $\sum_{m, \bm{k}_m, n}$ is executed through the procedure described in Section \[Sum\_88\]. The results are added to the array $\hat{S}_2$. Now, it holds $\hat{S}_2 = \tilde{\mathcal{A}} + \sum_{m, \bm{k}_m, n} t_{m \bm{k}_m n \bm{k}_n}^{a \bm{k}_a b \bm{k}_b} \tilde{\mathcal{Y}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j}$.
### Computation of the term $\hat{\mathcal{A}}$ – The second term
In computation of the third item in the right-hand side of (\[A\_4th\]), no communication between processes occurs. When this computation is finished, it holds $$\begin{aligned}
\hat{S}_2 =& \tilde{\mathcal{A}}
+ \sum_{m, \bm{k}_m, n} t_{m \bm{k}_m n \bm{k}_n}^{a \bm{k}_a b \bm{k}_b} \tilde{\mathcal{Y}}_{m \bm{k}_m n \bm{k}_n i \bm{k}_i j \bm{k}_j} \notag \\
&- P_-(ij) \sum_n \left( t_{i \bm{k}_i n \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} \left( \tilde{\mathcal{F}}_{n \bm{k}_j j \bm{k}_j} + \frac{1}{2} \sum_{c} t_{j \bm{k}_j}^{c \bm{k}_j} \tilde{\mathcal{F}}_{n \bm{k}_j}^{c \bm{k}_j} \right) \right) .
\end{aligned}$$
### Computation of the term $\hat{\mathcal{A}}$ – The third and the fourth terms
The third term and the intermediate $\hat{\mathcal{G}}$ are computed. The results are subtracted from or added to $\hat{S}_2$ and the intermediate $\hat{\mathcal{G}}$ is stored in another array. After this procedure, it holds $$\hat{S}_2 = \tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} + \hat{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} + \check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{b \bm{k}_b a \bm{k}_a}.$$ The procedure described in Section \[DiagSwap\] for operation of the operator $P_-(ab)$. The elements $\check{\mathcal{G}}_{i \bm{k}_i j \bm{k}_j}^{b \bm{k}_b a \bm{k}_a}$ stored in the processes whose process numbers are $\bm{k}_a + \bm{k}_b N_{\bm{k}}$ are subtracted from the array $\hat{S}_2$. After this procedure, it holds $\hat{S}_2 = \tilde{\mathcal{A}} + \hat{\mathcal{A}}$.
### Addition of the term $( \tilde{\mathcal{A}} + \hat{\mathcal{A}})$ to the array $S_2$
The elements of the term $( \tilde{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} + \hat{\mathcal{A}}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} )$ are stored in processes whose numbers are $\bm{k}_b + \bm{k}_a N_{\bm{k}}$. To add them to the array $S_2$, redistribution of them is necessary. The elements should be redistributed according to the indices $\bm{k}_i$ and $\bm{k}_a$ to processes whose numbers are $\bm{k}_i + \bm{k}_a N_{\bm{k}}$. For this purpose, the elements of the array $\hat{S}_2$ are exchanged between processes by the procedure described in Section \[IndexChange\]. See also Fig. \[Fig\_ExcRow\]. The redistributed elements are added to the array $S_2$. Now, it holds $S_2 = \bar{\mathcal{A}} + \check{\mathcal{A}} + \tilde{\mathcal{A}} + \hat{\mathcal{A}}$.
### Convergence judgment to the solutions
Since the correct single-excitation and double-excitation amplitudes satisfy (\[criterion\_single\]) and (\[criterion\_double\]), respectively, the simplest convergence criterion for these amplitudes should be the one such that $$\begin{gathered}
\left| ( R_1 )_{g \bar{\bm{k}}}^{p \bar{\bm{k}}} \right| < \varepsilon_1, \label{criterion_single} \\
\left| ( R_2 )_{i {\bm{k}}_i j {\bm{k}}_j}^{a {\bm{k}}_a b {\bm{k}}_b} \right| < \varepsilon_2, \label{criterion_double}
\end{gathered}$$ where $\varepsilon_1$ and $\varepsilon_2$ are some constants, for all the indices. We continue to update both of single-excitation and double-excitation amplitudes via successive substitution until this criterion is satisfied.
### Update of single-excitation and double-excitation amplitudes
We introduce the following quantities defined as $$\begin{gathered}
D_{i \bar{\bm{k}}}^{a \bar{\bm{k}}} = f_{i \bar{\bm{k}} i \bar{\bm{k}}} - f^{a \bar{\bm{k}} a \bar{\bm{k}}}, \\
D_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} = f_{i \bm{k}_i i \bm{k}_i} + f_{j \bm{k}_j j \bm{k}_j} - f^{a \bm{k}_a a \bm{k}_a} - f^{b \bm{k}_b b \bm{k}_b}.
\end{gathered}$$ Let us denote the updated single-excitation and double-excitation amplitudes by $\breve{t}_{i \bar{\bm{k}}}^{a \bar{\bm{k}}}$ and $\breve{t}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}$, respectively. Then, we update the amplitudes as $$\begin{gathered}
\breve{t}_{i \bar{\bm{k}}}^{a \bar{\bm{k}}} = t_{i \bar{\bm{k}}}^{a \bar{\bm{k}}} + \left( D_{i \bar{\bm{k}}}^{a \bar{\bm{k}}} \right) ^{-1} ( R_1 )_{g \bar{\bm{k}}}^{p \bar{\bm{k}}}, \\
\breve{t}_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} = t_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}
+ \left( D_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b} \right) ^{-1} ( R_2 )_{i \bm{k}_i j \bm{k}_j}^{a \bm{k}_a b \bm{k}_b}.
\end{gathered}$$ They are used in the next iteration.
Conclusion {#Conclusion}
==========
A parallel computing method for the Coupled-Cluster Singles and Doubles (CCSD) in periodic systems is presented. The presented method uses $N_{\bm{k}}^2$ for the number of $k$ points since two indices which represent momentum attached to quantities which have eight indices are identified through a process number in parallel computing. The orders of computational cost and required memory space in each process are reduced by $N_{\bm{k}}^2$ compared with a sequential method. In implementation of the presented method, communication between processes in parallel computing appears in the outmost loop in a nested loop but does not appear inner nested loop.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by MEXT as Exploratory Challenge on Post-K computer” (Frontiers of Basic Science: Challenging the Limits, Challenge of Basic Science: Fundamental Quantum Mechanics and Informatics).
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|
---
author:
- 'Aleksandr Y. Aravkin[^1]'
- 'James V. Burke[^2]'
- 'Dmitriy Drusvyatskiy[^3]'
- 'Michael P. Friedlander[^4]'
- 'Scott Roy[^5]'
bibliography:
- 'dima\_bib.bib'
- 'references\_sasha.bib'
date: 'February 3, 2016'
title: 'Level-set methods for convex optimization'
---
> **Abstract**
>
> Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the objective and constraint functions, and instead approximately solves a sequence of parametric level-set problems. A zero-finding procedure, based on inexact function evaluations and possibly inexact derivative information, leads to an efficient solution scheme for the original problem. We describe the theoretical and practical properties of this approach for a broad range of problems, including low-rank semidefinite optimization, sparse optimization, and generalized linear models for inference.
Introduction {#sec:intro}
============
Root-finding with inexact oracles {#sec:alg_fram}
=================================
Some problem classes {#sec:pr_class}
====================
There is a surprising variety of useful problems that can be treated by the root-finding approach. These include problems from sparse optimization, with applications in compressed sensing and sparse recovery, generalized linear models, which feature prominently in statistical applications, and conic optimization, which includes semidefinite programming. The following sections are in some sense a “cookbook” that describes how features of particular problems can be combined to apply the root-finding approach. In some cases, such as with conic optimization, we have the opportunity to derive unexpected algorithms.
Case studies {#case:stud}
============
Robust elastic net regularization {#sec:mixed_norm}
---------------------------------
[^1]: Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA; [sites.google.com/site/saravkin/](sites.google.com/site/saravkin/); Research supported by the Washington Research Foundation Data Science Professorship.
[^2]: Department of Mathematics, University of Washington, Seattle, WA 98195, USA; [www.math.washington.edu/\~burke/ ](www.math.washington.edu/~burke/ ); Research supported in part by the NSF award DMS-1514559.
[^3]: Department of Mathematics, University of Washington, Seattle, WA 98195, USA; [www.math.washington.edu/\~ddrusv/](www.math.washington.edu/~ddrusv/); Research supported by the AFOSR YIP award FA9550-15-1-0237.
[^4]: Department of Mathematics, UC Davis, One Shields Ave, Davis, CA 95616; [www.math.ucdavis.edu/\~mpf/ ](www.math.ucdavis.edu/~mpf/ ); Research supported by the ONR award N00014-16-1-2242.
[^5]: Department of Mathematics, University of Washington, Seattle, WA 98195, USA; Research supported in part by the AFOSR YIP award FA9550-15-1-0237.
|
---
abstract: 'The distributed shared memory (DSM) architecture is widely used in today’s computer design to mitigate the ever-widening processing-memory gap, and inevitably exhibits non-uniform memory access (NUMA) to shared-memory parallel applications. Failure to achieve full NUMA-awareness can significantly downgrade application performance, especially on today’s manycore platforms with tens to hundreds of cores. Yet traditional approaches such as first-touch and memory policy fail short in either false page-sharing, fragmentation, or ease-of-use. In this paper, we propose a partitioned shared memory approach which allows multi-threaded applications to achieve full NUMA-awareness with only minor code changes and develop a companying NUMA-aware heap manager which eliminates false page-sharing and minimizes fragmentation. Experiments on a 256-core cc-NUMA computing node show that the proposed approach achieves true NUMA-awareness and improves the performance of typical multi-threaded scientific applications up to 4.3 folds with the increased use of cores.'
author:
- Zhang Yang
- Aiqing Zhang
- Zeyao Mo
title: 'JArena: Partitioned Shared Memory for NUMA-awareness in Multi-threaded Scientific Applications'
---
Partitioned Shared Memory, NUMA-awareness, Heap Manager, Multithread, Manycore.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Dr. Linping WU from High Performance Computing Center of Institute of Applied Physics and Computational Mathematics for his help on understanding the OS interferences on cc-NUMA systems. Dr. Xu LIU and Dr. Xiaowen XU contribute several key ideas to the refinement of this paper.
|
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